Research ArticleA Hidden Semi-Markov Model with Duration-Dependent StateTransition Probabilities for Prognostics

Ning Wang1 Shu-dong Sun2 Zhi-qiang Cai2 Shuai Zhang2 and Can Saygin3

1 Department of Automobile Changrsquoan University Xirsquoan 710064 China2Department of Industrial Engineering Northwestern Polytechnical University Xirsquoan 710072 China3Mechanical Engineering Department University of Texas San Antonio San Antonio TX 78249 USA

Correspondence should be addressed to Ning Wang ningwangchdeducn

Received 5 January 2014 Accepted 7 March 2014 Published 14 April 2014

Academic Editor Manyu Xiao

Copyright copy 2014 Ning Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Realistic prognostic tools are essential for effective condition-based maintenance systems In this paper a Duration-DependentHidden Semi-MarkovModel (DD-HSMM) is proposed which overcomes the shortcomings of traditional HiddenMarkovModels(HMM) including the Hidden Semi-Markov Model (HSMM) (1) it allows explicit modeling of state transition probabilitiesbetween the states (2) it relaxes observationsrsquo independence assumption by accommodating a connection between consecutiveobservations and (3) it does not follow the unrealistic Markov chainrsquos memoryless assumption and therefore it provides a morepowerful modeling and analysis capability for real world problems To facilitate the computation of the proposed DD-HSMMmethodology new forward-backward algorithm is developed The demonstration and evaluation of the proposed methodology iscarried out through a case studyThe experimental results show that the DD-HSMMmethodology is effective for equipment healthmonitoring and management

1 Introduction

Fault is a change from the normal operating conditionof a system to an abnormal condition which occurs asa result of system performance degradation over time [1]Diagnostics indicates the occurrence of a fault and its rootcause Prognostics is fault prediction method it involvesdetection of a pending fault before it occurs identifyingits root cause and estimating the remaining useful life(RUL) which is also known as time-to-failure [2] Condition-based Maintenance (CBM) is a maintenance program thatrecommends maintenance actions based on the informationcollected through condition monitoring

A CBM program can be used to do diagnostics orprognostics however regardless of the application it followsthree steps [3ndash5] First data relevant to events and systemhealth are collected through data acquisition techniquesData acquisition in CBM includes event-type data (ieinformation of what happened) and condition monitoringdata which are the measurements related to system health

Second event and condition monitoring data are interpretedfor better understanding in the data processing step Finallymaintenance decisions are made based on the interpretationand analysis of data In particular to identify the weakestcomponents and states and improve the efficiency of CBMthe integrated importance measure of multistate system wasintroduced by Si et al [6 7] An extensive survey on machinediagnostics and prognostics implementing condition-basedmonitoring can be found in Jardine et al [8] and Heng et al[9]

Data analysis for event data only is reliability analysiswhich maps the event data over a time axis to determine theprobability of events and uses the probability distribution topredict failures On the other hand data acquisition in CBMprovides event and condition monitoring data Thereforeit is more effective to combine events and conditions in amodel in order to do diagnostics or prognostics HiddenMarkov model (HMM) is a technique for modeling andanalyzing event and condition monitoring data together Itconsists of two stochastic processes (1) a Markov chain with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 632702 10 pageshttpdxdoiorg1011552014632702

2 Mathematical Problems in Engineering

finite number of states describing an underlying mechanismand (2) an observation process depending on the hiddenstate [10ndash12] An HMM contains finite states connectedby transitions Each state is characterized by a transitionprobability and an observation probability [13]

Researchers have proposed a number of techniquesto address these limitations Continuous variable durationHMM is adopted in the speech recognition Compared tostandard HMM results show that the absence of a correctduration model increases the error rate by 50 [14ndash16]Another example is in handwritten word recognition areadue to the inherent ambiguity related to the segmentationprocess in handwritten words it is a practical idea to usethe variable duration model for the states in a HMM-based handwrittenword recognition system [17 18] Recentlysome researchers apply HMM in the area of diagnostics andprognostics in machining process [19 20] However thesestudies use only ordinary HMM technique The inherentlimitation of HMM as mentioned above still exists in thesemodels

Prognostic methods used in CBM are often a combi-nation of statistical inference and machine learning meth-ods [4 21ndash24] Model-based methods assume that mea-sured information is stochastically correlated with the actualmachine condition HMM identifies the actual machineconditions from observable monitored data through a sta-tistical approach HMM has been very effective in variousapplications ranging from speech recognition [10 14 25ndash27]to tool wear monitoring and machining [3 20 28 29]

The primary advantage of HMM is its robust mathemati-cal foundation that can allow for many practical applicationsand different areas of use An added benefit of employingHMMs is the ease ofmodel interpretation in comparisonwithpure ldquoblack-boxrdquo modeling methods such as artificial neuralnetworks that are often employed in advanced diagnosticmodels [28] However an inherent limitation of HMMapproach is that its state duration follows an exponential dis-tribution In other words HMM does not provide adequaterepresentation of temporal structure

To overcome the limitations of HMM in prognosisDong and He [30 31] propose a Hidden Semi-MarkovModel-based (HSMM) methodology by adding an explicittemporary structure intoHMMto predict RULof equipmentIn this model the states of HSMM are used to representthe health status of equipment The trained HSMM can beused to diagnose the health state of equipment Throughparameter estimation of the health-state duration probabilitydistribution and the proposed backward recursive equationsthe RULof the equipment can be predicted [32] Although theresults from HSMM are promising the deterioration in thesame state of the system is not taken into consideration in thismodel It assumed that the state transition probabilities staythe same in the same state which assumes all observationsare independent which typically does not hold in real worldapplications

This paper presents a new approach that expands theHSMM methodology [30 31 33] with duration-dependentstate transition probabilities Different from HSMM theproposed Duration-Dependent Hidden Semi-MarkovModel

(DD-HSMM) does not follow the unrealistic Markov chainrsquosmemoryless assumption and therefore provides a more real-istic and powerful modeling and analysis capability Themajor contribution of the DD-HSMM methodology is thatit allows explicit modeling of the transition probabilitieswhich (1) do not only depend on the state but also (2)

vary with the duration of each state and (3) it provides thecapability to relax observationsrsquo independence assumptionby accommodating a link between consecutive observationswhich makes it more realistic in real world applications

2 Theoretical Background

21 Description of General HSMM A Hidden Semi-MarkovModel (HSMM) is an extension of HMM by allowing theunderlying process to be a semi-Markov chain with a variableduration or sojourn time for each state The HSMMmodel isan ideal mathematical model for estimating the unobservablehealth states with observable sensor signal For example asmall change in a bearing alignment could cause a small nickin the bearing which could cause scratches in the bearingrace and additional nicks leading to complete bearing failureThis process can be well described by the HSMM Let 119904

119905be

the hidden state at time 119905 and 119874 the observation sequence aHSMM is characterized by its parameters The parameters ofa HSMM are as follows the initial state distribution (denotedby 120587) the transition model (denoted by 119860) the observationmatrix (denoted by 119862) and the state duration distribution(denoted by 119863) Thus a HSMM can be written as 120582 =

(120587 119860 119862119863) For a given state 119878 119862 is the probability matrixof observation being 119900

119905at time 119905 and 119900

119905+1at time 119905 + 1

22 State Transitions In HSMM there are 119873 states and thetransitions between the states are according to the transitionmatrix 119860 that is 119875(119894 rarr 119895) = 119886

119894119895 Similar to standard HMM

the state 1199040at time 119905 = 0 is a special state ldquoSTARTrdquo

Although the distinct health-state transition 119904119902119897minus1

rarr 119904119902119897

is Markov

119875 [119904119902119897= 119895 | 119904

119902119897minus1= 119894] = 119886

119894119895 (1)

the state transition 119904119905minus1

rarr 119904119905is usually not Markov It is the

reason why the model is called ldquosemi-Markovrdquo [32] whichmeans in the HSMM case the conditional independencebetween the past and the future is only ensured when theprocessmoves fromone health state to another distinct healthstate

23 Inference Procedures Similar to HMM HSMM also hasbasic problems to deal with that is evaluation recognitionand training problems

(1) evaluation (also called classification) given the obser-vation sequence119874 = 119900

11199002sdot sdot sdot 119900119879and a HSMM 120582 what

is the probability of the observation sequence giventhe model that is 119875(119874 | 120582)

(2) decoding (also called recognition) given the observa-tion sequence 119874 = 119900

11199002sdot sdot sdot 119900119879and a HSMM 120582 what

Mathematical Problems in Engineering 3

sequence of hidden states 119878 = 11990411199042sdot sdot sdot 119904119879most proba-

bly generates the given sequence of observations

(3) learning (also called training) how do we adjust themodel parameters 120582 = (120587 119860119863 119861) tomaximize119875(119874 |

120582)

Different algorithms have been developed for abovethree problems The most straightforward way of solvingthe evaluation problem is enumerating every possible statesequence of length119879 (the number of observations) Howeverthe computation burden for this exhaustive enumerationis prohibitively high Fortunately there is a more efficientalgorithm that is based on dynamic programming calledforward-backward procedureThe goal for decoding problemis to find the optimal state sequence associated with thegiven observation sequenceThemost widely used optimalitycriterion is to find the single best state sequence (path) thatis to maximize 119875(119878 | 119874 120582) that is equivalent to maximizing119875(119878 119874 | 120582) Viterbi algorithm is used to find this singlebest state sequence which is based on dynamic programmingmethods For learning problem there is no known wayto obtain analytical solution However we can adjust themodel parameter 120582 = (120587 119860119863 119861) such that 119875(119874 | 120582)

is locally maximized using an iterative procedure such asthe Baum-Welch method (or equivalently the Expectation-Maximization algorithm)

3 Inference and Learning Mechanisms ofDD-HSMM

31 Model Structure Although HSMM has explicit stateduration probability distribution 119875

119894(119889) the sate transi-

tion probabilities 119886119894119895are duration invariant In this paper

we replace duration-invariant state transition probabilitieswith duration-dependent state transition probabilities Theparameters for a DD-HSMM are as follows the initial statedistribution denoted by 120587 = 120587

119894 1 le 119894 le 119873 (120587

119894= 119875[119904

1=

119894] 1 le 119894 le 119873) the transition model denoted by 119860 =

119886119894119895(119889) (1 le 119894 119895 le 119873 1 le 119889 le 119863) the observation matrix

denoted by 119861 = 119887119894(119896) (119887

119894(119896) = 119875[V

119896| 119904119905= 119894] where 1 le

119894 le 119873 1 le 119896 le 119872 119872 is the observation number in 119894 andV119894= V1 V2 V

119872 is the observation symbols in state 119894)

and the state duration distribution denoted by 119863(119875119894(119889) (119894 =

1 2 119873)) Thus a DD-HSMM can be written as 1205821015840 =

(120587 119860119863 119861) Here for the duration in given state 119878 is119889119905(119894) = 119889

the state transition probability 119886119894119895(119889) = 119875(119902

119905+1= 119895 | 119902

119905=

119894 119889119905(119894) = 119889) 1 le 119894 119895 le 119873 1 le 119889 le 119863

119894(119873 is the state number

119863119894is the max staying time in state 119894) And the state transition

probabilities satisfy the constraint sum119873119895=1

119886119894119895(119889) = 1 1 le 119889 le

119863119894 1 le 119894 le 119873

32 Duration-Dependent State Transition Probability In DD-HSMM the state transition probability distribution 119860 =

119886119894119895(119889) 1 le 119894 119895 le 119873 1 le 119889 le 119863 We define duration-

dependents state transition probabilities as follows

119886119894119895(119889) = 119875 (119902

119905+1= 119895119902119905= 119894 119889119905(119894) = 119889) (2)

where 119873 and 119863 are the number of states and the maximumduration in any states respectively Equation (2) representsthe transition from state 119894 to state 119895 given that the durationin state 119894 at time 119905 is 119889

119905(119894) = 119889 It indicated that in the DD-

HSMM case the state transition probability is not only statedependent but also duration variant

33 Inference Procedures Similar to HSMM DD-HSMMalso has basic problems to deal with that is evaluation recog-nition and training problems To facilitate the computationin the proposed DD-HSMM-based health prediction modelin the following new forward-backward variables are definedand modified forward-backward algorithm is developed

A dynamic programming scheme is employed for the effi-cient computation of the inference procedures To implementthe inference procedures a forward variable120572

119905(119894 119889) is defined

as the probability of generating 1199001 1199002 119900

119905and ending in

state 119894 and the duration 119889119905(119894) = 119889

120572119905(119894 119889) = 119875 (119900

1 1199002 119900

119905 119902

119905= 119894 119889119905(119894) = 119889 | 120582) (3)

The initial conditions are established at time 119905 = 1 as follows

1205721(1 1198891) = 1

1205721(119895 1) = 120572

1(1 1198891) 1198861119895(1198891) 119887119895(1199001)

1205721(119873 119889119873) =

119873

sum119894=2

1205721(119894 1) 119875

119894(1) 119886119894119873(1)

(4)

All unspecified 120572 values are zero For time 119905 = 2 119879

120572119905(119895 1) =

119873minus1

sum119894=2

119894 = 119895

119863119894

sum119889=1

120572119905minus1

(119894 119889) 119875119894(119889) 119886119894119895(119889) 119887119895(119900119905)

120572119905(119895 119889) = 120572

119905minus1(119895 119889 minus 1) 119887

119895(119900119905)

120572119905(119873 119889119873) =

119873minus1

sum119894=2

119863119894

sum119889=1

120572119905(119894 119889) 119875

119894(119889) 119886119894119873(119889)

(5)

where 119886119894119895(119889) is the state transition probability from state 119894 to

state 119895 given that the duration in state 119894 at time 119905 is 119889119905(119894) = 119889

119887119895(119900119905) is the output probability of observation vector 119900

119905from

state 119895 and 119901119894(119889) is the state duration probability of state 119894119873

is the number of states inDD-HSMMand119863119894is themaximum

duration in state 119894Similar to the forward variable the backward variable can

be written as

120573119905(119894 119889) = 119875 (119900

119905 119900

119879119902119905= 119894 119889119905(119894) = 119889 120582) (6)

For the backward probability the initial conditions are setat time 119905 = 119879 as follows

120573119879(119873 119889119873) = 1

120573119879(119894 119889) = 119875

119894(119889) 119886119894119873(119889)

120573119879(1 1198891) =

119873minus1

sum119895=2

1205721119895(1198891) 119887119895(119900119879) 120573119879(119895 1)

(7)

4 Mathematical Problems in Engineering

For time 0 lt 119905 lt 119879

120573119905(119894 119889) = 119875

119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+ 119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873

(8)

120573119905(1 1198891) =

119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) (9)

Then the total probability can be computed by

119875119903= 119875 (119874 | 120582) =

119873

sum119894=1

119863119894

sum119889=1

120572119905(119894 119889) 120573

119905(119894 119889) (10)

34 Modified Forward-Backward Algorithm for DD-HSMMIn order to give reestimation formulas for all variable of theDD-HSMM one DD-HSMM-featured forward-backwardvariable is defined

120585119905(119894 119895 119889) = 119901 (119905 = 119902

119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894)

= 119889 | 119874 1205821015840)

(11)

In this equation 120585119905(119894 119895 119889) is the probability of state transition

from state 119894 to state 119895 at time 119905 + 1 after being in state 119894 fora duration of 119889

119905(119894) = 119889 given the model 120582 and observation

119874 From the definition of the forward-backward variables wecan derive 120585

119905(119894 119895 119889) as follows

1205851199051199051015840 (119894 119895 119889)

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 | 119874

119879

0 120582)

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 119874

119879

0| 120582)

times (119875(119874119879

0| 120582))minus1

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 119874

119905

0 1198741199051015840

119905+1 119874119879

1199051015840+1| 120582)

times (119875(119874119879

0| 120582))minus1

= 119875 (119905 = 119902119899 119904119905= 119894 119889119905(119894) = 119889 119874

119905

0| 120582)

times 119875 (1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119874

1199051015840

119905+1

119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894 119889119905(119894) = 119889 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905

= 119902119899 119904119905= 119894 119889119905(119894) = 119889 119874

119905

0 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905

= 119902119899 119904119905= 119894 119889119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 | 119905 = 119902

119899

119904119905= 119894 119889119905(119894) = 119889 120582)

times 119875 (1198741199051015840

119905+1 119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894

1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119886

119894119895(119889) 119875 (119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894

1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

=

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1)

119875 (1198741198790| 120582)

119894 = 1

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1)

times(119875(1198741198790| 120582))minus1

2 le 119894 119895 lt 119873

120572119905(119894 119889) 119901

119894(119889)

times119886119894119873(119889) 120573119905(119894 119889)

times(119875(1198741198790| 120582))minus1

2 le 119894 lt 119873 119895 = 119873

(12)

Then we have

120585119905(119894 119895 119889)

= 119901 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840

= 119895 119889119905(119894) = 119889 | 119874 120582

1015840)

Mathematical Problems in Engineering 5

=

1

119875119903

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1

1

119875119903

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1) 2 le 119894 119895 lt 1198731

119875119903

120572119905(119894 119889) 119901

119894(119889) 2 le 119894 lt 119873 119895 = 119873

times119886119894119873(119889) 120573119905(119894 119889)

(13)

and the probability in state 119894 at time 119905 with duration of 119889 isdefined as 120574

119905(119894 119889) and from the definition of the forward-

backward variables we can easily derive 120574119905(119894 119889) as follows

120574119905(119894 119889) = 119901 (119902

119905= 119894 119889119894= 119889119874 120582) =

1

119875119903

120572119905(119894 119889) 120573

119905(119894 119889) (14)

The forward-Backward algorithm computes the followingprobabilities

Forward Pass The forward pass of the algorithm computes120572119905(119894 119889)

Step 1 (initialization (119905 = 1)) The forward variable is shownas follows

1205721(119894 119889) =

1 119895 = 1 119889 = 1198891

1205721(1 1198891) 1198861119895(1198891) 119887119895(1199001) 1 lt 119895 lt 119873 119889 = 1

119873

sum119894=2

1205721(119894 1) 119875

119894(1) 119886119894119873(1) 119895 = 119873 119889 = 119889

119873

(15)

Step 2 (forward recursion (119905 gt 1)) For 119905 = 2 119879

120572119905(119895 119889) =

119873minus1

sum119894=2

119894 = 119895

119863119894

sum119889=1

120572119905minus1

(119894 119889) 119875119894(119889) 119886119894119895(119889) 119887119895(119900119905) 119889 = 1

120572119905minus1

(119895 119889 minus 1) 119887119895(119900119905)

119873minus1

sum119894=2

119863119894

sum119889=1

120572119905(119894 119889) 119875

119894(119889) 119886119894119873(119889) 119895 = 119873

(16)

Backward Pass The backward pass computes 120573119905(119894 119889)

Step 1 (initialization (119905 = 119879)) Thebackward variable is shownas follows

120573119879(119894 119889)

=

1 119894 = 119873 119889 = 119889119873

119875119894(119889) 119886119894119873(119889) 1 lt 119894 lt 119873 119889

1lt 119889 lt 119889

119873

119873minus1

sum119895=2

1205721119895(1198891) 119887119895(119900119879) 120573119879(119895 1) 119894 = 1 119889 = 119889

1

(17)

Step 2 (backward recursion (119905 lt 119879)) For 119905 = 2 119879

120573119905(119894 119889) =

119875119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1 119889 = 119889

119894

(18)

35 Parameter Reestimation for DD-HSMM The reestima-tion formula for initial state distribution is the probability thatstate119894was the first state given 119874

120587119894=120587119894[sum119863

119889=1120573 (119894 119889) 119875 (119889 | 119894) (119888

119894

1199000119900111988811989411990001199001

sdot sdot sdot 119888119894119900119889minus1119900119889

)]

119875 (119874 | 120582) (19)

The reestimation formula of state transition probabilities isthe ratio of expected number of transition from state 119894 to state119895 to the expected number of transitions from state 119894

119886119894119895(119889) =

sum119879

119905=1120585119905(119894 119895 119889)

sum119879

119905=1120574119905(119894 119889)

(20)

36 Training of State Duration Models Using Parametric Prob-ability Distributions In this paper state duration densitiesare modeled by single Gaussian distribution estimated fromtraining data The existing state duration estimation methodis through the simultaneous training DD-HSMM and theirduration densities However these techniques are inefficientbecause the training process requires huge storage andcomputational load Therefore a new approach is adoptedfor training state duration models In this approach stateduration probabilities are estimated on the lattice (or trellis)of observations and states which is obtained in the DD-HSMM training stage

The mean 120583119894and variance 120590(119894) of duration probability of

state 119894 are determined by

120583 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

120590 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

2

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

minus [120583 (119894)]2

(21)

In these equations 120594119905(119894 119889) is the probability of state 119894 at time

119905 with the duration of 119889119905(119894) = 119889 and 120594

119905(119894 119889) can present as

120594119905(119894 119889) =

1

119901119903

120572119905(119894 119889) 119901

119894(119889)

[[[

[

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119886119894119873(119889) 120573119905(119873 119889119873)]]]

]

(22)

6 Mathematical Problems in Engineering

Table 1 Prognostics results

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3 Recognition accuracyNormal 0 29 1 0 0 967Degradation 1 2 27 1 0 90Degradation 2 0 1 28 1 933Failure 3 0 0 1 29 967Total accuracy 942

Table 2 State transition probability (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 06231 03473 00296Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

4 DD-HSMM Based Health Prognostic

Many applications in the actuarial econometric engineeringand medical literature involve the use of the hazard ratefunction [33] The mathematical properties of HR functioncan reveal a variety of features in the data

Let 119879 denote the time to failure of an item underconsideration with lifetime distribution function 119865(119905) andreliability function 119877(119905) where 119865(119905) + 119877(119905) = 1 and 119865(0) = 0Assume that 119865(0) = 0 and density function119891(119905) = 119865

1015840(119905) existthen the HR function can be defined as

120582 (119905) = lim119873rarrinfin

Δ119905rarr0

Δ119898 (119905)

[119873 minus 119898 (119905)] Δ119905=

119889119898 (119905)

[119873 minus 119898 (119905)] 119889119905

=119889119898 (119905) 119872

1 minus 119865 (119905)=119891 (119905)

119877 (119905)

(23)

In which 119872 is the total number of sample items 119898(119905) isthe number of items that fail before time 119905 and Δ119898(119905) is thenumber of items that fail during the time interval (119905 119905 + Δ119905)The ERL function 120583(119905) is the expected time remaining tofailure given that the system has survived to time 119905 then120583(119905) = 119864(119879 minus 119905 | 119879 gt 119905) = (1119877(119905)) int

infin

119905119877(119909)119889119909 for 119905

such that 119877(119905) gt 0 Therefore 120582(119905) can be approximated asthe conditional probability of failure during the time interval(119905 119905 + Δ119905) given survival to time 119905

Suppose that a machine will go through health states119894 (119894 = 1 2 119873minus 1) before entering failure state119873 Let119863(119894)denote the expected duration of themachine staying at healthstate 119894 based on the parameters estimated above we can get119863(119894) as follows

119863 (119894) = 120583 (119894) + 1205881205752(119894) (24)

And 120588 can be denoted by

120588 =(119879 minus sum

119871minus1

119897=0120583 (119894))

sum119871minus1

119897=01205752 (119894)

(25)

Then once the machine has entered the health state 119894 itsexpected residual life equals the summation of the expected

residual duration of the machine staying at health state 119894

and the total remaining staying in the future health statesbefore failure Denote119863(119894119889) as the expected residual durationof the machine staying in the health state 119894 for 119889 Whenthe equipment entered state 119894 at time 119905

119894 the conditional

probability of failure during (119905119894+ 119889 119905

119894+ (119889 + Δ119905)) can be

defined as the probability that the machine will transit to anyother state during the coming Δ119905 and the probability that themachine still stay at state 119894 It can be seen from (9) and (10)that (119905 + 119889)Δ119905 can be denoted as follows

(119905 + 119889) Δ119905 =120585119905(119894 119895 119889)

120574119905(119894 119889)

(26)

Then

119863(119894119889) = 119863 (119894) (1 minus (119905 + 119889) Δ119905)

= 119863 (119894) (1 minus120585119905(119894 119895 119889)

120574119905(119894 119889)

)

(27)

The DD-HSMM equipment health prediction procedure isgiven as follows

Step 1 From the DD-HSMM training procedure (ie param-eter estimation) the state transition probability for the DD-HSMM can be obtained

Step 2 Through the DD-HSMM parameter estimation theduration probability density function for each health-statecan be obtained Therefore the duration mean and variancecan be calculated

Step 3 By classification identify the current health status ofthe equipment

Step 4 The remaining useful life (RUL) of equipment canbe predicted by the following formula (suppose that theequipment currently stays at health state 119894 with duration of119889)

RUL(119889)119894

= 119863(119894119889) +

119873minus1

sum119895=119894+1

119863(119895) (28)

Mathematical Problems in Engineering 7

Table 3 Mean and variance of duration in each state (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 72105 71435 33435Variance 13953 07429 07924 05452

Table 4 State transition probability (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 04728 04154 01118Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

5 Case Study

In this case study long-term wear experiments on rollingelement bearings were conducted [1] In order to collectadequate amount of data sets for the validation of theproposed scheme three experiments with normal operatingconditions three experiments with cage defect fault andthree experiments each of inner and outer race defect faultswere performed until the bearing reached a complete failurestate and stopped operating Bearing characteristic frequen-cies in the frequency domain are extracted from the vibrationsignals corresponding to different degrees of the health statesof the bearing acquired during experiments

During the test running under each condition vibrationsignals were collected These signals were extracted usinga Mahalanobis-Taguchi System (MTS) based model in theoriginal paper [1] and used for the proposed DD-HSMMmethodology in this paper The expert judgment is made offour integer numbers ranging from 0 to 3 representing 4system states as follows

0rarr the bearing is operating normally

1rarr the bearing is operating and shows signs of deteriora-tion it is advisable to take some preventive action atthe next planned maintenance

2rarr the bearing is operating but requires immediate atten-tion

3rarr the bearing has failed

51 Operation State Identification In order to identify theaccuracy of the operation state identification method pro-posed in this paper experimental data with normal operatingcondition were obtained The experimental data set included50 samples for each state (denoted by 0 1 2 and 3) Of thesedata points 20 of them were used to train the model and theremaining 30 samples were used to validate the model

In theDD-HSMMmixtureGaussian distribution and thesingle Gaussian distribution were used to model the outputprobability distribution and the state duration densitiesseparately in which the number of states is 4 The maximumnumber of iterations in training process is set to 100 and theconvergence error to 0000001

minus80

minus90

minus100

minus110

minus120

minus130

minus140

0 5 10 15 20 25 30 35 40

Log-

likel

ihoo

d

Iterations

Normal (0)Contamination (1)

Contamination (2)Fail (3)

Figure 1 Training curve of the DD-HSMMmodel

The DD-HSMM-based training model is shown asFigure 1 The x-axis shows the training steps and the y-axisrepresents the likelihood probability of different states Ascan be seen from Figure 1 the progression of the four statesreaches the set error in less than 40 steps This demonstratesthe potential of the model to have a strong real-time signalprocessing capability

The classification results obtained on the remaining 30data samples are shown in Table 1 As indicated in the resultsthe accuracy of the DD-HSMMmethod is 942

52 Health Prediction for RUL As described before a four-state DD-HSMM prediction model is constructed In thetraining process even if the device is in the same runningcondition the dwell time is different transition probabilitiesbetween states and the mean or variance of duration ineach state are not the same Tables 2 and 3 show the statetransition probability the mean and variance of duration ineach state when 119889119905(1) = 1 representing the bearing in state1 with duration of 1 Tables 4 and 5 show the state transitionprobability the mean and variance of duration in each statewhen 119889119905(1) = 4 representing the bearing in state 1 withduration of 4

First the state 119894 of the current operating state based onthe recognition results is determined then the residence

8 Mathematical Problems in Engineering

Table 5 Mean and variance of duration in each state (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 53105 71435 33435Variance 13953 11328 07924 05452

Table 6 Comparison of DD-HSMM versus HSMM

Actual RUL DD-HSMMmodel HSMMmodelPredicted RUL Error () Predicted RUL Error ()

270000 264027 2212269852

00548240000 236596 1418 12438220000 220856 0389 2266170000 176142 3613 170734 0437150000 157246 4831 13829120000 116945 2546

10519912334

110000 100602 8544 436590000 91944 216 1688850000 51253 2506 36104 2779230000 30122 0407 20347

time sum119873minus1

119895=119894+1119863(119895) is calculated according to the duration

parameters of the operating state in training process Thenthe remaining effective life in the current operational stateis calculated using (25) Finally the RUL of the bearing canbe calculated using (26) Suppose that the bearing is now atstate 1 with a duration of 1 then the following can be obtained119863(2)+119863(3) = 109426119863(11) = 60875 by (25) and RUL(1)

1=

170211 by (26)

53 Prediction Comparison In order to compare the prog-nostic method based on the DD-HSMM with the prognosticmethod based on the HSMM (29) is used to evaluate thelife error In (29) RULactual represents the actual life of thecomponent and RULforecasted represents the expected lifepredicted by DD-HSMM or HSMM

Error =100 times

1003816100381610038161003816RULactual minus RULforecasted1003816100381610038161003816

RULactual (29)

Table 6 shows the prediction comparison of DD-HSMMversus HSMM Failure prediction of the HSMM method isonly state dependent while the DD-HSMM method usesboth state dependency and duration dependency The DD-HSMM method has a self-updating capability in which thehistorical data on states are used in the calculation of statetransition probability matrix As indicated in the resultsthe DD-HSMM method is more accurate than the HSMMmethod

6 Conclusion

This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics As opposed tothe Hidden Semi-MarkovModel (HSMM) failure predictioncapability of the DD-HSMM method uses state dependency

and duration dependency The two important aspects ofequipment health monitoring which are the stages and therate of aging are taken into consideration in an integratedmanner in the proposed DD-HSMM model The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant toreal world applications

In order to facilitate the computational procedure anew forward-backward algorithm and reestimation approachare developed By using autoregression the interdependencybetween observations is established in themodel By incorpo-rating an explicitly defined temporal structure into themodelthe DD-HSMM is capable of predicting the remaining usefullife of equipment more accurately

The demonstration of the proposed model is carried outusing experimental data on rolling element bearings Theproposed model provides a powerful state recognition capa-bility and very accurate results in terms of remaining usefullife prediction In order to draw general conclusion on thecapabilities of the proposed DD-HSMM more experimentaldata in various prognostics areas are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support forthis research from the National High Technology Researchand Development Program of China (no 2012AA040914)the National Natural Science Foundation of China (Grant no71101116) and the Basic Research Foundation of NPU (Grantno JC20120228)

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

finite number of states describing an underlying mechanismand (2) an observation process depending on the hiddenstate [10ndash12] An HMM contains finite states connectedby transitions Each state is characterized by a transitionprobability and an observation probability [13]

Researchers have proposed a number of techniquesto address these limitations Continuous variable durationHMM is adopted in the speech recognition Compared tostandard HMM results show that the absence of a correctduration model increases the error rate by 50 [14ndash16]Another example is in handwritten word recognition areadue to the inherent ambiguity related to the segmentationprocess in handwritten words it is a practical idea to usethe variable duration model for the states in a HMM-based handwrittenword recognition system [17 18] Recentlysome researchers apply HMM in the area of diagnostics andprognostics in machining process [19 20] However thesestudies use only ordinary HMM technique The inherentlimitation of HMM as mentioned above still exists in thesemodels

Prognostic methods used in CBM are often a combi-nation of statistical inference and machine learning meth-ods [4 21ndash24] Model-based methods assume that mea-sured information is stochastically correlated with the actualmachine condition HMM identifies the actual machineconditions from observable monitored data through a sta-tistical approach HMM has been very effective in variousapplications ranging from speech recognition [10 14 25ndash27]to tool wear monitoring and machining [3 20 28 29]

The primary advantage of HMM is its robust mathemati-cal foundation that can allow for many practical applicationsand different areas of use An added benefit of employingHMMs is the ease ofmodel interpretation in comparisonwithpure ldquoblack-boxrdquo modeling methods such as artificial neuralnetworks that are often employed in advanced diagnosticmodels [28] However an inherent limitation of HMMapproach is that its state duration follows an exponential dis-tribution In other words HMM does not provide adequaterepresentation of temporal structure

To overcome the limitations of HMM in prognosisDong and He [30 31] propose a Hidden Semi-MarkovModel-based (HSMM) methodology by adding an explicittemporary structure intoHMMto predict RULof equipmentIn this model the states of HSMM are used to representthe health status of equipment The trained HSMM can beused to diagnose the health state of equipment Throughparameter estimation of the health-state duration probabilitydistribution and the proposed backward recursive equationsthe RULof the equipment can be predicted [32] Although theresults from HSMM are promising the deterioration in thesame state of the system is not taken into consideration in thismodel It assumed that the state transition probabilities staythe same in the same state which assumes all observationsare independent which typically does not hold in real worldapplications

This paper presents a new approach that expands theHSMM methodology [30 31 33] with duration-dependentstate transition probabilities Different from HSMM theproposed Duration-Dependent Hidden Semi-MarkovModel

(DD-HSMM) does not follow the unrealistic Markov chainrsquosmemoryless assumption and therefore provides a more real-istic and powerful modeling and analysis capability Themajor contribution of the DD-HSMM methodology is thatit allows explicit modeling of the transition probabilitieswhich (1) do not only depend on the state but also (2)

vary with the duration of each state and (3) it provides thecapability to relax observationsrsquo independence assumptionby accommodating a link between consecutive observationswhich makes it more realistic in real world applications

2 Theoretical Background

21 Description of General HSMM A Hidden Semi-MarkovModel (HSMM) is an extension of HMM by allowing theunderlying process to be a semi-Markov chain with a variableduration or sojourn time for each state The HSMMmodel isan ideal mathematical model for estimating the unobservablehealth states with observable sensor signal For example asmall change in a bearing alignment could cause a small nickin the bearing which could cause scratches in the bearingrace and additional nicks leading to complete bearing failureThis process can be well described by the HSMM Let 119904

119905be

the hidden state at time 119905 and 119874 the observation sequence aHSMM is characterized by its parameters The parameters ofa HSMM are as follows the initial state distribution (denotedby 120587) the transition model (denoted by 119860) the observationmatrix (denoted by 119862) and the state duration distribution(denoted by 119863) Thus a HSMM can be written as 120582 =

(120587 119860 119862119863) For a given state 119878 119862 is the probability matrixof observation being 119900

119905at time 119905 and 119900

119905+1at time 119905 + 1

22 State Transitions In HSMM there are 119873 states and thetransitions between the states are according to the transitionmatrix 119860 that is 119875(119894 rarr 119895) = 119886

119894119895 Similar to standard HMM

the state 1199040at time 119905 = 0 is a special state ldquoSTARTrdquo

Although the distinct health-state transition 119904119902119897minus1

rarr 119904119902119897

is Markov

119875 [119904119902119897= 119895 | 119904

119902119897minus1= 119894] = 119886

119894119895 (1)

the state transition 119904119905minus1

rarr 119904119905is usually not Markov It is the

reason why the model is called ldquosemi-Markovrdquo [32] whichmeans in the HSMM case the conditional independencebetween the past and the future is only ensured when theprocessmoves fromone health state to another distinct healthstate

23 Inference Procedures Similar to HMM HSMM also hasbasic problems to deal with that is evaluation recognitionand training problems

(1) evaluation (also called classification) given the obser-vation sequence119874 = 119900

11199002sdot sdot sdot 119900119879and a HSMM 120582 what

is the probability of the observation sequence giventhe model that is 119875(119874 | 120582)

(2) decoding (also called recognition) given the observa-tion sequence 119874 = 119900

11199002sdot sdot sdot 119900119879and a HSMM 120582 what

Mathematical Problems in Engineering 3

sequence of hidden states 119878 = 11990411199042sdot sdot sdot 119904119879most proba-

bly generates the given sequence of observations

(3) learning (also called training) how do we adjust themodel parameters 120582 = (120587 119860119863 119861) tomaximize119875(119874 |

120582)

Different algorithms have been developed for abovethree problems The most straightforward way of solvingthe evaluation problem is enumerating every possible statesequence of length119879 (the number of observations) Howeverthe computation burden for this exhaustive enumerationis prohibitively high Fortunately there is a more efficientalgorithm that is based on dynamic programming calledforward-backward procedureThe goal for decoding problemis to find the optimal state sequence associated with thegiven observation sequenceThemost widely used optimalitycriterion is to find the single best state sequence (path) thatis to maximize 119875(119878 | 119874 120582) that is equivalent to maximizing119875(119878 119874 | 120582) Viterbi algorithm is used to find this singlebest state sequence which is based on dynamic programmingmethods For learning problem there is no known wayto obtain analytical solution However we can adjust themodel parameter 120582 = (120587 119860119863 119861) such that 119875(119874 | 120582)

is locally maximized using an iterative procedure such asthe Baum-Welch method (or equivalently the Expectation-Maximization algorithm)

3 Inference and Learning Mechanisms ofDD-HSMM

31 Model Structure Although HSMM has explicit stateduration probability distribution 119875

119894(119889) the sate transi-

tion probabilities 119886119894119895are duration invariant In this paper

we replace duration-invariant state transition probabilitieswith duration-dependent state transition probabilities Theparameters for a DD-HSMM are as follows the initial statedistribution denoted by 120587 = 120587

119894 1 le 119894 le 119873 (120587

119894= 119875[119904

1=

119894] 1 le 119894 le 119873) the transition model denoted by 119860 =

119886119894119895(119889) (1 le 119894 119895 le 119873 1 le 119889 le 119863) the observation matrix

denoted by 119861 = 119887119894(119896) (119887

119894(119896) = 119875[V

119896| 119904119905= 119894] where 1 le

119894 le 119873 1 le 119896 le 119872 119872 is the observation number in 119894 andV119894= V1 V2 V

119872 is the observation symbols in state 119894)

and the state duration distribution denoted by 119863(119875119894(119889) (119894 =

1 2 119873)) Thus a DD-HSMM can be written as 1205821015840 =

(120587 119860119863 119861) Here for the duration in given state 119878 is119889119905(119894) = 119889

the state transition probability 119886119894119895(119889) = 119875(119902

119905+1= 119895 | 119902

119905=

119894 119889119905(119894) = 119889) 1 le 119894 119895 le 119873 1 le 119889 le 119863

119894(119873 is the state number

119863119894is the max staying time in state 119894) And the state transition

probabilities satisfy the constraint sum119873119895=1

119886119894119895(119889) = 1 1 le 119889 le

119863119894 1 le 119894 le 119873

32 Duration-Dependent State Transition Probability In DD-HSMM the state transition probability distribution 119860 =

119886119894119895(119889) 1 le 119894 119895 le 119873 1 le 119889 le 119863 We define duration-

dependents state transition probabilities as follows

119886119894119895(119889) = 119875 (119902

119905+1= 119895119902119905= 119894 119889119905(119894) = 119889) (2)

where 119873 and 119863 are the number of states and the maximumduration in any states respectively Equation (2) representsthe transition from state 119894 to state 119895 given that the durationin state 119894 at time 119905 is 119889

119905(119894) = 119889 It indicated that in the DD-

HSMM case the state transition probability is not only statedependent but also duration variant

33 Inference Procedures Similar to HSMM DD-HSMMalso has basic problems to deal with that is evaluation recog-nition and training problems To facilitate the computationin the proposed DD-HSMM-based health prediction modelin the following new forward-backward variables are definedand modified forward-backward algorithm is developed

A dynamic programming scheme is employed for the effi-cient computation of the inference procedures To implementthe inference procedures a forward variable120572

119905(119894 119889) is defined

as the probability of generating 1199001 1199002 119900

119905and ending in

state 119894 and the duration 119889119905(119894) = 119889

120572119905(119894 119889) = 119875 (119900

1 1199002 119900

119905 119902

119905= 119894 119889119905(119894) = 119889 | 120582) (3)

The initial conditions are established at time 119905 = 1 as follows

1205721(1 1198891) = 1

1205721(119895 1) = 120572

1(1 1198891) 1198861119895(1198891) 119887119895(1199001)

1205721(119873 119889119873) =

119873

sum119894=2

1205721(119894 1) 119875

119894(1) 119886119894119873(1)

(4)

All unspecified 120572 values are zero For time 119905 = 2 119879

120572119905(119895 1) =

119873minus1

sum119894=2

119894 = 119895

119863119894

sum119889=1

120572119905minus1

(119894 119889) 119875119894(119889) 119886119894119895(119889) 119887119895(119900119905)

120572119905(119895 119889) = 120572

119905minus1(119895 119889 minus 1) 119887

119895(119900119905)

120572119905(119873 119889119873) =

119873minus1

sum119894=2

119863119894

sum119889=1

120572119905(119894 119889) 119875

119894(119889) 119886119894119873(119889)

(5)

where 119886119894119895(119889) is the state transition probability from state 119894 to

state 119895 given that the duration in state 119894 at time 119905 is 119889119905(119894) = 119889

119887119895(119900119905) is the output probability of observation vector 119900

119905from

state 119895 and 119901119894(119889) is the state duration probability of state 119894119873

is the number of states inDD-HSMMand119863119894is themaximum

duration in state 119894Similar to the forward variable the backward variable can

be written as

120573119905(119894 119889) = 119875 (119900

119905 119900

119879119902119905= 119894 119889119905(119894) = 119889 120582) (6)

For the backward probability the initial conditions are setat time 119905 = 119879 as follows

120573119879(119873 119889119873) = 1

120573119879(119894 119889) = 119875

119894(119889) 119886119894119873(119889)

120573119879(1 1198891) =

119873minus1

sum119895=2

1205721119895(1198891) 119887119895(119900119879) 120573119879(119895 1)

(7)

4 Mathematical Problems in Engineering

For time 0 lt 119905 lt 119879

120573119905(119894 119889) = 119875

119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+ 119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873

(8)

120573119905(1 1198891) =

119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) (9)

Then the total probability can be computed by

119875119903= 119875 (119874 | 120582) =

119873

sum119894=1

119863119894

sum119889=1

120572119905(119894 119889) 120573

119905(119894 119889) (10)

34 Modified Forward-Backward Algorithm for DD-HSMMIn order to give reestimation formulas for all variable of theDD-HSMM one DD-HSMM-featured forward-backwardvariable is defined

120585119905(119894 119895 119889) = 119901 (119905 = 119902

119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894)

= 119889 | 119874 1205821015840)

(11)

In this equation 120585119905(119894 119895 119889) is the probability of state transition

from state 119894 to state 119895 at time 119905 + 1 after being in state 119894 fora duration of 119889

119905(119894) = 119889 given the model 120582 and observation

119874 From the definition of the forward-backward variables wecan derive 120585

119905(119894 119895 119889) as follows

1205851199051199051015840 (119894 119895 119889)

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 | 119874

119879

0 120582)

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 119874

119879

0| 120582)

times (119875(119874119879

0| 120582))minus1

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 119874

119905

0 1198741199051015840

119905+1 119874119879

1199051015840+1| 120582)

times (119875(119874119879

0| 120582))minus1

= 119875 (119905 = 119902119899 119904119905= 119894 119889119905(119894) = 119889 119874

119905

0| 120582)

times 119875 (1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119874

1199051015840

119905+1

119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894 119889119905(119894) = 119889 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905

= 119902119899 119904119905= 119894 119889119905(119894) = 119889 119874

119905

0 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905

= 119902119899 119904119905= 119894 119889119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 | 119905 = 119902

119899

119904119905= 119894 119889119905(119894) = 119889 120582)

times 119875 (1198741199051015840

119905+1 119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894

1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119886

119894119895(119889) 119875 (119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894

1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

=

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1)

119875 (1198741198790| 120582)

119894 = 1

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1)

times(119875(1198741198790| 120582))minus1

2 le 119894 119895 lt 119873

120572119905(119894 119889) 119901

119894(119889)

times119886119894119873(119889) 120573119905(119894 119889)

times(119875(1198741198790| 120582))minus1

2 le 119894 lt 119873 119895 = 119873

(12)

Then we have

120585119905(119894 119895 119889)

= 119901 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840

= 119895 119889119905(119894) = 119889 | 119874 120582

1015840)

Mathematical Problems in Engineering 5

=

1

119875119903

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1

1

119875119903

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1) 2 le 119894 119895 lt 1198731

119875119903

120572119905(119894 119889) 119901

119894(119889) 2 le 119894 lt 119873 119895 = 119873

times119886119894119873(119889) 120573119905(119894 119889)

(13)

and the probability in state 119894 at time 119905 with duration of 119889 isdefined as 120574

119905(119894 119889) and from the definition of the forward-

backward variables we can easily derive 120574119905(119894 119889) as follows

120574119905(119894 119889) = 119901 (119902

119905= 119894 119889119894= 119889119874 120582) =

1

119875119903

120572119905(119894 119889) 120573

119905(119894 119889) (14)

The forward-Backward algorithm computes the followingprobabilities

Forward Pass The forward pass of the algorithm computes120572119905(119894 119889)

Step 1 (initialization (119905 = 1)) The forward variable is shownas follows

1205721(119894 119889) =

1 119895 = 1 119889 = 1198891

1205721(1 1198891) 1198861119895(1198891) 119887119895(1199001) 1 lt 119895 lt 119873 119889 = 1

119873

sum119894=2

1205721(119894 1) 119875

119894(1) 119886119894119873(1) 119895 = 119873 119889 = 119889

119873

(15)

Step 2 (forward recursion (119905 gt 1)) For 119905 = 2 119879

120572119905(119895 119889) =

119873minus1

sum119894=2

119894 = 119895

119863119894

sum119889=1

120572119905minus1

(119894 119889) 119875119894(119889) 119886119894119895(119889) 119887119895(119900119905) 119889 = 1

120572119905minus1

(119895 119889 minus 1) 119887119895(119900119905)

119873minus1

sum119894=2

119863119894

sum119889=1

120572119905(119894 119889) 119875

119894(119889) 119886119894119873(119889) 119895 = 119873

(16)

Backward Pass The backward pass computes 120573119905(119894 119889)

Step 1 (initialization (119905 = 119879)) Thebackward variable is shownas follows

120573119879(119894 119889)

=

1 119894 = 119873 119889 = 119889119873

119875119894(119889) 119886119894119873(119889) 1 lt 119894 lt 119873 119889

1lt 119889 lt 119889

119873

119873minus1

sum119895=2

1205721119895(1198891) 119887119895(119900119879) 120573119879(119895 1) 119894 = 1 119889 = 119889

1

(17)

Step 2 (backward recursion (119905 lt 119879)) For 119905 = 2 119879

120573119905(119894 119889) =

119875119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1 119889 = 119889

119894

(18)

35 Parameter Reestimation for DD-HSMM The reestima-tion formula for initial state distribution is the probability thatstate119894was the first state given 119874

120587119894=120587119894[sum119863

119889=1120573 (119894 119889) 119875 (119889 | 119894) (119888

119894

1199000119900111988811989411990001199001

sdot sdot sdot 119888119894119900119889minus1119900119889

)]

119875 (119874 | 120582) (19)

The reestimation formula of state transition probabilities isthe ratio of expected number of transition from state 119894 to state119895 to the expected number of transitions from state 119894

119886119894119895(119889) =

sum119879

119905=1120585119905(119894 119895 119889)

sum119879

119905=1120574119905(119894 119889)

(20)

36 Training of State Duration Models Using Parametric Prob-ability Distributions In this paper state duration densitiesare modeled by single Gaussian distribution estimated fromtraining data The existing state duration estimation methodis through the simultaneous training DD-HSMM and theirduration densities However these techniques are inefficientbecause the training process requires huge storage andcomputational load Therefore a new approach is adoptedfor training state duration models In this approach stateduration probabilities are estimated on the lattice (or trellis)of observations and states which is obtained in the DD-HSMM training stage

The mean 120583119894and variance 120590(119894) of duration probability of

state 119894 are determined by

120583 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

120590 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

2

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

minus [120583 (119894)]2

(21)

In these equations 120594119905(119894 119889) is the probability of state 119894 at time

119905 with the duration of 119889119905(119894) = 119889 and 120594

119905(119894 119889) can present as

120594119905(119894 119889) =

1

119901119903

120572119905(119894 119889) 119901

119894(119889)

[[[

[

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119886119894119873(119889) 120573119905(119873 119889119873)]]]

]

(22)

6 Mathematical Problems in Engineering

Table 1 Prognostics results

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3 Recognition accuracyNormal 0 29 1 0 0 967Degradation 1 2 27 1 0 90Degradation 2 0 1 28 1 933Failure 3 0 0 1 29 967Total accuracy 942

Table 2 State transition probability (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 06231 03473 00296Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

4 DD-HSMM Based Health Prognostic

Many applications in the actuarial econometric engineeringand medical literature involve the use of the hazard ratefunction [33] The mathematical properties of HR functioncan reveal a variety of features in the data

Let 119879 denote the time to failure of an item underconsideration with lifetime distribution function 119865(119905) andreliability function 119877(119905) where 119865(119905) + 119877(119905) = 1 and 119865(0) = 0Assume that 119865(0) = 0 and density function119891(119905) = 119865

1015840(119905) existthen the HR function can be defined as

120582 (119905) = lim119873rarrinfin

Δ119905rarr0

Δ119898 (119905)

[119873 minus 119898 (119905)] Δ119905=

119889119898 (119905)

[119873 minus 119898 (119905)] 119889119905

=119889119898 (119905) 119872

1 minus 119865 (119905)=119891 (119905)

119877 (119905)

(23)

In which 119872 is the total number of sample items 119898(119905) isthe number of items that fail before time 119905 and Δ119898(119905) is thenumber of items that fail during the time interval (119905 119905 + Δ119905)The ERL function 120583(119905) is the expected time remaining tofailure given that the system has survived to time 119905 then120583(119905) = 119864(119879 minus 119905 | 119879 gt 119905) = (1119877(119905)) int

infin

119905119877(119909)119889119909 for 119905

such that 119877(119905) gt 0 Therefore 120582(119905) can be approximated asthe conditional probability of failure during the time interval(119905 119905 + Δ119905) given survival to time 119905

Suppose that a machine will go through health states119894 (119894 = 1 2 119873minus 1) before entering failure state119873 Let119863(119894)denote the expected duration of themachine staying at healthstate 119894 based on the parameters estimated above we can get119863(119894) as follows

119863 (119894) = 120583 (119894) + 1205881205752(119894) (24)

And 120588 can be denoted by

120588 =(119879 minus sum

119871minus1

119897=0120583 (119894))

sum119871minus1

119897=01205752 (119894)

(25)

Then once the machine has entered the health state 119894 itsexpected residual life equals the summation of the expected

residual duration of the machine staying at health state 119894

and the total remaining staying in the future health statesbefore failure Denote119863(119894119889) as the expected residual durationof the machine staying in the health state 119894 for 119889 Whenthe equipment entered state 119894 at time 119905

119894 the conditional

probability of failure during (119905119894+ 119889 119905

119894+ (119889 + Δ119905)) can be

defined as the probability that the machine will transit to anyother state during the coming Δ119905 and the probability that themachine still stay at state 119894 It can be seen from (9) and (10)that (119905 + 119889)Δ119905 can be denoted as follows

(119905 + 119889) Δ119905 =120585119905(119894 119895 119889)

120574119905(119894 119889)

(26)

Then

119863(119894119889) = 119863 (119894) (1 minus (119905 + 119889) Δ119905)

= 119863 (119894) (1 minus120585119905(119894 119895 119889)

120574119905(119894 119889)

)

(27)

The DD-HSMM equipment health prediction procedure isgiven as follows

Step 1 From the DD-HSMM training procedure (ie param-eter estimation) the state transition probability for the DD-HSMM can be obtained

Step 2 Through the DD-HSMM parameter estimation theduration probability density function for each health-statecan be obtained Therefore the duration mean and variancecan be calculated

Step 3 By classification identify the current health status ofthe equipment

Step 4 The remaining useful life (RUL) of equipment canbe predicted by the following formula (suppose that theequipment currently stays at health state 119894 with duration of119889)

RUL(119889)119894

= 119863(119894119889) +

119873minus1

sum119895=119894+1

119863(119895) (28)

Mathematical Problems in Engineering 7

Table 3 Mean and variance of duration in each state (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 72105 71435 33435Variance 13953 07429 07924 05452

Table 4 State transition probability (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 04728 04154 01118Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

5 Case Study

In this case study long-term wear experiments on rollingelement bearings were conducted [1] In order to collectadequate amount of data sets for the validation of theproposed scheme three experiments with normal operatingconditions three experiments with cage defect fault andthree experiments each of inner and outer race defect faultswere performed until the bearing reached a complete failurestate and stopped operating Bearing characteristic frequen-cies in the frequency domain are extracted from the vibrationsignals corresponding to different degrees of the health statesof the bearing acquired during experiments

During the test running under each condition vibrationsignals were collected These signals were extracted usinga Mahalanobis-Taguchi System (MTS) based model in theoriginal paper [1] and used for the proposed DD-HSMMmethodology in this paper The expert judgment is made offour integer numbers ranging from 0 to 3 representing 4system states as follows

0rarr the bearing is operating normally

1rarr the bearing is operating and shows signs of deteriora-tion it is advisable to take some preventive action atthe next planned maintenance

2rarr the bearing is operating but requires immediate atten-tion

3rarr the bearing has failed

51 Operation State Identification In order to identify theaccuracy of the operation state identification method pro-posed in this paper experimental data with normal operatingcondition were obtained The experimental data set included50 samples for each state (denoted by 0 1 2 and 3) Of thesedata points 20 of them were used to train the model and theremaining 30 samples were used to validate the model

In theDD-HSMMmixtureGaussian distribution and thesingle Gaussian distribution were used to model the outputprobability distribution and the state duration densitiesseparately in which the number of states is 4 The maximumnumber of iterations in training process is set to 100 and theconvergence error to 0000001

minus80

minus90

minus100

minus110

minus120

minus130

minus140

0 5 10 15 20 25 30 35 40

Log-

likel

ihoo

d

Iterations

Normal (0)Contamination (1)

Contamination (2)Fail (3)

Figure 1 Training curve of the DD-HSMMmodel

The DD-HSMM-based training model is shown asFigure 1 The x-axis shows the training steps and the y-axisrepresents the likelihood probability of different states Ascan be seen from Figure 1 the progression of the four statesreaches the set error in less than 40 steps This demonstratesthe potential of the model to have a strong real-time signalprocessing capability

The classification results obtained on the remaining 30data samples are shown in Table 1 As indicated in the resultsthe accuracy of the DD-HSMMmethod is 942

52 Health Prediction for RUL As described before a four-state DD-HSMM prediction model is constructed In thetraining process even if the device is in the same runningcondition the dwell time is different transition probabilitiesbetween states and the mean or variance of duration ineach state are not the same Tables 2 and 3 show the statetransition probability the mean and variance of duration ineach state when 119889119905(1) = 1 representing the bearing in state1 with duration of 1 Tables 4 and 5 show the state transitionprobability the mean and variance of duration in each statewhen 119889119905(1) = 4 representing the bearing in state 1 withduration of 4

First the state 119894 of the current operating state based onthe recognition results is determined then the residence

8 Mathematical Problems in Engineering

Table 5 Mean and variance of duration in each state (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 53105 71435 33435Variance 13953 11328 07924 05452

Table 6 Comparison of DD-HSMM versus HSMM

Actual RUL DD-HSMMmodel HSMMmodelPredicted RUL Error () Predicted RUL Error ()

270000 264027 2212269852

00548240000 236596 1418 12438220000 220856 0389 2266170000 176142 3613 170734 0437150000 157246 4831 13829120000 116945 2546

10519912334

110000 100602 8544 436590000 91944 216 1688850000 51253 2506 36104 2779230000 30122 0407 20347

time sum119873minus1

119895=119894+1119863(119895) is calculated according to the duration

parameters of the operating state in training process Thenthe remaining effective life in the current operational stateis calculated using (25) Finally the RUL of the bearing canbe calculated using (26) Suppose that the bearing is now atstate 1 with a duration of 1 then the following can be obtained119863(2)+119863(3) = 109426119863(11) = 60875 by (25) and RUL(1)

1=

170211 by (26)

53 Prediction Comparison In order to compare the prog-nostic method based on the DD-HSMM with the prognosticmethod based on the HSMM (29) is used to evaluate thelife error In (29) RULactual represents the actual life of thecomponent and RULforecasted represents the expected lifepredicted by DD-HSMM or HSMM

Error =100 times

1003816100381610038161003816RULactual minus RULforecasted1003816100381610038161003816

RULactual (29)

Table 6 shows the prediction comparison of DD-HSMMversus HSMM Failure prediction of the HSMM method isonly state dependent while the DD-HSMM method usesboth state dependency and duration dependency The DD-HSMM method has a self-updating capability in which thehistorical data on states are used in the calculation of statetransition probability matrix As indicated in the resultsthe DD-HSMM method is more accurate than the HSMMmethod

6 Conclusion

This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics As opposed tothe Hidden Semi-MarkovModel (HSMM) failure predictioncapability of the DD-HSMM method uses state dependency

and duration dependency The two important aspects ofequipment health monitoring which are the stages and therate of aging are taken into consideration in an integratedmanner in the proposed DD-HSMM model The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant toreal world applications

In order to facilitate the computational procedure anew forward-backward algorithm and reestimation approachare developed By using autoregression the interdependencybetween observations is established in themodel By incorpo-rating an explicitly defined temporal structure into themodelthe DD-HSMM is capable of predicting the remaining usefullife of equipment more accurately

The demonstration of the proposed model is carried outusing experimental data on rolling element bearings Theproposed model provides a powerful state recognition capa-bility and very accurate results in terms of remaining usefullife prediction In order to draw general conclusion on thecapabilities of the proposed DD-HSMM more experimentaldata in various prognostics areas are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support forthis research from the National High Technology Researchand Development Program of China (no 2012AA040914)the National Natural Science Foundation of China (Grant no71101116) and the Basic Research Foundation of NPU (Grantno JC20120228)

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

sequence of hidden states 119878 = 11990411199042sdot sdot sdot 119904119879most proba-

bly generates the given sequence of observations

(3) learning (also called training) how do we adjust themodel parameters 120582 = (120587 119860119863 119861) tomaximize119875(119874 |

120582)

Different algorithms have been developed for abovethree problems The most straightforward way of solvingthe evaluation problem is enumerating every possible statesequence of length119879 (the number of observations) Howeverthe computation burden for this exhaustive enumerationis prohibitively high Fortunately there is a more efficientalgorithm that is based on dynamic programming calledforward-backward procedureThe goal for decoding problemis to find the optimal state sequence associated with thegiven observation sequenceThemost widely used optimalitycriterion is to find the single best state sequence (path) thatis to maximize 119875(119878 | 119874 120582) that is equivalent to maximizing119875(119878 119874 | 120582) Viterbi algorithm is used to find this singlebest state sequence which is based on dynamic programmingmethods For learning problem there is no known wayto obtain analytical solution However we can adjust themodel parameter 120582 = (120587 119860119863 119861) such that 119875(119874 | 120582)

is locally maximized using an iterative procedure such asthe Baum-Welch method (or equivalently the Expectation-Maximization algorithm)

3 Inference and Learning Mechanisms ofDD-HSMM

31 Model Structure Although HSMM has explicit stateduration probability distribution 119875

119894(119889) the sate transi-

tion probabilities 119886119894119895are duration invariant In this paper

we replace duration-invariant state transition probabilitieswith duration-dependent state transition probabilities Theparameters for a DD-HSMM are as follows the initial statedistribution denoted by 120587 = 120587

119894 1 le 119894 le 119873 (120587

119894= 119875[119904

1=

119894] 1 le 119894 le 119873) the transition model denoted by 119860 =

119886119894119895(119889) (1 le 119894 119895 le 119873 1 le 119889 le 119863) the observation matrix

denoted by 119861 = 119887119894(119896) (119887

119894(119896) = 119875[V

119896| 119904119905= 119894] where 1 le

119894 le 119873 1 le 119896 le 119872 119872 is the observation number in 119894 andV119894= V1 V2 V

119872 is the observation symbols in state 119894)

and the state duration distribution denoted by 119863(119875119894(119889) (119894 =

1 2 119873)) Thus a DD-HSMM can be written as 1205821015840 =

(120587 119860119863 119861) Here for the duration in given state 119878 is119889119905(119894) = 119889

the state transition probability 119886119894119895(119889) = 119875(119902

119905+1= 119895 | 119902

119905=

119894 119889119905(119894) = 119889) 1 le 119894 119895 le 119873 1 le 119889 le 119863

119894(119873 is the state number

119863119894is the max staying time in state 119894) And the state transition

probabilities satisfy the constraint sum119873119895=1

119886119894119895(119889) = 1 1 le 119889 le

119863119894 1 le 119894 le 119873

32 Duration-Dependent State Transition Probability In DD-HSMM the state transition probability distribution 119860 =

119886119894119895(119889) 1 le 119894 119895 le 119873 1 le 119889 le 119863 We define duration-

dependents state transition probabilities as follows

119886119894119895(119889) = 119875 (119902

119905+1= 119895119902119905= 119894 119889119905(119894) = 119889) (2)

where 119873 and 119863 are the number of states and the maximumduration in any states respectively Equation (2) representsthe transition from state 119894 to state 119895 given that the durationin state 119894 at time 119905 is 119889

119905(119894) = 119889 It indicated that in the DD-

HSMM case the state transition probability is not only statedependent but also duration variant

33 Inference Procedures Similar to HSMM DD-HSMMalso has basic problems to deal with that is evaluation recog-nition and training problems To facilitate the computationin the proposed DD-HSMM-based health prediction modelin the following new forward-backward variables are definedand modified forward-backward algorithm is developed

A dynamic programming scheme is employed for the effi-cient computation of the inference procedures To implementthe inference procedures a forward variable120572

119905(119894 119889) is defined

as the probability of generating 1199001 1199002 119900

119905and ending in

state 119894 and the duration 119889119905(119894) = 119889

120572119905(119894 119889) = 119875 (119900

1 1199002 119900

119905 119902

119905= 119894 119889119905(119894) = 119889 | 120582) (3)

The initial conditions are established at time 119905 = 1 as follows

1205721(1 1198891) = 1

1205721(119895 1) = 120572

1(1 1198891) 1198861119895(1198891) 119887119895(1199001)

1205721(119873 119889119873) =

119873

sum119894=2

1205721(119894 1) 119875

119894(1) 119886119894119873(1)

(4)

All unspecified 120572 values are zero For time 119905 = 2 119879

120572119905(119895 1) =

119873minus1

sum119894=2

119894 = 119895

119863119894

sum119889=1

120572119905minus1

(119894 119889) 119875119894(119889) 119886119894119895(119889) 119887119895(119900119905)

120572119905(119895 119889) = 120572

119905minus1(119895 119889 minus 1) 119887

119895(119900119905)

120572119905(119873 119889119873) =

119873minus1

sum119894=2

119863119894

sum119889=1

120572119905(119894 119889) 119875

119894(119889) 119886119894119873(119889)

(5)

where 119886119894119895(119889) is the state transition probability from state 119894 to

state 119895 given that the duration in state 119894 at time 119905 is 119889119905(119894) = 119889

119887119895(119900119905) is the output probability of observation vector 119900

119905from

state 119895 and 119901119894(119889) is the state duration probability of state 119894119873

is the number of states inDD-HSMMand119863119894is themaximum

duration in state 119894Similar to the forward variable the backward variable can

be written as

120573119905(119894 119889) = 119875 (119900

119905 119900

119879119902119905= 119894 119889119905(119894) = 119889 120582) (6)

For the backward probability the initial conditions are setat time 119905 = 119879 as follows

120573119879(119873 119889119873) = 1

120573119879(119894 119889) = 119875

119894(119889) 119886119894119873(119889)

120573119879(1 1198891) =

119873minus1

sum119895=2

1205721119895(1198891) 119887119895(119900119879) 120573119879(119895 1)

(7)

4 Mathematical Problems in Engineering

For time 0 lt 119905 lt 119879

120573119905(119894 119889) = 119875

119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+ 119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873

(8)

120573119905(1 1198891) =

119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) (9)

Then the total probability can be computed by

119875119903= 119875 (119874 | 120582) =

119873

sum119894=1

119863119894

sum119889=1

120572119905(119894 119889) 120573

119905(119894 119889) (10)

34 Modified Forward-Backward Algorithm for DD-HSMMIn order to give reestimation formulas for all variable of theDD-HSMM one DD-HSMM-featured forward-backwardvariable is defined

120585119905(119894 119895 119889) = 119901 (119905 = 119902

119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894)

= 119889 | 119874 1205821015840)

(11)

In this equation 120585119905(119894 119895 119889) is the probability of state transition

from state 119894 to state 119895 at time 119905 + 1 after being in state 119894 fora duration of 119889

119905(119894) = 119889 given the model 120582 and observation

119874 From the definition of the forward-backward variables wecan derive 120585

119905(119894 119895 119889) as follows

1205851199051199051015840 (119894 119895 119889)

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 | 119874

119879

0 120582)

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 119874

119879

0| 120582)

times (119875(119874119879

0| 120582))minus1

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 119874

119905

0 1198741199051015840

119905+1 119874119879

1199051015840+1| 120582)

times (119875(119874119879

0| 120582))minus1

= 119875 (119905 = 119902119899 119904119905= 119894 119889119905(119894) = 119889 119874

119905

0| 120582)

times 119875 (1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119874

1199051015840

119905+1

119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894 119889119905(119894) = 119889 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905

= 119902119899 119904119905= 119894 119889119905(119894) = 119889 119874

119905

0 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905

= 119902119899 119904119905= 119894 119889119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 | 119905 = 119902

119899

119904119905= 119894 119889119905(119894) = 119889 120582)

times 119875 (1198741199051015840

119905+1 119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894

1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119886

119894119895(119889) 119875 (119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894

1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

=

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1)

119875 (1198741198790| 120582)

119894 = 1

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1)

times(119875(1198741198790| 120582))minus1

2 le 119894 119895 lt 119873

120572119905(119894 119889) 119901

119894(119889)

times119886119894119873(119889) 120573119905(119894 119889)

times(119875(1198741198790| 120582))minus1

2 le 119894 lt 119873 119895 = 119873

(12)

Then we have

120585119905(119894 119895 119889)

= 119901 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840

= 119895 119889119905(119894) = 119889 | 119874 120582

1015840)

Mathematical Problems in Engineering 5

=

1

119875119903

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1

1

119875119903

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1) 2 le 119894 119895 lt 1198731

119875119903

120572119905(119894 119889) 119901

119894(119889) 2 le 119894 lt 119873 119895 = 119873

times119886119894119873(119889) 120573119905(119894 119889)

(13)

and the probability in state 119894 at time 119905 with duration of 119889 isdefined as 120574

119905(119894 119889) and from the definition of the forward-

backward variables we can easily derive 120574119905(119894 119889) as follows

120574119905(119894 119889) = 119901 (119902

119905= 119894 119889119894= 119889119874 120582) =

1

119875119903

120572119905(119894 119889) 120573

119905(119894 119889) (14)

The forward-Backward algorithm computes the followingprobabilities

Forward Pass The forward pass of the algorithm computes120572119905(119894 119889)

Step 1 (initialization (119905 = 1)) The forward variable is shownas follows

1205721(119894 119889) =

1 119895 = 1 119889 = 1198891

1205721(1 1198891) 1198861119895(1198891) 119887119895(1199001) 1 lt 119895 lt 119873 119889 = 1

119873

sum119894=2

1205721(119894 1) 119875

119894(1) 119886119894119873(1) 119895 = 119873 119889 = 119889

119873

(15)

Step 2 (forward recursion (119905 gt 1)) For 119905 = 2 119879

120572119905(119895 119889) =

119873minus1

sum119894=2

119894 = 119895

119863119894

sum119889=1

120572119905minus1

(119894 119889) 119875119894(119889) 119886119894119895(119889) 119887119895(119900119905) 119889 = 1

120572119905minus1

(119895 119889 minus 1) 119887119895(119900119905)

119873minus1

sum119894=2

119863119894

sum119889=1

120572119905(119894 119889) 119875

119894(119889) 119886119894119873(119889) 119895 = 119873

(16)

Backward Pass The backward pass computes 120573119905(119894 119889)

Step 1 (initialization (119905 = 119879)) Thebackward variable is shownas follows

120573119879(119894 119889)

=

1 119894 = 119873 119889 = 119889119873

119875119894(119889) 119886119894119873(119889) 1 lt 119894 lt 119873 119889

1lt 119889 lt 119889

119873

119873minus1

sum119895=2

1205721119895(1198891) 119887119895(119900119879) 120573119879(119895 1) 119894 = 1 119889 = 119889

1

(17)

Step 2 (backward recursion (119905 lt 119879)) For 119905 = 2 119879

120573119905(119894 119889) =

119875119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1 119889 = 119889

119894

(18)

35 Parameter Reestimation for DD-HSMM The reestima-tion formula for initial state distribution is the probability thatstate119894was the first state given 119874

120587119894=120587119894[sum119863

119889=1120573 (119894 119889) 119875 (119889 | 119894) (119888

119894

1199000119900111988811989411990001199001

sdot sdot sdot 119888119894119900119889minus1119900119889

)]

119875 (119874 | 120582) (19)

The reestimation formula of state transition probabilities isthe ratio of expected number of transition from state 119894 to state119895 to the expected number of transitions from state 119894

119886119894119895(119889) =

sum119879

119905=1120585119905(119894 119895 119889)

sum119879

119905=1120574119905(119894 119889)

(20)

36 Training of State Duration Models Using Parametric Prob-ability Distributions In this paper state duration densitiesare modeled by single Gaussian distribution estimated fromtraining data The existing state duration estimation methodis through the simultaneous training DD-HSMM and theirduration densities However these techniques are inefficientbecause the training process requires huge storage andcomputational load Therefore a new approach is adoptedfor training state duration models In this approach stateduration probabilities are estimated on the lattice (or trellis)of observations and states which is obtained in the DD-HSMM training stage

The mean 120583119894and variance 120590(119894) of duration probability of

state 119894 are determined by

120583 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

120590 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

2

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

minus [120583 (119894)]2

(21)

In these equations 120594119905(119894 119889) is the probability of state 119894 at time

119905 with the duration of 119889119905(119894) = 119889 and 120594

119905(119894 119889) can present as

120594119905(119894 119889) =

1

119901119903

120572119905(119894 119889) 119901

119894(119889)

[[[

[

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119886119894119873(119889) 120573119905(119873 119889119873)]]]

]

(22)

6 Mathematical Problems in Engineering

Table 1 Prognostics results

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3 Recognition accuracyNormal 0 29 1 0 0 967Degradation 1 2 27 1 0 90Degradation 2 0 1 28 1 933Failure 3 0 0 1 29 967Total accuracy 942

Table 2 State transition probability (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 06231 03473 00296Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

4 DD-HSMM Based Health Prognostic

Many applications in the actuarial econometric engineeringand medical literature involve the use of the hazard ratefunction [33] The mathematical properties of HR functioncan reveal a variety of features in the data

Let 119879 denote the time to failure of an item underconsideration with lifetime distribution function 119865(119905) andreliability function 119877(119905) where 119865(119905) + 119877(119905) = 1 and 119865(0) = 0Assume that 119865(0) = 0 and density function119891(119905) = 119865

1015840(119905) existthen the HR function can be defined as

120582 (119905) = lim119873rarrinfin

Δ119905rarr0

Δ119898 (119905)

[119873 minus 119898 (119905)] Δ119905=

119889119898 (119905)

[119873 minus 119898 (119905)] 119889119905

=119889119898 (119905) 119872

1 minus 119865 (119905)=119891 (119905)

119877 (119905)

(23)

In which 119872 is the total number of sample items 119898(119905) isthe number of items that fail before time 119905 and Δ119898(119905) is thenumber of items that fail during the time interval (119905 119905 + Δ119905)The ERL function 120583(119905) is the expected time remaining tofailure given that the system has survived to time 119905 then120583(119905) = 119864(119879 minus 119905 | 119879 gt 119905) = (1119877(119905)) int

infin

119905119877(119909)119889119909 for 119905

such that 119877(119905) gt 0 Therefore 120582(119905) can be approximated asthe conditional probability of failure during the time interval(119905 119905 + Δ119905) given survival to time 119905

Suppose that a machine will go through health states119894 (119894 = 1 2 119873minus 1) before entering failure state119873 Let119863(119894)denote the expected duration of themachine staying at healthstate 119894 based on the parameters estimated above we can get119863(119894) as follows

119863 (119894) = 120583 (119894) + 1205881205752(119894) (24)

And 120588 can be denoted by

120588 =(119879 minus sum

119871minus1

119897=0120583 (119894))

sum119871minus1

119897=01205752 (119894)

(25)

Then once the machine has entered the health state 119894 itsexpected residual life equals the summation of the expected

residual duration of the machine staying at health state 119894

and the total remaining staying in the future health statesbefore failure Denote119863(119894119889) as the expected residual durationof the machine staying in the health state 119894 for 119889 Whenthe equipment entered state 119894 at time 119905

119894 the conditional

probability of failure during (119905119894+ 119889 119905

119894+ (119889 + Δ119905)) can be

defined as the probability that the machine will transit to anyother state during the coming Δ119905 and the probability that themachine still stay at state 119894 It can be seen from (9) and (10)that (119905 + 119889)Δ119905 can be denoted as follows

(119905 + 119889) Δ119905 =120585119905(119894 119895 119889)

120574119905(119894 119889)

(26)

Then

119863(119894119889) = 119863 (119894) (1 minus (119905 + 119889) Δ119905)

= 119863 (119894) (1 minus120585119905(119894 119895 119889)

120574119905(119894 119889)

)

(27)

The DD-HSMM equipment health prediction procedure isgiven as follows

Step 1 From the DD-HSMM training procedure (ie param-eter estimation) the state transition probability for the DD-HSMM can be obtained

Step 2 Through the DD-HSMM parameter estimation theduration probability density function for each health-statecan be obtained Therefore the duration mean and variancecan be calculated

Step 3 By classification identify the current health status ofthe equipment

Step 4 The remaining useful life (RUL) of equipment canbe predicted by the following formula (suppose that theequipment currently stays at health state 119894 with duration of119889)

RUL(119889)119894

= 119863(119894119889) +

119873minus1

sum119895=119894+1

119863(119895) (28)

Mathematical Problems in Engineering 7

Table 3 Mean and variance of duration in each state (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 72105 71435 33435Variance 13953 07429 07924 05452

Table 4 State transition probability (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 04728 04154 01118Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

5 Case Study

In this case study long-term wear experiments on rollingelement bearings were conducted [1] In order to collectadequate amount of data sets for the validation of theproposed scheme three experiments with normal operatingconditions three experiments with cage defect fault andthree experiments each of inner and outer race defect faultswere performed until the bearing reached a complete failurestate and stopped operating Bearing characteristic frequen-cies in the frequency domain are extracted from the vibrationsignals corresponding to different degrees of the health statesof the bearing acquired during experiments

During the test running under each condition vibrationsignals were collected These signals were extracted usinga Mahalanobis-Taguchi System (MTS) based model in theoriginal paper [1] and used for the proposed DD-HSMMmethodology in this paper The expert judgment is made offour integer numbers ranging from 0 to 3 representing 4system states as follows

0rarr the bearing is operating normally

1rarr the bearing is operating and shows signs of deteriora-tion it is advisable to take some preventive action atthe next planned maintenance

2rarr the bearing is operating but requires immediate atten-tion

3rarr the bearing has failed

51 Operation State Identification In order to identify theaccuracy of the operation state identification method pro-posed in this paper experimental data with normal operatingcondition were obtained The experimental data set included50 samples for each state (denoted by 0 1 2 and 3) Of thesedata points 20 of them were used to train the model and theremaining 30 samples were used to validate the model

In theDD-HSMMmixtureGaussian distribution and thesingle Gaussian distribution were used to model the outputprobability distribution and the state duration densitiesseparately in which the number of states is 4 The maximumnumber of iterations in training process is set to 100 and theconvergence error to 0000001

minus80

minus90

minus100

minus110

minus120

minus130

minus140

0 5 10 15 20 25 30 35 40

Log-

likel

ihoo

d

Iterations

Normal (0)Contamination (1)

Contamination (2)Fail (3)

Figure 1 Training curve of the DD-HSMMmodel

The DD-HSMM-based training model is shown asFigure 1 The x-axis shows the training steps and the y-axisrepresents the likelihood probability of different states Ascan be seen from Figure 1 the progression of the four statesreaches the set error in less than 40 steps This demonstratesthe potential of the model to have a strong real-time signalprocessing capability

The classification results obtained on the remaining 30data samples are shown in Table 1 As indicated in the resultsthe accuracy of the DD-HSMMmethod is 942

52 Health Prediction for RUL As described before a four-state DD-HSMM prediction model is constructed In thetraining process even if the device is in the same runningcondition the dwell time is different transition probabilitiesbetween states and the mean or variance of duration ineach state are not the same Tables 2 and 3 show the statetransition probability the mean and variance of duration ineach state when 119889119905(1) = 1 representing the bearing in state1 with duration of 1 Tables 4 and 5 show the state transitionprobability the mean and variance of duration in each statewhen 119889119905(1) = 4 representing the bearing in state 1 withduration of 4

First the state 119894 of the current operating state based onthe recognition results is determined then the residence

8 Mathematical Problems in Engineering

Table 5 Mean and variance of duration in each state (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 53105 71435 33435Variance 13953 11328 07924 05452

Table 6 Comparison of DD-HSMM versus HSMM

Actual RUL DD-HSMMmodel HSMMmodelPredicted RUL Error () Predicted RUL Error ()

270000 264027 2212269852

00548240000 236596 1418 12438220000 220856 0389 2266170000 176142 3613 170734 0437150000 157246 4831 13829120000 116945 2546

10519912334

110000 100602 8544 436590000 91944 216 1688850000 51253 2506 36104 2779230000 30122 0407 20347

time sum119873minus1

119895=119894+1119863(119895) is calculated according to the duration

parameters of the operating state in training process Thenthe remaining effective life in the current operational stateis calculated using (25) Finally the RUL of the bearing canbe calculated using (26) Suppose that the bearing is now atstate 1 with a duration of 1 then the following can be obtained119863(2)+119863(3) = 109426119863(11) = 60875 by (25) and RUL(1)

1=

170211 by (26)

53 Prediction Comparison In order to compare the prog-nostic method based on the DD-HSMM with the prognosticmethod based on the HSMM (29) is used to evaluate thelife error In (29) RULactual represents the actual life of thecomponent and RULforecasted represents the expected lifepredicted by DD-HSMM or HSMM

Error =100 times

1003816100381610038161003816RULactual minus RULforecasted1003816100381610038161003816

RULactual (29)

Table 6 shows the prediction comparison of DD-HSMMversus HSMM Failure prediction of the HSMM method isonly state dependent while the DD-HSMM method usesboth state dependency and duration dependency The DD-HSMM method has a self-updating capability in which thehistorical data on states are used in the calculation of statetransition probability matrix As indicated in the resultsthe DD-HSMM method is more accurate than the HSMMmethod

6 Conclusion

This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics As opposed tothe Hidden Semi-MarkovModel (HSMM) failure predictioncapability of the DD-HSMM method uses state dependency

and duration dependency The two important aspects ofequipment health monitoring which are the stages and therate of aging are taken into consideration in an integratedmanner in the proposed DD-HSMM model The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant toreal world applications

In order to facilitate the computational procedure anew forward-backward algorithm and reestimation approachare developed By using autoregression the interdependencybetween observations is established in themodel By incorpo-rating an explicitly defined temporal structure into themodelthe DD-HSMM is capable of predicting the remaining usefullife of equipment more accurately

The demonstration of the proposed model is carried outusing experimental data on rolling element bearings Theproposed model provides a powerful state recognition capa-bility and very accurate results in terms of remaining usefullife prediction In order to draw general conclusion on thecapabilities of the proposed DD-HSMM more experimentaldata in various prognostics areas are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support forthis research from the National High Technology Researchand Development Program of China (no 2012AA040914)the National Natural Science Foundation of China (Grant no71101116) and the Basic Research Foundation of NPU (Grantno JC20120228)

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

For time 0 lt 119905 lt 119879

120573119905(119894 119889) = 119875

119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+ 119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873

(8)

120573119905(1 1198891) =

119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) (9)

Then the total probability can be computed by

119875119903= 119875 (119874 | 120582) =

119873

sum119894=1

119863119894

sum119889=1

120572119905(119894 119889) 120573

119905(119894 119889) (10)

34 Modified Forward-Backward Algorithm for DD-HSMMIn order to give reestimation formulas for all variable of theDD-HSMM one DD-HSMM-featured forward-backwardvariable is defined

120585119905(119894 119895 119889) = 119901 (119905 = 119902

119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894)

= 119889 | 119874 1205821015840)

(11)

In this equation 120585119905(119894 119895 119889) is the probability of state transition

from state 119894 to state 119895 at time 119905 + 1 after being in state 119894 fora duration of 119889

119905(119894) = 119889 given the model 120582 and observation

119874 From the definition of the forward-backward variables wecan derive 120585

119905(119894 119895 119889) as follows

1205851199051199051015840 (119894 119895 119889)

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 | 119874

119879

0 120582)

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 119874

119879

0| 120582)

times (119875(119874119879

0| 120582))minus1

= 119875 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840 = 119895

119889119905(119894) = 119889 119874

119905

0 1198741199051015840

119905+1 119874119879

1199051015840+1| 120582)

times (119875(119874119879

0| 120582))minus1

= 119875 (119905 = 119902119899 119904119905= 119894 119889119905(119894) = 119889 119874

119905

0| 120582)

times 119875 (1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119874

1199051015840

119905+1

119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894 119889119905(119894) = 119889 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905

= 119902119899 119904119905= 119894 119889119905(119894) = 119889 119874

119905

0 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905

= 119902119899 119904119905= 119894 119889119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119875 (119905

1015840= 119902119899+1

1199041199051015840 = 119895 | 119905 = 119902

119899

119904119905= 119894 119889119905(119894) = 119889 120582)

times 119875 (1198741199051015840

119905+1 119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894

1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

= 120572119905(119894 119889) 119886

119894119895(119889) 119875 (119874

1199051015840

119905+1 119874119879

1199051015840+1| 119905 = 119902

119899 119904119905= 119894

1199051015840= 119902119899+1

1199041199051015840 = 119895 119889

119905(119894) = 119889 119900

119905 120582)

times (119875(119874119879

0| 120582))minus1

=

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1)

119875 (1198741198790| 120582)

119894 = 1

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1)

times(119875(1198741198790| 120582))minus1

2 le 119894 119895 lt 119873

120572119905(119894 119889) 119901

119894(119889)

times119886119894119873(119889) 120573119905(119894 119889)

times(119875(1198741198790| 120582))minus1

2 le 119894 lt 119873 119895 = 119873

(12)

Then we have

120585119905(119894 119895 119889)

= 119901 (119905 = 119902119899 119904119905= 119894 1199051015840= 119902119899+1

1199041199051015840

= 119895 119889119905(119894) = 119889 | 119874 120582

1015840)

Mathematical Problems in Engineering 5

=

1

119875119903

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1

1

119875119903

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1) 2 le 119894 119895 lt 1198731

119875119903

120572119905(119894 119889) 119901

119894(119889) 2 le 119894 lt 119873 119895 = 119873

times119886119894119873(119889) 120573119905(119894 119889)

(13)

and the probability in state 119894 at time 119905 with duration of 119889 isdefined as 120574

119905(119894 119889) and from the definition of the forward-

backward variables we can easily derive 120574119905(119894 119889) as follows

120574119905(119894 119889) = 119901 (119902

119905= 119894 119889119894= 119889119874 120582) =

1

119875119903

120572119905(119894 119889) 120573

119905(119894 119889) (14)

The forward-Backward algorithm computes the followingprobabilities

Forward Pass The forward pass of the algorithm computes120572119905(119894 119889)

Step 1 (initialization (119905 = 1)) The forward variable is shownas follows

1205721(119894 119889) =

1 119895 = 1 119889 = 1198891

1205721(1 1198891) 1198861119895(1198891) 119887119895(1199001) 1 lt 119895 lt 119873 119889 = 1

119873

sum119894=2

1205721(119894 1) 119875

119894(1) 119886119894119873(1) 119895 = 119873 119889 = 119889

119873

(15)

Step 2 (forward recursion (119905 gt 1)) For 119905 = 2 119879

120572119905(119895 119889) =

119873minus1

sum119894=2

119894 = 119895

119863119894

sum119889=1

120572119905minus1

(119894 119889) 119875119894(119889) 119886119894119895(119889) 119887119895(119900119905) 119889 = 1

120572119905minus1

(119895 119889 minus 1) 119887119895(119900119905)

119873minus1

sum119894=2

119863119894

sum119889=1

120572119905(119894 119889) 119875

119894(119889) 119886119894119873(119889) 119895 = 119873

(16)

Backward Pass The backward pass computes 120573119905(119894 119889)

Step 1 (initialization (119905 = 119879)) Thebackward variable is shownas follows

120573119879(119894 119889)

=

1 119894 = 119873 119889 = 119889119873

119875119894(119889) 119886119894119873(119889) 1 lt 119894 lt 119873 119889

1lt 119889 lt 119889

119873

119873minus1

sum119895=2

1205721119895(1198891) 119887119895(119900119879) 120573119879(119895 1) 119894 = 1 119889 = 119889

1

(17)

Step 2 (backward recursion (119905 lt 119879)) For 119905 = 2 119879

120573119905(119894 119889) =

119875119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1 119889 = 119889

119894

(18)

35 Parameter Reestimation for DD-HSMM The reestima-tion formula for initial state distribution is the probability thatstate119894was the first state given 119874

120587119894=120587119894[sum119863

119889=1120573 (119894 119889) 119875 (119889 | 119894) (119888

119894

1199000119900111988811989411990001199001

sdot sdot sdot 119888119894119900119889minus1119900119889

)]

119875 (119874 | 120582) (19)

The reestimation formula of state transition probabilities isthe ratio of expected number of transition from state 119894 to state119895 to the expected number of transitions from state 119894

119886119894119895(119889) =

sum119879

119905=1120585119905(119894 119895 119889)

sum119879

119905=1120574119905(119894 119889)

(20)

36 Training of State Duration Models Using Parametric Prob-ability Distributions In this paper state duration densitiesare modeled by single Gaussian distribution estimated fromtraining data The existing state duration estimation methodis through the simultaneous training DD-HSMM and theirduration densities However these techniques are inefficientbecause the training process requires huge storage andcomputational load Therefore a new approach is adoptedfor training state duration models In this approach stateduration probabilities are estimated on the lattice (or trellis)of observations and states which is obtained in the DD-HSMM training stage

The mean 120583119894and variance 120590(119894) of duration probability of

state 119894 are determined by

120583 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

120590 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

2

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

minus [120583 (119894)]2

(21)

In these equations 120594119905(119894 119889) is the probability of state 119894 at time

119905 with the duration of 119889119905(119894) = 119889 and 120594

119905(119894 119889) can present as

120594119905(119894 119889) =

1

119901119903

120572119905(119894 119889) 119901

119894(119889)

[[[

[

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119886119894119873(119889) 120573119905(119873 119889119873)]]]

]

(22)

6 Mathematical Problems in Engineering

Table 1 Prognostics results

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3 Recognition accuracyNormal 0 29 1 0 0 967Degradation 1 2 27 1 0 90Degradation 2 0 1 28 1 933Failure 3 0 0 1 29 967Total accuracy 942

Table 2 State transition probability (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 06231 03473 00296Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

4 DD-HSMM Based Health Prognostic

Many applications in the actuarial econometric engineeringand medical literature involve the use of the hazard ratefunction [33] The mathematical properties of HR functioncan reveal a variety of features in the data

Let 119879 denote the time to failure of an item underconsideration with lifetime distribution function 119865(119905) andreliability function 119877(119905) where 119865(119905) + 119877(119905) = 1 and 119865(0) = 0Assume that 119865(0) = 0 and density function119891(119905) = 119865

1015840(119905) existthen the HR function can be defined as

120582 (119905) = lim119873rarrinfin

Δ119905rarr0

Δ119898 (119905)

[119873 minus 119898 (119905)] Δ119905=

119889119898 (119905)

[119873 minus 119898 (119905)] 119889119905

=119889119898 (119905) 119872

1 minus 119865 (119905)=119891 (119905)

119877 (119905)

(23)

In which 119872 is the total number of sample items 119898(119905) isthe number of items that fail before time 119905 and Δ119898(119905) is thenumber of items that fail during the time interval (119905 119905 + Δ119905)The ERL function 120583(119905) is the expected time remaining tofailure given that the system has survived to time 119905 then120583(119905) = 119864(119879 minus 119905 | 119879 gt 119905) = (1119877(119905)) int

infin

119905119877(119909)119889119909 for 119905

such that 119877(119905) gt 0 Therefore 120582(119905) can be approximated asthe conditional probability of failure during the time interval(119905 119905 + Δ119905) given survival to time 119905

Suppose that a machine will go through health states119894 (119894 = 1 2 119873minus 1) before entering failure state119873 Let119863(119894)denote the expected duration of themachine staying at healthstate 119894 based on the parameters estimated above we can get119863(119894) as follows

119863 (119894) = 120583 (119894) + 1205881205752(119894) (24)

And 120588 can be denoted by

120588 =(119879 minus sum

119871minus1

119897=0120583 (119894))

sum119871minus1

119897=01205752 (119894)

(25)

Then once the machine has entered the health state 119894 itsexpected residual life equals the summation of the expected

residual duration of the machine staying at health state 119894

and the total remaining staying in the future health statesbefore failure Denote119863(119894119889) as the expected residual durationof the machine staying in the health state 119894 for 119889 Whenthe equipment entered state 119894 at time 119905

119894 the conditional

probability of failure during (119905119894+ 119889 119905

119894+ (119889 + Δ119905)) can be

defined as the probability that the machine will transit to anyother state during the coming Δ119905 and the probability that themachine still stay at state 119894 It can be seen from (9) and (10)that (119905 + 119889)Δ119905 can be denoted as follows

(119905 + 119889) Δ119905 =120585119905(119894 119895 119889)

120574119905(119894 119889)

(26)

Then

119863(119894119889) = 119863 (119894) (1 minus (119905 + 119889) Δ119905)

= 119863 (119894) (1 minus120585119905(119894 119895 119889)

120574119905(119894 119889)

)

(27)

The DD-HSMM equipment health prediction procedure isgiven as follows

Step 1 From the DD-HSMM training procedure (ie param-eter estimation) the state transition probability for the DD-HSMM can be obtained

Step 2 Through the DD-HSMM parameter estimation theduration probability density function for each health-statecan be obtained Therefore the duration mean and variancecan be calculated

Step 3 By classification identify the current health status ofthe equipment

Step 4 The remaining useful life (RUL) of equipment canbe predicted by the following formula (suppose that theequipment currently stays at health state 119894 with duration of119889)

RUL(119889)119894

= 119863(119894119889) +

119873minus1

sum119895=119894+1

119863(119895) (28)

Mathematical Problems in Engineering 7

Table 3 Mean and variance of duration in each state (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 72105 71435 33435Variance 13953 07429 07924 05452

Table 4 State transition probability (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 04728 04154 01118Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

5 Case Study

In this case study long-term wear experiments on rollingelement bearings were conducted [1] In order to collectadequate amount of data sets for the validation of theproposed scheme three experiments with normal operatingconditions three experiments with cage defect fault andthree experiments each of inner and outer race defect faultswere performed until the bearing reached a complete failurestate and stopped operating Bearing characteristic frequen-cies in the frequency domain are extracted from the vibrationsignals corresponding to different degrees of the health statesof the bearing acquired during experiments

During the test running under each condition vibrationsignals were collected These signals were extracted usinga Mahalanobis-Taguchi System (MTS) based model in theoriginal paper [1] and used for the proposed DD-HSMMmethodology in this paper The expert judgment is made offour integer numbers ranging from 0 to 3 representing 4system states as follows

0rarr the bearing is operating normally

1rarr the bearing is operating and shows signs of deteriora-tion it is advisable to take some preventive action atthe next planned maintenance

2rarr the bearing is operating but requires immediate atten-tion

3rarr the bearing has failed

51 Operation State Identification In order to identify theaccuracy of the operation state identification method pro-posed in this paper experimental data with normal operatingcondition were obtained The experimental data set included50 samples for each state (denoted by 0 1 2 and 3) Of thesedata points 20 of them were used to train the model and theremaining 30 samples were used to validate the model

In theDD-HSMMmixtureGaussian distribution and thesingle Gaussian distribution were used to model the outputprobability distribution and the state duration densitiesseparately in which the number of states is 4 The maximumnumber of iterations in training process is set to 100 and theconvergence error to 0000001

minus80

minus90

minus100

minus110

minus120

minus130

minus140

0 5 10 15 20 25 30 35 40

Log-

likel

ihoo

d

Iterations

Normal (0)Contamination (1)

Contamination (2)Fail (3)

Figure 1 Training curve of the DD-HSMMmodel

The DD-HSMM-based training model is shown asFigure 1 The x-axis shows the training steps and the y-axisrepresents the likelihood probability of different states Ascan be seen from Figure 1 the progression of the four statesreaches the set error in less than 40 steps This demonstratesthe potential of the model to have a strong real-time signalprocessing capability

The classification results obtained on the remaining 30data samples are shown in Table 1 As indicated in the resultsthe accuracy of the DD-HSMMmethod is 942

52 Health Prediction for RUL As described before a four-state DD-HSMM prediction model is constructed In thetraining process even if the device is in the same runningcondition the dwell time is different transition probabilitiesbetween states and the mean or variance of duration ineach state are not the same Tables 2 and 3 show the statetransition probability the mean and variance of duration ineach state when 119889119905(1) = 1 representing the bearing in state1 with duration of 1 Tables 4 and 5 show the state transitionprobability the mean and variance of duration in each statewhen 119889119905(1) = 4 representing the bearing in state 1 withduration of 4

First the state 119894 of the current operating state based onthe recognition results is determined then the residence

8 Mathematical Problems in Engineering

Table 5 Mean and variance of duration in each state (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 53105 71435 33435Variance 13953 11328 07924 05452

Table 6 Comparison of DD-HSMM versus HSMM

Actual RUL DD-HSMMmodel HSMMmodelPredicted RUL Error () Predicted RUL Error ()

270000 264027 2212269852

00548240000 236596 1418 12438220000 220856 0389 2266170000 176142 3613 170734 0437150000 157246 4831 13829120000 116945 2546

10519912334

110000 100602 8544 436590000 91944 216 1688850000 51253 2506 36104 2779230000 30122 0407 20347

time sum119873minus1

119895=119894+1119863(119895) is calculated according to the duration

parameters of the operating state in training process Thenthe remaining effective life in the current operational stateis calculated using (25) Finally the RUL of the bearing canbe calculated using (26) Suppose that the bearing is now atstate 1 with a duration of 1 then the following can be obtained119863(2)+119863(3) = 109426119863(11) = 60875 by (25) and RUL(1)

1=

170211 by (26)

53 Prediction Comparison In order to compare the prog-nostic method based on the DD-HSMM with the prognosticmethod based on the HSMM (29) is used to evaluate thelife error In (29) RULactual represents the actual life of thecomponent and RULforecasted represents the expected lifepredicted by DD-HSMM or HSMM

Error =100 times

1003816100381610038161003816RULactual minus RULforecasted1003816100381610038161003816

RULactual (29)

Table 6 shows the prediction comparison of DD-HSMMversus HSMM Failure prediction of the HSMM method isonly state dependent while the DD-HSMM method usesboth state dependency and duration dependency The DD-HSMM method has a self-updating capability in which thehistorical data on states are used in the calculation of statetransition probability matrix As indicated in the resultsthe DD-HSMM method is more accurate than the HSMMmethod

6 Conclusion

This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics As opposed tothe Hidden Semi-MarkovModel (HSMM) failure predictioncapability of the DD-HSMM method uses state dependency

and duration dependency The two important aspects ofequipment health monitoring which are the stages and therate of aging are taken into consideration in an integratedmanner in the proposed DD-HSMM model The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant toreal world applications

In order to facilitate the computational procedure anew forward-backward algorithm and reestimation approachare developed By using autoregression the interdependencybetween observations is established in themodel By incorpo-rating an explicitly defined temporal structure into themodelthe DD-HSMM is capable of predicting the remaining usefullife of equipment more accurately

The demonstration of the proposed model is carried outusing experimental data on rolling element bearings Theproposed model provides a powerful state recognition capa-bility and very accurate results in terms of remaining usefullife prediction In order to draw general conclusion on thecapabilities of the proposed DD-HSMM more experimentaldata in various prognostics areas are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support forthis research from the National High Technology Researchand Development Program of China (no 2012AA040914)the National Natural Science Foundation of China (Grant no71101116) and the Basic Research Foundation of NPU (Grantno JC20120228)

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

=

1

119875119903

120572119905(1 1198891) 1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1

1

119875119903

120572119905(119894 119889) 119901

119894(119889) 119886119894119895(119889)

times119887119895(119900119905+1) 120573119905+1

(119895 1) 2 le 119894 119895 lt 1198731

119875119903

120572119905(119894 119889) 119901

119894(119889) 2 le 119894 lt 119873 119895 = 119873

times119886119894119873(119889) 120573119905(119894 119889)

(13)

and the probability in state 119894 at time 119905 with duration of 119889 isdefined as 120574

119905(119894 119889) and from the definition of the forward-

backward variables we can easily derive 120574119905(119894 119889) as follows

120574119905(119894 119889) = 119901 (119902

119905= 119894 119889119894= 119889119874 120582) =

1

119875119903

120572119905(119894 119889) 120573

119905(119894 119889) (14)

The forward-Backward algorithm computes the followingprobabilities

Forward Pass The forward pass of the algorithm computes120572119905(119894 119889)

Step 1 (initialization (119905 = 1)) The forward variable is shownas follows

1205721(119894 119889) =

1 119895 = 1 119889 = 1198891

1205721(1 1198891) 1198861119895(1198891) 119887119895(1199001) 1 lt 119895 lt 119873 119889 = 1

119873

sum119894=2

1205721(119894 1) 119875

119894(1) 119886119894119873(1) 119895 = 119873 119889 = 119889

119873

(15)

Step 2 (forward recursion (119905 gt 1)) For 119905 = 2 119879

120572119905(119895 119889) =

119873minus1

sum119894=2

119894 = 119895

119863119894

sum119889=1

120572119905minus1

(119894 119889) 119875119894(119889) 119886119894119895(119889) 119887119895(119900119905) 119889 = 1

120572119905minus1

(119895 119889 minus 1) 119887119895(119900119905)

119873minus1

sum119894=2

119863119894

sum119889=1

120572119905(119894 119889) 119875

119894(119889) 119886119894119873(119889) 119895 = 119873

(16)

Backward Pass The backward pass computes 120573119905(119894 119889)

Step 1 (initialization (119905 = 119879)) Thebackward variable is shownas follows

120573119879(119894 119889)

=

1 119894 = 119873 119889 = 119889119873

119875119894(119889) 119886119894119873(119889) 1 lt 119894 lt 119873 119889

1lt 119889 lt 119889

119873

119873minus1

sum119895=2

1205721119895(1198891) 119887119895(119900119879) 120573119879(119895 1) 119894 = 1 119889 = 119889

1

(17)

Step 2 (backward recursion (119905 lt 119879)) For 119905 = 2 119879

120573119905(119894 119889) =

119875119894(119889)

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119887119895(119900119905+1) 120573119905+1

(119894 119889 + 1) 1 lt 119894 lt 119873119873minus1

sum119895=2

1198861119895(1198891) 119887119895(119900119905) 120573119905(119895 1) 119894 = 1 119889 = 119889

119894

(18)

35 Parameter Reestimation for DD-HSMM The reestima-tion formula for initial state distribution is the probability thatstate119894was the first state given 119874

120587119894=120587119894[sum119863

119889=1120573 (119894 119889) 119875 (119889 | 119894) (119888

119894

1199000119900111988811989411990001199001

sdot sdot sdot 119888119894119900119889minus1119900119889

)]

119875 (119874 | 120582) (19)

The reestimation formula of state transition probabilities isthe ratio of expected number of transition from state 119894 to state119895 to the expected number of transitions from state 119894

119886119894119895(119889) =

sum119879

119905=1120585119905(119894 119895 119889)

sum119879

119905=1120574119905(119894 119889)

(20)

36 Training of State Duration Models Using Parametric Prob-ability Distributions In this paper state duration densitiesare modeled by single Gaussian distribution estimated fromtraining data The existing state duration estimation methodis through the simultaneous training DD-HSMM and theirduration densities However these techniques are inefficientbecause the training process requires huge storage andcomputational load Therefore a new approach is adoptedfor training state duration models In this approach stateduration probabilities are estimated on the lattice (or trellis)of observations and states which is obtained in the DD-HSMM training stage

The mean 120583119894and variance 120590(119894) of duration probability of

state 119894 are determined by

120583 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

120590 (119894) =sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889) 119889

2

sum119879

119905=1sum119863119894

119889=1120594119905(119894 119889)

minus [120583 (119894)]2

(21)

In these equations 120594119905(119894 119889) is the probability of state 119894 at time

119905 with the duration of 119889119905(119894) = 119889 and 120594

119905(119894 119889) can present as

120594119905(119894 119889) =

1

119901119903

120572119905(119894 119889) 119901

119894(119889)

[[[

[

119873minus1

sum119895=2

119895 = 119894

119886119894119895(119889) 119887119895(119900119905+1) 120573119905+1

(119895 1)

+119886119894119873(119889) 120573119905(119873 119889119873)]]]

]

(22)

6 Mathematical Problems in Engineering

Table 1 Prognostics results

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3 Recognition accuracyNormal 0 29 1 0 0 967Degradation 1 2 27 1 0 90Degradation 2 0 1 28 1 933Failure 3 0 0 1 29 967Total accuracy 942

Table 2 State transition probability (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 06231 03473 00296Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

4 DD-HSMM Based Health Prognostic

Many applications in the actuarial econometric engineeringand medical literature involve the use of the hazard ratefunction [33] The mathematical properties of HR functioncan reveal a variety of features in the data

Let 119879 denote the time to failure of an item underconsideration with lifetime distribution function 119865(119905) andreliability function 119877(119905) where 119865(119905) + 119877(119905) = 1 and 119865(0) = 0Assume that 119865(0) = 0 and density function119891(119905) = 119865

1015840(119905) existthen the HR function can be defined as

120582 (119905) = lim119873rarrinfin

Δ119905rarr0

Δ119898 (119905)

[119873 minus 119898 (119905)] Δ119905=

119889119898 (119905)

[119873 minus 119898 (119905)] 119889119905

=119889119898 (119905) 119872

1 minus 119865 (119905)=119891 (119905)

119877 (119905)

(23)

In which 119872 is the total number of sample items 119898(119905) isthe number of items that fail before time 119905 and Δ119898(119905) is thenumber of items that fail during the time interval (119905 119905 + Δ119905)The ERL function 120583(119905) is the expected time remaining tofailure given that the system has survived to time 119905 then120583(119905) = 119864(119879 minus 119905 | 119879 gt 119905) = (1119877(119905)) int

infin

119905119877(119909)119889119909 for 119905

such that 119877(119905) gt 0 Therefore 120582(119905) can be approximated asthe conditional probability of failure during the time interval(119905 119905 + Δ119905) given survival to time 119905

Suppose that a machine will go through health states119894 (119894 = 1 2 119873minus 1) before entering failure state119873 Let119863(119894)denote the expected duration of themachine staying at healthstate 119894 based on the parameters estimated above we can get119863(119894) as follows

119863 (119894) = 120583 (119894) + 1205881205752(119894) (24)

And 120588 can be denoted by

120588 =(119879 minus sum

119871minus1

119897=0120583 (119894))

sum119871minus1

119897=01205752 (119894)

(25)

Then once the machine has entered the health state 119894 itsexpected residual life equals the summation of the expected

residual duration of the machine staying at health state 119894

and the total remaining staying in the future health statesbefore failure Denote119863(119894119889) as the expected residual durationof the machine staying in the health state 119894 for 119889 Whenthe equipment entered state 119894 at time 119905

119894 the conditional

probability of failure during (119905119894+ 119889 119905

119894+ (119889 + Δ119905)) can be

defined as the probability that the machine will transit to anyother state during the coming Δ119905 and the probability that themachine still stay at state 119894 It can be seen from (9) and (10)that (119905 + 119889)Δ119905 can be denoted as follows

(119905 + 119889) Δ119905 =120585119905(119894 119895 119889)

120574119905(119894 119889)

(26)

Then

119863(119894119889) = 119863 (119894) (1 minus (119905 + 119889) Δ119905)

= 119863 (119894) (1 minus120585119905(119894 119895 119889)

120574119905(119894 119889)

)

(27)

The DD-HSMM equipment health prediction procedure isgiven as follows

Step 1 From the DD-HSMM training procedure (ie param-eter estimation) the state transition probability for the DD-HSMM can be obtained

Step 2 Through the DD-HSMM parameter estimation theduration probability density function for each health-statecan be obtained Therefore the duration mean and variancecan be calculated

Step 3 By classification identify the current health status ofthe equipment

Step 4 The remaining useful life (RUL) of equipment canbe predicted by the following formula (suppose that theequipment currently stays at health state 119894 with duration of119889)

RUL(119889)119894

= 119863(119894119889) +

119873minus1

sum119895=119894+1

119863(119895) (28)

Mathematical Problems in Engineering 7

Table 3 Mean and variance of duration in each state (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 72105 71435 33435Variance 13953 07429 07924 05452

Table 4 State transition probability (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 04728 04154 01118Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

5 Case Study

In this case study long-term wear experiments on rollingelement bearings were conducted [1] In order to collectadequate amount of data sets for the validation of theproposed scheme three experiments with normal operatingconditions three experiments with cage defect fault andthree experiments each of inner and outer race defect faultswere performed until the bearing reached a complete failurestate and stopped operating Bearing characteristic frequen-cies in the frequency domain are extracted from the vibrationsignals corresponding to different degrees of the health statesof the bearing acquired during experiments

During the test running under each condition vibrationsignals were collected These signals were extracted usinga Mahalanobis-Taguchi System (MTS) based model in theoriginal paper [1] and used for the proposed DD-HSMMmethodology in this paper The expert judgment is made offour integer numbers ranging from 0 to 3 representing 4system states as follows

0rarr the bearing is operating normally

1rarr the bearing is operating and shows signs of deteriora-tion it is advisable to take some preventive action atthe next planned maintenance

2rarr the bearing is operating but requires immediate atten-tion

3rarr the bearing has failed

51 Operation State Identification In order to identify theaccuracy of the operation state identification method pro-posed in this paper experimental data with normal operatingcondition were obtained The experimental data set included50 samples for each state (denoted by 0 1 2 and 3) Of thesedata points 20 of them were used to train the model and theremaining 30 samples were used to validate the model

In theDD-HSMMmixtureGaussian distribution and thesingle Gaussian distribution were used to model the outputprobability distribution and the state duration densitiesseparately in which the number of states is 4 The maximumnumber of iterations in training process is set to 100 and theconvergence error to 0000001

minus80

minus90

minus100

minus110

minus120

minus130

minus140

0 5 10 15 20 25 30 35 40

Log-

likel

ihoo

d

Iterations

Normal (0)Contamination (1)

Contamination (2)Fail (3)

Figure 1 Training curve of the DD-HSMMmodel

The DD-HSMM-based training model is shown asFigure 1 The x-axis shows the training steps and the y-axisrepresents the likelihood probability of different states Ascan be seen from Figure 1 the progression of the four statesreaches the set error in less than 40 steps This demonstratesthe potential of the model to have a strong real-time signalprocessing capability

The classification results obtained on the remaining 30data samples are shown in Table 1 As indicated in the resultsthe accuracy of the DD-HSMMmethod is 942

52 Health Prediction for RUL As described before a four-state DD-HSMM prediction model is constructed In thetraining process even if the device is in the same runningcondition the dwell time is different transition probabilitiesbetween states and the mean or variance of duration ineach state are not the same Tables 2 and 3 show the statetransition probability the mean and variance of duration ineach state when 119889119905(1) = 1 representing the bearing in state1 with duration of 1 Tables 4 and 5 show the state transitionprobability the mean and variance of duration in each statewhen 119889119905(1) = 4 representing the bearing in state 1 withduration of 4

First the state 119894 of the current operating state based onthe recognition results is determined then the residence

8 Mathematical Problems in Engineering

Table 5 Mean and variance of duration in each state (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 53105 71435 33435Variance 13953 11328 07924 05452

Table 6 Comparison of DD-HSMM versus HSMM

Actual RUL DD-HSMMmodel HSMMmodelPredicted RUL Error () Predicted RUL Error ()

270000 264027 2212269852

00548240000 236596 1418 12438220000 220856 0389 2266170000 176142 3613 170734 0437150000 157246 4831 13829120000 116945 2546

10519912334

110000 100602 8544 436590000 91944 216 1688850000 51253 2506 36104 2779230000 30122 0407 20347

time sum119873minus1

119895=119894+1119863(119895) is calculated according to the duration

parameters of the operating state in training process Thenthe remaining effective life in the current operational stateis calculated using (25) Finally the RUL of the bearing canbe calculated using (26) Suppose that the bearing is now atstate 1 with a duration of 1 then the following can be obtained119863(2)+119863(3) = 109426119863(11) = 60875 by (25) and RUL(1)

1=

170211 by (26)

53 Prediction Comparison In order to compare the prog-nostic method based on the DD-HSMM with the prognosticmethod based on the HSMM (29) is used to evaluate thelife error In (29) RULactual represents the actual life of thecomponent and RULforecasted represents the expected lifepredicted by DD-HSMM or HSMM

Error =100 times

1003816100381610038161003816RULactual minus RULforecasted1003816100381610038161003816

RULactual (29)

Table 6 shows the prediction comparison of DD-HSMMversus HSMM Failure prediction of the HSMM method isonly state dependent while the DD-HSMM method usesboth state dependency and duration dependency The DD-HSMM method has a self-updating capability in which thehistorical data on states are used in the calculation of statetransition probability matrix As indicated in the resultsthe DD-HSMM method is more accurate than the HSMMmethod

6 Conclusion

This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics As opposed tothe Hidden Semi-MarkovModel (HSMM) failure predictioncapability of the DD-HSMM method uses state dependency

and duration dependency The two important aspects ofequipment health monitoring which are the stages and therate of aging are taken into consideration in an integratedmanner in the proposed DD-HSMM model The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant toreal world applications

In order to facilitate the computational procedure anew forward-backward algorithm and reestimation approachare developed By using autoregression the interdependencybetween observations is established in themodel By incorpo-rating an explicitly defined temporal structure into themodelthe DD-HSMM is capable of predicting the remaining usefullife of equipment more accurately

The demonstration of the proposed model is carried outusing experimental data on rolling element bearings Theproposed model provides a powerful state recognition capa-bility and very accurate results in terms of remaining usefullife prediction In order to draw general conclusion on thecapabilities of the proposed DD-HSMM more experimentaldata in various prognostics areas are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support forthis research from the National High Technology Researchand Development Program of China (no 2012AA040914)the National Natural Science Foundation of China (Grant no71101116) and the Basic Research Foundation of NPU (Grantno JC20120228)

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 1 Prognostics results

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3 Recognition accuracyNormal 0 29 1 0 0 967Degradation 1 2 27 1 0 90Degradation 2 0 1 28 1 933Failure 3 0 0 1 29 967Total accuracy 942

Table 2 State transition probability (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 06231 03473 00296Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

4 DD-HSMM Based Health Prognostic

Many applications in the actuarial econometric engineeringand medical literature involve the use of the hazard ratefunction [33] The mathematical properties of HR functioncan reveal a variety of features in the data

Let 119879 denote the time to failure of an item underconsideration with lifetime distribution function 119865(119905) andreliability function 119877(119905) where 119865(119905) + 119877(119905) = 1 and 119865(0) = 0Assume that 119865(0) = 0 and density function119891(119905) = 119865

1015840(119905) existthen the HR function can be defined as

120582 (119905) = lim119873rarrinfin

Δ119905rarr0

Δ119898 (119905)

[119873 minus 119898 (119905)] Δ119905=

119889119898 (119905)

[119873 minus 119898 (119905)] 119889119905

=119889119898 (119905) 119872

1 minus 119865 (119905)=119891 (119905)

119877 (119905)

(23)

In which 119872 is the total number of sample items 119898(119905) isthe number of items that fail before time 119905 and Δ119898(119905) is thenumber of items that fail during the time interval (119905 119905 + Δ119905)The ERL function 120583(119905) is the expected time remaining tofailure given that the system has survived to time 119905 then120583(119905) = 119864(119879 minus 119905 | 119879 gt 119905) = (1119877(119905)) int

infin

119905119877(119909)119889119909 for 119905

such that 119877(119905) gt 0 Therefore 120582(119905) can be approximated asthe conditional probability of failure during the time interval(119905 119905 + Δ119905) given survival to time 119905

Suppose that a machine will go through health states119894 (119894 = 1 2 119873minus 1) before entering failure state119873 Let119863(119894)denote the expected duration of themachine staying at healthstate 119894 based on the parameters estimated above we can get119863(119894) as follows

119863 (119894) = 120583 (119894) + 1205881205752(119894) (24)

And 120588 can be denoted by

120588 =(119879 minus sum

119871minus1

119897=0120583 (119894))

sum119871minus1

119897=01205752 (119894)

(25)

Then once the machine has entered the health state 119894 itsexpected residual life equals the summation of the expected

residual duration of the machine staying at health state 119894

and the total remaining staying in the future health statesbefore failure Denote119863(119894119889) as the expected residual durationof the machine staying in the health state 119894 for 119889 Whenthe equipment entered state 119894 at time 119905

119894 the conditional

probability of failure during (119905119894+ 119889 119905

119894+ (119889 + Δ119905)) can be

defined as the probability that the machine will transit to anyother state during the coming Δ119905 and the probability that themachine still stay at state 119894 It can be seen from (9) and (10)that (119905 + 119889)Δ119905 can be denoted as follows

(119905 + 119889) Δ119905 =120585119905(119894 119895 119889)

120574119905(119894 119889)

(26)

Then

119863(119894119889) = 119863 (119894) (1 minus (119905 + 119889) Δ119905)

= 119863 (119894) (1 minus120585119905(119894 119895 119889)

120574119905(119894 119889)

)

(27)

The DD-HSMM equipment health prediction procedure isgiven as follows

Step 1 From the DD-HSMM training procedure (ie param-eter estimation) the state transition probability for the DD-HSMM can be obtained

Step 2 Through the DD-HSMM parameter estimation theduration probability density function for each health-statecan be obtained Therefore the duration mean and variancecan be calculated

Step 3 By classification identify the current health status ofthe equipment

Step 4 The remaining useful life (RUL) of equipment canbe predicted by the following formula (suppose that theequipment currently stays at health state 119894 with duration of119889)

RUL(119889)119894

= 119863(119894119889) +

119873minus1

sum119895=119894+1

119863(119895) (28)

Mathematical Problems in Engineering 7

Table 3 Mean and variance of duration in each state (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 72105 71435 33435Variance 13953 07429 07924 05452

Table 4 State transition probability (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 04728 04154 01118Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

5 Case Study

In this case study long-term wear experiments on rollingelement bearings were conducted [1] In order to collectadequate amount of data sets for the validation of theproposed scheme three experiments with normal operatingconditions three experiments with cage defect fault andthree experiments each of inner and outer race defect faultswere performed until the bearing reached a complete failurestate and stopped operating Bearing characteristic frequen-cies in the frequency domain are extracted from the vibrationsignals corresponding to different degrees of the health statesof the bearing acquired during experiments

During the test running under each condition vibrationsignals were collected These signals were extracted usinga Mahalanobis-Taguchi System (MTS) based model in theoriginal paper [1] and used for the proposed DD-HSMMmethodology in this paper The expert judgment is made offour integer numbers ranging from 0 to 3 representing 4system states as follows

0rarr the bearing is operating normally

1rarr the bearing is operating and shows signs of deteriora-tion it is advisable to take some preventive action atthe next planned maintenance

2rarr the bearing is operating but requires immediate atten-tion

3rarr the bearing has failed

51 Operation State Identification In order to identify theaccuracy of the operation state identification method pro-posed in this paper experimental data with normal operatingcondition were obtained The experimental data set included50 samples for each state (denoted by 0 1 2 and 3) Of thesedata points 20 of them were used to train the model and theremaining 30 samples were used to validate the model

In theDD-HSMMmixtureGaussian distribution and thesingle Gaussian distribution were used to model the outputprobability distribution and the state duration densitiesseparately in which the number of states is 4 The maximumnumber of iterations in training process is set to 100 and theconvergence error to 0000001

minus80

minus90

minus100

minus110

minus120

minus130

minus140

0 5 10 15 20 25 30 35 40

Log-

likel

ihoo

d

Iterations

Normal (0)Contamination (1)

Contamination (2)Fail (3)

Figure 1 Training curve of the DD-HSMMmodel

The DD-HSMM-based training model is shown asFigure 1 The x-axis shows the training steps and the y-axisrepresents the likelihood probability of different states Ascan be seen from Figure 1 the progression of the four statesreaches the set error in less than 40 steps This demonstratesthe potential of the model to have a strong real-time signalprocessing capability

The classification results obtained on the remaining 30data samples are shown in Table 1 As indicated in the resultsthe accuracy of the DD-HSMMmethod is 942

52 Health Prediction for RUL As described before a four-state DD-HSMM prediction model is constructed In thetraining process even if the device is in the same runningcondition the dwell time is different transition probabilitiesbetween states and the mean or variance of duration ineach state are not the same Tables 2 and 3 show the statetransition probability the mean and variance of duration ineach state when 119889119905(1) = 1 representing the bearing in state1 with duration of 1 Tables 4 and 5 show the state transitionprobability the mean and variance of duration in each statewhen 119889119905(1) = 4 representing the bearing in state 1 withduration of 4

First the state 119894 of the current operating state based onthe recognition results is determined then the residence

8 Mathematical Problems in Engineering

Table 5 Mean and variance of duration in each state (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 53105 71435 33435Variance 13953 11328 07924 05452

Table 6 Comparison of DD-HSMM versus HSMM

Actual RUL DD-HSMMmodel HSMMmodelPredicted RUL Error () Predicted RUL Error ()

270000 264027 2212269852

00548240000 236596 1418 12438220000 220856 0389 2266170000 176142 3613 170734 0437150000 157246 4831 13829120000 116945 2546

10519912334

110000 100602 8544 436590000 91944 216 1688850000 51253 2506 36104 2779230000 30122 0407 20347

time sum119873minus1

119895=119894+1119863(119895) is calculated according to the duration

parameters of the operating state in training process Thenthe remaining effective life in the current operational stateis calculated using (25) Finally the RUL of the bearing canbe calculated using (26) Suppose that the bearing is now atstate 1 with a duration of 1 then the following can be obtained119863(2)+119863(3) = 109426119863(11) = 60875 by (25) and RUL(1)

1=

170211 by (26)

53 Prediction Comparison In order to compare the prog-nostic method based on the DD-HSMM with the prognosticmethod based on the HSMM (29) is used to evaluate thelife error In (29) RULactual represents the actual life of thecomponent and RULforecasted represents the expected lifepredicted by DD-HSMM or HSMM

Error =100 times

1003816100381610038161003816RULactual minus RULforecasted1003816100381610038161003816

RULactual (29)

Table 6 shows the prediction comparison of DD-HSMMversus HSMM Failure prediction of the HSMM method isonly state dependent while the DD-HSMM method usesboth state dependency and duration dependency The DD-HSMM method has a self-updating capability in which thehistorical data on states are used in the calculation of statetransition probability matrix As indicated in the resultsthe DD-HSMM method is more accurate than the HSMMmethod

6 Conclusion

This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics As opposed tothe Hidden Semi-MarkovModel (HSMM) failure predictioncapability of the DD-HSMM method uses state dependency

and duration dependency The two important aspects ofequipment health monitoring which are the stages and therate of aging are taken into consideration in an integratedmanner in the proposed DD-HSMM model The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant toreal world applications

In order to facilitate the computational procedure anew forward-backward algorithm and reestimation approachare developed By using autoregression the interdependencybetween observations is established in themodel By incorpo-rating an explicitly defined temporal structure into themodelthe DD-HSMM is capable of predicting the remaining usefullife of equipment more accurately

The demonstration of the proposed model is carried outusing experimental data on rolling element bearings Theproposed model provides a powerful state recognition capa-bility and very accurate results in terms of remaining usefullife prediction In order to draw general conclusion on thecapabilities of the proposed DD-HSMM more experimentaldata in various prognostics areas are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support forthis research from the National High Technology Researchand Development Program of China (no 2012AA040914)the National Natural Science Foundation of China (Grant no71101116) and the Basic Research Foundation of NPU (Grantno JC20120228)

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 3 Mean and variance of duration in each state (119889119905(1) = 1)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 72105 71435 33435Variance 13953 07429 07924 05452

Table 4 State transition probability (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Normal 0 08813 00454 00732 00001Degradation 1 00000 04728 04154 01118Degradation 2 00000 00000 09263 00737Failure 3 00000 00000 00000 10000

5 Case Study

In this case study long-term wear experiments on rollingelement bearings were conducted [1] In order to collectadequate amount of data sets for the validation of theproposed scheme three experiments with normal operatingconditions three experiments with cage defect fault andthree experiments each of inner and outer race defect faultswere performed until the bearing reached a complete failurestate and stopped operating Bearing characteristic frequen-cies in the frequency domain are extracted from the vibrationsignals corresponding to different degrees of the health statesof the bearing acquired during experiments

During the test running under each condition vibrationsignals were collected These signals were extracted usinga Mahalanobis-Taguchi System (MTS) based model in theoriginal paper [1] and used for the proposed DD-HSMMmethodology in this paper The expert judgment is made offour integer numbers ranging from 0 to 3 representing 4system states as follows

0rarr the bearing is operating normally

1rarr the bearing is operating and shows signs of deteriora-tion it is advisable to take some preventive action atthe next planned maintenance

2rarr the bearing is operating but requires immediate atten-tion

3rarr the bearing has failed

51 Operation State Identification In order to identify theaccuracy of the operation state identification method pro-posed in this paper experimental data with normal operatingcondition were obtained The experimental data set included50 samples for each state (denoted by 0 1 2 and 3) Of thesedata points 20 of them were used to train the model and theremaining 30 samples were used to validate the model

In theDD-HSMMmixtureGaussian distribution and thesingle Gaussian distribution were used to model the outputprobability distribution and the state duration densitiesseparately in which the number of states is 4 The maximumnumber of iterations in training process is set to 100 and theconvergence error to 0000001

minus80

minus90

minus100

minus110

minus120

minus130

minus140

0 5 10 15 20 25 30 35 40

Log-

likel

ihoo

d

Iterations

Normal (0)Contamination (1)

Contamination (2)Fail (3)

Figure 1 Training curve of the DD-HSMMmodel

The DD-HSMM-based training model is shown asFigure 1 The x-axis shows the training steps and the y-axisrepresents the likelihood probability of different states Ascan be seen from Figure 1 the progression of the four statesreaches the set error in less than 40 steps This demonstratesthe potential of the model to have a strong real-time signalprocessing capability

The classification results obtained on the remaining 30data samples are shown in Table 1 As indicated in the resultsthe accuracy of the DD-HSMMmethod is 942

52 Health Prediction for RUL As described before a four-state DD-HSMM prediction model is constructed In thetraining process even if the device is in the same runningcondition the dwell time is different transition probabilitiesbetween states and the mean or variance of duration ineach state are not the same Tables 2 and 3 show the statetransition probability the mean and variance of duration ineach state when 119889119905(1) = 1 representing the bearing in state1 with duration of 1 Tables 4 and 5 show the state transitionprobability the mean and variance of duration in each statewhen 119889119905(1) = 4 representing the bearing in state 1 withduration of 4

First the state 119894 of the current operating state based onthe recognition results is determined then the residence

8 Mathematical Problems in Engineering

Table 5 Mean and variance of duration in each state (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 53105 71435 33435Variance 13953 11328 07924 05452

Table 6 Comparison of DD-HSMM versus HSMM

Actual RUL DD-HSMMmodel HSMMmodelPredicted RUL Error () Predicted RUL Error ()

270000 264027 2212269852

00548240000 236596 1418 12438220000 220856 0389 2266170000 176142 3613 170734 0437150000 157246 4831 13829120000 116945 2546

10519912334

110000 100602 8544 436590000 91944 216 1688850000 51253 2506 36104 2779230000 30122 0407 20347

time sum119873minus1

119895=119894+1119863(119895) is calculated according to the duration

parameters of the operating state in training process Thenthe remaining effective life in the current operational stateis calculated using (25) Finally the RUL of the bearing canbe calculated using (26) Suppose that the bearing is now atstate 1 with a duration of 1 then the following can be obtained119863(2)+119863(3) = 109426119863(11) = 60875 by (25) and RUL(1)

1=

170211 by (26)

53 Prediction Comparison In order to compare the prog-nostic method based on the DD-HSMM with the prognosticmethod based on the HSMM (29) is used to evaluate thelife error In (29) RULactual represents the actual life of thecomponent and RULforecasted represents the expected lifepredicted by DD-HSMM or HSMM

Error =100 times

1003816100381610038161003816RULactual minus RULforecasted1003816100381610038161003816

RULactual (29)

Table 6 shows the prediction comparison of DD-HSMMversus HSMM Failure prediction of the HSMM method isonly state dependent while the DD-HSMM method usesboth state dependency and duration dependency The DD-HSMM method has a self-updating capability in which thehistorical data on states are used in the calculation of statetransition probability matrix As indicated in the resultsthe DD-HSMM method is more accurate than the HSMMmethod

6 Conclusion

This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics As opposed tothe Hidden Semi-MarkovModel (HSMM) failure predictioncapability of the DD-HSMM method uses state dependency

and duration dependency The two important aspects ofequipment health monitoring which are the stages and therate of aging are taken into consideration in an integratedmanner in the proposed DD-HSMM model The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant toreal world applications

In order to facilitate the computational procedure anew forward-backward algorithm and reestimation approachare developed By using autoregression the interdependencybetween observations is established in themodel By incorpo-rating an explicitly defined temporal structure into themodelthe DD-HSMM is capable of predicting the remaining usefullife of equipment more accurately

The demonstration of the proposed model is carried outusing experimental data on rolling element bearings Theproposed model provides a powerful state recognition capa-bility and very accurate results in terms of remaining usefullife prediction In order to draw general conclusion on thecapabilities of the proposed DD-HSMM more experimentaldata in various prognostics areas are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support forthis research from the National High Technology Researchand Development Program of China (no 2012AA040914)the National Natural Science Foundation of China (Grant no71101116) and the Basic Research Foundation of NPU (Grantno JC20120228)

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 5 Mean and variance of duration in each state (119889119905(1) = 4)

System state Normal 0 Degradation level 1 Degradation level 2 Failure 3Mean 79577 53105 71435 33435Variance 13953 11328 07924 05452

Table 6 Comparison of DD-HSMM versus HSMM

Actual RUL DD-HSMMmodel HSMMmodelPredicted RUL Error () Predicted RUL Error ()

270000 264027 2212269852

00548240000 236596 1418 12438220000 220856 0389 2266170000 176142 3613 170734 0437150000 157246 4831 13829120000 116945 2546

10519912334

110000 100602 8544 436590000 91944 216 1688850000 51253 2506 36104 2779230000 30122 0407 20347

time sum119873minus1

119895=119894+1119863(119895) is calculated according to the duration

parameters of the operating state in training process Thenthe remaining effective life in the current operational stateis calculated using (25) Finally the RUL of the bearing canbe calculated using (26) Suppose that the bearing is now atstate 1 with a duration of 1 then the following can be obtained119863(2)+119863(3) = 109426119863(11) = 60875 by (25) and RUL(1)

1=

170211 by (26)

53 Prediction Comparison In order to compare the prog-nostic method based on the DD-HSMM with the prognosticmethod based on the HSMM (29) is used to evaluate thelife error In (29) RULactual represents the actual life of thecomponent and RULforecasted represents the expected lifepredicted by DD-HSMM or HSMM

Error =100 times

1003816100381610038161003816RULactual minus RULforecasted1003816100381610038161003816

RULactual (29)

Table 6 shows the prediction comparison of DD-HSMMversus HSMM Failure prediction of the HSMM method isonly state dependent while the DD-HSMM method usesboth state dependency and duration dependency The DD-HSMM method has a self-updating capability in which thehistorical data on states are used in the calculation of statetransition probability matrix As indicated in the resultsthe DD-HSMM method is more accurate than the HSMMmethod

6 Conclusion

This paper presents a Duration-Dependent Hidden Semi-Markov Model (DD-HSMM) for prognostics As opposed tothe Hidden Semi-MarkovModel (HSMM) failure predictioncapability of the DD-HSMM method uses state dependency

and duration dependency The two important aspects ofequipment health monitoring which are the stages and therate of aging are taken into consideration in an integratedmanner in the proposed DD-HSMM model The duration-dependent state transition probability in the Hidden Semi-Markov model makes the decision-making more relevant toreal world applications

In order to facilitate the computational procedure anew forward-backward algorithm and reestimation approachare developed By using autoregression the interdependencybetween observations is established in themodel By incorpo-rating an explicitly defined temporal structure into themodelthe DD-HSMM is capable of predicting the remaining usefullife of equipment more accurately

The demonstration of the proposed model is carried outusing experimental data on rolling element bearings Theproposed model provides a powerful state recognition capa-bility and very accurate results in terms of remaining usefullife prediction In order to draw general conclusion on thecapabilities of the proposed DD-HSMM more experimentaldata in various prognostics areas are needed

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors gratefully acknowledge the financial support forthis research from the National High Technology Researchand Development Program of China (no 2012AA040914)the National Natural Science Foundation of China (Grant no71101116) and the Basic Research Foundation of NPU (Grantno JC20120228)

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

References

[1] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobistaguchi system (MTS) as a prognostics tool for rolling elementbearing failuresrdquo Journal of Manufacturing Science and Engi-neering vol 132 no 5 Article ID 051014 2010

[2] A Soylemezoglu S Jagannathan and C Saygin ldquoMahalanobis-Taguchi system as a multi-sensor based decision making prog-nostics tool for centrifugal pump failuresrdquo IEEE Transactions onReliability vol 60 no 4 pp 864ndash878 2011

[3] C Bunks DMcCarthy and T Al-Ani ldquoCondition-basedmain-tenance of machines using hiddenMarkovmodelsrdquoMechanicalSystems and Signal Processing vol 14 no 4 pp 597ndash612 2000

[4] POrth S Yacout and L Adjengue ldquoAccuracy and robustness ofdecision making techniques in condition based maintenancerdquoJournal of Intelligent Manufacturing vol 23 no 2 pp 255ndash2642012

[5] S Ambani L Li and J Ni ldquoCondition-based maintenancedecision-making for multiple machine systemsrdquo Journal ofManufacturing Science and Engineering vol 131 no 3 pp0310091ndash0310099 2009

[6] S Si H Dui Z Cai S Sun and Y Zhang ldquoJoint integratedimportance measure for multi-state transition systemsrdquo Com-munications in StatisticsTheory andMethods vol 41 no 21 pp3846ndash3862 2012

[7] S Shubin G Levitin D Hongyan and S Shudong ldquoCompo-nent state-based integrated importance measure for multi-statesystemsrdquo Reliability Engineering and System Safety vol 116 pp75ndash83 2013

[8] A K S JardineD Lin andD Banjevic ldquoA review onmachinerydiagnostics and prognostics implementing condition-basedmaintenancerdquoMechanical Systems and Signal Processing vol 20no 7 pp 1483ndash1510 2006

[9] A Heng S Zhang A C C Tan and J Mathew ldquoRotatingmachinery prognostics State of the art challenges and oppor-tunitiesrdquo Mechanical Systems and Signal Processing vol 23 no3 pp 724ndash739 2009

[10] L R Rabiner ldquoTutorial on hiddenMarkov models and selectedapplications in speech recognitionrdquo Proceedings of the IEEE vol77 no 2 pp 257ndash286 1989

[11] R J Elliott L Aggoun and J B MooreHiddenMarkovModelsEstimation and Control vol 29 Springer New York NY USA1995

[12] M D Le andCM Tan ldquoOptimalmaintenance strategy of dete-riorating system under imperfect maintenance and inspectionusing mixed inspection schedulingrdquo Reliability Engineering ampSystem Safety vol 113 pp 21ndash29 20132013

[13] Y Xu and M Ge ldquoHidden Markov model-based processmonitoring systemrdquo Journal of IntelligentManufacturing vol 15no 3 pp 337ndash350 2004

[14] M Ostendorf and S Roukos ldquoStochastic segment model forphoneme-based continuous speech recognitionrdquo IEEE Transac-tions on Acoustics Speech and Signal Processing vol 37 no 12pp 1857ndash1869 1989

[15] A Ljolje and S E Levinson ldquoDevelopment of an acoustic-phonetic hiddenMarkovmodel for continuous speech recogni-tionrdquo IEEE Transactions on Signal Processing vol 39 no 1 pp29ndash39 1991

[16] A Kannan and M Ostendorf ldquoComparison of trajectory andmixture modeling in segment-based word recognitionrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing pp 327ndash330 April 1993

[17] M Y Chen A Kundu and J Zhou ldquoOff-line handwrittenwork recognition using a hiddenMarkov model type stochasticnetworkrdquo IEEE Transactions on Pattern Analysis and MachineIntelligence vol 16 no 5 pp 481ndash496 1994

[18] M Y Chen A Kundu and S N Srihari ldquoVariable durationhidden Markov model and morphological segmentation forhandwritten word recognitionrdquo IEEE Transactions on ImageProcessing vol 4 no 12 pp 1675ndash1688 1995

[19] L Atlas M Ostendorf and G D Bernard ldquoHidden Markovmodels for monitoring machining tool-wearrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 3887ndash3890 June 2000

[20] L Wang M G Mehrabi and E Kannatey-Asibu Jr ldquoHiddenMarkov model-based tool wear monitoring in turningrdquo Journalof Manufacturing Science and Engineering vol 124 no 3 pp651ndash658 2002

[21] S Lee L Li and J Ni ldquoOnline degradation assessment andadaptive fault detection using modified hidden markov modelrdquoJournal of Manufacturing Science and Engineering vol 132 no2 pp 0210101ndash02101011 2010

[22] S Si H Dui Z Cai and S Sun ldquoThe Integrated ImportanceMeasure of Multi-State Coherent Systems for MaintenanceProcessesrdquo IEEE Transactions on Reliability vol 61 no 2 pp266ndash273 2012

[23] E Zio and M Compare ldquoEvaluating maintenance policies byquantitative modeling and analysisrdquo Reliability Engineering ampSystem Safety vol 109 no 203 pp 53ndash65 2013

[24] Z Cai S Sun S Si and B Yannou ldquoIdentifying product failurerate based on a conditional Bayesian network classifierrdquo ExpertSystems with Applications vol 38 no 5 pp 5036ndash5043 2011

[25] K Tokuda H Zen and A W Black ldquoAn HMM-based speechsynthesis system applied to Englishrdquo in Proceedings of the IEEEWorkshop on Speech Synthesis pp 227ndash230 2002

[26] H Zen K Tokuda T Masuko T Kobayasih and T KitamuraldquoA hidden semi-Markovmodel-based speech synthesis systemrdquoIEICE Transactions on Information and Systems vol 90 no 5pp 825ndash834 2007

[27] K Hashimoto Y Nankaku and K Tokuda ldquoA Bayesianapproach to hidden semi-Markov model based speech syn-thesisrdquo in Proceedings of the 10th Annual Conference of theInternational Speech Communication Association pp 1751ndash1754September 2009

[28] P Baruah and R B Chinnam ldquoHMMs for diagnostics andprognostics in machining processesrdquo International Journal ofProduction Research vol 43 no 6 pp 1275ndash1293 2005

[29] T Boutros and M Liang ldquoDetection and diagnosis of bearingand cutting tool faults using hidden Markov modelsrdquoMechan-ical Systems and Signal Processing vol 25 no 6 pp 2102ndash21242011

[30] M Dong and D He ldquoA segmental hidden semi-Markov model(HSMM)-based diagnostics and prognostics framework andmethodologyrdquoMechanical Systems and Signal Processing vol 21no 5 pp 2248ndash2266 2007

[31] M Dong and D He ldquoHidden semi-Markov model-basedmethodology for multi-sensor equipment health diagnosis andprognosisrdquo European Journal of Operational Research vol 178no 3 pp 858ndash878 2007

[32] S Yu ldquoHidden semi-MarkovmodelsrdquoArtificial Intelligence vol174 no 2 pp 215ndash243 2010

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

[33] M Dong ldquoA tutorial on nonlinear time-series data mining inengineering asset health and reliability prediction conceptsmodels and algorithmsrdquo Mathematical Problems in Engineer-ing vol 2010 Article ID 175936 22 pages 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of