The Scientific World Journal

Theory and Application on Rough Set, Fuzzy Logic, and Granular Computing

Guest Editors: Xibei Yang, Weihua Xu, and Yanhong She

Theory and Application on Rough Set,Fuzzy Logic, and Granular Computing

The Scientific World Journal

Theory and Application on Rough Set,Fuzzy Logic, and Granular Computing

Guest Editors: Xibei Yang, Weihua Xu, and Yanhong She

Copyright 2015 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in The ScientificWorld Journal. All articles are open access articles distributed under the Creative Com-mons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Contents

Theory and Application on Rough Set, Fuzzy Logic, and Granular Computing, Xibei Yang, Weihua Xu,and Yanhong SheVolume 2015, Article ID 967350, 1 page

Fault Detection and Diagnosis for Gas Turbines Based on a Kernelized Information Entropy Model,Weiying Wang, Zhiqiang Xu, Rui Tang, Shuying Li, and Wei WuVolume 2014, Article ID 617162, 13 pages

On Distribution Reduction and Algorithm Implementation in Inconsistent Ordered InformationSystems, Yanqin ZhangVolume 2014, Article ID 307468, 9 pages

Further Study of Multigranulation -Fuzzy Rough Sets, Wentao Li, Xiaoyan Zhang, and Wenxin SunVolume 2014, Article ID 927014, 18 pages

Approximation Set of the Interval Set in Pawlaks Space, Qinghua Zhang, Jin Wang, Guoyin Wang,and Feng HuVolume 2014, Article ID 317387, 12 pages

ANovel Method of the Generalized Interval-Valued Fuzzy Rough Approximation Operators,Tianyu Xue, Zhanao Xue, Huiru Cheng, Jie Liu, and Tailong ZhuVolume 2014, Article ID 783940, 14 pages

-Cut Decision-Theoretic Rough Set Approach: Model and Attribute Reductions, Hengrong Ju,Huili Dou, Yong Qi, Hualong Yu, Dongjun Yu, and Jingyu YangVolume 2014, Article ID 382439, 12 pages

EditorialTheory and Application on Rough Set, Fuzzy Logic, andGranular Computing

Xibei Yang,1 Weihua Xu,2 and Yanhong She3,4

1School of Computer Science and Engineering, Jiangsu University of Science and Technology, Jiangsu 212003, China2School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China3College of Science, Xian Shiyou University, Shaanxi 710065, China4Department of Computer Science, University of Regina, Regina, SK, Canada S4S 0A2

Correspondence should be addressed to Weihua Xu; chxuwh@gmail.com

Received 8 June 2015; Accepted 8 June 2015

Copyright 2015 Xibei Yang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently, the rough set and fuzzy set theory have generateda great deal of interest among more and more researchers.Granular computing (GrC) is an emerging computingparadigm of information processing and an approach forknowledge representation and data mining. The purpose ofgranular computing is to seek for an approximation schemewhich can effectively solve a complex problem at a certainlevel of granulation. This issue on the theory and applicationabout rough set, fuzzy logic and granular computing, mostof which are very meticulously performed reviews of theavailable current literature.

Four models of fuzzy or rough sets that are leading toa greater understanding of rough sets and fuzzy sets arediscussed. These include multigranulation T-fuzzy roughsets, the so called approximation set of the interval set, thegeneralized interval-valued fuzzy rough set, and the -cutdecision-theoretic rough set. Based on a kernelized infor-mation entropy model, an application on the fault detectionand diagnosis for gas turbines is presented. The methods forreductions and their relevant algorithms are addressed intwo manuscripts. Y. Zhang studies the distribution reductionin the inconsistent ordered information systems and furtherprovides its algorithm. H. Ju et al. firstly give the model of-cut decision-theoretic rough set and then investigate theattribute reductions in this new decision-theoretic rough setmodel.

From the view of GrC, the optimistic multigranulationT-fuzzy rough set model was established based on multiple

granulations under T-fuzzy approximation space by W. Xu.The manuscript of W. Li et al. improves the optimisticmultigranulation T-fuzzy rough set deeply by investigatingsome further properties. And the relationships betweenmultigranulation and classical T-fuzzy rough sets have beenstudied carefully. The interval set is a special fuzzy set, whichdescribes uncertainty of an uncertain concept with its twocrisp boundaries. Q. Zhang et al. review the similarity degreesbetween an interval-valued set and its two approximationsand propose disadvantages of using upper approximationset or lower approximation as approximation sets of theuncertain set and present a new method for looking for abetter approximation set of the interval set. T. Xue et al. alsoconstruct a novel model of the generalized fuzzy rough setunder interval-valued fuzzy relation.

The aim of this special issue is to encourage researchers inrelated areas to discuss and communicate the latest advance-ments of rough set, fuzzy logic, and GrC, which covers boththeoretical and practical results.

Xibei YangWeihua Xu

Yanhong She

Hindawi Publishing Corporatione Scientific World JournalVolume 2015, Article ID 967350, 1 pagehttp://dx.doi.org/10.1155/2015/967350

http://dx.doi.org/10.1155/2015/967350

Research ArticleFault Detection and Diagnosis for Gas Turbines Based on aKernelized Information Entropy Model

Weiying Wang,1,2 Zhiqiang Xu,1,2 Rui Tang,2 Shuying Li,2 and Wei Wu3

1 College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China2Harbin Marine Boiler & Turbine Research Institute, Harbin 150036, China3Harbin Institute of Technology, Harbin 150001, China

Correspondence should be addressed to Zhiqiang Xu; xuzhiqiang@hit.edu.cn

Received 28 April 2014; Accepted 19 June 2014; Published 28 August 2014

Academic Editor: Xibei Yang

Copyright 2014 Weiying Wang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Gas turbines are considered as one kind of the most important devices in power engineering and have been widely used in powergeneration, airplanes, and naval ships and also in oil drilling platforms. However, they are monitored without man on duty in themost cases. It is highly desirable to develop techniques and systems to remotely monitor their conditions and analyze their faults.In this work, we introduce a remote system for online condition monitoring and fault diagnosis of gas turbine on offshore oil welldrilling platforms based on a kernelized information entropy model. Shannon information entropy is generalized for measuringthe uniformity of exhaust temperatures, which reflect the overall states of the gas paths of gas turbine. In addition, we also extendthe entropy to compute the information quantity of features in kernel spaces, which help to select the informative features for acertain recognition task. Finally, we introduce the information entropy based decision tree algorithm to extract rules from faultsamples. The experiments on some real-world data show the effectiveness of the proposed algorithms.

1. Introduction

Gas turbines, mechanical systems operating on a thermo-dynamic cycle, usually with air as the working fluid, areconsidered as one kind of the most important devices inpower engineering, where the air is compressed, mixed withfuel, and burnt in a combustor, with the generated hot gasexpanded through a turbine to generate power, which is usedfor driving the compressor and for providing the means toovercome external loads. Gas turbines play an increasinglyimportant role in the domains of mechanical drives in the oiland gas sectors, electricity generation in the power sector, andpropulsion systems in the aerospace and marine sectors.

Safety and economy are always two fundamentally impor-tant factors in designing, producing, and operating gasturbine systems.Once amalfunction occurs to a gas turbine, aserious accident, even disaster,may take place. It was reportedthat about 25 accidents take place every year due to jetmalfunctioning. In 1989, 111 were killed in a plane crash dueto an engine fault. Although great progress has been madethese years in the area of condition monitoring and fault

diagnosis, how to predict and detect malfunctions is still anopen problem for the complex systems. In some cases, suchas offshore oil well drilling platforms, the main power systemis self-monitoring without man on duty. So the reliabilityand stabilization are of critical importance to these systems.There are hundreds of offshore platforms with gas turbinesproviding electricity and powers in China.There is an urgentrequirement to design and develop online remotemonitoringand health management techniques for these systems.

More than two hundred sensors are installed in eachgas turbine for monitoring the state of a gas turbine. Thedata gathered by these sensors reflects the state and trendof the system. If we build a center to monitor two hundredgas turbine systems, we should watch the data coming frommore than forty thousand sensors. Obviously, it is infeasibleto manually analyze them. Techniques on intelligent dataanalysis have been employed in gas turbine monitoring anddiagnosis. In 2007, Wang et al. designed a conceptual systemfor remote monitoring and fault diagnosis of gas turbine-based power generation systems [1]. In 2008, Donat et al.discussed the issue of data visualization, data reduction,

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 617162, 13 pageshttp://dx.doi.org/10.1155/2014/617162

http://dx.doi.org/10.1155/2014/617162

2 The Scientific World Journal

and ensemble learning for intelligent fault diagnosis in gasturbine engines [2]. In 2009, Li and Nilkitsaranont describeda prognostic approach to estimating the remaining useful lifeof gas turbine engines before their next major overhaul basedon a combined regression technique with both linear andquadratic models [3]. In the same year, Bassily et al. proposeda technique, which assessed whether or not the multivariateautocovariance functions of two independently sampledsignals coincide, to detect faults in a gas turbine [4]. In 2010,Young et al. presented an offline fault diagnosis method forindustrial gas turbines in a steady-state using Bayesian dataanalysis. The authors employed multiple Bayesian modelsvia model averaging for improving the performance of theresulted system [5]. In 2011, Yu et al. designed a sensorfault diagnosis technique for Micro-Gas Turbine Enginebased on wavelet entropy, where wavelet decomposition wasutilized to decompose the signal in different scales, and thenthe instantaneous wavelet energy entropy and instantaneouswavelet singular entropy are computed based on the previouswavelet entropy theory [6].

In recent years, signal processing and data miningtechniques are combined to extract knowledge and buildmodels for fault diagnosis. In 2012, Wu et al. studied theissue of bearing fault diagnosis based on multiscale per-mutation entropy and support vector machine [7]. In 2013,they designed a technique for defecting diagnostics basedon multiscale analysis and support vector machines [8].Nozari et al. presented a model-based robust fault detectionand isolation method with a hybrid structure, where time-delay multilayer perceptron models, local linear neurofuzzymodels, and linear model tree were used in the system [9].Sarkar et al. [10] designed symbolic dynamic filtering byoptimally partitioning sensor observation, and the objectiveis to reduce the effects of sensor noise level variation andmagnify the system fault signatures. Feature extraction andpattern classification are used for fault detection in aircraftgas turbine engines.

Entropy is a fundamental concept in the domains ofinformation theory and thermodynamics. It was first definedto be a measure of progressing towards thermodynamicequilibrium; then it was introduced in information theory byShannon [11] as a measure of the amount of information thatis missing before reception.This concept gets popular in bothdomains [1216]. Now it is widely used in machine learningand data driven modeling [17, 18]. In 2011, a new measure-ment, called maximal information coefficient, was reported.This function can be used to discover the association betweentwo random variables [19]. However, it cannot be used tocompute the relevance between feature sets.

In this work, we will develop techniques to detect abnor-mality and analyze faults based on a generalized informationentropy model. Moreover, we also describe a system forstate monitoring of gas turbines on offshore oil well drillingplatforms. First we will describe a system developed forremote and online condition monitoring and fault diagnosisof gas turbines installed on oil drilling platforms. As vastamount of historical records is gathered in this system, it isan urgent task to design algorithms for automatically onlinedetecting abnormality of the data and analyze the data to

obtain the causes and sources of faults. Due to the complexityof gas turbine systems, we focus on the gas-path subsystemin this work.The function of entropy is employed to measurethe uniformity of exhaust temperatures, which is a key factorreflecting the health of the gas path of a gas turbine and alsoreflecting the performance of the gas turbine.Thenwe extractfeatures from the healthy and abnormal records. An extendedinformation entropy model is introduced to evaluate thequality of these features for selecting informative attributes.Finally, the selected features are used to build models forautomatic fault recognition, where support vector machines[20] and C4.5 are considered. Real-world data are collectedto show the effectiveness of the proposed techniques.

The remainder of the work is organized as follows.Section 2 describes the architecture of the remotemonitoringand fault diagnosis center for gas turbines installed on theoil drilling platforms. Section 3 designs an algorithm fordetecting abnormality of the exhaust temperatures. Thenwe extract features from the exhaust temperature data andselect informative ones based on evaluating the informationbottlenecks with extend information entropy in Section 4.Support vector machines and C4.5 are introduced for build-ing fault diagnosis models in Section 5. In addition, numer-ical experiments are also described in this section. Finally,conclusions and future work are given in Section 6.

2. Framework of Remote Monitoring andFault Diagnosis Center for Gas Turbine

Gas turbines are widely used as power and electric powersources. The structure of a general gas turbine is presentedin Figure 1. This system transforms chemical energy intothermal power, then mechanical energy, and finally electricenergy. Gas turbines are usually considered as the hearts of alot of mechanical systems.

As the offshore oil well drilling platforms are usuallyunattended, an online and remote state monitoring systemis much useful in this area, which can help find abnormalitybefore serious faults occur. However, the sensor data cannotbe sent into a center with ground based internet.The data canonly be transmitted via telecommunication satellite, whichwas too expensive in the past. Now this is available.

The system consists of four subsystems: data acquisitionand local monitoring subsystem (DALM), data commu-nication subsystem (DAC), data management subsystem(DMS), and intelligent diagnosis system (IDS). The firstsubsystem gathers the outputs from different sensors andchecks whether there is any abnormality in the system. Thesecond one packs the acquired data and transforms theminto the monitoring center. Users in the center can also senda message to this subsystem to ask for some special data ifabnormality or fault occurs.The datamanagement subsystemstores the historic information and also fault data and faultcases. A data compression algorithm is embedded in thesystem. As most of the historic data are useless for the finalanalysis, they will be compressed and removed for savingstorage space. Finally, IDS watches the alarm informationfrom different unit assemblies and starts the correspondingmodule to analyze the related information. This system gives

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Exhaust

Turbine

Combustor

Compressor

Output shaft and gearbox

Titan 130Single shaft gas turbine forpower generation applications

Figure 1: Prototype structure of a gas turbine.

some decision and explains how the decision has been made.The structure of the system is shown in Figure 2.

One of the webpages of the system is given in Figure 3,where we can see the rose figure of exhaust temperatures,and some statistical parameters varying with time are alsopresented.

3. Abnormality Detection inExhaust Temperatures Based onInformation Entropy

Exhaust temperature is one of the most critical parameters ina gas turbine as excessive turbine temperatures may lead tolife reduction or catastrophic failures. In the current gener-ation of machines, temperatures at the combustor dischargeare too high for the type of instrumentation available. Exhausttemperature is also used as an indicator of turbine inlettemperature.

As the temperature profile out of a gas turbine is notuniform, a number of probes will help pinpoint disturbancesor malfunctions in the gas turbine by highlighting the shiftsin the temperature profile. Thus there are usually a set ofthermometers fixed on the exhaust. If the system is normallyoperating, all the thermometers give similar outputs. How-ever, if a fault occurs to some components of the turbine,different temperatures will be observed. The uniformity ofexhaust temperatures reflects the state of the system. So weshould develop an index to measure the uniformity of theexhaust temperatures. In this work, we consider the entropyfunction for it is widely used in measuring uniformity ofrandomvariables. However, to the best of our knowledge, thisfunction has not been used in this domain.

Assume that there are thermometers and their outputsare , = 1, . . . , , respectively. Then we define the unifor-

mity of these outputs as

() =

=1

log2

, (1)

where = . As

0, we define 0 log 0 = 0.

Obviously, we have log2 () 0. () = log

2 if and

only if 1= 2= =

. In this case, all the thermometers

produce the same output. So the uniformity of the sensorsis maximal. In another extreme case, if

1= 2= 1

=

+1

= = 0 and

= , then () = 0.

It is notable that the value of entropy is independentof the values of thermometers, while it depends on thedistribution of the temperatures. The entropy is maximal ifall the thermometers output the same values.

Now we show two sets of real exhaust temperatures mea-sured on an oil well drilling platform, where 13 thermometersare fixed. In the first set, the gas turbine starts from a timepoint and then runs for several minutes; finally the systemstops.

Observing the curves in Figure 4, we can see that the13 thermometers give the almost the same outputs at thebeginning. In fact, the outputs are the room temperature inthis case, as shown in Figure 6(a). Thus, the entropy reachesthe peak value.

Some typical samples are presented in Figure 6, wherethe temperature distributions around the exhaust at timepoints = 5,130,250,400, and 500 are given. Obviously, thedistributions at = 130,250, and 400 are not desirable. It canbe derived that some abnormality occurs to the system. Theentropy of temperature distribution is given in Figure 5.

4 The Scientific World Journal

Local monitoring Data acquiring Inner firewallOuter firewall

TelstarSatellite signal generator Satellite signal receiver

Inner firewall Outer firewall Internet

Online database subsystem

Video conference system

Internet

Client

Client

Client

Intelligent diagnosis subsystem

Human-system interface

Data communication

Oil drilling platforms

Gas turbine

Diagnosis center

Figure 2: Structure of the remote system of condition monitoring and fault analysis.

Figure 3: A typical webpage for monitoring of the subsystem.

0 50 100 150 200 250 300 350 400 450 500

0

100

200

300

400

500

600

700

Time

Tem

pera

ture

Figure 4: Exhaust temperatures from a set of thermometers.

0 50 100 150 200 250 300 350 400 450 500

Time

2.55

2.6

2.65

2.7

2.75

2.8

Info

rmat

ion

entro

py

Figure 5: Uniformity of the temperatures (red dash line is the idealcase; blue line is the real case).

The Scientific World Journal 5

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(b) Rose map of exhaust temperatures at =130

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(e) Rose map of exhaust temperatures at =500

Figure 6: Samples of temperature distribution in different times.

Another example is also given in Figures 7 to 9. In thisexample, there is significant difference between the outputsof 13 thermometers even when the gas turbine is not running,just as shown in Figure 9(a).Thus the entropy of temperaturedistribution is a little lower than the ideal case, as shown inFigure 8. Besides, some representative samples are also givenin Figure 9.

Considering the above examples, we can see that the func-tion of entropy is an effective measurement of uniformity. Itcan be used to reflect the uniformity of exhaust temperatures.

If the uniformity is less than a threshold, some faults possiblyoccur to the gas path of the gas turbine. Thus the entropyfunction is used as an index of the health of the gas path.

4. Fault Feature Quality Evaluation withGeneralized Entropy

The above section gives an approach to detecting the abnor-mality in the exhaust temperature distribution. However, thefunction of entropy cannot distinguish what kind of faults

6 The Scientific World Journal

0 500 1000 1500

0

100

200

300

400

500

600

700

Time

Tem

pera

ture

Figure 7: Exhaust temperatures from another set of thermometers.

0 500 1000 1500

Time

2.45

2.5

2.55

2.6

2.65

2.7

2.75

2.8

Info

rmat

ion

entro

py

Figure 8: Entropy of the temperature distribution, where the reddash line is the ideal case and the blue one is the real case.

occurs to the system although it detects abnormality. In orderto analyze why the temperature distribution is not uniform,we should develop some algorithms to recognize the fault.

Before training an intelligent model, we should constructsome features and select the most informative subsets torepresent different faults. In this section, we will discuss thisissue.

Intuitively, we know that the temperatures of all ther-mometers reflect the state of the system. Besides, the tem-perature difference between neighboring thermometers alsoindicates the source of faults, which are considered as spaceneighboring information. Moreover, we know the temper-ature change of a thermometer necessarily gives hints tostudy the faults, which can be viewed as time neighboringinformation. In fact, the inlet temperature

0is also an impor-

tant factor. In summary, we can use exhaust temperaturesand their neighboring information along time and space torecognize different faults. If there are ( = 13 in our system)thermometers, we can form a feature vector to describe thestate of the exhaust system as

= {0, 1, 2, . . . ,

, 1 2, 2 3, . . . ,

1,

1,

2, . . . ,

} ,

(2)

where = ()

( 1).

() is the temperature at time

of the th thermometer.Apart from the above features, we can also construct other

attributes to reflect the conditions of the gas turbine. In thiswork, we consider a gas turbine with 13 thermometers aroundthe exhaust. So we can form a 40-attribute vector finally.

There are some questions whether all the extractedfeatures are useful for finalmodeling and howwe can evaluatethe features and find the most informative features. In fact,there are a number of measures to estimate feature quality,such as dependency in the rough set theory [21], consistency[22], mutual information in the information theory [23], andclassification margin in the statistical learning theory [24].However, all these measures are computed in the originalinput space, while the effective classification techniquesusually implement a nonlinear mapping of the original spaceto a feature space by a kernel function. In this case, we requirea new measure to reflect the classification information ofthe feature space. Now we extend the traditional informationentropy to measure it.

Given a set of samples = {1, 2, . . . ,

}, each sample

is described with features = {1, 2, . . . ,

}. As to

classification learning, each training sample is associated

with a decision . As to an arbitrary subset and a

kernel function, we can calculate a kernel matrix

=[[

[

11

. . . 1

... d...

1

. . .

]]

]

, (3)

where = (

, ). The Gaussian function is a representa-

tive kernel function:

= exp(

2

) . (4)

A number of kernel functions have the properties(1) [0, 1]; (2)

= .

Kernel matrix plays a bottleneck role in kernel basedlearning [25]. All the information that a classification algo-rithm can use is hidden in this matrix. In the same time, wecan also calculate a decision kernel matrix as

=[[

[

11

. . . 1

... d...

1

. . .

]]

]

, (5)

where = 1 if

= ; otherwise,

= 0. In fact, the matrix

is a matching kernel.

Definition 1. Given a set of samples = {1, 2, . . . ,

}, each

sample is described with features = {1, 2, . . . ,

}.

, is a kernelmatrix over in terms of.Then the entropyof is defined as

() = 1

=1

log2

, (6)

where=

=1.

As to the above entropy function, if we use Gaussianfunction as the kernel, we have log

2 () 0. () = 0

if and only if = 1 , . () = log

2 if and only if

= 0,

= . () = 0 means that any pair of samples cannot be

The Scientific World Journal 7

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(a) Rose map of exhaust temperatures at = 500

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(c) Rose map of exhaust temperatures at = 820

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270

120

300

150

330

180 0

400

300

200

100

(d) Rose map of exhaust temperatures at =1220

Figure 9: Samples of temperature distribution in different moments.

distinguished with the current features, while () = log2

means any pair of samples is different from each other. Sothey can be distinguished. These are two extreme cases. Inreal-world applications, part of samples can be discernedwiththe available features, while others are not. In this case, theentropy function takes value in the interval [0, log

2].

Moreover, it is easy to show that if 1 2, (1)

(2), where

1 2means

1(, ) 2(, ), , .

Definition 2. Given a set of samples = {1, 2, . . . ,

},

each sample is described with features = {1, 2, . . . ,

}.

1, 2 .

1and

2are two kernel matrices induced by

1

and 2. is a new function computed with

1 2. Then the

joint entropy of 1and

2is defined as

(1, 2) = () =

1

=1

log2

, (7)

where=

=1.

As to the Gaussian function, (, ) =

1(, )

2(, ). Thus

1and

2. In this case, ()

(1) and () (

2).

Definition 3. Given a set of samples = {1, 2, . . . ,

},

each sample is described with features = {1, 2, . . . ,

}.One has

1, 2 .

1and

2are two kernel matrices

induced by 1and

2. is a new kernel function computed

with 1 2. Knowning

1, the condition entropy of

2is

defined as

(1| 2) = () (1) . (8)

As to the Gaussian kernel, () (1) and () (

2),

so (1| 2) 0 and (

2| 1) 0.

Definition 4. Given a set of samples = {1, 2, . . . ,

},

each sample is described with features = {1, 2, . . . ,

}.

One has 1, 2

. 1and

2are two kernel matrices

induced by 1and

2. is a new kernel function computed

with 1 2. Then the mutual information of

1and

2is

defined as

MI (1, 2) = (

1) + (

2) () . (9)

As to Gaussian kernel, MI(1, 2) = MI(

2, 1). If

1

2, we have MI(

1, 2) = (

2) and if

2 1, we have

MI(1, 2) = (

1).

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0 5 10 15 20 25 30 35 40 450

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Feature index

Dep

ende

ncy

Figure 10: Fuzzy dependency between a single feature and decision.

0 5 10 15 20 25 30 35 40 45

Feature index

Dep

ende

ncy

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 11: Kernelized mutual information between a single featureand decision.

Please note that if 1 2, we have

2 1. However,

2 1does not mean

1 2.

Definition 5. Given a set of samples = {1, 2, . . . ,

},

each sample is described with features = {1, 2, . . . ,

}.

, is a kernelmatrix over in terms of, and is the

kernel matrix computed with the decision. Then the featuresignificance related to the decision is defined as

MI (,) = () + () (,) . (10)

MI (,) measures the importance of feature subset in the kernel space to distinguish different classes. It can beunderstood as a kernelized version of Shannon informationentropy, which is widely used feature evaluation selection.In fact, it is easy to derive the equivalence between thisentropy function and Shannon entropy in the condition thatthe attributes are discrete and the matching kernel is used.

Now we show an example in gas turbine fault diagnosis.We collect 3581 samples from two sets of gas turbine systems.1440 samples are healthy and the others belong to four kindsof faults: load rejection, sensor fault, fuel switching, and saltspray corrosion. The numbers of samples are 45, 588, 71, and1437, respectively. Thirteen thermometers are installed in theexhaust. According to the approach described above, we forma 40-dimensional vector to represent the state of the exhaust.

1 2 3 4

0

0.2

0.4

0.6

0.8

1

Selected feature number

Fuzz

y de

pend

ency

0.3475

0.9006

0.99960.9969

Figure 12: Fuzzy dependency between the selected features anddecisions (Features 5, 37, 2, and 3 are selected sequentially).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Kern

eliz

ed m

utua

l inf

orm

atio

n

1.267

1.6121.5771.510

1 2 3 4

Number of selected features

Figure 13: Kernelized mutual information between the selectedfeatures and decisions (Features 39, 31, 38, and 40 are selectedsequentially).

Obviously, the classification task is not understandable insuch high dimensional space. Moreover, some features maybe redundant for classification learning, which may confusethe learning algorithm and reduce modeling performance.So it is a key preprocessing step to select the necessary andsufficient subsets.

Here we compare the fuzzy rough set based featureevaluation algorithm with the proposed kernelized mutualinformation. Fuzzy dependency has been widely discussedand applied in feature selection and attribute reduction theseyears [2628]. Fuzzy dependency can be understood as theaverage distance from the samples and their nearest neighborbelonging to different classes, while the kernelized mutualinformation reflects the relevance between features anddecision in the kernel space.

Comparing Figures 10 and 11, significant difference isobtained. As to fuzzy rough sets, Feature 5 produces thelargest dependency and then Feature 38. However, Feature39 gets the largest mutual information, and Feature 2 is thesecond one. Thus different feature evaluation functions willlead to completely different results.

Figures 10 and 11 present the significance of singlefeatures. In applications, we should combine a set of features.Now we consider a greedy search strategy. Starting from anempty set and the best features are added one by one. In

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0 0.5 1

0

0.5

1

0 0.5 1

0

0.5

1

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0

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Figure 14: Scatter plots in 2D space expended with feature pairs selected by fuzzy dependency.

each round, we select a feature which produces the largestsignificance increment with the selected subset. Both fuzzydependency and kernelized mutual information increasemonotonically if new attributes are added. If the selectedfeatures are sufficient for classification, these two functionswill keep invariant by adding any new attributes. So we canstop the algorithm if the increment of significance is less thana given threshold. The significances of the selected featuresubset are shown in Figures 12 and 13, respectively.

In order to show the effectiveness of the algorithm, wegive the scatter plots in 2D spaces, as shown in Figures 14 to16, which are expended by the feature pairs selected by fuzzydependency, kernelized mutual information, and Shannonmutual information. As to fuzzy dependency, we selectFeatures 5, 37, 2, and 3.Then there are 44 = 16 combinationsof feature pairs. The subplot in the th row and th column inFigure 14 gives the scatters of samples in 2D space expandedby the th selected feature and the th selected feature.

Observing the 2nd subplots in the first row of Figure 14,we can find that the classification task is nonlinear. The firstclass is dispersed and the third class is also located at different

regions, which leads to the difficulty in learning classificationmodels.

However, in the corresponding subplot of Figure 15, wecan see that each class is relatively compact, which leads to asmall intraclass distance.Moreover, the samples in five classescan be classified with some linear models, which also bringbenefit for learning a simple classification model.

Comparing Figures 15 and 16, we can find that differentclasses are overlapped in feature spaces selected by Shannonmutual information or get entangled, which leads to the badclassification performance.

5. Diagnosis Modeling with InformationEntropy Based Decision Tree Algorithm

After selecting the informative features, we now go to clas-sification modeling. There are a great number of learningalgorithms for building a classificationmodel. Generalizationcapability and interpretability are the two most importantcriteria in evaluating an algorithm. As to fault diagnosis, adomain expert usually accepts a model which is consistent

10 The Scientific World Journal

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Figure 15: Scatter in 2D space expended with feature pairs selected by kernelized mutual information.

with his common knowledge. Thus, he expects the modelis understandable; otherwise, he will not believe the outputsof the model. In addition, if the model is understandable, adomain expert can adapt it according to his prior knowledge,which makes the model suitable for different diagnosisobjects.

Decision tree algorithms, including CART [29], ID3[17], and C4.5 [18], are such techniques for training anunderstandable classification model. The learned model canbe transformed into a set of rules. All these algorithms builda decision tree from training samples. They start from a rootnode and select one of the features to divide the samples withcuts into different branches according to their feature values.This procedure is interactively conducted until the branch ispure or a stopping criterion is satisfied.The key difference liesin the evaluation function in selecting attributes or cuts. InCART, splitting rules GINI and Twoing are adopted, whileID3 uses information gain and C4.5 takes information gainratio.Moreover, C4.5 can deal with numerical attributes com-pared with ID3. Competent performance is usually observedwith C4.5 in real-world applications compared with somepopular algorithms, including SVM and Baysian net. In this

work, we introduce C4.5 to train classification models. Thepseudocode of C4.5 is formulated as follows.

Decision tree algorithm C4.5Input: a set of training samples = {

1, 2, . . . ,

}

with features = {1, 2, . . . ,

}

Stopping criterion Output: decision tree

(1) Check for sample set(2) For each attribute compute the normalized infor-

mation gain ratio from splitting on (3) Let f best be the attribute with the highest normalized

information gain(4) Create a decision node that splits on f best(5) Recurse on the sublists obtained by splitting on

f best, and add those nodes as children of node untilstopping criterion is satisfied

(6) Output .

The Scientific World Journal 11

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Figure 16: Scatter in 2D space expended with feature pairs selected by Shannon mutual information.

F2

0.50>0.50

F37 F2

0.41

Class 2

>0.18

Class 5

0.49

Class 1

>0.49

Class 4 F3

Class 3

>0.41

0.18

Figure 17: Decision tree trained on the features selected with fuzzyrough sets.

We input the data sets into C4.5 and build the followingtwo decision trees. Features 5, 37, 2, and 3 are included in thefirst dataset, and Features 39, 31, 38, and 40 are selected in thesecond dataset. The two trees are given in Figures 17 and 18,respectively.

F390.17>0.17

F39 F40

0.42

Class 3

>0.42

Class 5

0.45

Class 2

>0.45

F380.80>0.80

Class 1Class 4

Figure 18: Decision tree trained on the features selected withkernelized mutual information.

We start from the root node to a leaf node along thebranch, and then a piece of rule is extracted from the tree.As to the first tree, we can get five decision rules:

(1) if F2 > 0.50 and F37 > 0.49, then the decision is Class4;

12 The Scientific World Journal

(2) if F2 > 0.50 and F37 0.49, then the decision is Class1;

(3) if 0.18 < F2 0.50 and F3 > 0.41, then the decision isClass 5;

(4) if 0.18 < F2 0.50 and F3 0.41, then the decision isClass 3;

(5) if F2 0.18, then the decision is Class 2.

As to the second decision tree, we can also obtain somerules as

(1) if F39 > 0.45 and F38 > 0.80, then the decision is Class4;

(2) if F39 > 0.45 and F38 0.80, then the decision is Class1;

(3) if 0.17 < F39 0.45, then the decision is Class 2;(4) if F39 0.17 and F40 > 0.42, then the decision is Class

5;(5) if F39 0.17, and F40 0.42, then the decision is Class

3.

We can see the derived decision trees are rather simpleand each can extract five pieces of rules. It is very easy fordomain experts to understand the rules and even revise therules. As the classification task is a little simple, the accuracyof each model is high to 97%. As new samples and faults arerecorded by the system,more andmore complex tasksmay bestored. In that case, the model may become more and morecomplex.

6. Conclusions and Future Works

Automatic fault detection and diagnosis are highly desirablein some industries, such as offshore oil well drilling plat-forms, for such systems are self-monitoring without manon duty. In this work, we design an intelligent abnormalitydetection and fault recognition technique for the exhaustsystem of gas turbines based on information entropy, whichis used in measuring the uniformity of exhaust temperatures,evaluating the significance of features in kernel spaces, andselecting splitting nodes for constructing decision trees. Themain contributions of the work are two parts. First, weintroduce the entropy function to measure the uniformity ofexhaust temperatures. The measurement is easy to computeand understand. Numerical experiments also show its effec-tiveness. Second, we extend Shannon entropy for evaluatingthe significance of attributes in kernelized feature spaces. Wecompute the relevance between a kernel matrix induced witha set of attributes and the matrix computed with the decisionvariable. Some numerical experiments are also presented.Good results are derived.

Although this work gives an effective framework forautomatic fault detection and recognition, the proposedtechnique is not tested on large-scale real tasks. We havedeveloped a remote state monitoring and fault diagnosis sys-tem. Large scale data are flooding into the center. In thefuture, we will improve these techniques and develop areliable diagnosis system.

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work is partially supported by National Natural Founda-tion under Grants 61222210 and 61105054.

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Research ArticleOn Distribution Reduction and Algorithm Implementation inInconsistent Ordered Information Systems

Yanqin Zhang

School of Economics, Xuzhou Institute of Technology, Xuzhou, Jiangsu 221008, China

Correspondence should be addressed to Yanqin Zhang; zhiyuan1111@163.com

Received 19 May 2014; Accepted 11 August 2014; Published 28 August 2014

Academic Editor: Weihua Xu

Copyright 2014 Yanqin Zhang.This is an open access article distributed under the Creative CommonsAttribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

As one part of our work in ordered information systems, distribution reduction is studied in inconsistent ordered informationsystems (OISs). Some important properties on distribution reduction are studied and discussed. The dominance matrix isrestated for reduction acquisition in dominance relations based information systems. Matrix algorithm for distribution reductionacquisition is stepped. And program is implemented by the algorithm. The approach provides an effective tool for the theoreticalresearch and the applications for ordered information systems in practices. For more detailed and valid illustrations, cases areemployed to explain and verify the algorithm and the program which shows the effectiveness of the algorithm in complicatedinformation systems.

1. Introduction

In Pawlaks original rough set theory [1], partition orequivalence (indiscernibility) is an important and primitiveconcept. However, partition or equivalence relation, as theindiscernibility relation in Pawlaks original rough set theory,is still restrictive for many applications. To address this issue,several interesting and meaningful extensions to equivalencerelation have been proposed in the past, such as neigh-borhood operators [2], tolerance relations [3], and others[410]. Moreover, the original rough set theory does notconsider attributes with preference ordered domain, that is,criteria. In many real life practices, we often face problems inwhich the ordering of properties of the considered attributesplays a crucial role. One such type of problem is theordering of objects. For this reason, Greco et al. proposedan extension rough set theory, called the dominance basedrough set approach (DRSA), to take into account the orderingproperties of criteria [1116]. This innovation is mainly basedon substitution of the indiscernibility relation by a dominancerelation. Moreover, Greco et al. characterizes the DRSAand decision rules induced from rough approximations,while the usefulness of the DRSA and its advantages overthe CRSA (classical rough set approach) are presented [1116]. In DRSA, condition attributes are criteria and classes

are preference ordered. Several studies have been madeabout properties and algorithmic implementations of DRSA[10, 1719].

Nevertheless, only a limited number of methods usingDRSA to acquire knowledge in inconsistent ordered infor-mation systems have been proposed and studied. Pioneeringwork on inconsistent ordered information systems with theDRSA has been proposed by Greco et al. [1116], but they didnot clearly point out the semantic explanation of unknownvalues. Shao and Zhang [20] further proposed an extensionof the dominance relation in incomplete ordered informationsystems. Their work was established on the basis of theassumption that all unknown values are lost. Despite this,they did not mention the underlying concept of attributereduction in inconsistent ordered decision system but theymentioned an approach to attribute reduction in consistentordered information systems. Therefore, the purpose ofthis paper is to develop approaches to attribute reductionsin inconsistent ordered information systems (IOIS). In thispaper, theories and approaches of distribution reduction areinvestigated in inconsistent ordered information systems.Furthermore, algorithm of matrix computation of distribu-tion reduction is introduced, from which we can provide anew approach to attributes reductions in inconsistent orderedinformation systems.

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 307468, 9 pageshttp://dx.doi.org/10.1155/2014/307468

http://dx.doi.org/10.1155/2014/307468

2 The Scientific World Journal

The rest of this paper is organized as follows. Somepreliminary concepts are briefly recalled in Section 2. InSection 3, theories and approaches of distribution reductionare investigated in IOIS. In Section 4, we restate the defini-tion of dominance matrix in ordered information systemsand step the matrix algorithm for distribution reductionacquisition. Preparations are implemented to place the algo-rithm and the program is designed. The algorithm andthe corresponding program we designed can provide a toolto theoretical research and applications of criterion basedinformation system. Cases are employed to illustrate thealgorithm and the program in Section 5. It is shown thatthe algorithm and program are effective in complicatedinformation system. Furthermore conclusions on what westudy in this paper are drawn to understand this paper briefly.

2. Ordered Information Systems

The following recalls necessary concepts and preliminariesrequired in the sequel of our work. Detailed description ofthe theory can be found in [1116].

An information system with decisions is an orderedquadruple I = (, , , ), where = {

1, 2, . . . ,

}

is a nonempty finite set of objects; is a nonempty finiteattributes set; = {

1, 2, . . . ,

} denotes the set of condition

attributes; = {1, 2, . . . ,

} denotes the set of decision

attributes, = ; = {|

, },

() is the

value of on ;

is the domain of

, ; = {

|

, },

() is the value of

on ,

is

the domain of , . In an information system, if the

domain of an attribute is ordered according to a decreasingor increasing preference, then the attribute is a criterion. Aninformation system is called an ordered information system(OIS) if all condition attributes are criterions.

Assume that the domain of a criterion is completelypreordered by an outranking relation

; then

means

that is at least as good as with respect to criterion .And we can say that dominates . In the following, withoutany loss of generality, we consider condition and decisioncriterions having a numerical domain; that is,

R (R

denotes the set of real numbers).We define by (, ) (, ) according to

increasing preference, where and , . For a subsetof attributes ,

means that

for any .

That is to say dominates with respect to all attributesin . Furthermore, we denote

by

. In general,

we indicate an ordered information system with decision byI = (, , , ). Thus the following definition can beobtained.

Let I = (, , , ) be an ordered informationsystem with decisions, for ; denote

= {(

, ) |

() () , } ;

= {(

, ) |

()

() ,

} .

(1)

and

are called dominance relations of information

systemI.

Table 1: An ordered information system.

1

2

3

1

1 2 1 32

3 2 2 23

1 1 2 14

2 1 3 25

3 3 2 36

3 2 3 1

If we denote

[]

= { | (

, )

}

= { |

() () , } ,

[]

= { | (

, )

}

= { |

()

() ,

} ,

(2)

then the following properties of a dominance relation aretrivial.

Let be a dominance relation.The following properties

hold.(1) is reflexive and transitive, but not symmetric, so it

is not an equivalence relation.(2) If , then

.

(3) If , then []

[]

.

(4) If []

, then [

]

[]

and [

]

= {[

]

|

[]

}.

(5) []

= []

if and only if (

, ) = (

, ) (

).(6) J = {[]

| } constitute a covering of .

For any subset of , and ofI, define

() = { []

} ,

() = { | []

= } .

(3)

() and

() are said to be the lower and upper approxi-

mations of with respect to a dominance relation . And

the approximations have also some properties which aresimilar to those of Pawlak approximation spaces.

For an ordered information system with decisions I =(, , , ), if

, then this information system is

consistent, otherwise, this information system is inconsistent(IOIS).

Example 1. An ordered information system is given inTable 1.

From the table, we have

[1]

= {1, 2, 5, 6} ; [

2]

= {2, 5, 6} ;

[3]

= {2, 3, 4, 5, 6} ; [

4]

= {4, 6} ;

The Scientific World Journal 3

[5]

= {5} ; [

6]

= {6} ;

[1]

= [5]

= {1, 5} ;

[2]

= [4]

= {1, 2, 4, 5} ;

[3]

= [6]

= {1, 2, 3, 4, 5, 6} .

(4)

Obviously, by the above, we have

, so the system

in Table 1 is inconsistent.For a simple description, the following information sys-

tem with decisions is based on dominance relations, that is,ordered information system.

3. Theories of Distribution Reduction inInconsistent Ordered Information Systems

Let I = (, , , ) be an information system withdecisions, and

,

dominance relations derived from

condition attributes set and decision attributes set ,respectively. For , denote

= {[]

| } ,

= {1, 2, . . . ,

} ,

() = (

1 []

||,

2 []

||, . . . ,

[]

||) ,

() = max{

1 []

||,

2 []

||, . . . ,

[]

||} ,

(5)

where []

= { | (, )

}. Furthermore, we let

() be a distribution function about attributions set and

() maximum distribution function about attributions set

.

Definition 2. Let = (1, 2, . . . ,

) and = (

1, 2, . . . ,

)

be two vectors with dimensions. If = , ( = 1, 2, . . . , ),

we say that is equal to and is denoted by = . If

, ( = 1, 2, . . . , ), we say that is less than and is denoted

by . Otherwise, if it exists 0(0 {1, 2, . . . , }) such that

0

> 0

, we say is not less than and it is denoted by ,such as (1, 2, 3) (1, 1, 4) and (1, 1, 4) (1, 2, 3).From the above, we can have the following propositions

immediately.

Proposition 3. Let I = (, , , ) be an inconsistentinformation system.

(1) If , then ()

(), .

(2) If , then ()

(), .

(3) If [] []

, then

()

(), , .

(4) If [] []

, then

()

(), , .

Definition 4. Let I = (, , , ) be an inconsistentinformation system. If

() =

(), for all , we

say that is a distribution consistent set of I. If is adistribution consistent set, and no proper subset of is adistribution consistent set, then is called a distributionconsistent reduction ofI.

Definition 5. Let I = (, , , ) be an inconsistentinformation system. If

() =

(), for all , we say

that is a maximum distribution consistent set of I. If is a maximum distribution set, and no proper subset of is a maximum distribution consistent set, then is called amaximum distribution consistent reduction ofI.

Example 6. For the system in Table 1, if we denote

1= [1]

= [5]

,

2= [2]

= [4]

,

3= [3]

= [6]

,

(6)

then we can have

(1) = (

1

3,1

2,2

3) ;

(2) = (

1

6,1

3,1

2) ;

(3) = (

1

6,1

2,5

6) ;

(4) = (0,

1

6,1

3) ;

(5) = (

1

6,1

6,1

6) ;

(6) = (0, 0,

1

6) ,

(1) =

2

3;

(2) =

1

2;

(3) =

5

6;

(4) =

1

3;

(5) =

1

6;

(6) =

1

6.

(7)

When = {2, 3}, it can be easily checked that []

= []

,

for all , so that() =

() and

() =

()

are true and = {2, 3} is a distribution consistent set

of I. Furthermore, we can examine that {2} and {

3}are

not consistent sets of I. That is to say = {2, 3}

is a distribution reduction and is a maximum distributionreduction ofI.

Moreover, it can easily be calculated that = {1, 3} and

= {1, 2} are not distribution consistent sets ofI. Thus

there exist only one distribution reduction and maximumdistribution reduction of I in the system of Table 1, whichare {

2, 3}.

4 The Scientific World Journal

The distribution consistent set and the maximum distri-bution consistent set are related in the following theorem.

Theorem 7. Let I = (, , , ) be an orderedinformation system and is a distribution consistent setofI if and only if is a maximum distribution consistent setofI.

Proof. It can be proved immediately from corresponding def-initions and properties. From the definitions of distributionand maximum distribution consistent set, the key results ofthe implication is that []

= []

always holds for any

while is a distribution consistent set or maximumdistribution consistent set.Thus, the theorem can be acquiredimmediately.

Theorem 8. Let I = (, , , ) be an orderedinformation system.

: is a distribution consistent set ofI.: While

()

(), []

[]

holds for any

, .

Then we have .

Proof. We will prove . Assume that when ()

(), []

[]

does not hold and that implies []

[]

. So we can obtain

()

() by Proposition 3(3). On

the other hand, since is a distribution consistent set ofI,we have

() =

() and

() =

(). Hence we can

get ()

(), which is a contradiction. The theorem is

proved.

The distribution consistent set requires that the classifi-cation ability of the consistent remains the same with theoriginal data table. That is, , which is a distributionconsistent set of, must satisfy the fact that []

= []

holds

for any . This is very strict and other reductions studiedin [21] may not reach this special condition.

4. Matrix Algorithm for DistributionReduction Acquisition in InconsistentOrdered Information Systems

In this section, the dominance matrices will be put as arestatement and matrices will be employed to realize thecalculation of distribution reductions.

Definition 9. Let I = (, , , ) be an orderedinformation system, and . Denote

= ()

, where

= {1,

[]

,

0, otherwise.(8)

The matrix is called dominance matrix of attributes set

. If || = , we say that the order of is .

Definition 10. Let I = (, , , ) be an orderedinformation system and

and

are dominancematrices

of attributes sets , . The intersection of and

is

defined by

= ()

(

)

= (min {,

})

.

(9)

The intersection defined above can be implemented bythe operator . in Matlab platform,

= . ,

that is, the product of elements in corresponding positions.Then the following properties are obvious.

Proposition 11. Let ,

be dominance matrices ofattributes sets , ; the following results always hold.

(1) = 1.

(2)

= .

From the above, we can see that a dominance relation ofobjects has one-one correspondence to a dominance matrix.The combination of dominance relations can be realizedby the corresponding matrices and the dominance relationscan be compared by the corresponding matrices from thefollowing definitions.

Definition 12. Let

= (1, 2, . . . ,

) and

= (1, 2,

. . . , ) be matrices with dimensions and

and

row

vectors, respectively. If holds, for any , we say

that is less than

and it is denoted by

.

By the definitions, dominance matrices have the follow-ing properties straightly.

Proposition 13. Let I = (, , , ) be an orderedinformation system and . The dominance matrices withrespect to and are, respectively,

and

. Then

.

In the following, we give the preparation of matrix com-putation for distribution reductions in ordered informationsystems.

Proposition 14. LetI = (, , , ) be an ordered infor-mation system and = {

1, 2, . . . ,

} and = {

1, 2, . . . ,

}. Then

=

=1

{}= (

11

12

1

21

22

2

...... d

...1

2

) (10)

and any vector = (1, 2, . . . ,

) represents the dominance

class of object by the values 0 and 1, where 0 means the object

not included in the class and 1 means the object included in theclass.

The Scientific World Journal 5

Input: An inconsistent ordered information systemI = (, ,, ), where = {1, 2, . . . ,

} and = {

1, 2, . . . ,

}.

Output: All distribution reductions ofI.Step 1. Load the ordered information system and simplify the system by combining the objects with same values of everyattribute,Step 2. Classify by every single criterion and store then in separate matrices

= (

11

12

1

21

22

2

...... d

...

1

2

),

= (

11

12

1

21

22

2

...... d

...1

2

).

Step 3. Check the consistence of the information system

=

=1

= 1.

2. .

= (

11

12

1

21

22

2

...... d

...1

2

).

where . is the operator in Matlab platform. If

, the system is consistent, terminate the algorithm.

Else the system is inconsistent, go to the next step.Step 4. Acquire the consistent set. Let = {

1, 2, ,

}

=

=1

= 1. 2. .

. = (

11

12

1

21

22

2

...... d

...1

2

).

If = , is a consistent set, store the set into the temporary storage cell. Else fetch another subset of and repeat this step.

Calculate till all subsets of are verified, then go to the next step.Step 5. Sort the consistent sets in the storage cell and find out the minimum consistent sets which are just the reductions.Output all reductions and terminate the algorithm.

Algorithm 1

Theorem 15. Let I = (, , , ) be an orderedinformation system and . is a consistent set if and onlyif = M.

Proof. As is known, []

[]

holds since .

() For is a distribution consistent set, one can have= . Then, for any and

, we have |

[]

| = |

[]

|. Since []

[]

, it is obvious that []

= []

.That is,

the row vectors inand

are correspondingly the same.

Then = .

() Since

= , we can easily obtain that []

=

[]

holds for any and

. Then |

[]

| = |

[]

|

holds for any and . We can obtain that

() =

()

holds for any . That is, is a distribution consistent set.To acquire reductions in inconsistent ordered informa-

tion system, the matrices can be the only forms of storagein computing. And we illustrate the progress to calculate thereductions as shown in Algorithm 1.

The algorithm and the distribution reduction allow usto calculate reductions which keep the classification ability

the same with the original system in a brief way. And we donot need to acquire every approximation of the decisions.Itshortens the computing time and provides an effective toolfor knowledge acquisition in criterion based rough set theory.The flow chart of the Algorithm 1 can be designed and it isplaced in Figure 1.Analysis to Time Complexity of Algorithm 1. Let I =(, , , ) be an ordered information system. ={1, 2, . . . ,

} is the simplified universe. The number of

objects in original information system not being simplifiedis denoted by

1. There are condition attributes in ; that

is, || = . The number of compressed decision classes is. We take a variable

to stand for the time complexity in

an implementation. In the next, we can analyze the timecomplexity of Algorithm 1 step by step.

The time complexity to simplify the original informationsystem is 2

1for any two objects being compared and is

denoted by 1

= 2

1. Since || = , || = , and || = 1,

the time complexities to be classified by condition attributesand decision are, respectively,

2= ||2 || and

3= ||2.

6 The Scientific World Journal

Inconsistent

End:

Consistent

Begin: Input data table

and simplify it.

Classify by every single criterion and store them in

separate matrices.

Is the system consistent?Term

inate the program.

Sort and outputall reductions.

Temporary storage of consistent sets.

Whether B is aconsistent set?

B = (b1, b2, , bm).

Figure 1: The flow chart of Algorithm 1.

For decision classes being merged by comparing classes ofany two objects, the time complexity is

4= ||

2. Now theconsistency of the information system needs to be checkedby comparing the condition class and decision class of anyobject. If the information system is consistent, the time com-plexity to check consistency is ||. If the information systemis inconsistent, the time complexity to check consistency isless than ||. Thus, the time complexity to check consistencyis no more than ||; that is, it is presented as

5 ||. Then,

the possible and compatible distribution functions can becalculated and the time complexity is

6= 2 ||. The time

complexity to calculate each of these two functions is ||and is denoted by

6=

6= ||. The analysis to Step 1 is

finished.For Step 2, the time complexity to calculate possible

and compatible distribution decision matrices, respectively,is denoted by

7=

7= ||

2. Thus, the time complexity tocalculate distribution decision matrices is

7= 2|

2|. The

time complexity of Step 2 is completed.The first two steps are preparations to calculate reduc-

tions. The next Step 3 to Step 5 are the steps which runthe operations. There are C1

= subsets {

} and the

dominance matrices are with dimensions . In addition,the representation C

is the combinatorial number which

means the number of selections to chose elements from

Table 2

Names Models ParametersCPU Intel Core i3-380U 1.33GHzMemory DDR3 SDRAM 2 2GB 1333MHzHard disk TOSHIBA 320GBSystem Windows 7 32 bitPlatform Matlab Leasehold

ones. We consider that the judgement of a vector if it is zeroruns one operation and the comparison of two vectors runsaccording to the dimension of the vectors.Therefore, the timecomplexities to compare

and

with

{}, respectively,

are ||2. And the time complexity to compare every linevector of

{}with zero is ||. The possible and compatible

distribution matrices are obtained by reassignment values times. And the time complexities to process possible andcompatible distribution matrices, respectively, are both .Then, we have that the total time complexity of Step 3 is8= C1(3||

2+3||).The judgement in Step 4 just need to

run according to the number of {} and the time complexity

is 9= 2C1.

Since we just need to compute the intersection of nonzero1st order possible (or compatible) distribution matrices, the

The Scientific World Journal 7

Table 3:I: An inconsistent ordered information system on animals sleep.

(, {}) 1

2

3

4

5

6

7

8

9

1: African giant pouched rat 1 6.6 6.3 2 8.3 4.5 42 3 1 3

2: Asian elephant 2547 4603 2.1 1.8 3.9 69 624 3 5 4

3: Baboon 10.55 179.5 9.1 0.7 9.8 27 180 4 4 4

4: Big brown bat 0.023 0.3 15.8 3.9 19.7 19 35 1 1 1

5: Brazilian tapir 160 169 5.2 1 6.2 30.4 392 4 5 4

6: Cat 3.3 25.6 10.9 3.6 14.5 28 63 1 2 1

7: Chimpanzee 52.16 440 8.3 1.4 9.7 50 230 1 1 1

8: Chinchilla 0.425 6.4 11 1.5 12.5 7 112 5 4 4

9: Cow 465 423 3.2 0.7 3.9 30 281 5 5 5

10: Eastern American mole 0.075 1.2 6.3 2.1 8.4 3.5 42 1 1 1

11: Echidna 3 25 8.6 0 8.6 50 28 2 2 2

12: European hedgehog 0.785 3.5 6.6 4.1 10.7 6 42 2 2 2

13: Galago 0.2 5 9.5 1.2 10.7 10.4 120 2 2 2

14: Goat 27.66 115 3.3 0.5 3.8 20 148 5 5 5

15: Golden hamster 0.12 1 11 3.4 14.4 3.9 16 3 1 2

16: Gray seal 85 325 4.7 1.5 6.2 41 310 1 3 1

17: Ground squirrel 0.101 4 10.4 3.4 13.8 9 28 5 1 3

18: Guinea pig 1.04 5.5 7.4 0.8 8.2 7.6 68 5 3 4

19: Horse 521 655 2.1 0.8 2.9 46 336 5 5 5

20: Lesser short-tailed shrew 0.005 0.14 7.7 1.4 9.1 2.6 21.5 5 2 4

21: Little brown bat 0.01 0.25 17.9 2 19.9 24 50 1 1 1

22: Man 62 1320 6.1 1.9 8 100 267 1 1 1

23: Mouse 0.023 0.4 11.9 1.3 13.2 3.2 19 4 1 3

24: Musk shrew 0.048 0.33 10.8 2 12.8 2 30 4 1 3

25: N. American opossum 1.7 6.3 13.8 5.6 19.4 5 12 2 1 1

26: Nine-banded armadillo 3.5 10.8 14.3 3.1 17.4 6.5 120 2 1 1

27: Owl monkey 0.48 15.5 15.2 1.8 17 12 140 2 2 2

28: Patas monkey 10 115 10 0.9 10.9 20.2 170 4 4 4

29: Phanlanger 1.62 11.4 11.9 1.8 13.7 13 17 2 1 2

30: Pig 192 180 6.5 1.9 8.4 27 115 4 4 4

31: Rabbit 2.5 12.1 7.5 0.9 8.4 18 31 5 5 5

32: Rat 0.28 1.9 10.6 2.6 13.2 4.7 21 3 1 3

33: Red fox 4.235 50.4 7.4 2.4 9.8 9.8 52 1 1 1

34: Rhesus monkey 6.8 179 8.4 1.2 9.6 29 164 2 3 2

35: Rock hyrax (Hetero.b) 0.75 12.3 5.7 0.9 6.6 7 225 2 2 2

36: Rock hyrax (Procavia hab) 3.6 21 4.9 0.5 5.4 6 225 3 2 3

37: Sheep 55.5 175 3.2 0.6 3.8 20 151 5 5 5

38: Tenrec 0.9 2.6 11 2.3 13.3 4.5 60 2 1 2

39: Tree hyrax 2 12.3 4.9 0.5 5.4 7.5 200 3 1 3

40: Tree shrew 0.104 2.5 13.2 2.6 15.8 2.3 46 3 2 2

41: Vervet 4.19 58 9.7 0.6 10.3 24 210 4 3 4

42: Water opossum 3.5 3.9 12.8 6.6 19.4 3 14 2 1 1

8 The Scientific World Journal

Table 4

1Body weight in kg;

2Brain weight in g;

3Show wave (nondreaming) sleep (hrs/day);

4Paradoxical (dreaming) sleep (hrs/day);

5Total sleep (hrs/day);

6Maximum life span (years);

7Gestation time (days);

8Predation index (15);

9Sleep exposure index (15);

Overall danger index (15).

maximum time complexities can be analyzed in the next stepsbut not the true ones in computing. Therefore, themaximumtime complexity relies on the number of attribute subsets 2||.The worst case is that no minimum reduction exists in theinformation system and all 2|| subsets are calculated in thealgorithm. Thus, the maximum time complexity of Step 5 is10

= 2C2

||2.

From the above analysis, we can know that the maximumtime complexity of the main part in the algorithm (Step 3 toStep 5) is main = 8 + 9 + 10 = || (||

2+ ||).

Hence, the maximum time complexity of the main algo-rithm is approximately ((||2 + 2||) ||).

5. Experimental Computing and Case Study

We design programs and employ two cases to demon-strate the effective of the method in the last section. Thisexperimental computing program is running on a personalcomputer with the following hardware and software config-uration. The configuration of the computer is a bit low butthe program runs well and fast. It also shows the advantage ofAlgorithm 1 and the corresponding computing program (seeTable 2).

An inconsistent ordered information system on animalssleep is presented in Table 3.

The information system is denoted by I = (, {}, , ), where is the condition attribute set and isthe single dominance decision. There are 42 objects whichrepresent the species of animals and 10 attributes withnumerical values in the ordered information system. Theanimals names are showed in Table 3 and the interpretationsof the attributes will be listed as follows. The interpretationsand the units of attributes are represented as shown in Table4.

By the experimental computing program, the distributionreductions of the system can be calculated and they arerepresented in the following. The operating time to computethis case is 0.158581 seconds.

The distribution reductions are

{1, 3, 4, 6, 7, 8, 9} ,

{2, 3, 4, 6, 7, 8, 9} ,

{1, 2, 3, 4, 6, 7, 8, 9} ,

{1, 3, 4, 5, 6, 7, 8, 9} ,

{2, 3, 4, 5, 6, 7, 8, 9} .

(11)

And it can be verified by taking the computer as anassistant that the above sets are reductions of the data table.Detailed progress of the verifying are not arranged here.From the results, we can easily see that the reductions studiedin this paper are different from the ones approached in[25], since these reductions are {

3, 4, 6, 7}, {4, 5, 6, 7},

{6, 8, 9}, and {

1, 2, 8, 9}. They are different kinds of

reductions in ordered information systems and can adapt todifferent needs in practices. From the definition of differentreductions, we can also easily obtain that possible andcompatible reductions are usually subsets of distributionreduction. This is not strict and should be studied andverified separately and theatrically . And the work may betaken into account as one part of the future studies in ourwork.

Finally, we take other inconsistent ordered informationsystem to acquire the distribution reduction, respectively.And the descriptions on the data tables are listed in Table 5.

From the results in Table 5, we can obtain that thealgorithm and the program we studied in this paper couldbe effective and useful to acquire distribution reductionsin practice. The numbers of objects and attributes canincrease the computing time. But the matrices storage hasthe ability to shorten the memory and computing time.And it can be helpful in research theoretically and it isapplicable.

6. Conclusions

As is known, many information systems are data tablesconsidering criteria for various factors in practise. There-fore, it is meaningful to study the attribute reductions ininconsistent information system on the basis of dominancerelations. In this paper, distribution reduction is restated ininconsistent ordered information systems. Some propertiesand theorems are studied and discussed. A fact is certifiedthat the distribution reduction is equivalent to themaximumdistribution reduction in ordered information systems. The-orems on distribution reduction are implemented to createpreparations for reduction acquisition and the dominancematrix is also restated to acquire distribution reductionsin criterion based information systems. The matrix algo-rithm for distribution reduction acquisition is stepped andprogrammed. The algorithm can provide an approach andthe program can be effective for theoretical research onknowledge reductions in criterion based inconsistent infor-mation systems. Dominance matrices are the only reliedparameters which need to be considered without otherssuch as approximations and subinformation systems beingbrought in. Furthermore, cases are employed to illustratethe validity of the matrix method and the program, whichshows that the effectiveness of the algorithm in complicatedinformation systems.

The Scientific World Journal 9

Table 5: Descriptions on the calculations.

Data name Values Objects Conditions Decisions Reductions Time OperationsBody fat Real 252 14 1 11 36.43723 s 10Glass Real 213 9 1 7 2.04624 s 10Animal sleep Real 42 9 1 5 0.13153 s 10

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (no. 61100116), the Natural ScienceFoundation Project of Jiangshu Province (no. BK2011492),and the Youth Foundation of Xuzhou Institute of Technology(no. xky2011201).

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Research ArticleFurther Study of Multigranulation -Fuzzy Rough Sets

Wentao Li, Xiaoyan Zhang, and Wenxin Sun

School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China

Correspondence should be addressed to Xiaoyan Zhang; zhangxyms@gmail.com

Received 15 May 2014; Revised 22 June 2014; Accepted 7 July 2014; Published 17 August 2014

Academic Editor: Xibei Yang

Copyright 2014 Wentao Li et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The optimistic multigranulation -fuzzy rough set model was established based on multiple granulations under -fuzzyapproximation space by Xu et al., 2012. From the reference, a natural idea is to consider pessimistic multigranulation model in-fuzzy approximation space. So, in this paper, the main objective is to make further studies according to Xu et al., 2012. Theoptimistic multigranulation -fuzzy rough set model is improved deeply by investigating some further properties. And a completemultigranulation -fuzzy