An Approach To Predicting Non-DeterministicNeural Network Behavior

Edgar Fullert, Sampath Yerramalla*, Bojan Cukic*, Srikanth Gururajant

*Lane Department of Computer Science and Electrical EngineeringWest Virginia University, Morgantown, WV 26506

sampath,cukic @csee.wvu.edu

tDepartment of MathematicsWest Virginia University, Morgantown, WV 26506

ef@math.wvu.edu

tDepartment of Mechanical and Aerospace EngineeringWest Virginia University, Morgantown, WV 26506

Abstract- This paper describes a methodology for generatingindicators of performance for the Dynamic Cell Structuresneural network, a type of growing self-organizing map. Theperformance indicators are based on the learning architecture ofthe neural network and are validated using correlation measuresof Murphys rule. Time estimates for neural network convergenceare generated based on the current data conditions and theconfidence in the neural network, which is provided by theperformance indicators. Analytical and experimental results arepresented for the Dynamic Cell Structures neural network duringits training from the Carnegie Mellon University two-spiralsbenchmark data.

I. INTRODUCTIONAn interesting feature of neural networks is the unconven-

tional, i.e. non-sequential approach of learning. While this fea-ture of neural networks enables them to solve high complexityproblems with relatively minimal computational effort whencompared to conventional computation methods and expertsystems, it also increases the difficulty of understanding itslearning behavior. Since it is a necessity to understand andpredict the nature of neural network (NN) behavior beforetheir deployment into safety-critical systems for example, NNbased aircraft control, one must develop a V&V methodologyfor NN evaluation [18] [13].

Traditional software V&V methods are based on the as-sumption of non-adaptation and fail to account for changesthat occur after deployment. Neural nets on the other handare sometimes designed for on-line adaptation, i.e. adaptationafter being deployed into a system. A change in behavior leadsto model-uncertainty, which is not acceptable by traditionalsoftware evaluation methods. This implies that there is a needfor the development of non-traditional V&V methods that canenable real-time NN evaluation.

Performance indicators and convergence-time estimates arefew of the many criteria that may be used as measures ofNN reliability. While performance provides a measure of the

ability of NN to perform a required function for a certain time,convergence time is a measure of the rate at which the NNlearning approaches a pre-specified error in learning. It is ingeneral hard to evaluate the performance and convergence ina neural net as the learning approach is stochastic and evolvesin an unpredictable manner over time. Some progress hasbeen made in understanding the stability properties of non-deterministic nature of neural nets using Lyapunov analysisapproach [15] [14]. A major problem in theoretical analy-sis of neural nets is confinement of the delineated stabilityboundaries for NN adaptation from stationary or fixed data.Learning from a stationary data manifold implies that once theNN learning algorithm is presented with the training data, theconfiguration of the data remains unchanged throughout thelearning process. It is observed that during on-line adaptationof neural nets, the learning may not always be based onstationary or fixed data but can also be based on non-stationaryor varying data. The real V&V challenge therefore is to predictthe nature of NN behavior even when learning in the NN isbased from varying data.

This paper endeavors to develop a non-traditional approachfor NN validation. In Section II the architecture of the NNrequiring an analysis, namely the Dynamic Cell Structures(DCS) is discussed. Section III provides mathematical formu-lations of the proposed performance indicators in the form ofstability-monitors. In section IV the correlation measure de-rived from using Murphys rule for the performance indicatorsare discussed. The convergence-time estimates for NN learningare provided in Section V, which provides understanding asto when the NN returns to a pre-specified learning errorafter encountering a disturbance in learning. Experimentalstudies of the proposed approach on the DCS are conductedin Section VI using the CMU twin-spiral benchmark data.Finally, conclusions are drawn in Section VII that are basedon the experimental study of the proposed V&V approach.

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II. NN ARCHITECTURES AND PERFORMANCEThe non-deterministic nature of the environment that most

neural networks encounter when deployed as real-time, adap-tive components of a control system means that establishingconfidence levels for the system cannot be accomplished usingtraditional means.

A. DefinitionsA topology representing self-organizing map can be used for

both supervised and un-supervised learning [10] [11]. Someapplications of SOMs include vector quantization, clustering,and dimensional reduction. Since some of the materials pre-sented in this paper are based on advanced SOM properties, adetailed overview of the basic properties of SOMs which uti-lize both the Kohonen rule and competitive Hebb adaptationsfollows.A SOM basically consists of neurons or nodes whose posi-

tions are represented by weight centers w i E {1, 2, ... N}and weighted lateral connections between neurons cij i, j C{1,2, ... N}, where N is the total number of neurons in theSOM. A SOM can be completely represented using a weightcenter matrix, W wi E W C (9 c RD, and a connectionstrength matrix, C cij E C C (9 C RD, where (9 is theoutput space in which the neural net is represented in. The mapgenerated by SOM can be considered as a graph consisting ofN nodes, G(W, C, N).A dynamically generated self-organizing map can be defined

as follows.Definition 2.1: SOM With Hebbian Learning

A self-organized mapping G(W, C, N) for a given featuremanifold, I C RD, is an Nth order neural network repre-sentation of the manifold I in the output space, C c RD.Within the neural network, the primary representative of agiven data point is referred to as the best matching unit forthe data value.

Definition 2.2: Best Matching Unit (BMU)A neuron, i = BMU of the network with weight centerWBMU E W C ( C RD is called the best matching unitfor a given input element m E I c RD if it is closer to theinput element (in terms of Euclidean distance) than any otherneuron of the network.

Im - WBMU 11 <_M -wi11, V i C {1, 2,.... N} (11.1)where N is the number of neurons in the network.The next best representative for a data value is the second bestunit.

Definition 2.3: Second Best Unit (SBU)A neuron i = SBU of the network with weight centerWSBU e W c ( C RD is called the second best unit fora given input element m E I C RD if it is second closest(closest being the BMU) to the input element (in terms ofEuclidean distance) than any other neuron of the network.

Im -WSBuI < lim- will,V i C {1, 2, . . . N},i 7/ BMU(1.2)

where N is the number of neurons in the network.

For topology preserving SOMs, it is necessary to vary theconnectivity to mirror the topology of the data being analyzed.As the network evolves, its connectivity develops in such away that it is topologically equivalent to the topology of thedata set being learned. For a given best matching unit of a dataelement, the neighborhood connected to the BMU reflects thistopological equivalence.

Definition 2.4: Neighbor and Neighborhood (nbr)Two neurons i, j of the network are said to be neighbors if thethey have a non-zero lateral connection strength cij existingbetween them. Neuron i is then said to be the neighbor of jand vice versa. The set of all neighbors of a neuron i is knownas its neighborhood {nbr(i)}.

j e {nbr(i)}, cij : 0,Vj E {1,2, ..., N} (11.3)

where N is the number of neurons in the network.

These four aspects of a topology preserving feature map arethe basic components of the network. The primary aspectsused in the present work to monitor network performanceare the error of the network determined by the Euclideandistance between a data value and its BMU and the dynamicsof the Kohonen update of a BMU thought of as WBMUnew =

WBMU(m) -E(M - WBMU(m)) where e is the Kohonenupdate constant. The distinguishing characteristic of the DCSSOM [3] is that it adds neurons as it learns in areas where thenetwork is least efficient and so grows dynamically over time.This growing SOM structure gives DCS a powerful methodfor adaptation in online environments [1].

III. MEASURES OF NN PERFORMANCEA standard methodology for evaluating the performance of

the DCS network is to compute its total and quantized errors.The total error, E, is defined to be the sum of the Euclideandistances between each data element of the presented input(training) data patten rm E I c RD and the closest node ofthe neural network, WBMU(m) E W c RD corresponding toeach m. The quantized error, Q, is this sum divided by thenumber N of nodes in the network. Symbolically, they arecomputed as

E = Z I|m-WBMU(m) QI= N E |nm-WBMU(m)HmEM mEM

(I.1)A necessary part of implementing NNs in safety or missioncritical applications is determining whether or not the NN isperforming as expected or desired [18] [17]. The followingfour error values numerically reflect the network's success inlearning the structure of the data set being trained.

Definition 3.1: Monitor #1, BMU Error: The BMU Erroris the total error of the network.

Monitor#l = E ln W-WBMU (Tm)IImEM

In order to get a second order estimate of performance, wecompute the SBU Error.

Definition 3.2: Monitor #2, SBU Error: SBU Error is theEuclidean distance between each data element of the presented

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input (training) data pattern m E M c RD and its secondclosest neuron in the neural network, WsBu(m) E W c RD.

Monitor#2 = S Irn -WSBU(m)IImEM

We then focus on local error in neighborhoods of the network.Definition 3.3: Monitor #3, Neighborhood (NBR) Error:

NBR Error is the mean Euclidean distance between each dataelement of the presented input (training) data pattern m EM c RD and the set of neighborhood neurons of the BMU ofneural network, known as the NBR-set W{NBR} (m, BMU) EWc RD.

Monitor#3 = 5 mean lm -W{NBR} (m) 11}mEM

Definition 3.4: Monitor #4, Non-Neighborhood (Non-NBR) Error: Non-NBR Error is the mean Euclidean distancebetween each data element of the presented input (training)data pattern m E M c RD and the set of laterally connected,non neighboring neurons of the BMU of neural network,known as the Non-NBR-set W{NOnNBR}(Im,BMU) E W CRD

Monitor#4 = mmean lm -W{Non-NBR} (m) 11}mEM

Neural network learning involves several components in-cluding Kohonen learning, Hebb update, resource update,growing neurons, pruning neurons. Therefore, having a singlemonitor to ensure correct online learning behavior may notbe a safe and reliable means of system assurance. This isnoticeable in cases where online learning neural nets encounterrapidly varying, divergent data manifolds [15]. The proposedonline monitoring system is composed of several monitors.Each monitor is essentially a Lyapunov-like function that isdesigned to analyze and capture instable behavior from aparticular aspect of online learning. These monitors providean observation of how well a set of associated neural centersof the OLNN are being overlayed over corresponding relativeelements of presented training data set. In other words, howstable is the behavior of OLNN when it encounters rapidlychanging data manifolds. Note that during the operation ofadaptive system under stressed performance conditions, OLNNwill likely encounter data representatives that are inconsistentwith each other. Making an online stability monitoring systemavailable for use in such conditions can significantly enhancesystem safety and reliability. In any given environment, it maynot be known apriori which of these monitors is an indicatorof faulty performance. In the following sections we contendwith this by fusing the monitor values into a single indicatorof confidence.

IV. MONITORS AND MURPHYS CORRELATION MEASURES

Murphy's rule is essentially a data fusion method basedon Dempster-Schafer theory [12]. The method has as inputsmultiple streams of output from online stability monitors. Theoutput of the fusion are minimum and maximum confidencevalues.

A. Pre-Processing Input DataPrior to implementing Murphy's rule of data fusion, the val-

ues from the multiple stability monitors are normalized so thatthe minimum and maximum values from the pre-processeddata should not exceed 0 and 1 respectively. The maximumvalue at the current time frame from each of these monitors isrecorded as the current maximum value. For each time framethe monitor values are divided by their corresponding currentmaxima. The pre-processed data obtained as such is consideredas the online monitor values and serve as real-time inputs tothe data fusion implementation.

B. Combination RuleWithin a given frame of discernment, theta is set. Two

propositions are of interest here: 1 How uncertain are thenetwork's outputs, i.e. how distant is the represented input dataform the actual data; 2 How trusted is the network's output, i.e.how confident we are in the network's output, known as theconfidence measure. The pre-processed input data is measureof probability of the uncertainty (error) of the network outputand serves as beliefs for proposition 1. The belief for theproposition 2 can be obtained from combining beliefs from1. Let m(1) be the belief for the proposition 1, obtained bycombining normalized online stability monitor values and letm(2) be the belief for proposition 2, then m(2) = 1 - m(1).

Let ma(l) and Mb(1) be two beliefs for proposition 1, i.e.two normalized online monitor values. Let mab(l) be the resultobtained from combining ma(l) and Mb(1), then

Mab(l) ~~((Mna(1)Mnb(1)Y1)fl)1mab(l)= (((Ma(1)Mb(1))n) + ((Ma(2)Mb(2)) )) (IV.l)

((ma(l)Mb(l) )n)Mab(l) = (((iMa()b(l))n) + (((1 -Ma(l))(l -Mb(l)))n))Mab(2) = 1 -Mab(l)

It is shown in [7] that the ordering of the combinations maybe chosen in such a way that this is computationally feasible.

C. Monitoring NN Behavior Using Murphy's RuleThe data that serves as the basis for the analysis in this

section and as the input to the online learning neural networkwas collected from an experimental F-15 flight simulator. Thetraining data for online learning for this case study consists of7 dimensions. Each of these 7 dimensions represents a specificcontrol parameter that needs to be learned and represented bythe online learning neural network. Twenty frames of new datais presented to the online learning neural network every secondwhich corresponds to a frequency of 20 Hz.

In order to simulate failed flight conditions, the flight isrun under nominal (no failure) conditions for the first 29seconds. Thereafter, a failure is simulated into the system thatchanges the operating condition into a failed flight condition.Meaning that the first 599 frames of data represents thedata that was generated from the simulator during nominalflight conditions. The data thereafter (600 to 800), representsdata generated under failed flight conditions. The failed flight

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condition in this case, is the aircraft's stuck left stabilator,which is simulated to stuck at an angle of +3 degree. Thedata after 599 frames can be thought of as the data that isgenerated by the adaptive system during its operation understressed performance conditions. As the OLNN is embeddedin the adaptive system, during these conditions, it will likelydeviate to an unstable state of learning. The goal of the stabilitymonitoring system is to detect this bifurcation away fromstable online learning behavior.

S?A"mm 1U... . . . . . re~~~1

SrA.1YuWICT2II

r*ia0

I_1f~

Fig. 2. Murphy's Rule Fusion of the Flight Simulator Monitors

9: ai iii 'm ti i-a: Z -ii uii emf 4O.

(a) Monitor # 1. BMU ErrornmL"N=3_

3ii -Ai a ia iii if ii i r iia

(b) Monitor # 2. SBU Errorg-- !!Tywft_4

:.IW

(c) Mi # 3. NBR Err *or

(c) Monitor # 3. NBR Error

S i i ia4iS*4 iSa ie iia

(d) Monitor # 4. Non-NBR Error

Fig. 1. On-line Stability Monitoring System: Monitors One Through FourDetect Error in Neuron Location

Figures 1(a),1(b),1(c), and l(d) show values from thefour monitors at this stage of online adaptation. Notice pre-dominant spikes in monitors # 3 and # 4 (Figures l(c) andl(d)) around the 600th frame of data. This is an indicationby the stability monitoring system that the state of onlinelearning has bifurcated away from stable learning behaviorafter presenting with the 600th frame of data. This is consistentwith the simulated failed flight condition, which was simulatedto occur at the end of 29th second using the simulator. Notethat 30 seconds of flying time corresponds to 600 frames ofdata with respect to a frequency of 20 Hz. Figure 2 showsthe Murphy's rule fusion of these four monitors for thissimulation. Lower and upper bounds of confidence for theneural network's performance are depicted in red and green.The stability monitoring system was successful in detectingan abnormal (unstable) learning behavior during the operationof adaptive system under stressed performance conditions andindicates this in the drop of both the minimum and maximumconfidence levels as well as the increase in the differencebetween the two.

V. CONVERGENCE TIMESGiven a method for detecting NN response to variations

in data streams, an important question is then how to relatethe process of convergence or stabilization to these measures

of performance and the confidence levels associated to them.In order to establish a methodology, we begin by analyzingthe effect of data perturbations on the Quantized and TotalErrors for DCS described in Section III. We further simplifythe analysis by ignoring network architectures other than theKohonen update for the best matching unit of a given datapoint. In what follows we attempt to anticipate the number oftraining epochs, defined to be the time between the insertion ofconsecutive new neurons, required to return the DCS networkto the previous error level when a single data point has beenadded to the data set after DCS has reached a prescribed levelof stability.A. Notational Conventions

For the rest of this paper we will use the following notation.Let e be the Kohonen constant for the BMU update, and letm be an element of the training data set M and WBMU(m) bethe best matching unit for m. Recall that, as in Section III,E is the total error for the DCS network measure and Q is thequantized error. Let Eo be the value of that error just beforethe new data element, denoted by mo, is added to the data set.Similarly, let Qo be the quantized error before the addition ofmo. Finally, let No be the number of network nodes just priorto mo being added to M and let Ep be the perturbation errorinduced by the addition of mo, namely

Ep = IMO-WBMU(mo) I IWe let N be the number of nodes in the network so thatQ = E/N.B. Returning to Prior Total Error LevelsThe first result in this situation is the following.Theorem 5.1: LetM be a data set being trained by the DCS

neural network and let the naming convention set above hold.If a data point mo is added to M following the addition ofnode No to the network, then DCS will return to the errorlevel Eo in

klog(NoEp)log(1-E) (V.l)

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1i . -01I 1,

--L *1

0O.

epochs.Proof: We wish for the total error E to return to the value

Eo. Since before the addition of mo each neuron contributedan average of -E to the total error, it is sufficient (though notNonecessary) for the error induced by the addition of mo to fallto the level of the average error before perturbation, E.l, inorder for the total error to return to the value Eo. We havethat after k additional epochs we need

(1 - E)kE, < E° (V.2)

klog(I-E)<log No

log (Eopk NoEg(p-)

log(1-,)This completes the proof. .

C. Quantized Error MeasuresOn the other hand, the quantized error, Q, is a commonly

used stopping criterion for implementations of the DCS net-work. For this case, we in effect compare the way in whichthe Kohonen rule decreases error relative to the epoch numberto the quantized error. We have the following result.

Theorem 5.2: If a data point mo is introduced into the dataset M after node No where the total error is Eo, then thequantized error Q will return to the value Qo within k epochsof learning provided

(1E)k <Eo (V.3)k NoP

Proof: Begin by noting that when mo is inserted, QoEl' and that after k epochs the new quantized error will beNo

Q E Eo + Ep(j _e)k (V.)N No +k

We wish to determine when Q < Qo or

Eo + Ep(j _ )k <KEo (V.5)No + k No

(1 _E)k Eok NoEp

which provides the upper bound for k implicitly.For this result, note that a closed form solution is not

possible so that the value of k must be determined by cyclingthrough values until the left hand side matches the right, whichis known. This complication arises due to the linear use of kin the denominator of the quantized error combined with theexponential use in the Kohonen update term. This emphasizesthe fact that the result for the total error in the previous sectionignores the growth of the network explicitly and as a resultshould yield longer times for returning to the unperturbedtotal error numbers whereas the latter result directly usesboth network growth and the Kohonen correction and shouldtherefore yield smaller re-stabilization times. This behavior iswhat is observed experimentally.

VI. EXPERIMENTAL STUDIES OF CONVERGENCE

In order to determine what, if any, patterns existed in therecovery behavior of DCS training on perturbed data, simu-lations were run on the two-spirals benchmark [4] data set.After the data point is inserted, the total error and quantizederror were evaluated to determine how many epochs wererequired for the network to recover in the sense that its totaland quantized errors had returned to their previous values.Examples of a DCS network training and the resulting totaland quantized errors for this network are shown in Figures3(a) and 3(b) before the addition of new data. In Figures 4(a)

DCS NN Representaton

(a) Learning Two-spirals

t0

0 10.. 0 400 t 0 20 30 40 so 60 70 80

1500 .

5000

t00 10 20 30 ..40 00 .,-0 20 3 4 o 0 70 s

(b) Quantized (top graph) and Total Error

Fig. 3. DCS and BMU Error Monitors for Two-spirals Training

and 4(b), the same training after the addition of a new datapoint at coordinate (130, 0) in the plane is shown along withits error plots. A spike is observed when there are 95 nodesin the network in these plots. Ten simulations were performedusing the same network characteristics. The Kohonen constantwas kept at e = .1 and the network was initialized with tenneurons and the DCS network was constructed until /\Q < .05at which time a data element was inserted either at coordinates(130, 0) or (0, 130) in the plane.The resulting induced error Ep averaged 123.68 and the

data insertion occurred when there were between 87 and 98nodes in the network. This variance is due to the stochasticnature of the Kohonen adaptation which gives a variabletraining efficiency depending on the sampling order in agiven simulation. The network required a minimum of 9

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H0 100 120

40 60 so 100 120

(b) Quantized (top graph) and Total Error

Fig. 4. DCS With BMU Error Monitors for Perturbed Data

and a maximum of 11 neuron insertions (depending on thesimulation) to recover to the original value of Q. It requireda minimum of 17 and a maximum of 25 neuron additions toreach the original, pre-perturbation, total error value. Thesevalues agree well with the predicted number of node creationsneeded of either k = 11 or k = 12 for Q re-stabilization.The predicted number of node insertions for total error re-

stabilization in a given simulation ranged from k = 34 tok = 37 which are a good deal larger than the observedvalues. The predicted values vary slightly from simulationto simulation. The improvement of the simlations over thepredicted values is likely due to the added effect of networkadaptation for other data values which is not considered in thecurrent work.

VII. CONCLUSIONSIn order to verify and validate a network architecture that

involves non-deterministic component such as an unsupervisedneural network, a methodology must be developed that eitherbuilds a picture of performance possibilities based on likelynetwork scenarios or one that analyzes network performanceas scenarios are encountered. For this work we have chosen thelatter of these two approaches and have constructed a systemof monitors that yield confidence measures for the DCS neuralnetwork performance in online implementations. Further, we

have a systematic way to approximate recovery times for theneural network when perturbed data are being encountered. Tomore precisely understand the connection between these twotools, a similar formal analysis of the three remaining monitorsmust be performed as well as a modification of the existing

analysis for total and quantized error that accommodates morecomplex scenarios such as multiple data value additions, DCSrecovery for more realistic data sets, and the inclusion ofconnectivity infornation in the analysis. These areas are thefocus of ongoing and future work.

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