Chapterhome.eng.iastate.edu/~julied/publications/FCM96.pdfstitute University of Southern California...

Chapter �

Virtual Worlds as

Fuzzy Dynamical Systems

Julie A� Dickerson and Bart Kosko

Electrical and Computer Engineering Department� Iowa State University� Ames�IA� ��

Department of Electrical Engineering�Systems� Signal and Image Processing In�stitute� University of Southern California� Los Angeles� CA� ��

Abstract

Fuzzy cognitive maps �FCMs� can structure virtual worlds that change withtime� A FCM links causal events� actors� values� goals� and trends in a fuzzyfeedback dynamical system� A fuzzy rule de�nes a fuzzy patch in the input�outputstate�space of a system� It links commonsense knowledge with state�space geometry�A FCM connects the fuzzy rules or causal �ow paths that relate events� It can guideactors in a virtual world as the actors move through a web of cause and e�ect andreact to events and to other actors� Experts draw FCM causal pictures of the virtualworld� Complex FCMs can give virtual worlds with �new or chaotic equilibriumbehavior� Simple FCMs give virtual worlds with periodic behavior� They map inputstates to limit�cycle equilibria� A FCM limit cycle repeats a sequence of events ora chain of actions and responses� Limit cycles can control the steady�state rhythmsand patterns in a virtual world� In nested FCMs each causal concept can controlits own FCM or fuzzy function approximator� Appendix A shows how an additivefuzzy system can uniformly approximate any continuous �or bounded measurable�function on a compact domain to any degree of accuracy� This gives levels of fuzzysystems that can choose goals and causal webs as well as move objects and guideactors in the webs� FCM matrices sum to give a combined FCM virtual world forany number of knowledge sources� Adaptive FCMs change their fuzzy causal web ascausal patterns change and as actors act and experts state their causal knowledge�Neural learning laws change the causal rules and the limit cycles� Actors learnnew patterns and reinforce old ones� In complex FCMs the user can choose thedynamical structure of the virtual world from a spectrum that ranges from mildlyto wildly nonlinear� We use an adaptive FCM to model an undersea virtual worldof dolphins� �sh� and sharks�

� Chapter ��

�� Fuzzy Virtual Worlds

What is a virtual world� It is what changes in a�virtual reality �

or �cy�berspace �� A virtual world links humans and computers in a causal mediumthat can trick the mind or senses�

At the broadest level a virtual world is a dynamical system� It changes withtime as the user or an actor moves through it� In the simplest case only the usermoves in the virtual world� In general both the user and the virtual world changeand they change each other�

Change in a virtual world is causal� Actors cause events to happen as theymove in a virtual world� They add new patterns of cause and e�ect and respondto old ones� In turn the virtual world acts on the actors or on their physical orsocial environments� The virtual world changes their behavior and can change itsown web of cause of e�ect� This feedback causality between actors and their virtualworld makes up a complex dynamical system that can model events� actors� actions�and data as they unfold in time�

Virtual worlds are fuzzy as well as fedback� Events occur and concepts holdonly to some degree� Events cause one another to some degree� In this sensevirtual worlds are fuzzy causal worlds� They are fuzzy dynamical systems� Afuzzy rule de�nes a fuzzy patch in the input�output state�space of a system andlinks commonsense knowledge with state�space geometry� An additive fuzzy systemapproximates a function by covering its graph with fuzzy patches in the input�output state space and averaging patches that overlap�

How do we model the fuzzy feedback causality� One way is to write downthe di�erential equations that show how the virtual ��ux or ��uid changes intime� This gives an exact model� The Navier�Stokes equations �� used in weathermodels give a �uid model of how actors move in a type of virtual world� They canshow how clouds or tornadoes form and dissolve in a changing atmosphere or howan airplane �ies through pockets of turbulence� The inverse kinematic equationsof robotics �� show how an actor moves through or grasps in a virtual joint space�The coupled di�erential equations of blood glucose and insulin �� cast the patientas a diabetic actor awash in a virtual world of sugar and hormones� Such mathmodels are hard to �nd� hard to solve� and hard to run in realtime� They paint too�ne a picture of the virtual world�

Fuzzy cognitive maps �FCMs� can model the virtual world in large fuzzychunks� They model the causal web as a fuzzy directed graph �� The nodesand edges show how causal concepts a�ect one another to some degree in the fuzzydynamical system� The �size of the nodes gives the chunk size� In a virtual worldthe concept nodes can stand for events� actions� values� moods� goals� or trends�The causal edges state fuzzy rules or causal �ows between concepts� In a predator�prey world survival threat increases prey runaway� The fuzzy rule states how muchone node grows or falls as some other node grows or falls�

Experts draw the FCMs as causal pictures� They do not state equations�They state concept nodes and link them to other nodes� The FCM system turnseach picture into a matrix of fuzzy rule weights� The system weights and adds theFCM matrices to combine any number of causal pictures� More FCMs tend to sumto a better picture of the causal web with rich tangles of feedback and fuzzy edgeseven if each expert gives binary �present or absent� edges� This makes it easy to

Technology for Multimedia �

add or delete actors or to change the background of a virtual world or to combinevirtual worlds that are disjoint or overlap� We can also let a FCM node control itsown FCM to give a nested FCM in a hierarchy of virtual worlds� The node FCMcan model the complex nonlinearities between the node�s input and output� It candrive the motions� sounds� actions� or goals of a virtual actor�

The FCM itself acts as a nonlinear dynamical system� Like a neural net itmaps inputs to output equilibrium states� Each input digs a path through thevirtual state space� In simple FCMs the path ends in a �xed point or limit cycle�In more complex FCMs the path may end in an aperiodic or �chaotic attractor�These �xed points and attractors represent meta�rules of the form �If input� thenattractor or �xed point� The rules are stored in the cube itself�

�� Additive Fuzzy Systems

A fuzzy system approximates a function by covering its graph with fuzzypatches and averaging patches that overlap� The approximation improves as thefuzzy patches grow in number and shrink in size� Figure � shows how fuzzy patchesin the input�output product space X � Y cover the real function f � X � Y � InFigure ��a� a few large patches approximate f� In Figure ��b� several smallerpatches better approximate f� The approximation improves as we add more smallpatches but storage and complexity costs increase� This section gives the algebraicdetails of the fuzzy approximation�

An additive fuzzy system adds the then�parts of �red if�then rules� Otherfuzzy systems combine the then�part sets with pairwise maxima� A fuzzy systemhas rules of the form �If input conditions hold� then output conditions hold or �IfX is A� then Y is B for fuzzy sets A and B� Each fuzzy rule de�nes a fuzzy patchor a Cartesian product A�B as shown in Figure �� The fuzzy system covers thegraph of a function with fuzzy patches and averages patches that overlap� Uncertain

��

X

Y f

X

Y f

Figure �� a� Four large fuzzy patches cover part of the graph of the unknownfunction f � X � Y � Fewer patches can decrease computation but decrease ap�proximation accuracy� �b� More smaller fuzzy patches better cover f but at greatercomputational cost� Each fuzzy rule de�nes a patch in the product space X � Y �A large but �nite number of fuzzy rules or precise rules can cover the graph witharbitrary accuracy�

� Chapter ��

X

Y

��

A1 A2 A3

B

1

B 2

B

3

A1 B1x

IF X=A1, THEN Y=B1

Figure �� The fuzzy rule patch �If X is fuzzy set A�� then Y is fuzzy set B�� isthe fuzzy Cartesian product A� �B� in the input�output product space X � Y �

fuzzy sets give a large patch or fuzzy rule� Small or more certain fuzzy sets givesmall patches�

Additive fuzzy systems �re all rules in parallel and average the scaled out�put sets B�j to get the output fuzzy set B as in Figure �� Correlation productinference scales each output set Bj by the degree mAj �x� �or aj�x�� that the rule�IF Aj � THEN Bj �res� Most rules �re to degree �� Defuzzi�cation of B givesa number or a control signal output� Centroidal defuzzi�cation with correlationproduct inference �� gives the output value yk at time k�

yk � F �xk� �

RymB�y�dyRmB�y�dy

��

�

Pmj�� V olume �B

�

j �Centroid�B�

j�Pmj�� V olume�B

�

j �

�

Pmj�� cyjVjmAj �xk�Pmj�� VjmAj �xk�

Vj is the volume of the j th output set Bj � We can always normalize the �nitevolumes Vj to unity to keep some rules from dominating others� cyj is the centroidof the j th output set� Fit value mAj �xk� scales the output set Bj � m is the numberof output fuzzy sets� In practice A is connected� It need not be� But then we couldview the rule �If X is A� then Y is A as two or more rules of the form �If X isA� then Y is B� and �If X is A� then Y is B� where B� and B� are two of thedisjoint components of A� So assume B is connected� Then the rule patch A�Bis connected and a patch proper�

The additive fuzzy system computes the conditional expectation E�Y jX � xif we view fuzzy sets as random sets �� if the curve mA � ��

� X is alocus of two point conditional densities� Then mA�x� is the probability of A giventhat X takes on x or mA�x� � p�x � A j X � x� and mA�x� � p�x �� A j X � x��The conditional mean E�Y jX is the mean�squared optimal estimate of Y given theinformation known about X�given the information in the random or fuzzy subsetsA of X�


•••

CentroidalDefuzzifier y

B2́

IF A1 THEN B1

IF A2 THEN B2

x

IF Am THEN Bm

B

1B´

1w

2w

wm B́m

Figure �� Additive fuzzy system architecture� The input xk acts as a deltapulse �or unit bit vector� and �res each rule to some degree� The system adds thescaled output fuzzy sets� The centroid of this combined set gives the output valueyk � The system computes the conditional expectation value EY jX xk��

In Appendix A we show that a fuzzy system can approximate any continuousreal function de�ned on a compact �closed and bounded inRn domain and show thateven a bivalent expert system can uniformly approximate a bounded measurablefunction� The fuzzy systems have a feedforward architecture that resembles thefeedforward multilayer neural systems used to approximate functions �

� Theuniform approximation of continuous functions allows us to replace each continuousfuzzy set with a �nite discretization or a point in a unit hypercube �� of highdimension�

Combining the scaled or ��red consequent fuzzy sets B�� B�

m in Fig�ure �� with pairwise maximum gives the envelope of the fuzzy sets and tendstowards the uniform distribution� Max combination ignores overlap in the fuzzysets Bj � Sum combination adds overlap to the peakedness of B� When the inputchanges slightly� the additive output B changes slightly� The max�combined outputmay ignore small input changes since for large sets of rules most change occurs inthe overlap regions of the fuzzy sets B�j � Here overlap problem arises since the cen�troid tends to stay the same for small changes in input� But the centroid smoothlytracks changes in the fuzzy�set sum ��

We now formally derive the standard additive model �SAM� in �� that weshall use in this chapter and show how an additive fuzzy system acts as a conditionalmean� A general additive fuzzy system is a map F � Rn � Rp� Both in practiceand in uniform approximation proofs we restrict the domain to a compact subsetU � Rn but we need not� Watkins �� has proved that an additive fuzzy systemwith just two rules can exactly represent any bounded function f � R� R even iff is not continuous� In this case the domain is the entire real line�

The additive fuzzy system stores m fuzzy patches Aj�Bj or rules of the form�IfXisAj � thenY isBj Here Aj � R

n and Bj � Rp multivalued or �fuzzy sets

with set functions aj � Rn � ��

and bj � Rp � ��

� We also use the membershipnotation mAj �x� and mBj �y� in this chapter for the set functions� For the followingderivation we use the t �fuzzy unit� notation aj and bj for simplicity�

In practice we de�ne the then�part set Aj by its n coordinate�projection setsA�j � � � � � A

nj and thus Aj � A

�j �A

�j � � � ��A

nj � How we de�ne this fuzzy Cartesian

product dictates the conjunctive �or t�norm� form of how we factor the joint set

� Chapter ��

function aj into its coordinate set functions a�j � � � � � anj � Minimum combination is

the most popular form

aj�x� � a�j�x�� a

�j �x�� a

�j �x�� a

nj �xn� ��

for input vector x � �xl� � � � � xn�� Product combination

aj �x� �nYi��

a ij �x i� ��

can simplify the analysis and computation of additive systems with Gaussian ��or radial�basis �� set functions of the form

aij �xi� � sji exp

��

�

�xi � �x

ji

�ji

�� for scaling constant � � sji � � The choice of combination operator does not a�ectthe structure of the standard model ��

The �rst step to show the conditional�mean property is to view each scalarfuzzy set aij as a random set �� Then a

ij�xi� is not the degree to which xi � A

ij but

the conditional probability p�xi � Aij j Xi � xi�� In the same way the complement

�t value � aij�xi� is just the dual conditional probability� p�xi �� Aij j Xi � xi��

So Aij is not a locus of membership degrees but a locus of two�point conditionaldensities�

The next step is the additive step� The m �t values aij�xi� ��re the then�

part sets Bj to give the �inferred sets B�j � Again the result combines aij�xi� and

Bj in some conjunctive �t�norm� way and again it depends on how we de�ne theCartesian patch Aj � Bj � Here min is less popular than product� The min �clipdiscards all information in Bj above the �t height a

ij�xi� and can thus change the

centroid of Bj if Bj is not symmetric� Product combination or correlation productdecoding �� keeps all relative information in Bj and does not change its centroid�

B�j � aij�x�Bj ��

We use �� as a default for a SAM� We can also view the inferred sets B�j as randomsets� An additive model �� then sums these inferred sets to produce the �nal outputset B�

B �mXj��

B�j ��

Each rule can have a weight wj that scales B�j in �� Learning can change theseweights or we can use them to model frequency or �usuality rule weights� Herewe take them as unity� wj � �

The only constraint on B or b is that it have a �nite integral or volume�

� � V �

Zb �y� dy � � ��


This means that each input x �res at least one rule to non zero degree� ThenB�V is a probability density function� Indeed it is a conditional probability sinceit depends on the fuzzy variable X taking on the input value x �the ratio of a jointto a marginal��

B

V� p�Y jX � x� ��

Note this does not require that we view the if�part sets as probability densityfunctions� They are not� Each is a locus of continuum�many two�point conditionaldensities� Formally the system accepts input x� as a delta pulse to produce the m�t values�

aj �x��

Z� �x� x�� aj �x� dx ��

Then the additive system output F �x� equals the centroid of B�

F �x� �

Ryb�x� y�dyRb�x� y�dy

��

�

Zyp�Y jX � x�dy �

�

� E�Y jX � x ��

What holds for one realization of a random vector holds for them all� Hence F �E�Y j X as claimed� The SAM model �� then computes the global conditionalmean value E�Y j X � x as a convex sum of local conditional means in ��

We now assume that the additive fuzzy system maps real vectors into scalarsF � Rn � R� Then put the additive assumption �� in the centroidal output ��to get the standard form of a additive model �� we use in this chapter�

F �x� �

R�

��y

mPj��

b�j�y�dy

R�

��

mPj��

b�j�y�dy��

�

mPj��

R�

��yaj�x�bj�y�dy

mPj��

R�

��aj�x�bj�y�dy

��

�

mPj��

aj�x�Vj

R�

��

ybj�y�dy

Vj

mPj��

aj�x�Vj

��

� Chapter ��

�

mPj��

aj�x�Vjcj

mPj��

aj�x�Vj

��

for then�part set volumes

Vj �

Z�

��

bj�y�dy ��

and then�part set centroids

cj �

R�

��ybj�y�dyR

�

��bj�y�dy

��

The model in �� is the standard additive model or SAM and the same as �� Itholds for F � Rn � Rp as well�

The standard model �� reduces to the Gaussian additive model of Wangand Mendel ��

F �x� �

mPj��

�zj�nQi��

�A

j

i

�xi��

mPj��

�nQi��

�A

j

i

�xi��

for the Gaussian if�part set in �� and Gaussian then�part sets with these identi��cations�

y � z ��

aj�x� �nYi��

aij�xi� ��

�nYi��

�A

j

i

�xi� ��

Vj � ��

cj � �zj ��

The choice of product combination �� gives �� and �� The unity volume followsin �� since Wang and Mendel integrate their m then�part Gaussian sets over allof R �and thus use the scaling constant in �� to account for the input truncation toa compact set�� follows because the mode of a Gaussian set equals its centroidand Wang and Mendel use the mode de�nition �is the point in R at which �Bj �z�achieves its maximum value� They used the Stone�Weierstrass Theorem to provethat additive Gaussian systems with all�product combination in �� are uniformapproximators of continuous maps on compact sets� This non�constructive result isa special case of the uniform approximation theorem for all additive systems� Wereview this general theorem and its constructive proof in Appendix A� It holds aswell for Gaussian sets with min combination �� of if�part �t values or min clippingof then�part sets Bj �


Next observe that taking the centroid of the additive B in �� leads to a setof convex coe�cients�

F �x� �

mPj��

aj �x� Vj cj

mPj��

aj �x� Vj

��

�mXj��

pj �x� cj ��

for the m convex coe�cients �or m terms of a discrete probability density�

pj �x� �aj �x� Vj

mPk��

ak �x� Vk

��

Wang and Mendel �� refer to the convex sum of centroids �� in the Gaussiancase as a �fuzzy basis function expansion even though the �basis functions pj�x�in �� are not orthogonal�

Feedforward fuzzy systems su�er exponential rule explosion as the numberof inputs increases� Optimal rules �� and function representation �� o�er twoways to deal with this �curse of dimensionality� Appendix B shows how supervisedlearning can tune the parameters of an additive fuzzy system� FCMs allow a fuzzysystem to approximate nonlinear dynamical systems with a �xed number of rules��

�� Fuzzy Cognitive Maps

Fuzzy cognitive maps �FCMs� are fuzzy signed digraphs with feedback ��An FCM is an additive fuzzy system with feedback� Nodes stand for fuzzy sets orevents that occur to some degree� The nodes are causal concepts� They can modelevents� actions� values� goals� or lumped�parameter processes�

Directed edges stand for fuzzy rules or the partial causal �ow between theconcepts� The sign �� or �� of an edge stands for causal increase or decrease� Thepositive edge rule in Figure ��a states that a survival threat increases runaway� Itis a positive causal connection� The runaway response grows or falls as the threatgrows or falls� The negative edge rule in Figure ��b states that running away froma predator decreases the survival threat� It is a negative causal connection� Thesurvival threat grows the less the prey runs away and falls the more the prey runsaway� The two rules in Figure ��c de�ne a minimal feedback loop in the FCMcausal web�

A FCM with n nodes has n� edges� The nodes Ci�t� are fuzzy sets andso take values in ��

� So a FCM state is the t �fuzzy unit� vector C�t� ��C��t�� Cn�t�� and thus a point in the fuzzy hypercube I

n � ��

n� A FCMinference is a path or point sequence in In� It is a fuzzy process or indexed familyof fuzzy sets C�t�� The FCM can only �forward chain �� to answer what�ifquestions� Nonlinearities do not permit reverse causality� FCMs cannot �backwardchain to answer why questions�

Chapter ��

Survival Threat Run Away+

�a�

Run Away Survival Threat–

�b�

Run AwaySurvival Threat

+

–

�c�

Figure �� Directed edges stand for fuzzy rules or the partial causal �ow betweenthe concepts� The sign � or �� of an edge stands for causal increase or decrease��a� A positive edge rule in states that a survival threat increases runaway� �b� Anegative edge rule states that running away from a predator decreases the survivalthreat� �c� Two rules de�ne a minimal feedback loop in the FCM causal web�

The FCM nonlinear dynamical system acts as a neural network� For eachinput state C�� it digs a trajectory in In that ends in an equilibrium attractor A�The FCM quickly converges or �settles down to a �xed point� limit cycle� limittorus� or chaotic attractor in the fuzzy cube� Figure �� shows three attractors ormeta�rules for a ��D dynamical FCM�

The output equilibrium is the answer to a causal what�if question� What ifC�� happens� In this sense each FCM stores a set of global rules of the form �IfC�� then equilibrium attractor A�

The size of the attractor regions in the fuzzy cube governs the number of theseglobal rules or �hidden patterns �� All points in the attractor region map to theattractor� A FCM with a global �xed point has only one global rule� All inputballs �roll down its �well� FCMs can have large and small attractor regions in thefuzzy cube� The attractor types can vary in complex FCMs with highly nonlinearconcepts and edges� Then one input state may lead to chaos and a more distantinput state may end in a �xed point or limit cycle�

Technology for Multimedia

F •

Limit CycleChaotic Attractor

Fixed Point

C 0•

(0,0)

(0,1) (1,1)

(1,0)

Figure �� The unit square is the state space for a FCM with two nodes� Thesystem has at most four fuzzy edge rules� In this case it has three fuzzy meta�rulesof the form �If input state vector C then attractorA�� The state C� converges toa �xed point F�

�� Simple FCMs

Simple FCMs have bivalent nodes and trivalent edges� Concept values Citake values in f��g� Causal edges take values in f��g� So for a concept eachsimple FCM state vector is one of the �n vertices of the fuzzy cube In� The FCMtrajectory hops from vertex to vertex� In ends in a �xed point or limit cycle at the�rst repeated vector�

We can draw simple FCMs from articles� editorials� or surveys� Most personscan state the sign of causal �ow between nodes� The hard part is to state its degreeor magnitude� We can average expert responses �� as in equation �� belowor use neural systems to learn fuzzy edge weights from data� The expert responsescan initialize the causal learning or modify it as a type of forcing function�

Figure �� shows a simple FCM with �ve concept nodes� The connection oredge matrix E lists the causal links between nodes�

� Chapter ��

C2: Fatigue

C3: Rest

C1: Herd Clustering

C4: Survival Threat

C5: Run away

++

+

+

+

—

—

—

——

—

—

+

Figure �� Simple FCM with �ve concept nodes� Edges show directed causal�ow between nodes�

E �

C� C� C� C� C�C� � � � �C� � � � �

C� � � � �

C� � � �

C� � � � �

The ith row lists the connection strength of the edges eik directed out from causalconcept Ci� The ith column lists the edges eki directed into Ci� Ci causally increasesCk if eik � �� decreases Ck if eik � �� and has no e�ect if eik � �� The causalconcept C� causally increases concepts C� and C�� It decreases C�� Concepts C�and C� decrease C�� Concept C� increases C��

�� FCM Recall

FCMs recall as the FCM dynamical system equilibrates� Simple FCM in�ference thresholds a matrix�vector multiplication �� State vectors Cn cyclethrough the FCM adjacency matrix E � C� � E � C� � E � C� � � � �� Thesystem nonlinearly transforms the weighted input to each node Ci

Ci �tn�� S

�NXk��

eki �tn� Ck �tn�

��

Here S�x� is a bounded signal function� For simple FCMs the sigmoid function

S �y� �

� e�c�y�T��

with large c � � approximates a binary threshold function�Simple threshold FCMs quickly converge to stable limit cycles or �xed points

�� These limit cycles show �hidden patterns in the causal web of the FCM�


The FCM in Figure �� gives a three�step limit cycle when input state C� �� res the FCM network� Equation �� and binary thresholdinggives the four step limit cycle C� � C� � C� � C� � C��

C� � ��

C�E � � � � �

� C� � � � � ��

C�E � �� C� � ��

C�E � ��

� C� � ��

C�E � ��

� C� � ��

In a virtual world the limit cycle might make in order wake up� go to work� comehome� then wake up again� Some complex actions such as walking break down intosimple cycles of movement ��

�

Each node in a simple FCM turns actions or goals on and o�� Each nodecan control its own FCM� fuzzy control system� goal�directed animation system�force feedback� or other input�output map� The FCM can control the temporalassociations or timing cycles that structure virtual worlds� These patterns establishthe rhythm of the world� �Grandmother nodes can control the time spent on eachstep in a FCM �avalanche �� This can change the update rate and thus thetiming for the network ��

�� Augmented FCMs

FCM matrices additively combine to form new FCMs �� This allows com�bination of FCMs for di�erent actors or environments in the virtual world� Thenew �augmented� FCM includes the union of the causal concepts for all the actorsand the environment in the virtual world� If a FCM does not include a concept�then those rows and columns are all zero� The sum of the augmented �zero�padded�FCM matrices for each actor forms the virtual world�

F �nXi��

wiFi ��

The wi are positive weights for the ith FCM Fi� The weights state the relativevalue of each FCM in the virtual world and can weight any subgraph of the FCM�Figure ��a shows three simple FCMs� Equation �� combines these FCMs to givethe new simple FCM in Figure ��b that has fuzzy or multivalued edges�

F �

��F� � F� �F��

�

��

� � � � � �

� � � � ��

� � � �

��

The FCM sum �� helps knowledge acquisition� Any number of experts candescribe their FCM virtual world views and �� will weight and combine them ��

� Chapter ��

C4

3C

2C

1C

+

+

++

5C

+

+

C4

3C

2C

6C

C42C

1C

6C

5C

+

+

+

+

+

+

–

+

+

+

FCM 1

FCM 2 FCM 3

–

–

+

–

+

–

+–

–

– –

+

––

+

�a�

C4

3C

2C

1C

6C

5C

– 13

– 23

1

13

23

– 13

23

23

13

23

– 13

13

– 13

13

�b�

Figure �� FCMs combine additively� �a� Three bivalent FCMs� �b� AugmentedFCM� The augmented FCM takes the union of the causal concepts of the smallerFCMs and sums the augmented connection matrices as shown in ��

The additive structure of combined FCMs also permits a Delphi �� or questionaireapproach to knowledge acquisition� In contrast an AI expert system �� is a binarytree with graph search� Two or more trees need not combine to a tree� CombinedFCMs tend to have feedback or closed loops and that precludes graph search withforward or backward �chaining� The strong law of large numbers �� ensures thatthe knowledge estimate F in �� improves with the expert sample size n if weview the experts as independent �unique� random knowledge sources with �nite


variance �bounded uncertainty� and identical distribution �same problem�domainfocus�� The sample FCM converges to the unknown population FCM as the numberof experts grows�

The FCM sum �� can lead to new limit cycles that are not found in theindividual summed FCMs� The limit cycles in the FCMs shown in Figure ��a aregiven below� FCM has the �xed point� ��

�� and the � step limit cycles�

��

��

��

FCM � has a � step limit cycle�

��

��

FCM � has one �xed point� �

�� The combined FCM has no �xed points andone � step limit cycle�

��

��

��

��

This limit cycle is distinct from the limit cycles of each of the summed FCMs�

�� Nested FCMs

FCMs can bring goals and intentions to virtual worlds as they de�ne dynamicphysical and social environments� This can give the �common representationneeded for a virtual world �� The FCM can combine simple actions to model�intelligent behavior ��

�� Each node in turn can control its own simple FCMin a nested FCM� Complex actions such as walking emerge from networks of sim�ple re�exes� Nested simple FCMs can mimic this process as a net of �nite statemachines with binary limit cycles�

The output of a simple FCM is a binary limit cycle that describes actionsor goalsKos��a� This holds even if the binary concept nodes change state asyn�chronously� Each output turns a function on or o� as in a robotic neural net ��

�This output can control smaller FCMs or fuzzy control systems� These systems candrive visual� auditory� or tactile outputs of the virtual world� The FCM can controlthe temporal associations or timing cycles that structure virtual worlds� The FCMstate vector drives the motion of each character as in a frame in a cartoon� Simpleequations of motion can move each actor between the states�

FCM nesting extends to any number of fuzzy sets for the inputs� A conceptcan divide into smaller fuzzy sets or subconcepts� The edges or rules link the sets�This leads to a discrete multivalued output for each node� Enough nodes allow thissystem to approximate any continuous function �

for signal functions of the form�� The subconcepts Qij partition the fuzzy concept Cj

Cj �

Nj

i��

Qij ��

Figure �� shows the concept of a SURVIVAL THREAT divided into subconcepts�Each subconcept is the degree of threat�

� Chapter ��

Avoid Predator

Evade PredatorSmall

Survival Threat

Medium Survival Threat

Large Survival Threat +

-

+

Figure �� Subconcepts map to other concepts� This gives a more varied re�sponse�

The FCM edges or rules map one subconcept to another� These subconceptmappings form a fuzzy system or set of fuzzy if�then rules that map inputs tooutputs� Each mapping is a fuzzy rule or state�space patch that links fuzzy sets�The patches cover the graph of some function in the input�output state space� Thefuzzy system then averages the patches that overlap to give an approximation ofa continuous function �� Figure �� shows how subconcepts can map to di�erentresponses in the FCM� This gives a more varied response to changes in the virtualworld�

�� Virtual Undersea World

Figure �� shows a simple FCM for a virtual dolphin� It lists a causal webof goals and actions in the life of a dolphin �� The connection matrix ED statesthese causal relations in numbers�

ED �

D� D� D� D� D� D D� D� D D��D� � � � � � � � � �D� � � � � � � � � �D� � � � � � � � �

D� � � � � � � � � �

D� � � � � � � � � �D � � � � � � � � �D� � � � � � � � � �D� � � � � � � � �D � � � � � � � �

D��

The ith row lists the connection strength of the edges eik directed out from causalconcept Di and the ith column lists the edges eki directed into Di � Row � showshow the concept SURVIVAL THREAT changes the other concepts� Column �shows the concepts that change SURVIVAL THREAT�

We can model the e�ect of a survival threat on the dolphin FCM as a sustainedinput to D� This means D � for all time tk� C� is the initial input state of thedolphin FCM�


D2: Companionship

D3: Fatigue

D4: Rest

D5: Herd Clustering

D6: Food Search

D7: Chase food

D8: Catch & Eat Food

D9: Survival Threat

D10: Run away

+

-

++

+

+

-

-

-

-

+

+

+

- -

+

++

-

-

-

-

-

- -

-

--

-

-

-

-

+

+

-

-

-

D1:Hunger

Figure �� Trivalent fuzzy cognitive map for the control of a dolphin actor ina fuzzy virtual world� The rules or edges connect causal concepts in a signedconnection matrix�

C� � � � � � � � � � � ��

ThenC�ED � ��

� C� � ��

�

The arrow stands for a threshold operation with �� as the threshold value� C�

keeps D� on since we want to study the e�ect of a sustained threat� C� shows thatwhen threatened the dolphins cluster in a herd and �ee the threat� The negativerules in the ninth row of ED show that a threat to survival turns o� other actions�The FCM converges to the limit cycle C� � C� � C� � C� � C� � C� � � � ifthe threat lasts�C�ED � ��

� C� � � � � � � � � �

�

� Chapter ��

C�ED � �� C� � ��

C�ED � � � � � � � � � � � �

� C� � � � � � � � ��

C�ED � � � � � � � � � � � � C� � � � � � � � � ��

C�ED � ��

� C� � ��

�

Flight causes fatigue �C�� The dolphin herd stops and rests staying closetogether �C�� All the activity causes hunger �C��C�� If the threat persists�they again try to �ee �C�� A threat surpresses hunger� This limit cycle shows a�hidden global pattern in the causal virtual world�

The FCM converges to the new limit cycle C � C� � C� � C � C�� C�� C�� C�� C � � � � when the shark gives up the chase or eats a dolphinand the threat ends �D � ��C � ��

C�ED � � � � � � � � � � � � � C� � � � � � � � � ��

C�ED � � � � � � � � � � �

� C� � � � � � � � � � � ��

CED � �� C � � � � � � � � � � ��

C��ED � �� C��

C��ED � �� C��

C��ED � �� C��

C��ED � �� C��

C��ED � ��

� C � ��

The dolphin herd rests from the previous chase �C�C�� Then they begina hunt of their own �C�C�� They eat �C�� and then they socialize and rest�C��C��C�� This makes them hungry and the feeding cycle repeats�

�� Augmented Virtual World

Figure �� shows an augmented FCM for an undersea virtual world� It com�bines �sh school� shark� and dolphin herd FCMs with� F � Ffish�Fshark�Fdolphin �The new links among these FCMs are those of predator and prey where the largereats the smaller� The actors chase� �ee� and eat one another� A hungry shark chasesthe dolphins and that leads to the limit cycle �C��C��C��C�� above� Augmentingthe FCM matrices gives a large but sparse FCM since the actors respond to eachother in few ways� Figure �

shows the connection matrix for the augmentedFCM in Figure �� The augmented FCM moves the actors in the virtual world�The binary output states of this FCM move the actors� Each FCM state maps toequations or function approximations for movement�

We used a simple update equation for position�

p �tn�� p �tn� � �tn� � tn� v �tn� ��


+

+

+

+

+

F1:Hunger

F2: Fatigue

F3: Rest

F4: School

+F5: Catch &

Eat Food

F7: Run Away

+ ++

+

S2: Fatigue

S3: Rest

S7: Catch & Eat Food

++

+-

S1:Hunger

F6: Survival Threat

+

S5: Chase Fish

S6: Chase Dolphins

.

+

++

++

+ D1:HungerD2:

Companionship

D3: Fatigue

D4: Rest

D5: Herd Clustering

D6: Food

Search

D7: Chase food

D8: Catch & Eat Food

+D9:

Survival Threat

D10: Run away +

+

+

-

++

++

S4: Food Search

++

--- -

----

--

-

----

-

- -- - - -

--

--

------

--

-

--

-

-

-

--

+

+

+

+

+

+

+

-

--

- -

Figure �� Augmented FCM for di�erent actors in a virtual world� The actorsinteract through linked common causal concepts such as chasing food and avoidinga threat�

The velocity v�t� does not change at time step �t � The FCM �nds the directionand magnitude of movement� The magnitude of the velocity depends on the FCMstate� If the FCM state is �run away� then the velocity is FAST� If the FCM stateis �rest� then the velocity is SLOW� The prey choose the direction that maximizesthe distance from the predator� The predator chases the prey� When a predatorsearches for food it swims at random �� Each state moves the actors through thesea�

The FCM in Figure �� encodes limit cycles between the actors� For example�if we start with a hungry shark and We set the causal link between concept S��FOOD SEARCH and S�� CHASE DOLPHINS equal to zero to look at sharkinteractions with the �sh school� Then the �rst state C� is

C� � �� This vector gives a ��step limit cycle after four transition steps�

C�EA � �� C� � ��

C�EA � �� C� � ��

C�EA � �� C� � ��

� Chapter ��

Dolphin Shark Fish

D�D�D�D�D�D�D�D�DD�� S� S�S� S� S� S� S� F� F�F� F� F� F�F�D� ��

D� � � � � � � � � � � � � � � � � � � � � � � � �

D� � � � � ��

D� � ��

D� � � � � � � � ��

D� � � � ��

D� � � � � � � � � � � � � � � � � � � � � � � � �

D� ��

D � � � � ��

D��

S� � � � � � � � � � � � � � � � � � � � � � � � �

S� � � � � � � � � � � � � � ��

S� � � � � � � � � � � ��

S� � � � � � � � � � � � � � � � ��

S� � � � � � � � � � � � � ��

S� � � � � � � � � � � � � ��

S� � � � � � � � � � � ��

F� � � � � � � � � � � � � � � � � � � � ��

F� � � � � � � � � � � � � � � � � � � � � ��

F� � � � � � � � � � � � � � � � � � ��

F� � � � � � � � � � � � � � � � � � � � � � ��

F� � � � � � � � � � � � � � � � � � ��

F� � � � � � � � � � � � � � � � � � � � � ��

F� � � � � � � ��

Figure �� AugmentedFCM connectionmatrix for the dolphinherd� �sh school�and shark� Figure �� shows the nodes and edges� The lines show the FCMs ofthe actors� The sparse region outside the lines shows the interaction space of theFCMs�

C�EA � ��

�C� � ��

�

C�EA � ��

�C � ��

�

CEA � �� C� � ��

C�EA � �� C� � ��

C�EA � �� C � ��

CEA � �� C��


⊗

⊗⊗

C

Α

Β

Figure �� FCMs control the virtual world� The augmented FCM controls theactions of the actors� In event A the hungry shark forces the dolphin herd to runaway� Each dashed line stands for a dolphin swim path� In event B the shark �ndsthe �sh and eats some� Each dashed line stands for the path of a �sh in the school�The cross shows the shark eating a �sh� In event C the �sh run into the dolphinsand su�er more losses� The solid lines are the dolphin paths� The dashes are the�sh swim paths� The cross shows a dolphin eating a �sh�

C��EA � �� C��

C��EA � ��

�C� � ��

�

In this limit cycle a shark searches for food �C��C��C�� The shark �nds some�sh �C�� chases the �sh �C�� and then eats some of the �sh �C�� To avoid theshark most �sh run away and then regroup as a school �C��C�C�� Then the �shrest and eat while the shark rests �C��C�� In time the shark gets hungry againand searches for �sh �C��C��

The result is a complex dance among the actors as they move in a ��D ocean�Figure �� shows these movements� The forcing function is a hungry shark �C��

�� The shark encounters the dolphins who cluster and then �ee the shark� Theshark chases but cannot keep up� The shark still searches for food and �nds the

�� Chapter ��

��

γ p

αm

��

Case 1 Case

2

VpV f

≤1

VpV f

>1

Figure �� Fish change their behavior as the degree of threat changes� �a� The�sh minimize time within the sighting angle of the predator� Case � shows theangle of escape when the �sh swim faster� Case � shows the desired angle whenthe predator swims faster� �b� The �sh maximize the distance between themselvesand the predator to evade the predator� The �sh swim straight ahead when the�sh swim faster than the predator� The �sh swim away at an angle if the predatorswims faster�

�sh� It catches a �sh and then rests with its hunger sated� Meanwhile the hungrydolphins search for food and eat more �sh� Each actor responds to the actions ofthe other�

�� Nested FCMs for Fish Schools

In a simple FCM the threat response concepts link as a rule� SURVIVALTHREAT implies RUN AWAY� Fish change their behavior as the degree of threatchanges� This rule does not model the e�ects of di�erent threats� For that weneed a nested FCM or a fuzzy function approximator that links the threat degreeto di�erent responses� The size of the threat is a function of the size� speed� andattack angle of the predator �� A small threat leads to avoidance behavior�Figure ��a shows how �sh avoid a predator� The �sh move in direction � tomaximize their distance from the predator��

cot� � cot�m �Vp

Vf sin�m��

Vp and Vf are the velocities of the predator and the �sh� �m is the angle thatminimizes the time in terms of the predator�s sighting angle p�

tan�m � � cot p ��

A large threat causes the �sh to evade the predator� The �sh try to maximizethe minimum distance from the predator Dp ��

D�p � ��Xo � Vp t� � Vf t cos �� Vf t sin��

� ��

X� is the initial distance between predator and prey� � is the escape angle of theprey� Vp and Vfare the velocities of the predator and the �sh� Figure ��b showshow �sh evade a predator� A fuzzy system can approximate these responses usinghand�picked rules or a neural�fuzzy learning �� These threat responses causethe �fountain e�ect and the �burst e�ect in �sh schools �� as each �sh tries to

Technology for Multimedia ��

F1:Hunger

F2: Fatigue

F3: Rest

F4: School

F5: Catch & Eat Food

F7: Large Survival Threat

F9: Avoid

+

+

+

+

+

+

+

F6: Small Survival Threat

F8: Evade

+

+

++

+

-

--

--

--

-

Figure �� Example of a nested FCM� The concept of a survival threat dividesinto two subconcepts that each map to a di�erent survival tactic�

increase its chances of survival� The fountain e�ect occurs when a predator movestowards a �sh school and the school splits and �ows around the predator� Theschool re�forms behind the predator� In the burst e�ect the school expands in theform of a sphere to evade the predator�

A small survival threat may be a slow�moving predator that either has notseen or decided to attack the �sh� A large survival threat may be a fast predatorsuch as a barracuda or shark that swims towards the center of the school� If weinsert this new sub�FCM into the Fish FCM in Figure �� we get the FCM inFigure �� Di�erent limit cycles appear for di�erent degrees of threat� For a smallthreat �F� the �sh avoid the predator �F� as they move out of the line�of�sightof the predator� Large threats �F�� cause the �sh to scatter quickly to evade thepredator F�� This leads to fatigue and rest �F� and F��

�� Adaptive Fuzzy Cognitive Maps

An adaptive FCM changes its causal web in time� The causal web learns fromdata� The causal edges or rules change in sign and magnitude� The additive schemeis a type of causal learning since it changes the FCM edge strengths� In general anedge eij changes with some �rst�order learning law�

�eij � fij �E�C� � gij �t� ��

Here gij is a forcing function� Data �res the concept nodes and in time this leavesa causal pattern in the edge� Causal learning is local in fij � It depends on just itsown value and on the node signals that it connects�

�eij � fij�eij � Ci� Cj� �Ci� �Cj

�� gij �t� ��

Correlation or Hebbian learning can encode some limit cycles in the FCMs ortemporal associative memories �TAMs� �� It adds pairwise correlation matrices


in �� This method can only store a few patterns� Di�erential Hebbian learningencodes changes in a concept in equation �� Both types of learning are local andlight in computation�

To encode binary limit cycles in connection matrix E the TAM method sumsthe weighted correlation matrices between successive states �� To encode the limitcycle C� � C� � C� � C� we �rst convert each binary state Ci into a bipolarstate vector Xi by replacing each � with a �� Then E is the weighted sum

E � q�XT�X� � q�X

T�X� � � � �� qn��X

Tn��Xn � qnX

TnX� ��

The length of the limit cycle should be less than the number of concepts� Elsecrosstalk can occur� Proper weighting of each correlation matrix pair can improvethe encoding �� and thus increase the FCM storage capacity� Correlation learningis a form of the unsupervised signal Hebbian learning law in neural networks ��

�eij � �eij �Ci�xi�Cj�xj� ��

A virtual world can encode an event sequence with �� or �� A simple chasecycle might be C� � C� � C��

C� � � � � � � � � �

C� � � � � � � �

C� � � � � � � � �

Then �� gives the FCM connection matrix E when qi � for all i�

E �

D� D� D� D� D� D D� D� D D��D� � ��

D� ��

D� � � � � � � � �

D� � ��

D� � � �� D ��

D� ��

D� ��

D � � � � � � � � �D��

ThenC�E � ��

� C� � � � � � � ��

C�E � �� C� � � � � � � � �

�

C�E � �� C� � � � � � � � � �

�

Correlation encoding treats negative and zero causal edges the same� It can encode�spurious causal implications between concepts such as e�� This meanssearching for food causes a desire to socialize� Correlation encoding is a poormodel of inferred causality� It says two concepts cause each other if they are on


at the same time� Di�erential Hebbian learning encodes causal changes to avoidspurious causality� The concepts must move in the same or opposite directions toinfer a causal link� They must come on and turn o� at the same time or one mustcome on as the other turns o�� Just being on does not lead to a new causal link�The patterns of turning on or o� must correlate positively or negatively�

The di�erential Hebbian learning law �� correlates concept changes or veloc�ities�

�eij � �eij � �Ci �xi� �Cj �xj� ��

So �Ci�xi� �Cj�xj� � � i� concepts Ci and Cj move in the same direction��Ci�xi� �Cj�xj� � � i� concepts Ci and Cj move in opposite directions� In this sense�� learns patterns of causal change� The �rst�order structure of �� implies thateij�t� is an exponentially weighted average of paired �or lagged� changes� The mostrecent changes have the most weight� The discrete change �Ci�t� � Ci�t��Ci�t��lies in f��g� The discrete di�erential Hebbian learning can take the form

eij �t� � �

eij �t� � ct ��Ci �xi��Cj �xj� � eij �t� if �Ci �xi� ��

eij �t� if �Ci �xi� � ��

Here ct is a learning coe�cient that decreases in time �� The sequence of learningcoe�cients fctg should decrease slowly �� in the sense of

�Xt��

ct ��

but not too slowly in the sense that

�Xt��

c�t ��

In practice ct

�t� �Ci�Cj � � i� concepts Ci and Cj move in the same direction�

�Ci�Cj � � i� concepts Ci and Cj move in opposite directions� E changes onlyif a concept changes� The changed edge slowly �forgets the old causal changes infavor of the new ones� This causal law can learn higher�order causal relations if itcorrelates multiple cause changes with e�ect changes�

We used di�erential Hebbian learning to encode a feeding sequence and achase sequence in a FCM� The concepts in the ith row learn only when �Ci�xi�equals or �� We used

ct �tk� � ��

�

�

tk

�N

��


The training data came from the rest� eat� play and the chase sequences in Sec�tion �� This gave the ED �

D� D� D� D� D� D D� D� D D��D� �� D� �� D� �� D� �� D� �� D �� D� �� D� �� D �� D��

This learned edge matrix ED resembles the FCM matrix in Figure �� Thecausal links it lacks between D�� and �D�D��D�� were not in the training set�The diagonal links terms for self�inhibition of each concept� This occurs since eachconcept is on for one cycle before the matrix transitions to the next state� Thehunger input CL� � � � � � � � � � � � with a threshold of �� now leads tothe limit cycle�CL�ED � ��

CL� � ��

CL�ED � ��

CL� � ��

CL�ED � ��

CL� � ��

CL�ED � ��

CL� � ��

CL�ED � ��

CL� � ��

CL�ED � ��

CL � ��

CLED � ��

CL� � � � � � � � � � � ��

Figure ��a� shows the hand�designed limit cycle from the previous section� Fig�ure ��b� shows the limit cycle from FCM found with di�erential Hebbian learn�ing� The DHL limit cycle is one step shorter� Both FCMs have just one limit cycle


and the null �xed point in the space of �� binary state vectors� The value of ED�does not change over � intervals� The learning law in �� learns only if there is achange in the node�

D10 Run Away

D9 Survival Threat

D8 Catch & Eat Food

D7 Chase Food

D6 Food Search

D5 Herd Clustering

D4 Rest

D3 Fatigue

D2 Companionship

D1 Hunger

Time Step 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

D10 Run Away

D9 Survival Threat

D8 Catch & Eat Food

D7 Chase Food

D6 Food Search

D5 Herd Clustering

D4 Rest

D3 Fatigue

D2 Companionship

D1 Hunger

Time Step 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

(a)

(b)

Figure �� Limit cycle comparison between the hand�designed system and theFCM found with di�erential Hebbian learning� Each column is a binary statevector� �a� Rest� feed� play� rest limit cycle for the FCM in Figure �� b� Limitcycle for the FCM found with ��

�� Conclusions

Fuzzy cognitive maps can model the causal web of a virtual world� The FCMcan control its local and global nonlinear behavior� The local fuzzy rules or edgesand the fuzzy concepts they connect model the causal links within and between


events� The global FCM nonlinear dynamics give the virtual world an �arrowof time� A user can change these dynamics at will and thus change the causalprocesses in the virtual world� FCMs let experts and users choose a causal web bydrawing causal pictures instead of by stating equations�

FCMs can also help visualize data� They show how variables relate to oneanother in the causal web� The FCM output states can guide a cartoon of thevirtual world as shown in Figure �� This cartoon shows the dolphin chase� rest�eat sequence described earlier� The cartoon animates the FCM dynamics as thesystem trajectory moves through the FCM state space� This can apply to modelsin economics� medicine� history� and politics ��

where the social and causal webcan change in complex ways that may arise from changing the sign or magnitudeof a single FCM causal rule or edge�

TIME STEP 0: THREAT APPEARS IN THE FORM OF A SHARK.

TIME STEP 1&2: DOLPHINS FLEE THE SHARK IN A TIGHTLY PACKED HERD.

TIME STEP 3&4: DOLPHINS CLUSTER TOGETHER AND REST

TIME STEP 5-7: DOLPHINS AVOID SHARK THEN REST.

TIME STEP 8&9: DOLPHINS START A SEARCH FOR FOOD.

TIME STEP 10: THE DOLPHINS FIND A SCHOOL OF FISH THEN BEGIN TO CHASE THEM

TIME STEP 11: THE DOLPHINS CATCH AND EAT SOME FOOD

TIME STEP 12-13 : THE DOLPHINS THEN PLAY AND REST. THEN THE CYCLE BEGINS AGAIN.

Figure �� The FCM output states can guide a cartoon of the virtual world�This cartoon shows the dolphin chase� rest� eat sequence described in section ��The cartoon animates the FCM dynamics as the system trajectory moves throughthe FCM state space�

REFERENCES ��

The additive structure of combined FCMs permits a Delphi �� or question�aire approach to knowledge acquisition� These new causal webs can change anadaptive FCM that learns its causal web as neural�like learning laws process time�series data� Experts can add their FCM matrices to the adaptive FCM to initializeor guide the learning� Such a causal web can learn the user�s values and actionhabits and perhaps can test them or train them�

More complex FCMs have more complex dynamics and can model more com�plex virtual worlds� Each concept node can �re on its own time scale and �re in itsown nonlinear way� The causal edge �ows or rules can have their own time scalestoo and may increase or decrease the causal �ow through them in nonlinear ways�This behavior does not �t in a simple FCM with threshold concepts and constantedge weights�

A FCM can model these complex virtual worlds if it uses more nonlinear mathto change its nodes and edges� The price paid may be a chaotic virtual world withunknown equilibrium behavior� Some users may want this to add novelty to theirvirtual world or to make it more exciting� A user might choose a virtual world thatis mildly nonlinear and has periodic equilibria� At the other extreme the user mightchoose a virtual world that is so wildly nonlinear it has only aperiodic equilibria�Think of a virtual game of tennis or raquetball where the gravitational potentialchanges at will or at random�

Fuzziness and nonlinearity are design parameters for a virtual world� Theymay give a better model of a real process�

REFERENCES

�

M� Krueger� Articial Reality II� Second ed� Addison�Wesley� ��

�� W� Gibson� Neuromancer� New York� Ace Books� ��

�� R� A� Brown� Fluid Mechanics of the Atmosphere� New York� Academic Press�

��

�� J� J� Craig� Introduction to Robotics� Reading� MA� Addison�Wesley� ��

�� E� Ackerman� L� Gatewood� J� Rosevear and G� Molnar� �Blood Glucose Regu�lation and Diabetes� in Concepts and Models of Biomathematics� F� Heinmets�Ed�� Marcel Dekker� ��

�� B� Kosko� �Fuzzy Cognitive Maps� International Journal Man�Machine Studies�Vol� �� No� � pp� ��

�� B� Kosko� �Hidden Patterns in Combined and Adaptive Knowledge Networks�International Journal of Approximate Reasoning� Vol� �� No� � pp� ��

�� B� Kosko� Neural Networks and Fuzzy Systems Englewood Cli�s� Prentice Hall�

��

�� B� Kosko� �Fuzzy Systems as Universal Approximators� IEEE Transactions onComputers� Vol� �� No�

� November� pp� ��

�� H� T� Nguyen� �On Random Sets and Belief Functions� Journal of MathematicalAnalysis and Applications� Vol� �� No� �� pp� ��

� Technology for Multimedia

�

K� Hornik� M� Stinchcombe and H� White� �Multilayer Feedforward Networks areUniversal Approximators� Neural Networks� Vol� �� No� � pp� ��

�� F� A� Watkins� �Fuzzy Engineering�Ph�D� Thesis� University of California atIrvine� ��

�� L� Wang and J� M� Mendel� �Fuzzy Basis Functions� Universal Approximation�and Orthogonal Least�Squares Learning� IEEE Transactions on Neural Networks�Vol� �� No� �� September� pp� ��

�� D� F� Specht� �A General Regression Neural Network� IEEE Transactions onNeural Networks� Vol� �� No� �� November� pp� ��

�� B� Kosko� �Optimal Fuzzy Rules Cover Extrema� International Journal of Intel�ligent Systems� Vol� �� No� �� pp� ��

�� F� A� Watkins� �The Representation Problem for Additive Fuzzy Systems� Pro�ceedings of the the �� IEEE International Conference on Fuzzy Systems �IEEEFUZZ��Vol� I� pp�

��

�� J� A� Dickerson and B� Kosko� �Virtual Worlds as Fuzzy Cognitive Maps� Pres�ence� Vol� �� No� �� Spring� pp� ��

�� P� H� Winston� Articial Intelligence� Second ed� Reading� MA� Addison�Wesley�

��

�� W� R� Taber and M� Siegel� �Estimation of Expert Weights with Fuzzy CognitiveMaps� Proceedings of the �st IEEE International Conference on Neural Networks�ICNN�� San Diego�Vol� II� pp� ��

�� B� Kosko� �Bidirectional Associative Memories� IEEE Transactions Systems�Man� and Cybernetics� Vol� �� No� � pp� ��

��

R� A� Brooks� �A Robot that Walks� Emergent Behaviors from a Carefully EvolvedNetwork� Neural Computation� Vol� � No� �� pp� ��

�� S� Grossberg� Studies of Mind and Brain� Boston� Reidel� ��

�� N� I� Badler� B� L� Webber� J� Kalita and J� Esakov� �Animation from Instruc�tions� in Making Them Move� Mechanics� Control� and Animation of ArticulatedFigures� N� I� Badler� B� A� Barsky and D� Zeltzer� Eds� San Mateo� CA� MorganKaufmann� pp� ��

�� J� H� Connell� Minimalist Mobile Robotics� A Colony�style Architecture for anArticial Creature Academic Press� Harcourt Brace Jovanovich� ��

�� S� H� Shane� �Comparison of Bottlenose Dolphin Behavior in Texas and Florida�with a Critique of Methods for Studying Dolphin Behavior� in The BottlenoseDolphin� S� Leatherwood and R� R� Reeves� Eds�� Academic Press� pp� ��

��

�� B� O� Koopman� Search and Screening� New York� Pergamon Press� ��

REFERENCES �

�� B� L� Partridge� �The Structure and Function of Fish Schools� Scientic Ameri�can� Vol� �� No� �� pp�

��

�� D� Weihs and W� P� W�� Optimal Avoidance and Evasion Tactics in Predator�Prey Interactions� Journal of Theoretical Biology� Vol� �� No� � pp� ��

��

�� J� A� Dickerson� �Fuzzy Function Approximation with Ellipsoidal Rules� Ph�D�Thesis� University of Southern California� ��

�� Y� F� Wang� J� B� Cruz and J� H� Mulligan� �Guaranteed Recall of All Train�ing Pairs for Bidirectional Associative Memory� IEEE Transactions on NeuralNetworks� Vol� �� No� �� pp� ��

��

W� R� Taber� �Knowledge Processing with Fuzzy Cognitive Maps� Expert Systemswith Applications� Vol� �� No� � pp� ��

�� J� P� Martino� Technological Forecasting for Decisionmaking� American Elsevier�

��

�� J� A� Dickerson and B� Kosko� �Fuzzy Function Approximation with SupervisedEllipsoidal Learning� Proceedings of the World Conference on Neural Networks�WCNN �� Portland� OR�Vol� II� pp� ��

�� J� A� Dickerson and B� Kosko� �Fuzzy Function Learning with Covariance El�lipsoids� Proceedings of the IEEE International Conference on Neural Networks�IEEE ICNN�� San Francisco� pp�

��

��

�� J� A� Dickerson and B� Kosko� �Fuzzy Function Approximation with EllipsoidalRules� IEEE Transactions on Systems� Man� and Cybernetics� No� August� pp�To Appear� ��

�� B� Kosko� �Stochastic Competitive Learning� IEEE Transactions on Neural Net�works� Vol� �� No� �� pp� ��

�� H� M� Kim and B� Kosko� �Fuzzy Prediction and Filtering in Impulsive Noise�Fuzzy Sets and Systems� Vol� �� No� � pp� ��

�� Technology for Multimedia

A Proof of the Fuzzy Approximation Theorem

Fuzzy Approximation Theorem An additive fuzzy system uniformly approxi�mates f � X � Y if X is compact and f is continuous�Proof�

Pick any small constant � �� We must show that jF �x�� f�x�j � for allx � X� X is a compact subset of Rn� F �x� is the centroidal output �� of theadditive fuzzy system F � Continuity of f on compact X gives uniform continuity�So there is a �xed distance � such that� for all x and z in X� jf�x�� f�z�j � �� ifjx� zj � �� Replace � by ��n for any Lp space with p � �� We can construct aset of open cubes M�� Mm that cover X and that have ordered overlap in theirn coordinates so that each cube corner lies at the midpoint cj of its neighbors Mj �Pick symmetric output fuzzy sets Bj centered on f�cj �� So the centroid of Bj isf�cj��

Pick u � X� Then by construction u lies in at most �j overlapping opencubes Mj � Pick any w in the same set of cubes� If u � Mj and w � Mk� then forall v � Mj �Mk� ju � vj � � and jv � wj � �� Uniform continuity implies thatjf�u� � f�w�j � jf�u� � f�v�j � jf�v� � f�w�j � �� So for cube centers cj and ck�jf�cj�� f�ck�j �

��

Pick x � X� Then x too lies in at most �j open cubes with centers cjand jf�cj� � f�x�j �

�� Along the kth coordinate of the range space R

p thekth component of the additive system centroid F �x� lies on or between the kthcomponents of the centroids of the Bj sets� So� since jf�cj� � f�ck�j �

�� for all

f�cj�� jF �x�� f�cj�j �� Then

jF �x�� f�x�j � jF �x�� f�cj�j� jf�cj�� f�x�j �

��

��

Q�E�D�

B Learning in SAMs� Unsupervised Clustering and Supervised

Gradient Descent

A fuzzy system learns if and only if its rule patches move or change shape inthe input�output product space X�Y � Learning can change the centers or widths oftriangle or trapezoidal sets� These changing sets then change the shape or positionof the Cartesian rule patches built out of them� The mean�value theorem and thecalculus of variations show �� that optimal lone rules cover the extrema or bumpsof the approximand� Good learning schemes �� tend to quickly move rulespatches to these bumps and then move extra rule patches between them as the rulebudget allows� Hybrid schemes use unsupervised clustering to learn the �rst setof fuzzy rule patches in position and number and to initialize gradient descent insupervised learning�

Learning changes system parameters with data� Unsupervised learning amountsto blind clustering in the system product space X � Y to learn and tune the mfuzzy rules or the sets that compose them� Then k quantization vectors qj � X�Ymove in the product space to �lter or approximate the stream of incoming datapairs �x�t�� y�t�� or the concatenated data points z�t� � �x�t�jy�t�T � The simplestform of such product space clustering �� centers a rule patch at each data point

REFERENCES ��

and thus puts k � m� In general both the data and the quantizing vectors greatlyoutnumber the rules and so k �� m�

A natural way to grow and tune rules is to identify a rule patch with theuncertainty ellipsoid �� that forms around each quantizing vector qj fromthe inverse of its positive de�nite covariance matrix Kj � Then sparse or noisy datagrows a patch larger and thus a less certain rule than does denser or less noisy data�Unsupervised competitive learning �� can learn these ellipsoidal rules in three steps�

kz�t�� qj�t�k � min�kz�t� � q��t�k� � � � � kz�t�� qk�t�k� �B��

qi�t� � �

qj�t� � �t�z�t�� qj�t� if i � jqi�t� if i �� j

�B��

Ki�t � � �

Kj�t� � vt��z�t�� qj�t��

T �z�t� � qj�t��Kj�t� if i � jKi�t� if i �� j

�B��for the Euclidean norm kzk� � z�� z

�np�

The �rst step �B�� is the competitive step �� It picks the nearest quantizingvector qj to the incoming data vector z�t� and ignores the rest� Some schemes maycount nearby vectors as lying in the winning subset� We used just one winnerper datum� This correlation matching approximates the competitive dynamics ofnonlinear neural networks� The second step updates the winning quantization or�synaptic vector and drives it toward the centroid of the sampled data patternclass �� The third step updates the covariance matrix of the winning quantizationvector� We initialize the quantization vector with sample data �qi�� z�i�� toavoid skewed groupings and to initialize the covariance matrix with small positivenumbers on its diagonal to keep it positive de�nite� Projection schemes �� can then convert the ellipsoids into fuzzy sets along each coordinate of the input�output space� Other schemes can use the unfactored joint set function directly��Supervised learning can also tune the eigenvalue parameters of the rule ellipsoids�

The sequences of learning coe�cients f�tg and fvtg should decrease slowly ��

in the sense of�Xt��

�t �� but not too slowly in the sense of�Xt��

��t �� In practice

�t

�t� The covariance coe�cients obey a like constraint as in our choice of vt �

�� t��N

where N is the total number of data points� The supervised learningschemes below also use a similar sequence of decreasing learning coe�cients�

Supervised learning changes SAM parameters with error data� The error ateach time t is the desired system output minus the actual SAM output� �t �dt � F �xt�� Unsupervised learning uses the blind data point z�t� instead of thedesired or labeled value dt� The teacher or supervisor supervises the learning processby giving the desired value dt at each training time t� Most supervised learningschemes perform stochastic gradient descent on the squared error and do so throughiterated use of the chain rule of di�erential calculus�

Supervised gradient descent can learn or tune SAM systems �� by chang�ing the rule weights wj in �B�� the then�part volumes Vj � the then�part centroidscj� or parameters of the if�part set functions aj� The rule weight wj enters the ratioform of the general SAM system

�� Technology for Multimedia

F �x� �

mXj��

wj aj�x� Vj cj

mXj��

wj aj�x� Vj

�B��

in the same way as does the then�part volume Vj in the SAM Theorem� Both cancelfrom �B�� if they have the same value if w� � � wm � � or if V� � � Vm �� So both have the same learning law if we replace the nonzero weight wj withthe nonzero volume Vj ��

wj�t� � � wj�t� � �t�Et�wj

�B��

� wj�t� � �t�Et�F

�F

�wj�B��

� wj�t� � �t �tpj�xt�

wj�t��cj � F �xt� �B��

for instantaneous squared error Et ��dt � F �xt��

� with desired�minus�actualerror �t � dt � F �xt�� We include the rule weights here for completeness� Ourfuzzy systems were unweighted and thus used w� � � wm � �� The volumesthen change in the same way if they are independent of the weights �which theymay not be in some ellipsoidal learning schemes��

Vj�t� � � Vj�t�� t�Et�Vj

�B��

� Vj�t� � �t �tpj�xt�

Vj�t��cj � F �xt� �B��

The learning law �B�� follows since �Et�F

� �� and since

�F

�wj�

aj�x� Vj cj

mXi��

wi ai�x� Vi � aj�x� Vj

mXi��

wi ai�x� Vi ci

�mXi��

wi ai�x� Vi��

�B��

�wj aj�x� Vj

wj

mXi��

wi ai�x� Vi

��cj

mXi��

wi ai�x� Vi

mXi��

wi ai�x� Vi

�

mXi��

wi ai�x� Vi ci

mXi��

wi ai�x� Vi

�� B�

�

�pj�x�

wj�cj � F �x� �B��

from the SAM Theorem�

REFERENCES ��

The centroid cj in the SAM Theorem has the simplest learning law�

cj�t � � � cj�t�� t�Et�F

�F

�cj�B��

� cj�t� � �t �t pj�xt�� B��

So the terms wj� Vj � and cj do not change when pj � and thus when thejthif�part set barely �res� aj�xt� ��

Tuning the if�part sets involves more computation since the update law con�tains an extra partial derivative� Suppose that the if�part set function aj is a func�tion of l parameters� aj � aj�m�j � � � � �m

lj�� Then we can update each parameter

with

mkj �t� � � mkj �t�� t

�Et�F

�F

�aj

�aj

�mkj�B��

� mkj �t� � �t �tpj�xt�

aj�xt��cj � F �xt�

�aj

�mkj� �B��

Exponential if�part set functions can reduce the learning complexity� They

have the form aj � efj �m

�

j ��mlj � and obey �aj

�mkj

� aj�fj�m

�

j ��mlj �

�mkj

� Then the param�

eter update �B�� simpli�es to

mkj �t� � � mkj �t� � �t �t pj�xt��cj � F �xt�

�fj

�mkj� �B��

This can arise for independent exponential or Gaussian sets aj�x� �nYi��

expff ij�xi�g �

expfnXi��

f ij�xi�g � expffj�x�g� The exponential set function

aj�x� � expfnXi��

uij�vij � xi�g �B��

has partial derivatives�fj

�ukj

� vkj � xk�t� and�fj

�vkj

� ukj �

The Gaussian set function

aj�x� � expf�

�

nXi��

�xi �mij

�ij��g �B��

has mean partial derivative�fj

�mkj

�xk�m

kj

��kj��

and variance partial derivative�fj

��kj

�

�xk�mkj �

�

��kj��

� Such Gaussian set functions reduce the SAM model to Specht�s ��

radial basis function network� We can use the smooth update law

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