Adaptation Of Vigilance Factor And Choice Parameter In Fuzzy ART System

Transcription

1 Adaptation Of Vigilance Factor And Choice Parameter In Fuzzy ART System M. R. Meybodi P. Bahri Soft Computing Laboratory Department of Computer Engineering Amirkabir University of Technology Tehran, Iran Abstract In adaptive resonance theory network, the choice of vigilance factor (VF) and choice parameter (CP) affects the performance of the network, such as the number of classes into which the data are classified. These parameters are typically chosen and adapted using human udgment, experience, and heuristic rules. Rather than choosing and optimizing these parameters manually, we use learning automata to automatically adapt these parameters. In an earlier paper [Bahri99] we examined the ability of P-model learning automata to only adapt the vigilance factor of fuzzy art network. In this paper we further study the effectiveness of LA in adaptation of VF and CP. This time we try to adapt both VF and CP simultaneously using different models of learning automata: P-model, Q-model, and S- model. The feasibility of the proposed method is shown through simulation on three problems: the circle in the square, nested spirals, and gaussian distributed two groups. Keywords. Neural Networks, Fuzzy ART, Vigilance Factor, Choice Parameter, Learning Automata. 1. Introduction. Fuzzy ART is a self organizing neural architecture which is capable of fast learning its tasks in any non-stationary environment. The architecture combines neural network and fuzzy logic and achieves excellent operation properties. One of the properties of the network is that it learns its tasks in an unsupervised mode, at the same time it can be trained in an supervised mode to adapt to the input /output environments. In our implementation of fuzzy ART the network has three layers, as well as an orienting subsystem. The in these layers process input, and the orienting subsystem guides the search for the fittest category that represents the input. The identity of the category of the input is determined by the third layer whose prototypes best fit the input. Two of the critical entities that determine the dynamics of the network are vigilance factor and choice parameter. The appropriate selection of these two parameters have large effect on the convergence of the algorithm. For example, if the vigilance parameter is too small, too much data compression may result and too broad classification made, and if it is too large, too many will be gened and good classification may not result. If the choice parameter is too small, the network may converge too early, and if it is too large, too many will be gened resulting in slower operation.

2 It is not easy to choose appropriate values for these parameters for a particular problem. These parameters are usually determined by trial and error and using past experience. To the author`s knowledge two different methods for adaptation of vigilance factor only have been reported in the literature. The first method is due to Arabshahi, et al.[arabshi96]. This method is based on the assumption that the number of clusters are known in advance, and for this reason is not relevant to us and will not be considered further. The second method is due to Fu[Fu94]. This method which is applied to ART-II neural network is a fuzzy adaptive vigilance algorithm, with vigilance factor optimally tailored in signal processing under noisy environment. By such an adaptation they have overcome the weakness of fixed VF which may cause the spurious memory. They have investigated the relationship between attractive basin and VF in ART-II, and analytically proved the features of self-stability of ART-II model and illustd the way that fuzzy adaptive VF is to adust the attractive basin. This method has been implemented and compared with methods presented in this paper. In this paper, by interconnection of learning automata and fuzzy ART, we apply learning automata for determining the choice parameter and vigilance factor. Three algorithms for adaptation of these parameters based on learning automata are presented. The first algorithm is used to adapt VF only and the second algorithm is used for simultaneous adaptation of VF and CP. In the third algorithm we associate a different vigilance factor to each long term memory trace and using a collection of learning automata try to find the best VF for each long term memory trace. Through simulation we have found that a 20 percent higher performance for fuzzy ART neural network in terms of recognition, reection, and the size of network gened can be obtained if vigilance factor and choice parameter is adapted simultaneously. In order to evaluate the performance of the proposed scheme, simulations are carried out on the following problems. The circle in the square: Adapted from [Carpenter95A], is an experiment in which the task of the network is to differentiate between the points inside circle and those outside, as shown in figure 1-c. The nested spirals: Adapted also from [Carpenter95A], is an experiment which is highly nonlinear. The task of the network is to differentiate between the points of the two nested spirals shown in figure 1-d We have added aliasing to the experiment, that is for each original point from either group we have added a gaussian noise of variance 0.15 and mean of zero in both X and Y directions sepaly, and submitted the point for recognition to the network, taking the original point as the reference point. The aliased points are shown in figure 1-a and 1-b. a) b) c) d) figure -1 figure - 2 Gaussian distributed two groups around a circle: Taken from [Carpenter95A] is an experiment in which there are two groups gaussian distributed around the central point and each part of the groups situated alternately. Each group has three representations. The benchmark is shown in figure 2. In this case both training and test samples are noisy. The rest of the paper is organized as follows. Section two explains the learning automaton. Section three discuses the augmented fuzzy ARTMAP in some detail. Section four shows the details of our methods, and section five discuses simulation results. Last section is the conclusion. 2. Learning Automata Learning automata operating in unknown random environments have been used as models of learning systems. These automata choose an action at each instant from a finite action set, observe the reaction of the environment to the action chosen and modify the selection of the next action on the basis of the response of the environment. A learning automaton is a quintuple.

3 α,β, Φ,F,G where: 1) α = (α 1,..., α R ) is the set of actions that it must choose from. 2) Φ = (Φ 1,..., Φ s ) is the set of states. 3) β is the set of inputs 4) F: Φ β Φ is a map called the transition map. It defines the transition of the state of the automaton on receiving an input, F may be stochastic. 5) G: Φ α is the output map and determines the action taken by the automaton if it is in state Φ. The selected action serves as the input to the environment which in turn emits a stochastic response β(n) at the time n. β(n) is an element of β and is the feedback response of the environment to the automaton. Deping on the nature of set β the automata can be classified into three classes. If β ={0,1} we have P model automata, if the input to the automata can assume multiple discrete values then we have Q model learning automata and if the input has a continuous range then we call the automata S model learning automata. On the basis of the response β(n), the state of the automaton Φ(n) is updated and a new action chosen at the time (n+1). It is desired that as a result of interaction with the environment the automaton arrives at the action which presents it with the minimum penalty response in an expected sense. If the probability of the transition from one state to another state and probabilities of correspondence of action and state are fixed, the automaton is called fixed structure automata and otherwise the automaton is called variable structure automata. Variable-structure automata which will be used in this paper is represented by sextuple β, φ, α, P, G, T, where β is a set of input actions, φ is a set of internal states, α is a set of outputs, P denotes the action probability vector governing the choice of the action at each stage k, G is the output mapping, and T is learning algorithm. The learning algorithm is used to modify the state probability vector. It is evident that the crucial factor affecting the performance of the variable structure learning automata, is the learning algorithm for updating the action probabilities. Various learning algorithms have been reported in the literature [Thatacher89]. Let α i be the action chosen at time k as a sample realization from distribution P(k). For the P model automata the linear reward-inaction algorithm LR I and linear ward-penalty algorithms LR P are among the earliest schemes in the literature. In an L scheme the recurrence equation for updating P is defined as R I p ( k) + θ (1 p (k)) p ( k) = p ( k) - p ( k) θ if i = if i if β is zero and P will remains unchanged if β is one(i is the chosen action). The parameter θ is called step length, it determines the amount of increases (decreases) of the action probabilities. In linear rewardpenalty algorithm L scheme the recurrence equation for updating P is defined as R P (4) p ( k) + θ (1 p (k)) p ( k) = p ( k) - p ( k) θ if i = if i if β =0 (5) p ( k) (1 - γ ) p ( k) = ( k) + (1 p p γ (k)) if i if i = if β = 1 (6) The parameters γ and θ represent step lengths. They determine the amount of increases (decreases) of the action probabilities. For Q model learning automata a linear algorithm is given below.

4 p ( k + 1) = p ( k) a(1 β ( k)) p ( k) if α( n) α p ( k + 1) = p ( k) + a(1 β ( k)) p ( k) if α( n) = α For S model learning automata a typical learning algorithm is as follows: i i p ( k + 1) = p ( k) + β ( k)[ a /( r 1) ap ( k)] [ 1 β ( k)] ap ( k) if α( n) α p ( k + 1) = p ( k) β ( k) ap ( k) + (1 β ( k)) a(1 p ( k)) if α( n) = α For more information about the theory and applications of learning automata refer to [Thatachar89], [Mars83], and [Mars98]. 3. Fuzzy ART Fuzzy art is a self organizing neural network that consists of three layers of and a supervising subsystem. The first layer, called F0, receives the input to be recognized. It preprocesses the input by a kind of normalization called complement coding. Suppose the original input is X=(x 1,x 2,,x m ) Then the complement coded of this vector will be X a =(x 1,x 2,,x m,1-x 1,1-x 2,,1-x m ). Doing this, the inputs are automatically normalized with respect to vector norm. The norm of X a denoted by X a is equal to M which is obtained by summing up all the elements of vector. After preprocessing, the new vector is submitted to the second layer called F1. This is the critical layer in which many actions is done. This layer first ss X a through an adaptive set of weight (called long term memory) to the third layer (F2). The weight of each link in this set is non increasing function of the bu td input. The weights are denoted by W=[W i ] (shown in figure 3 and 4 by W i = Wi )where i {1,2, 2m}, {1,2,..,n}. n is the number of the of the F2 layer. Each of the F2 node values are computed according to the following formula X a W T ( X a ) = (7) α + W. denotes the vector norm, is the fuzzy and (minimum), W is the th column of vector W, and α is called the choice parameter, which is one of the factors that determines the dynamics of the network. figure 3 figure - 4 F2 is a competition layer, that is, each time during the iteration of the network the node with maximum value is chosen. Afterwards the top down expectations are sent to F1 through weights W td, which is a vector of weights with the chosen category J denoted by W J. At F1 the following quantity is computed.

5 Xa WJ (8) Xa and compared with the value of vigilance factor ρ. If this value is greater than or equal to ρ, resonance occurs and the category that J represents is the recognized category. Otherwise the supervising subsystem initiates a reset to F2, the previously chosen are put aside and a new node with maximum value is chosen, and the corresponding expectations are sent. Sometimes for all in the F2 layer reset occurs. if this happens during training a new F2 node is gened with weight equal to input which automatically entails satisfaction of vigilance criteria, and if this happens during normal operation the input is classified as not recognized. 4. The proposed method In the proposed methods, we use variable structure learning automata for adusting the vigilance factor and/or the choice parameter. The interconnection of learning automata and fuzzy ART is shown in figure 4. The learning Automata according to some criteria adusts the vigilance factor and/or choice parameter of the fuzzy ART. The actions of the automata correspond to the values of vigilance factor or choice parameter and the input to the automata is some function of the error in the output of fuzzy ART. The method is described by the following algorithms. The first algorithm is used to adapt VF only and Algorithm ART_LA(Model) Algorithm ART_Simultaneous_LA_!(Model) β= 0 β= 0 s bottom up signals s bottom up signals for I=0 to number of F2 do for I=0 to number of F2 do compute F2 node with maximum value compute F2 node with maximum value s top down expectations s top down expectations If vigilance criteria is met Then If vigilance criteria is met Then If input is classified correctly Then If input is classified correctly Then Reinforcement(Model,VF, β) else Reinforcement(Model,VF, β) Reinforcement(Model,VF, 1) Reinforcement(Model,CP, β) else else discard current F2 node Reinforcement(Model,VF,1) if(model=q_model) then Reinforcement(Model,VF, 1) Reinforcement(Model, 0.6) β= i / number of F2 else {for} If no correct classification is made Then discard current F2 node Reinforcement( Model,VF, 1) if(model=q_model) then Algorithm Reinforcement(Model, VF, 0.6) Reinforcement(Model, CP, 0.6) Algorithm 1: Algorithm for adaptation of VF β= i / number of F2 { for} If no correct classification is made Then Reinforcement( Model, VF, 1) Reinforcement( Model, CP, 1) Algorithm Algorithm 2: Algorithm for adaptation of VF and CP

6 Algorithm ART_Simultaneous_LA_2 s bottom up signals for I=0 to number of F2 do J=index of F2 node with maximum value s top down expectations for K=0 to A_Const do choose a value for the th vigilance factor if vigilance criteria met then if input is correctly classified then Reinforcement(J, 0) Break inner loop else Reinforcement(J, 1) else continue the inner loop if correct classification is made then break the outer loop else discard current F2 node Algorithm Algorithm - 3 Another algorithm for simultaneous adaptation of VF and CP the second algorithm is used for simultaneous adaptation of VF and CP. In the third algorithm we have one learning automata for each F2 node which determines the vigilance factor of each group of weights emanating from all F1 to one of F2. The rational behind having one vigilance factor for each group is to better cover the region with prototypes. In the following three paragraphs we describe the first algorithm. The second algorithm has a similar description. The first algorithm is much like the standard fuzzy art algorithm except that the vigilance factor varies among distinct and predetermined values between zero and one. After preprocessing the input (i.e. Complement coding ), the resulting input values which are now in the second layer (i.e. F1 ) are sent to the third layer (i.e. F2 ) through bottom up adaptive weights. In this layer, the values of the are determined using formula ( 5 ). Then the algorithm s searching for the best prototype ( Top down weights which are equal to the bottom up weights ). Using formula ( 6 ) which is compared to the current value of the vigilance factor, and using the known type of the input, the current value of the vigilance factor is tested to be the most correct value representing the characteristics of the problem space. If the vigilance criteria is met, and if the recognition is correct a positive reinforcement is made and the weights are adusted and the algorithm breaks the search loop and s. But if an incorrect assignment is made, the algorithm breaks the search loop and terminates. In case that the vigilance criteria is not met, the current F2 node is discarded and search continues. At the of search if no recognition ( correct or incorrect ) is made, the algorithm dreates a new F2 node ( a new prototype ) accompanying a negative reinforcement. The above description is for the P model LA. For the learning with Q model LA the overall algorithm is nearly the same except that we use three levels for β ( reinforcement reward penalty parameter ) i.e. 0, 0.6, and 1. We use zero in the case of correct classification, 1 in the case of incorrect classification, and 0.6 when the vigilance criteria is not met each time during search. The value 0.9 is chosen in order to punish a little the choice of the current VF and favor direct access, i.e. the first chosen F2 node be the correct selected prototype. In S model LA we have chosen β to be equal to ( number of iterations of the search ) divided by (number of F2 ) the motivation behind choosing such a β is again to favor direct access and prevent more search. The third algorithm is a little different from the standard algorithm. This algorithm in addition to the

7 search loop has another loop, which is ited each time that the search is done, at most equal to a constant multiplied by the number of classes. The constant is chosen here to be 1.5. The algorithm has another difference from standard algorithm and it is that for each F2 node we have chosen a sepa vigilance factor. This is done because we want to better approximate the input space with prototypes, and determine the size of basin of attraction for each prototype sepaly. At each iterations of the main loop an F2 node with maximum index is chosen, afterwards the inner loop s with choosing a VF value for this node according to the vigilance probabilities for the node. If the vigilance criteria is met and the class J is the correct class, a positive reinforcement is made and the algorithm goes out of both loops. Otherwise a negative reinforcement is made and the algorithm continues. If the inner loop is unsuccessful ( no classification ), the F2 node J is discarded and search for another prototype continues. 5 Simulations We have conducted two sets of experiments. In the first set of experiments we have adapted the vigilance factor only using different models of learning automata and compared the results with Fu s [Fu94] method. In the second set of experiment both vigilance factor and choice parameter are adapted simultaneously, again using different models of learning automata. For these experiments as the actions of the LA s we choose 11 different values for VF, and 11 different values for CP. The values for VF range from 0.66 to 0.96 in equally spaced intervals. The choice parameter values vary from 0.1 too 100 with increasing interval magnitudes, that is, the points are dense in the vicinity of zero and rare in the vicinity of 100. Each point of the curves given below is obtained by holding CP constant and adapting VF. We have used 450 points for training and 450 other points for testing. Figure 5a, 5b, and 5c show the results of experiments for the noisy circle in the square problem with noiseless training file. The number of gened by Fu s method is higher than the number of gened by our methods. Among the P, Q, and S models the P model has produced the least number of. With respect to the of reection of inputs, Q model has performed the best and Fu`s method the worst. The Fu`s method has nearly produced the highest of recognition. Figure 6a, 6b, and 6c show the result for the noisy nested spirals. For this problem Fu s method has produced the highest number of and also highest of reection. Q, S, and P models come after Fu s method in terms of number of gened. Q model has the least of reection, and P and S model have more or less the same with respect to reection but with respect to recognition at all points they have the same performance. figure 5 - a figure 5 - b Figure 7a, 7b and 7c show the results of experiments for gaussian distributed groups. For this problem Q model has gened the highest number of, after Q model come the S and P models. The Fu s method in terms of number of gened is best and in terms of recognition is worst. Regarding other performance measures the proposed methods perform comparatively equal or better than the Fu s method. Looking at figures 5 (a - c), it can be seen that our methods have outperformed fuzzy method in all three criteria. The number of gened is the lowest for P model, after P model come S and Q

8 models. The performance regarding reection and recognition s are nearly the same for the three proposed models, all being better than the fuzzy method. figure 5 - c figure 6 - a figure 6 - b figure 6 - c Figures 6 (a - c) show the results for the nested spirals. Again much better performance is obtained for our method regarding all criteria. The number of gened is minimal for the P model, after which come S, and Q models, and lastly comes the fuzzy method. Reection s are the highest for the fuzzy method and the least for S and Q models, the highest recognition is obtained for the Q model. Figures 7 (a - c) show the results of the experiment for the gaussian distributed two groups problem. In the case of number of gened the score is the lowest for the fuzzy method, and next comes P, S, and Q models. Fuzzy adaptation method is especially tailor maid for noisy environments and for this reason it performs good for this problem. Reection is the lowest for fuzzy method and the P model, having large fluctuations in the case of S and Q models. figure 7 - a figure 7 - b

9 figure 7 - c Summarizing the results of the above mentioned figures we can conclude that in most cases when the training file is not noisy, our method performs better than the fuzzy adaptation method. Also the figures show a lot of variation in the degree of performance when the choice parameter is varied. This motivated us to adapt choice parameter and vigilance factor simultaneously. Tables 1- a through 3- d show the results obtained for the simultaneous adaptation of vigilance factor and choice parameter using algorithm 2 with P, Q, and S model LA, and algorithm 3 (which uses P model LA). We also observed that algorithm 2 with P model automata, and algorithm 3 were sensitive to the values of step lengths. So we conducted the experiments with different values of the step lengths. Tables 1 (a - d) indicates that the number of gened is highest for algorithm 3, and lowest for S, and Q models. The recognition s and reection s are nearly the same for all cases. Tables 2 (a - d) show that the number of gened for algorithm 3 and algorithm 2 with P model LA is the highest. For the S model the recognition is higher and the reection s are the same. In tables 3 (a - d) we see that the recognition is much higher for algorithm 3 ( about 16 percent higher), after which comes P model. The reection s are nearly zero in all cases and the number of gened is higher in the case of algorithm 3 and P model. Table 1- a Algorithm 2- P model- Circle in the square Table 1- b Algorithm 2- Q model- Circle in the square LA Parameters re re Conclusion In this paper we have studied the effectiveness of different models of learning automata in adaptation of VF and CP in fuzzy ART. The effectiveness of the proposed methods have been shown through simulation on three problems: the circle in the square, nested spirals, and gaussian distributed two groups. The result of simulations indicate that if both VF and CP are adapted simultaneously, we get higher performance than when only VF is adapted. The results of simulations also show that : One of the proposed algorithms ( Algorithm3 ) produces high of recognition especially when the environment is noisy. In order to get faster response with reasonable of recognition and reection, simultaneous adaptation of VF and CP using S model LA is better.

10 Adaptation of VF using learning automata perform better than the methods reported in the litrature. Table 1- c Algorithm2- Smodel- Circle in the square Re Table 1- d Algorithm 3 Circle in the square re Table 2- a Algorithm2- P model- nested spirals re Table 2- b Algorithm2 S model- nested spirals re Table 2- c Algorithm2- Q model- nested spirals Nodes re Table 2- d Algorithm3- nested spirals re Table 3- a Algorithm2- P model- gaussian distributed two groups re Table 3- b Algorithm2- S model- gaussian distributed two groups re

11 Table 3- c Table 3- d Algorithm2- Q model- gaussian Algorithm3 - gaussian distributed two groups distributed two groups re. Rate re References [Thatachar89] M. A. L. Thatachar and K. S. Narra, Learning Automata: An Introduction, Printice Hall inc, [Carpenter95a] G. A. Carpenter and W. D., ARTMAP: ART_MAP: A Neural Network Architecture For Obect Recognition By Evidence Accumulation, IEEE Transaction on Neural Networks, Vol. 6, No. 4, July [Carpenter95b] G. A. Carpenter, S. Grossberg and J. H. Reynolds, A Fuzzy ARTMAP Nonparametric Probability Estimator For Nonstationary Pattern Recognition Problems, IEEE Transaction on Neural Networks Vol.6, No. 6, November [Carpenter92] G. Carpenter, S. Grossberg, N. Markuzon, J. Reynolds and D. Rosed, Fuzzy ARTMAP: A Neural Network Architeture for Incremental Supervised Learning Of Analog Multidimensional Maps, IEEE Transaction on Neural Networks Vol.3, [Arabshahi96] P. Arabshahi et. Al., Fuzzy Parameter Adaptation in Optimazation: Some Neural Net Training examples, IEEE Computational Science & Engineering, Spring 1996, pp [Bahri99] P. Bahri and M. R. Meybodi,, A Method for Adaptation of Vigilance Factor and Choice Parameter in Fuzzy Art System, Proceedings of The 7th Iranian Conference on Electrical Engineering Iran Telecommunication Research Center, Tehran, Iran., May 1999, pp, [Healy97] M. J. Healy and T.P. Caudell, Aquiring Rule Sets as a Product of Learning in a Logical Neural Architecture, IEEE Transaction on Neural Networks, May [Fu94] L. Fu and J. Zhan, Fuzzy Adapting Vigilance Parameter of ARTII Neural Nets, IEEE World Congress on Computational Intelligence,Vol.3, 1994, pp [Hashim86] A. Hashim, S. Amir, and P. Mars, Application of Learning Automata to Data Compression, In Adaptive and Learning systems, K. S. Narra (Ed), New York: Plenum press, pp , [Mars83] P. Mars, K. S. Narra, and M. Chrystall,. Learning Automata Control of Computer Communication Networks, Proc. of Third Yale Workshop on Applications of Adaptive Systems Theory,Yale University., [Mars98] P. Mars, J. R. Chen,and R. Nambir Learning Algorithms: Theory and Application in Signal Processing, Control, and Communication, CRC press, New York,1998. [Meybodi98] M. R. Meybodi and H. Beigy, New Class of Learning Automata Based Scheme For Adaptation of Backpropagation Algorithm Parameters, Proceedings of EUFIT-98, Achen, Germany, pp ,1998.