Radial-Basis Function Networks. Radial-Basis Function Networks

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1 Radial-Basis Function Networks November 00 Michel Verleysen Radial-Basis Function Networks - Radial-Basis Function Networks p Origin: Cover s theorem p Interpolation problem p Regularization theory p Generalized RBFN p Universal approximation p RBFN kernel regression p Learning p Centers vector quantization p Widths p Multiplying factors p Other forms p Vector quantization Michel Verleysen Radial-Basis Function Networks -

2 Radial-Basis Function Networks p Origin: Cover s theorem p Interpolation problem p Regularization theory p Generalized RBFN p Universal approximation p RBFN kernel regression p Learning p Centers vector quantization p Widths p Multiplying factors p Other forms p Vector quantization Michel Verleysen Radial-Basis Function Networks - 3 Origin: Covers theorem p Covers theorem on separability of patterns (965) p x, x,, x P assigned to two classes C C p ϕ-separability: w T w ϕ T w ϕ ( x ) ( x ) > 0 < 0 x C x C p Cover s theorem: p non-linear functions ϕ(x) p dimension hidden space > dimension input space probability of separability closer to p Example linear quadratic Michel Verleysen Radial-Basis Function Networks - 4

3 Radial-Basis Function Networks p Origin: Cover s theorem p Interpolation problem p Regularization theory p Generalized RBFN p Universal approximation p RBFN kernel regression p Learning p Centers vector quantization p Widths p Multiplying factors p Other forms p Vector quantization Michel Verleysen Radial-Basis Function Networks - 5 Interpolation problem p Given points (x p, t p ), x p R D, t p R, p P : p Find F : R D Rthat satisfies p p ( ) KP F x = t, p = p RBF technique (Powell, 988): ( ) F P p ( x ) = w p ϕ( x x ) p= p p ϕ x x p as many functions as data points p centers fixed at known points x p are arbitrary non-linear functions (RBF) Michel Verleysen Radial-Basis Function Networks - 6 3

4 4 Michel Verleysen Radial-Basis Function Networks - 7 Interpolation problem p Into matrix form: p Vital question: is Φ non-singular? ( ) p p F = t x ( ) ( ) = ϕ = P p p w k F x x x = ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ P P PP P P P P t t t w w w M M L M O M M L L ( ) l k kl x x = ϕ ϕ where x w = Φ x w = Φ Michel Verleysen Radial-Basis Function Networks - 8 Michelli s theorem p If points x k are distinct, Φ is non-singular (regardless of the dimension of the input space) p Valid for a large class of RBF functions: ( ), + l = ϕ c x c x ( ) l, + = ϕ c x c x (l > 0) ( ) σ = ϕ exp c x c x, (σ > 0) non-localized function localized functions

5 Radial-Basis Function Networks p Origin: Cover s theorem p Interpolation problem p Regularization theory p Generalized RBFN p Universal approximation p RBFN kernel regression p Learning p Centers vector quantization p Widths p Multiplying factors p Other forms p Vector quantization Michel Verleysen Radial-Basis Function Networks - 9 Learning: ill-posed problem t x p Necessity for regularization p Error criterion: E P p p ( F ) ( t F( x ) + λ C( w ) = P p= MSE regularization Michel Verleysen Radial-Basis Function Networks - 0 5

6 Solution to the regularization problem p Poggio & Girosi (990): p if C(w) is a (problem-dependent) linear differential operator, the solution to P p p E( F ) ( t F ( x ) + λ C( w ) is of the following form: where F P p ( x ) = w p G( x, x ) p= G() is a Green s function, w = = P p= ( G + λi ) t G kl = G(x k,x l ) Michel Verleysen Radial-Basis Function Networks - Interpolation - Regularization p Interpolation F P p ( x ) = w p ϕ( x x ) w = Φ p= x p Exact interpolator p Possible RBF: p x x p ϕ( x, x ) = exp σ p Regularization F P p ( x ) = w p G( x, x ) w = p= ( G + λi ) t p Exact interpolator p Equal to the «interpolation» solution iff λ=0 p Example of Green s function: G p ( x, x ) p x x = exp σ One RBF / Green s function for each learning pattern! Michel Verleysen Radial-Basis Function Networks - 6

7 Radial-Basis Function Networks p Origin: Cover s theorem p Interpolation problem p Regularization theory p Generalized RBFN p Universal approximation p RBFN kernel regression p Learning p Centers vector quantization p Widths p Multiplying factors p Other forms p Vector quantization Michel Verleysen Radial-Basis Function Networks - 3 Generalized RBFN (GRBFN RBFN) p As many radial functions as learning patterns: p computationally (too) intensive (inversion of PxP matrix grows with P 3 ) p ill-conditioned matrix p regularization not easy (problem-specific) Generalized RBFN approach! Typically: p K << P p F ϕ ( x ) w i ϕ( x ci ) = K i = ( x c ) i = exp x c σ i i Parameters: c i, σ i, w i Michel Verleysen Radial-Basis Function Networks - 4 7

8 Radial-Basis Function Networks (RBFN) F ( x ) w i ϕ( x ci ) = K i = x 0 (bias) x ϕ( x c i ) σ w i F(x) ϕ ( x c ) i = exp x c i σ i c ij σk if several outputs x d st layer nd layer p Possibilities: p several outputs (common hidden layer) p bias (recommended) (see extensions) Michel Verleysen Radial-Basis Function Networks - 5 RBFN: universal approximation p Park & Sandberg 99: p For any continuous input-output mapping function f(x) F K ( x ) = wiϕ( x ci ) Lp( f ( x ),F ( x )) < ε ( ε > 0,p [, ] ) i = p The theorem is stronger (radial symmetry not needed) p K not specified p Provides a theoretical basis for practical RBFN! Michel Verleysen Radial-Basis Function Networks - 6 8

9 RBFN and kernel regression p non-linear regression model p p p p p ( ) P t = f x + ε = y + ε, p p estimation of f(x): average of t around x. More precisely: f ( x ) = E[ y x] = = yf yf Y ( y x ) X Y, f x dy ( x,y ) ( x ) dy p Need for estimates of fx, Y ( x,y ) and f X ( x ) Parzen-Rosenblatt density estimator Michel Verleysen Radial-Basis Function Networks - 7 Parzen-Rosenblatt density estimator fˆ x P x x K d Ph p= h ( x) = p with K() continuous, bounded, symmetric about the origin, with maximum value at 0, and with unit integral, is consistent (asymptotically unbiased). p Estimation of f (,y ) fˆ X,Y ( x,y ) X, Y x P x x = K d Ph p= h y y K + h p p Michel Verleysen Radial-Basis Function Networks - 8 9

10 RBFN and kernel regression fˆ ( x) yfˆ = X,Y fˆ X ( x,y ) ( x ) dy P p p x x y K p= h = P p x x K p= h f ( x ) = yfx Y, fx ( x,y ) ( x ) dy p Weighted average of y i p called Nadaraya-Watson estimator (964) p equivalent to Normalized RBFN in the unregularized context Michel Verleysen Radial-Basis Function Networks - 9 RBFN MLP p RBFN p single hidden layer p non-linear hidden layer linear output layer p argument of hidden units: Euclidean norm p universal approximation property p local approximators p splitted learning p MLP p single or multiple hidden layers p non-linear hidden layer linear or non-linear output layer p argument of hidden units: scalar product p universal approximation property p global approximators p global learning Michel Verleysen Radial-Basis Function Networks - 0 0

11 Radial-Basis Function Networks p Origin: Cover s theorem p Interpolation problem p Regularization theory p Generalized RBFN p Universal approximation p RBFN kernel regression p Learning p Centers vector quantization p Widths p Multiplying factors p Other forms p Vector quantization Michel Verleysen Radial-Basis Function Networks - RBFN: learning strategies F ( x ) w i ϕ( x ci ) ϕ( x ci ) = K i = = exp x c σ i i p Parameters to be determined: c i, σ i, w i p Traditional learning strategy: splitted computation. centers c i. widths σ i 3. weights w i Michel Verleysen Radial-Basis Function Networks -

12 RBFN: computation of centers p Idea: centers c i must have the (density) properties of learning points x k vector quantization p seen in details hereafter p This phase only uses the x k information, not the t k Michel Verleysen Radial-Basis Function Networks - 3 RBFN: computation of widths p Universal approximation property: valid with identical widths p In practice (limited learning set): variable widths σ i p Idea: RBFN use local clusters p choose σ i according to standard deviation of clusters Michel Verleysen Radial-Basis Function Networks - 4

13 RFBN: computation of weights F K ( x ) = w i ϕ( x ci ) i = ϕ ( x c ) i = exp x c i σ i p Problem becomes linear! p Solution of least square criterion leads to T T w = Φ + t = ( Φ Φ) Φ where k Φ ϕki = ϕ x ci p In practise: use SVD! ( ) constants! E P p p ( F ) = ( t F( x ) P p= Michel Verleysen Radial-Basis Function Networks - 5 p 3-steps method: RBFN: gradient descent F ( x ) = K w i exp i = 3 supervised x c i σ i unsupervised p Once c i, σ i, w i have been set by the previous method, possibility of gradient descent on all parameters p Some improvement, but p learning speed p local minima p risk of non-local basis functions p etc. Michel Verleysen Radial-Basis Function Networks - 6 3

14 More elaborated models p Add constant and linear terms K D i F( ) w exp x c x = i + w' x w' i i + 0 i = σi i = good idea (very difficult to approximate a constant with kernels) p Use normalized RBFN x ci exp K ( ) σi F x = w i = i K x c j exp = σ j j basis functions are bouded [0,] can be interpreted as probability values (classification) Michel Verleysen Radial-Basis Function Networks - 7 Back to the widths p choose σ i according to standard deviation of clusters p In the literature: p σ = dmax K where d max = maximum distance between centroids [] q p σ = c c where index j scans the q nearest centroids to c i [] i i q j = ( ) j p σi = r min ci c j where r is an overlap constant [3] j p.. [] S. Haykin, "Neural Networks a Comprehensive Foundation", Prentice-Hall Inc, second edition, 999. [] J. Moody and C. J. Darken, "Fast learning in networks of locally-tuned processing units", Neural Computation, pp. 8-94, 989. [3] A. Saha and J. D. Keeler, ''Algorithms for Better Representation and Faster Learning in Radial Basis Function Networks", Advances in Neural Information Processing Systems, Edited by David S. Touretzky, pp , 989. Michel Verleysen Radial-Basis Function Networks - 8 4

15 Basic example p Approximation of f(x) = with a d-dimensional RBFN p In theory: identical w i p Experimentally: side effects only middle taken into account p Error versus width Michel Verleysen Radial-Basis Function Networks - 9 Basic example: erros vs space dimension Michel Verleysen Radial-Basis Function Networks

16 Basic example: local decomposition? Michel Verleysen Radial-Basis Function Networks - 3 Multiple local minima in error curve p Choose the first minimum to preserve the locality of clusters p The first local minimum is usually less sensitive to variability Michel Verleysen Radial-Basis Function Networks - 3 6

17 Some concluding comments p RBFN: easy learning (compared to MLP) p in a cross-validation scheme: important! p Many RBFN models p Even more RBFN learning schemes p Results not very sensitive to unsupervised part of learning (c i, σ i ) p Open work for a priori (proble-dependent) choice of widths σ i Michel Verleysen Radial-Basis Function Networks - 33 Radial-Basis Function Networks p Origin: Cover s theorem p Interpolation problem p Regularization theory p Generalized RBFN p Universal approximation p RBFN kernel regression p Learning p Centers vector quantization p Widths p Multiplying factors p Other forms p Vector quantization Michel Verleysen Radial-Basis Function Networks

18 Back to the centers: Vector quantization p Aim and principle - What is a vector quantizer? p Vector scalar quantization p Lloyd s principle and algorithm p Initialization p Neural algorithms p Competitive learning p Frequency Sensitive Learning p Winner-take-all winner-take-most p Soft Competition Scheme p Stochastic Relaxation Scheme p Neural gas Michel Verleysen Radial-Basis Function Networks - 35 Vector quantization p Aim and principle - What is a vector quantizer? p Vector scalar quantization p Lloyd s principle and algorithm p Initialization p Neural algorithms p Competitive learning p Frequency Sensitive Learning p Winner-take-all winner-take-most p Soft Competition Scheme p Stochastic Relaxation Scheme p Neural gas Michel Verleysen Radial-Basis Function Networks

19 Aim of vector quantization p To reduce the size of a database N features N features P vectors Q vectors Q < P Michel Verleysen Radial-Basis Function Networks - 37 Principle of vector quantization p To project a continuous input space on a discrete output space, while minimizing the loss of information Q vectors P points Q vectors Michel Verleysen Radial-Basis Function Networks

20 Principle of vector quantization p To define zones in the space, the set of points contained in each zone being projected on a representative vector (centroid) p Example: -dimensional spaces Michel Verleysen Radial-Basis Function Networks - 39 What is a vector quantizer? p Vector quantizer =. A codebook (set of centroids, or codewords) { y j, j Q} m =. A quantization function q: i j ( x ) y i q x q = Usually: q is defined by the nearest neighbour rule (according to some distance measure) Michel Verleysen Radial-Basis Function Networks

21 p Least square error d p r-norm error dr Distance measures D i j i j i j i j T i j ( x, y ) ( x y ) = x y = ( x y ) ( x y ) = k k = D i j i j r ( x, y ) ( x y ) = k k= k k r p Mahalanobis distance (Γ is the covariance matrix of the inputs) d W ( i j i j T i j x, y ) = ( x y ) Γ ( x y ) Michel Verleysen Radial-Basis Function Networks - 4 Vector quantization p Aim and principle - What is a vector quantizer? p Vector scalar quantization p Lloyd s principle and algorithm p Initialization p Neural algorithms p Competitive learning p Frequency Sensitive Learning p Winner-take-all winner-take-most p Soft Competition Scheme p Stochastic Relaxation Scheme p Neural gas Michel Verleysen Radial-Basis Function Networks - 4

22 Vector scalar quantization p Shannon: a vector quantizer always gives better results than the product of scalar quantizers, even if the probability densities are independent p Example #: uniform -D distribution a scalar quantization [ ( ) 0 d x y ] a E = 36 vector quantization [ ( ) 6 E d x y ] = a 36 Michel Verleysen Radial-Basis Function Networks - 43 Vector scalar quantization p Shannon: a vector quantizer always gives better results than the product of scalar quantizers, even if the probability densities are independent p Example #: uniform -D distribution scalar quantization 4 [ d( x y ) ] c 6 E SQ = vector quantization 4 [ d( x y ) ]. 08h E VQ = c same ratio #centroids/unit surface: E VQ = 0.96 E SQ h Michel Verleysen Radial-Basis Function Networks - 44

23 Vector quantization p Aim and principle - What is a vector quantizer? p Vector scalar quantization p Lloyd s principle and algorithm p Initialization p Neural algorithms p Competitive learning p Frequency Sensitive Learning p Winner-take-all winner-take-most p Soft Competition Scheme p Stochastic Relaxation Scheme p Neural gas Michel Verleysen Radial-Basis Function Networks - 45 Lloyd s principle x i encoder j decoder y j p 3 properties:. The first one gives the best encoder, once the decoder is known. The second one gives the best decoder, once the encoder is known 3. There is no point on the borders between Voronoï regions (probability = 0) p Optimal quantizer: properties are bad necessary, but not sufficient: good Michel Verleysen Radial-Basis Function Networks

24 Lloyd: property # x i encoder j decoder y j p For a given decoder β, the best encoder is given by: α i i j j ( x ) = argmin [ d( x, y )] where y = β( j ) j nearest-neighbor rule! Michel Verleysen Radial-Basis Function Networks - 47 Lloyd: property # x i encoder j decoder y j p For a given encoder α, the best decoder is given by: β i j i ( j ) = argmin [ E[ d( x, y ) α( x ) j ] y j = center-of-gravity rule! Michel Verleysen Radial-Basis Function Networks

25 Lloyd: property #3 p The probability to find a point xi on a border (between Voronoï regions) is zero! probability = 0 Michel Verleysen Radial-Basis Function Networks - 49 Lloyd s algorithm. Choice of an initial codebook.. All points x i are encoded; E VQ is evaluated. 3. If E VQ is small enough, then stop. 4. All centroids y j are replaced by the center-of-gravity of the data x i associated to y j in step. 5. Back to step. Michel Verleysen Radial-Basis Function Networks

26 Lloyd: example. Initialization of the codeboook y y y 3 y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks - 5 Lloyd: example. Encoding (nearest-neighbor) y y y 3 y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks - 5 6

27 Lloyd: example 4. Decoding (center-of-gravity) y y 3 y y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks - 53 Lloyd: example. Encoding (nearest-neighbor) new borders y y y 3 y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks

28 Lloyd: example 4. Decoding (center-of-gravity) y y 3 new positions of centroids y y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks - 55 Lloyd: example. Encoding (nearest-neighbor) new borders y y 3 y y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks

29 Lloyd: example 4. Decoding (center-of-gravity) new positions of centroids y y 3 y y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks - 57 Lloyd: example. Encoding (nearest-neighbor) final borders (convergence) y y 3 y y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks

30 Lloyd s algorithm: the names p Lloyd s algorithm p Generalized Lloyd s algorithm p Linde-Buzzo-Gray (LBG) algorithm p K-means p ISODATA p All based on the same principle! Michel Verleysen Radial-Basis Function Networks - 59 Lloyd s algorithm: properties p The codebook is modified only after the presentation of the whole dataset p The mean square error (E VQ ) decreases at each iteration p The risk of getting trapped in local minima is high p The final quantizer depends on the initial one Michel Verleysen Radial-Basis Function Networks

31 Vector quantization p Aim and principle - What is a vector quantizer? p Vector scalar quantization p Lloyd s principle and algorithm p Initialization p Neural algorithms p Competitive learning p Frequency Sensitive Learning p Winner-take-all winner-take-most p Soft Competition Scheme p Stochastic Relaxation Scheme p Neural gas Michel Verleysen Radial-Basis Function Networks - 6 How to initialize Lloyd s algorithm?. randomly in the input space. the Q first data points x i 3. Q randomly chosen data points x i 4. Product codes : the product of scalar quantizers 5. growing initial set: p a first centroid y is randomly chosen (in the data set) p a second centroid y is randomly chosen (in the data set); if d(y,y ) > threshold, y is kept p a third centroid y 3 is randomly chosen (in the data set); if d(y,y 3 ) > threshold AND d(y,y 3 ) > threshold, y 3 is kept p Michel Verleysen Radial-Basis Function Networks - 6 3

32 How to initialize Lloyd s algorithm? 6. pairwise nearest neighbor : p a first codebook is built with all data points x i p the two centroids y j nearest one from another are merged (center-of-gravity) p Variant: p the increase of distortion (E VQ ) is evaluated for the merge of each pair of centroids y j ; the pair gicing the lowest increase is merged. Michel Verleysen Radial-Basis Function Networks - 63 How to initialize Lloyd s algorithm? 7. Splitting p a first centroid y is randomly chosen (in the data set) p a second centroid y +ε is created; Lloyd s algorithm is applied to the new codebook p two new centroids are created by perturbing the two existing ones; Lloyd s algorithm is applied to this 4-centroids codebook p Michel Verleysen Radial-Basis Function Networks

33 Vector quantization p Aim and principle - What is a vector quantizer? p Vector scalar quantization p Lloyd s principle and algorithm p Initialization p Neural algorithms p Competitive learning p Frequency Sensitive Learning p Winner-take-all winner-take-most p Soft Competition Scheme p Stochastic Relaxation Scheme p Neural gas Michel Verleysen Radial-Basis Function Networks - 65 Vector quantization: «neural» algorithms p Principle: the codebook is (partly or fully) modified at each presentation of one data vector x i p Advantages: p simplicity p adaptive algorithm (with varying data) p possible parallelisation p speed? p avoids local minima? Michel Verleysen Radial-Basis Function Networks

34 Competitive learning p Algorithm: p For each input vector x i, the «winner» is selected: y y 3 d i k i j ( x, y ) d( x, y ), j,k Q p Adaptation rule: the winner is moved towards the input vector: y k t k i k ( + ) = y ( t) + α( x y ) y y k (t) y k (t+) x i y 4 x i : data y j : centroids Michel Verleysen Radial-Basis Function Networks - 67 Competitive learning p Adaptation rule: stochastic gradient descent on j ( ( x) E = x y ) p( x ) dx convergence to (local) minimum y y 3 p Robbins-Monro conditions on α: α() t = and α () t < t = 0 t = 0 p Local minima! p Some centroids may be «lost»! y never «winner»! y 4 Michel Verleysen Radial-Basis Function Networks

35 Frequency sensitive learning p Competitive learning; some centroids may be lost during learning centroids often chosen (as winners) are penalized! p Choice of winner is replaced by u k d i k j i j ( x, y ) u d( x, y ), j,k Q where u j, u k are incremented each time they are chosen as winner (starting at ) Michel Verleysen Radial-Basis Function Networks - 69 Frequency sensitive learning y y 3 u d u d i ( x, y ) i ( x, y ) y x i possible «winner»! y 4 (with u = and u =3 for example) Michel Verleysen Radial-Basis Function Networks

36 Vector quantization p Aim and principle - What is a vector quantizer? p Vector scalar quantization p Lloyd s principle and algorithm p Initialization p Neural algorithms p Competitive learning p Frequency Sensitive Learning p Winner-take-all winner-take-most p Soft Competition Scheme p Stochastic Relaxation Scheme p Neural gas Michel Verleysen Radial-Basis Function Networks - 7 Winner-take-all Winner-take-most p Most VQ algorithms: if two centroids are close, it will be hard to separate them! p (Competitive learning and LVQ: one or two winners are adapted) p Solution: to adapt the winner and other centroids: p (with respect to the distance between the centroid and x i (SCS) ) p stochastically with respect to the distance (SRS) p with respect to the order of proximity with x i (neural-gas) p those in a neighborhood (on a grid) of the winner (Kohonen) These algorithms accelerate the VQ too! Michel Verleysen Radial-Basis Function Networks

37 Soft Competition Scheme (SCS) p Adaptation rule on all centroids: k y k i i k ( t + ) = y ( t) + αg( k, x )( x y () t ) with i ( x ) G k, i k x y () t e T = i j x y () t Q T e j= (T is made decreasing over time) Michel Verleysen Radial-Basis Function Networks - 73 Stochastic Relaxation Scheme (SRS) p Adaptation rule on all centroids: k y k i i k ( t + ) = y ( t) + αg( k, x )( x y () t ) with i ( x ) G k, = 0 i k x y () t e T P = i j x y () t Q T e j = with probabilit y P with probabilit y P (T is made decreasing over time) Michel Verleysen Radial-Basis Function Networks

38 Neural gas p Principle: centroids are ranked according to their distance with x i : h(j, x i ) = c if y j is the c-st nearest centroid from x i p Adaptation rule: k y k ( t ) = y ( t) i ( k, x ) h + + αe λ i k ( x y () t ) Michel Verleysen Radial-Basis Function Networks - 75 Neural gas p Properties: p if λ=0: competitive learning p partial sorting possible p improvement: prossibility of frequency sensitive learning p The adaptation rule is a stochastic gradient descent on i ( ) Q h j, x E = λ () e Q h l λ j = e l = i j x y () t p( x ) dx Michel Verleysen Radial-Basis Function Networks

39 Sources and references (RBFN) p Most of the basic concepts developed in these slides come from the excellent book: p Neural networks a comprehensive foundation, S. Haykin, Macmillan College Publishing Company, 994. p Some supplementary comments come from the tutorial on RBF: p An overview of Radial Basis Function Networks, J. Ghosh & A. Nag, in: Radial Basis Function Networks, R.J. Howlett & L.C. Jain eds., Physica-Verlag, 00. p The results on the RBFN basic exemple were generated by my colleague N. Benoudjit, and are submitted for publication. Michel Verleysen Radial-Basis Function Networks - 77 Sources and references (VQ) p A classical tutorial on vector quantization p Vector quantization, R.M. Gray, IEEE ASSP Mag., vol., pp. 4-9, April 984. p Most concepts in this chapter come from the following (specialized) papers: p Vector quantization is speech coding, K. Makhoul, S. Roucos, H. Gish, Proceedings IEEE, vol. 73, n., November 985. p Neural-gas network for vector quantization and its application to timeseries prediction, T.M. Martinetz, S.G. Berkovich, K.J. Schulten, IEEE T. Neural Networks, vol.4, n.4, July 993. p Habituation in Learning Vector Quantization, T.? Gestzi, I. Csabai, Complex Systems, n.6, 99. p DVQ: Dynamic Vector Quantization an incremental LVQ, F. Poirier, A. Ferrieux, in: Artificial Neural Networks, T. Kohonen et al. eds., Elsevier, 99. p Representation of nonlinear data structures thourgh a fast VQP neural network, P. Demartines, J. Hérault, Proc. Neuro-Nîmes 993. Michel Verleysen Radial-Basis Function Networks