Artificial Neural Networks Lect7: Neural networks based on competition

1

CS407 Neural Computation

Lecture 7: Neural Networks Based on

Competition.

Lecturer: A/Prof. M. Bennamoun

2

Neural Networks Based on Competition.

IntroductionFixed weight competitive nets– Maxnet– Mexican Hat– Hamming Net

Kohonen Self-Organizing Maps (SOM)SOM in MatlabReferences and suggested reading



3

Introduction…Competitive netsFausett

The most extreme form of competition among a group of neurons is called “Winner Take All”.As the name suggests, only one neuron in the competing group will have a nonzero output signal when the competition is completed.A specific competitive net that performs Winner Take All (WTA) competition is the Maxnet. A more general form of competition, the “Mexican Hat” will also be described in this lecture (instead of a non-zero o/p for the winner

and zeros for all other competing nodes, we have a bubble around the winner ).All of the other nets we discuss in this lecture use WTA competition as part of their operation.With the exception of the fixed-weight competitive nets (namely Maxnet, Mexican Hat, and Hamming net) all of the other nets combine competition with some form of learningto adjusts the weights of the net (i.e. the weights that are not part of any interconnections in the competitive layer).

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The form of learning depends on the purpose for which the net is being trained:

– LVQ and counterpropagation net are trained to perform mappings. The learning in this case is supervised.

– SOM (used for clustering of input data): a common use of unsupervised learning.

– ART are also clustering nets: also unsupervised.Several of the nets discussed use the same learning algorithm known as “Kohonen learning”: where the units that update their weights do so by forming a new weight vector that is a linear combination of the old weight vector and the current input vector. Typically, the unit whose weight vector was closest to the input vector is allowed to learn.

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The weight update for output (or cluster) unit j is given as:

where x is the input vector, w.j is the weight vector for unit j, and α the learning rate, decreases as learning proceeds.

Two methods of determining the closest weight vector to a pattern vector are commonly used for self-organizing nets.Both are based on the assumption that the weight vector for each cluster (output) unit serves as an exemplar for the input vectors that have been assigned to that unit during learning.

– The first method of determining the winner uses the squared Euclidean distance between the I/P vector

[ ]old)()1(

old)()old()new(

.

...

j

jjj

wx

wxww

αα

α

−+

−+=

6


and the weight vector and chooses the unit whose weight vector has the smallest Euclidean distance from the I/P vector.

– The second method uses the dot product of the I/P vector and the weight vector.The dot product can be interpreted as giving the correlation between the I/P and weight vector.

In general and for consistency, we will use Euclidean distance squared.Many NNs use the idea of competition among neurons to enhance the contrast in activations of the neuronsIn the most extreme situation, the case of the Winner-Take-All, only the neuron with the largest activation is allowed to remain “on”.

8

Introduction…Competitive nets

1. Euclidean distance between two vectors

2. Cosine of the angle between two vectors

( ) ( )itii xxxxxx −−=−

i

it

xxxx

=ψcos

21ψψψ <<

T

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10

MAXNET

[ ]kM

k yWy Γ=+1

−−

−−−−

=

1 . . . .

. . 1 . . 1

εε

εεεε

MW

( )

≥<

=0 ,0 ,0

netnetnet

netf

p10 << ε

11

Hamming Network and MAXNETMAXNET– A recurrent network involving both excitatory

and inhibitory connections• Positive self-feedbacks and negative

cross-feedbacks– After a number of recurrences, the only non-

zero node will be the one with the largest initializing entry from i/p vector

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• Maxnet Lateral inhibition between competitorsMaxnet Fixed-weight competitive net

Fausett

>

=otherwise 0

0 if )(

:Maxnet for thefunction activationxx

xf

Competition: • iterative process until the net stabilizes (at most one

node with positive activation)

– Step 0:• Initialize activations and weight (set )where m is the # of nodes (competitors)

,/10 m<< ε

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Algorithm

Step 1: While stopping condition is false do steps 2-4

– Step 2: Update the activation of each node:

– Step 3: Save activation for use in next iteration:

– Step 4: Test stopping condition:If more than one node has a nonzero activation, continue; otherwise, stop.

−

==

=

otherwise if 1

:weights

node input to )0(

εji

w

Aa

ij

jj

mj(old)a(old) –a f(new) ajk

kjj ,...,1for =

= ∑

≠

ε

mj(new) a(old) a jj ,...,1for ==

the total i/p to node Aj from all nodes, including itself

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ExampleNote that in step 2, the i/p to the function f is the total i/p to node Aj from all nodes, including itself.Some precautions should be incorporated to handle the situation in which two or more units have the same, maximal, input.Example:ε = .2 and the initial activations (input signals) are:a1(0) = .2, a2(0) = .4 a3(0) = .4 a4(0) = .8As the net iterates, the activations are:

a1(1) = f{a1(0) -.2 [a2(0) + a3(0) + a4(0)]} = f(-.12)=0a1(1) = .0, a2(1) = .08 a3(1) = .32 a4(1) = .56a1(2) = .0, a2(2) = .0 a3(2) = .192 a4(2) = .48a1(3) = .0, a2(3) = .0 a3(3) = .096 a4(3) = .442a1(4) = .0, a2(4) = .0 a3(4) = .008 a4(4) = .422

a1(5) = .0, a2(5) = .0 a3(5) = .0 a4(5) = .421

The only nodeto remain on

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16

Mexican Hat Fixed-weight competitive netArchitecture

Each neuron is connected with excitatory links (positively weighted) to a number of “cooperative neighbors” neurons that are in close proximity.Each neuron is also connected with inhibitory links (with negative weights) to a number of “competitive neighbors” neurons that are somewhat further away.There may also be a number of neurons, further away still, to which the neurons is not connected.All of these connections are within a particular layer of a neural net.The neurons receive an external signal in addition to these interconnections signals (just like Maxnet).This pattern of interconnections is repeated for each neuron in the layer. The interconnection pattern for unit Xi is as follows:

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Mexican Hat Fixed-weight competitive net

The size of the region of cooperation (positive connections) and the region of competition (negative connections) may vary, as may vary the relative magnitudes of the +ve and –ve weights and the topology of the regions (linear, rectangular, hexagonal, etc..)

si = external activation

w1 = excitatory connectionw2 = inhibitory connection

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Mexican Hat Fixed-weight competitive netThe contrast enhancement of the signal si received by unit Xiis accomplished by iteration for several time steps. The activation of unit Xi at time t is given by:

Where the terms in the summation are the weighted signalfrom other units cooperative and competitive neighbors) at the previous time step.

1

−+= ∑ +

kkikii )(txw(t) s f(t) x

inputs

+

-

-

-

-

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Mexican Hat Fixed-weight competitive netNomenclature

R2=Radius of region of interconnections; Xi is connected to units Xi+kand Xi-k for k=1, …R2

R1=Radius of region with +ve reinforcement; R1< R2

wk = weights on interconnections between Xi and units Xi+k and Xi-k; wk is positive for 0 ≤ k ≤ R1

wk is negative for R1 < k ≤ R2

x = vector of activationsx_old = vector of activations at previous time stept_max = total number of iterations of contrast enhancements = external signal

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Mexican Hat Fixed-weight competitive netAlgorithm

As presented, the algorithm corresponds to the external signal being given only for the first iteration (step 1).Step 0 Initialize parameters t_max, R1, R2 as desired

Initialize weights:

Step 1 Present external signal s:

Set iteration counter: t = 1

0x_old to Initialize...,1for0

...,0for0

212

11

+=<=>

=RRkC

RkCwij

ii xoldx ==

=

_:n)..., 1,i(for array in activation Save x_old

sx

21

Mexican Hat Fixed-weight competitive netAlgorithm

Step 2 While t is less than t_max, do steps 3-7Step 3 Compute net input (i = 1, …n)

Step 4 Apply activation function f (ramp from 0 to x_max, slope 1):

∑∑ ∑+−=

+−=

−−

−=++ ++=

2

1

1

1

1

2 12

1

21 ___R

Rkki

R

Rk

R

Rkkikii oldxColdxColdxCx

>≤≤

<=

maxmax_max0

00)(

xifxxifx

xifxf

nixxx ii ,...,1)),0max(max,_min( ==

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Mexican Hat Fixed-weight competitive netStep 5 Save current activations in x_old:

Step 6 Increment iteration counter:

Step 7 Test stopping condition:If t < t_max, continue; otherwise, stop.

Note:The +ve reinforcement from nearby units and –ve reinforcement from units that are further away have the effect of – increasing the activation of units with larger initial activations,– reducing the activations of those that had a smaller external

signal.

nixoldx ii ,...,1_ ==

1+= tt

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Mexican Hat ExampleLet’s consider a simple net with 7 units. The activation function for this net is:

Step 0 Initialize parameters R1=1, R2 =2, C1 =.6, C2 =-.4

Step 1 t=0 The external signal s = (.0, .5, .8, 1, .8, .5, .0)

Step 2 t=1, the update formulas used in step 3, are listed as follows for reference:

)0.0,5.0,8.0,0.1,8.0,5.0,0.0()1(old_:_oldin Save

)0.0,5.0,8.0,0.1,8.0,5.0,0.0()0(

=

=

xx

x

>≤≤

<=

2220

00)(

xifxifx

xifxf

The node

to be enhanced

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Mexican Hat Example

7657

76546

765435

654324

543213

43212

3211

_6.0_6.0_4.0_6.0_6.0_6.0_4.0

_4.0_6.0_6.0_6.0_4.0_4.0_6.0_6.0_6.0_4.0_4.0_6.0_6.0_6.0_4.0

_4.0_6.0_6.0_6.0_4.0_6.0_6.0

oldxoldxoldxxoldxoldxoldxoldxx

oldxoldxoldxoldxoldxxoldxoldxoldxoldxoldxxoldxoldxoldxoldxoldxx

oldxoldxoldxoldxxoldxoldxoldxx

++−=+++−=

−+++−=−+++−=−+++−=

−++=−+=

2.0)0.0(6.0)5.0(6.0)8.0(4.038.0)0.0(6.0)5.0(6.0)8.0(6.0)0.1(4.0

06.1)0.0(4.0)5.0(6.0)8.0(6.0)0.1(6.0)8.0(4.016.1)5.0(4.0)8.0(6.0)0.1(6.0)8.0(6.0)5.0(4.006.1)8.0(4.0)0.1(6.0)8.0(6.0)5.0(6.0)0.0(4.0

38.0)0.1(4.0)8.0(6.0)5.0(6.0)0.0(6.02.0)8.0(4.0)5.0(6.0)0.0(6.0

7

6

5

4

3

2

1

−=++−==+++−=

=−+++−==−+++−==−+++−=

=−++=−=−+=

xxxxxxxStep 3 t=1

25

Mexican Hat ExampleStep 4

Step 5-7 Bookkeeping for next iteration

Step 3: t=2

Step 4:

Step 5-7 Bookkeeping for next iteration

196.0)0.0(6.0)38.0(6.0)06.1(4.039.0)0.0(6.0)38.0(6.0)06.1(6.0)16.1(4.0

14.1)0.0(4.0)38.0(6.0)06.1(6.0)16.1(6.0)06.1(4.066.1)38.0(4.0)06.1(6.0)16.1(6.0)06.1(6.0)38.0(4.0

14.1)06.1(4.0)16.1(6.0)06.1(6.0)38.0(6.0)0.0(4.039.0)16.1(4.0)06.1(6.0)38.0(6.0)0.0(6.0

196.0)06.1(4.0)38.0(6.0)0.0(6.0

7

6

5

4

3

2

1

−=++−==+++−=

=−+++−==−+++−=

=−+++−==−++=

−=−+=

xxxxxxx

)0.0,38.0,06.1,16.1,06.1,38.0,0.0(=x

)0.0,39.0,14.1,66.1,14.1,39.0,0.0(=x

The node

Which has been enhanced

26






27

Hamming Network Fixed-weight competitive netHamming distance of two vectors is the number of components in which the vector differ. For 2 bipolar vectors, of dimension n,

The Hamming net uses Maxnet as a subnet to find the unit with the largest input (see figure on next slide).

yx and

distance Hammingshorterlarger larger)(5.0

2),HD( distanceHammingandindifferentbitsofnumberis

andinagreementinbitsofnumberis:where

⇒⇒⋅+⋅=

−=⋅=−=

−=⋅

ana

naandda

da

yxyx

yxyx

yxyx

yx

010

and 011

28

Hamming Network Fixed-weight competitive netArchitecture

The sample architecture shown assumes input vectors are 4-tuples, to be categorized as belonging to one of 2 classes.

29

Hamming Network Fixed-weight competitive netAlgorithm

Given a set of m bipolar exemplar vectors e(1), e(2), …, e(m), the Hamming net can be used to find the exemplar that is closest to the bipolar i/p vector x.

The net i/p y_inj to unit Yj gives the number of components in which the i/p vector and the exemplar vector for unit Yje(j) agree (n minus the Hamming distance between these vectors).

Nomenclature:

n number of input nodes, number of components of any I/p vector

m number of o/p nodes, number of exemplar vectors

e(j) the jth exemplar vector:

( ))(,),(),()( 1 jejejej ni LL=e

30

Hamming Network Fixed-weight competitive netAlgorithm

Step 0 To store the m exemplar vectors, initialise the weights and biases:

Step 1 For each vector x, do steps 2-4

Step 2 Compute the net input to each unit Yj:

Step 3 Initialise activations for Maxnet

Step 4 Maxnet iterates to find the best match exemplar.

nbjew jiij 5.0 ),(5.0 ==

mjwxbinyi

ijijj ,,1for _ L=+= ∑

mjinyy jj ,,1for _)0( L==

31

Hamming Network Fixed-weight competitive net

Upper layer: Maxnet– it takes the y_in as its initial value, then

iterates toward stable state (the best match exemplar)

– one output node with highest y_in will be the winner because its weight vector is closest to the input vector

jY

32

Hamming Network Fixed-weight competitive netExample (see architecture n=4)

[ ][ ] patterns)(exemplar 2 case in this m 1-1,-1,-1,)2(

1-1,-1,-1,)1(:ectorsexemplar v Given the

=−==

ee

The Hamming net can be used to find the exemplar that is closest to each of the bipolar i/p patterns, (1, 1, -1, -1), (1, -1, -1, -1), (-1, -1, -1, 1), and (-1, -1, 1, 1)

Step 1: Store the m exemplar vectors in the weights:

case) (in this 4nodesinput ofnumber 2/2

:biases theInitialize

2)( re whe

5.5.5.5.5.5.5.5.

21

=====

=

−−−−−−−

=

nnbb

jewW iij

33

Hamming Network Fixed-weight competitive netExample

For the vector x=(1,1, -1, -1), let’s do the following steps:

These values represent the Hamming similarity because (1, 1, -1, -1) agrees with e(1)= (1,-1, -1, -1) in the 1st , 3rd, and 4th components and because (1,1, -1, -1) agrees with e(2)=(-1,-1,-1,1) in only the 3rd component

Since y1(0) > y2(0), Maxnet will find that unit Y1 has the best match exemplar for input vector x = (1, 1, -1, -1).

112_

312_

222

111

=−=+=

=+=+=

∑

∑

iii

iii

wxbiny

wxbiny

1)0(3)0(

2

1

==

yy

34


For the vector x=(1,-1, -1, -1), let’s do the following steps:

Note that the input agrees with e(1) in all 4 components and agrees with e(2) in the 2nd and 3rd components

Since y1(0) > y2(0), Maxnet will find that unit Y1 has the best match exemplar for input vector x = (1, -1, -1, -1).

202_

422_

222

111

=+=+=

=+=+=

∑

∑

iii

iii

wxbiny

wxbiny

2)0(4)0(

2

1

==

yy

35


For the vector x=(-1,-1, -1, 1), let’s do the following steps:

Note that the input agrees with e(1) in the 2nd and 3rd

components and agrees with e(2) in all four components

Since y2(0) > y1(0), Maxnet will find that unit Y2 has the best match exemplar for input vector x = (-1, -1, -1, 1).

422_

202_

222

111

=+=+=

=+=+=

∑

∑

iii

iii

wxbiny

wxbiny

4)0(2)0(

2

1

==

yy

36


For the vector x=(-1,-1, 1, 1), let’s do the following steps:

Note that the input agrees with e(1) in the 2nd component and agrees with e(2) in the 1st, 2nd, and 4th components

Since y2(0) > y1(0), Maxnet will find that unit Y2 has the best match exemplar for input vector x = (-1, -1, 1, 1).

312_

112_

222

111

=+=+=

=−=+=

∑

∑

iii

iii

wxbiny

wxbiny

3)0(1)0(

2

1

==

yy

37






38

Kohonen Self-Organizing Maps (SOM)

Fausett

Teuvo Kohonen, born: 11-07-34, Finland Author of the book: “Self-organizing maps”, Springer (1997)SOM = “Topology preserving maps”SOM assume a topological structure among the cluster unitsThis property is observed in the brain, but not found in otherANNs

39


Fausett

40

Fausett

Kohonen Sefl-Organizing Maps (SOM)

Architecture:

41

Fausett


Architecture:

* * {* (* [#] *) *} * *

{ } R=2 ( ) R=1 [ ] R=0

Linear array of cluster units

This figure represents the neighborhood of the unit designated by # in a 1D topology (with 10 cluster units).

************************#************************

R=2

R=1

R=0

Neighbourhoods for rectangular grid (Each unit has 8 neighbours)

42

Fausett


Architecture:

************

********#***

************

******R=2

R=1

R=0

Neighbourhoods for hexagonal grid (Each unit has 6 neighbours).

In each of these illustrations, the “winning unit” is indicated by the symbol # and the other units are denoted by *

NOTE: What if the winning unit is on the edge of the grid?

• Winning units that are close to the edge of the grid will have some neighbourhoods that have fewer units than those shown in the respective figures

• Neighbourhoods do not “wrap around” from one side of the grid to the other; “missing”units are simply ignored.

43

Training SOM NetworksWhich nodes are trained?How are they trained?– Training functions– Are all nodes trained the same amount?

Two basic types of training sessions?– Initial formation– Final convergence

How do you know when to stop?– Small number of changes in input mappings– Max # epochs reached

44

Fausett


Algorithm:Step 0. Initialise weights wij (randomly)

Set topological neighbourhood parameters

Set learning rate parameters

Step 1 While stopping condition is false, do steps 2-8.

Step 2 For each i/p vector x, do steps 3-5

Step 3 For each j, compute:

Step 4 Find index J such that D(J) is a minimum

Step 5 For all units j within a specified neighbourhood of J, and for all i:

Step 6 Update learning rate.

Step 7 Reduce radius of topological neighbourhood at specified timeStep 8 Test stopping condition

∑ −=i

iij xwJD 2)()(

)]([)()( oldwxoldwneww ijiijij −+= α

45

Fausett


Alternative structure for reducing R and α:

The learning rate α is a slowly decreasing function of time (or training epochs). Kohonen indicates that a linearly decreasing function is satisfactory for practical computations; a geometric decrease would produce similarresults.

The radius of the neighbourhood around a cluster unit also decreases as the clustering process progresses.

The formation of a map occurs in 2 phases:

The initial formation of the correct order and

the final convergence

The second phase takes much longer than the first and requires a small value for the learning rate.

Many iterations through the training set may be necessary, at least in some applications.

46

Fausett


Applications and Examples:

The SOM have had may applications:

Computer generated music (Kohonen 1989)

Traveling salesman problem

Examples: A Kohonen SOM to cluster 4 vectors:

– Let the 4 vectors to be clustered be:

s(1) = (1, 1, 0, 0) s(2) = (0, 0, 0, 1)

s(3) = (1, 0, 0, 0) s(4) = (0, 0, 1, 1)

- The maximum number of clusters to be formed is m=2 (4 input nodes, 2 output nodes)

Suppose the learning rate is α(0)=0.6 and α(t+1) = 0.5 α(t)

With only 2 clusters available, the neighborhood of node J (step 4) is set so that only one cluster update its weights at each step (i.e. R=0)

47

Fausett


Examples:

Step 0 Initial weight matrix

Initial radius: R=0

Initial learning rate: α(0)=0.6

Step 1 Begin training

Step 2 For s(1) = (1, 1, 0, 0) , do steps 3-5

Step 3

=

3. 9.7. 5.4. 6.8. 2.

W

98.0)03(.)07(.)14(.)18(.)2(86.1)09(.)05(.)16(.)12(.)1(

2222

2222

=−+−+−+−=

=−+−+−+−=

DD

48

Fausett


Examples:

Step 4 The i/p vector is closest to o/p node 2, so J=2

Step 5 The weights on the winning unit are updated

This gives the weight matrix:

ii

iiii

xoldwoldwxoldwneww

6.0)(4.0)]([6.0)()(

2

222

+−+=

This set of weights has notbeen modified.

=

12. 9.28. 5.76. 6.92. 2.

W

49

Fausett


Examples:


Step 3

2768.2)112(.)028(.)076(.)092(.)2(66.0)19(.)05(.)06(.)02(.)1(2222

2222

=−+−+−+−=

=−+−+−+−=

DD




=

12. 96.28. 20.76. 24.92. 08.

W


50

Fausett


Examples:


Step 3

6768.0)012(.)028(.)076(.)192(.)2(8656.1)096(.)02(.)024(.)108(.)1(

2222

2222

=−+−+−+−=

=−+−+−+−=

DD




=

048. 96.112. 20.304. 24.968. 08.

W


51

Fausett


Examples:


Step 3

724.2)1048(.)1112(.)0304(.)0968(.)2(7056.0)196(.)12(.)024(.)008(.)1(

2222

2222

=−+−+−+−=

=−+−+−+−=

DD




Step 6 Reduce the learning rate α = 0.5(0.6) = 0.3

=

048. 984.112. 680.304. 096.968. 032.

WThis set of weights has notbeen modified.

52

Fausett


Examples:

The weights update equations are now

The weight matrix after the 2nd epoch of training is:

Modifying the learning rate and after 100 iterations (epochs), the weight matrix appears to be converging to:

iij

ijiijij

xoldw

oldwxoldwneww

3.0)(7.0

)]([3.0)()(

+

−+=

=

024. 999.055. 630.360. 047.980. 016.

W

=

0.0 0.10.0 5.5.0 0.0.1 0.

WThe 1st column is the average of the 2 vects in cluster 1, and the 2nd col is the average of the 2 vects in cluster 2


53

Example of a SOM That Use a 2-D Neighborhood

Character Recognition – Fausett, pp. 176-178

54

Traveling Salesman Problem (TSP) by SOMThe aim of the TSP is to find a tour of a given set of cities that is of minimum length.A tour consists of visiting each city exactly once and returning to the starting city.The net uses the city coordinates as input (n=2); there are as many cluster units as there are cities to be visited. The net has a linear topology (with the first and last unit also connected).

Fig. shows the Initial position of cluster units and location of cities

55

Traveling Salesman Problem (TSP) by SOMFig. Shows the result after 100 epochs of training with R=1 (learning rate decreasing from 0.5 to 0.4).

Fig. Shows the final tour after 100 epochs of training with R=0

Two candidate solutions:

ADEFGHIJBC

ADEFGHIJCB

56






57

Mat

lab

SO

M E

xam

ple

58

Architecture

59

Cre

atin

g a

SOM

newsom -- Create a self-organizing map

Syntax

net = newsom

net = newsom(PR,[D1,D2,...],TFCN,DFCN,OLR,OSTEPS,TLR,TND)

Description

net = newsom creates a new network with a dialog box.

net = newsom (PR,[D1,D2,...],TFCN,DFCN,OLR,OSTEPS,TLR,TND) takes,

PR - R x 2 matrix of min and max values for R input elements.

Di - Size of ith layer dimension, defaults = [5 8].

TFCN - Topology function, default ='hextop'.

DFCN - Distance function, default ='linkdist'.

OLR - Ordering phase learning rate, default = 0.9.

OSTEPS - Ordering phase steps, default = 1000.

TLR - Tuning phase learning rate, default = 0.02;

TND - Tuning phase neighborhood distance, default = 1.

and returns a new self-organizing map.

The topology function TFCN can be hextop, gridtop, or randtop. The distance function can be linkdist, dist, or mandist.

60






61

Suggested Reading.L. Fausett,

“Fundamentals of Neural Networks”, Prentice-Hall, 1994, Chapter 4.

J. Zurada, “Artificial Neural systems”, West Publishing, 1992, Chapter 7.

62

References:These lecture notes were based on the references of the

previous slide, and the following references

1. Berlin Chen Lecture notes: Normal University, Taipei, Taiwan, ROC. http://140.122.185.120

2. Dan St. Clair, University of Missori-Rolla, http://web.umr.edu/~stclair/class/classfiles/cs404_fs02/Misc/CS404_fall2001/Lectures/ Lecture 14.

http://140.122.185.120/

http://140.122.185.120/

http://web.umr.edu/~stclair/class/classfiles/cs404_fs02/Misc/CS404_fall2001/Lectures/Lect21_102301/

http://web.umr.edu/~stclair/class/classfiles/cs404_fs02/Misc/CS404_fall2001/Lectures/Lect21_102301/

Artificial Neural Networks Lect7: Neural networks based on competition

Engineering

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