x0 xn w0 wn o Threshold units SOM

Self-Organizing Maps

Inputs Neurons Teuvo Kohonen

Self-Organizing Maps : Origins Ideas first introduced by C. von der Malsburg (1973), developed and refined by T. Kohonen (1982) Neural network algorithm using unsupervised competitive learning Primarily used for organization and visualization of complex data Biological basis: ‘brain maps’

Self-Organizing Maps SOM - Architecture 2d array of neurons Lattice of neurons (‘nodes’) accepts and responds to set of input signals Responses compared; ‘winning’ neuron selected from lattice Selected neuron activated together with ‘neighbourhood’ neurons Adaptive process changes weights to more closely resemble inputs 2d array of neurons Set of input signals (connected to all neurons in lattice) Weighted synapses x1 x2 x3 xn ... wj1 wj2 wj3 wjn j

Self-Organizing Maps Randomly initialise all weights SOM – Algorithm Overview Randomly initialise all weights Select input vector x = [x1, x2, x3, … , xn] Compare x with weights wj for each neuron j to determine winner Update winner so that it becomes more like x, together with the winner’s neighbours Adjust parameters: learning rate & ‘neighbourhood function’ Repeat from (2) until the map has converged (i.e. no noticeable changes in the weights) or pre-defined no. of training cycles have passed

Initialisation Randomly initialise the weights

Finding a Winner Euclidean distance Find the best-matching neuron w(x), usually the neuron whose weight vector has smallest Euclidean distance from the input vector x The winning node is that which is in some sense ‘closest’ to the input vector ‘Euclidean distance’ is the straight line distance between the data points, if they were plotted on a (multi-dimensional) graph Euclidean distance between two vectors a and b, a = (a1,a2,…,an), b = (b1,b2,…bn), is calculated as: Euclidean distance

wj(t +1) = wj(t) + (t) (x)(j,t) [x - wj(t)] Weight Update SOM Weight Update Equation wj(t +1) = wj(t) + (t) (x)(j,t) [x - wj(t)] “The weights of every node are updated at each cycle by adding Current learning rate × Degree of neighbourhood with respect to winner × Difference between current weights and input vector to the current weights” Example of (t) Example of (x)(j,t) L. rate No. of cycles x-axis shows distance from winning node y-axis shows ‘degree of neighbourhood’ (max. 1)

Kohonen’s Algorithm jth input Winner ith

Neighborhoods Square and hexagonal grid with neighborhoods based on box distance Grid-lines are not shown

i One-dimensional i Two-dimensional Neighborhood of neuron i

A neighborhood function (i, k) indicates how closely neurons i and k in the output layer are connected to each other. Usually, a Gaussian function on the distance between the two neurons in the layer is used:  position of i position of k

Clustering of the Self Organising Map A simple toy example Clustering of the Self Organising Map

However, instead of updating only the winning neuron i However, instead of updating only the winning neuron i*, all neurons within a certain neighborhood Ni* (d), of the winning neuron are updated using the Kohonen rule. Specifically, we adjust all such neurons i Ni* (d), as follow Here the neighborhood Ni* (d), contains the indices for all of the neurons that lie within a radius d of the winning neuron i*.

Topologically Correct Maps The aim of unsupervised self-organizing learning is to construct a topologically correct map of the input space.

Self Organizing Map Determine the winner (the neuron of which the weight vector has the smallest distance to the input vector) Move the weight vector w of the winning neuron towards the input i Before learning i w After learning i w

Network Features Input nodes are connected to every neuron The “winner” neuron is the one whose weights are most “similar” to the input Neurons participate in a “winner-take-all” behavior The winner output is set to 1 and all others to 0 Only weights to the winner and its neighbors are adapted

P 1 2 3 4 5 6 7 8 9 wi

P1 P2 1 2 3 4 5 6 7 8 9 wi2 wi1

output input (n-dimensional) winner

Example I: Learning a one-dimensional representation of a two-dimensional (triangular) input space: 20 100 10000 25000 1000

Some nice illustrations

Some nice illustrations

Some nice illustrations

Self Organizing Map Impose a topological order onto the competitive neurons (e.g., rectangular map) Let neighbors of the winner share the “prize” (The “postcode lottery” principle) After learning, neurons with similar weights tend to cluster on the map

Conclusion Advantages SOM is Algorithm that projects high-dimensional data onto a two-dimensional map. The projection preserves the topology of the data so that similar data items will be mapped to nearby locations on the map. SOM still have many practical applications in pattern recognition, speech analysis, industrial and medical diagnostics, data mining Disadvantages Large quantity of good quality representative training data required No generally accepted measure of ‘quality’ of a SOM e.g. Average quantization error (how well the data is classified)

Topologies (gridtop, hextop, randtop) pos = gridtop(3,2) pos = 0 1 0 1 0 1 0 0 1 1 2 2 plotsom (pos) pos = gridtop(2,3) pos = 0 1 0 1 0 1 0 0 1 1 2 2 plotsom (pos)

pos = gridtop(8,10); plotsom(pos)

pos = hextop(2,3) pos = 0 1.0000 0.5000 1.5000 0 1.0000 0 0 0.8660 0.8660 1.7321 1.7321

pos = hextop(3,2) pos = 0 1.0000 2.0000 0.5000 1.5000 2.5000 0 0 0 0.8660 0.8660 0.8660 plotsom(pos)

pos = hextop(8,10); plotsom(pos)

pos = randtop(2,3) pos = 0 0.7787 0.4390 1.0657 0.1470 0.9070 0 0.1925 0.6476 0.9106 1.6490 1.4027

pos = randtop(3,2) pos = 0 0.7787 1.5640 0.3157 1.2720 2.0320 0.0019 0.1944 0 0.9125 1.0014 0.7550

pos = randtop(8,10); plotsom(pos)

Distance Funct. (dist, linkdist, mandist, boxdist) pos2 = [ 0 1 2; 0 1 2] pos2 = 0 1 2 D2 = dist(pos2) D2 = 0 1.4142 2.8284 1.4142 0 1.4142 2.8284 1.4142 0

pos = gridtop(2,3) pos = 0 1 0 1 0 1 0 0 1 1 2 2 plotsom(pos) d = boxdist(pos) d = 0 1 1 1 2 2 1 0 1 1 2 2 1 1 0 1 1 1 1 1 1 0 1 1 2 2 1 1 0 1 2 2 1 1 1 0

pos = gridtop(2,3) pos = 0 1 0 1 0 1 0 0 1 1 2 2 plotsom(pos) d=linkdist(pos) d = 0 1 1 2 2 3 1 0 2 1 3 2 1 2 0 1 1 2 2 1 1 0 2 1 2 3 1 2 0 1 3 2 2 1 1 0

The Manhattan distance between two vectors x and y is calculated as D = sum(abs(x-y)) Thus if we have W1 = [ 1 2; 3 4; 5 6] W1 = 1 2 3 4 5 6 and P1= [1;1] P1 = 1 then we get for the distances Z1 = mandist(W1,P1) Z1 = 5 9

A One-dimensional Self-organizing Map angles = 0:2*pi/99:2*pi; P = [sin(angles); cos(angles)]; plot(P(1,:),P(2,:),'+r')

net = newsom([-1 1;-1 1],[30]); net.trainParam.epochs = 100; net = train(net,P); plotsom(net.iw{1,1},net.layers{1}.distances) The map can now be used to classify inputs, like [1; 0]: Either neuron 1 or 10 should have an output of 1, as the above input vector was at one end of the presented input space. The first pair of numbers indicate the neuron, and the single number indicates its output. p = [1;0]; a = sim (net, p) a = (1,1) 1

x = -4:0.01:4 P = [x;x.^2]; plot(P(1,:),P(2,:),'+r') net = newsom([-10 10;0 20],[10 10]); net.trainParam.epochs = 100; net = train(net,P); plotsom(net.iw{1,1},net.layers{1}.distances)