%SOM_DEMO1 Basic properties and behaviour of the Self-Organizing Map. % Contributed to SOM Toolbox 2.0, February 11th, 2000 by Juha Vesanto % http://www.cis.hut.fi/projects/somtoolbox/ % Version 1.0beta juuso 071197 % Version 2.0beta juuso 030200 clf reset; figure(gcf) echo on clc % ========================================================== % SOM_DEMO1 - BEHAVIOUR AND PROPERTIES OF SOM % ========================================================== % som_make - Create, initialize and train a SOM. % som_randinit - Create and initialize a SOM. % som_lininit - Create and initialize a SOM. % som_seqtrain - Train a SOM. % som_batchtrain - Train a SOM. % som_bmus - Find best-matching units (BMUs). % som_quality - Measure quality of SOM. % SELF-ORGANIZING MAP (SOM): % A self-organized map (SOM) is a "map" of the training data, % dense where there is a lot of data and thin where the data % density is low. % The map constitutes of neurons located on a regular map grid. % The lattice of the grid can be either hexagonal or rectangular. subplot(1,2,1) som_cplane('hexa',[10 15],'none') title('Hexagonal SOM grid') subplot(1,2,2) som_cplane('rect',[10 15],'none') title('Rectangular SOM grid') % Each neuron (hexagon on the left, rectangle on the right) has an % associated prototype vector. After training, neighboring neurons % have similar prototype vectors. % The SOM can be used for data visualization, clustering (or % classification), estimation and a variety of other purposes. pause % Strike any key to continue... clf clc % INITIALIZE AND TRAIN THE SELF-ORGANIZING MAP % ============================================ % Here are 300 data points sampled from the unit square: D = rand(300,2); % The map will be a 2-dimensional grid of size 10 x 10. msize = [10 10]; % SOM_RANDINIT and SOM_LININIT can be used to initialize the % prototype vectors in the map. The map size is actually an % optional argument. If omitted, it is determined automatically % based on the amount of data vectors and the principal % eigenvectors of the data set. Below, the random initialization % algorithm is used. sMap = som_randinit(D, 'msize', msize); % Actually, each map unit can be thought as having two sets % of coordinates: % (1) in the input space: the prototype vectors % (2) in the output space: the position on the map % In the two spaces, the map looks like this: subplot(1,3,1) som_grid(sMap) axis([0 11 0 11]), view(0,-90), title('Map in output space') subplot(1,3,2) plot(D(:,1),D(:,2),'+r'), hold on som_grid(sMap,'Coord',sMap.codebook) title('Map in input space') % The black dots show positions of map units, and the gray lines % show connections between neighboring map units. Since the map % was initialized randomly, the positions in in the input space are % completely disorganized. The red crosses are training data. pause % Strike any key to train the SOM... % During training, the map organizes and folds to the training % data. Here, the sequential training algorithm is used: sMap = som_seqtrain(sMap,D,'radius',[5 1],'trainlen',10); subplot(1,3,3) som_grid(sMap,'Coord',sMap.codebook) hold on, plot(D(:,1),D(:,2),'+r') title('Trained map') pause % Strike any key to view more closely the training process... clf clc % TRAINING THE SELF-ORGANIZING MAP % ================================ % To get a better idea of what happens during training, let's look % at how the map gradually unfolds and organizes itself. To make it % even more clear, the map is now initialized so that it is away % from the data. sMap = som_randinit(D,'msize',msize); sMap.codebook = sMap.codebook + 1; subplot(1,2,1) som_grid(sMap,'Coord',sMap.codebook) hold on, plot(D(:,1),D(:,2),'+r'), hold off title('Data and original map') % The training is based on two principles: % % Competitive learning: the prototype vector most similar to a % data vector is modified so that it it is even more similar to % it. This way the map learns the position of the data cloud. % % Cooperative learning: not only the most similar prototype % vector, but also its neighbors on the map are moved towards the % data vector. This way the map self-organizes. pause % Strike any key to train the map... echo off subplot(1,2,2) o = ones(5,1); r = (1-[1:60]/60); for i=1:60, sMap = som_seqtrain(sMap,D,'tracking',0,... 'trainlen',5,'samples',... 'alpha',0.1*o,'radius',(4*r(i)+1)*o); som_grid(sMap,'Coord',sMap.codebook) hold on, plot(D(:,1),D(:,2),'+r'), hold off title(sprintf('%d/300 training steps',5*i)) drawnow end title('Sequential training after 300 steps') echo on pause % Strike any key to continue with 3D data... clf clc % TRAINING DATA: THE UNIT CUBE % ============================ % Above, the map dimension was equal to input space dimension: both % were 2-dimensional. Typically, the input space dimension is much % higher than the 2-dimensional map. In this case the map cannot % follow perfectly the data set any more but must find a balance % between two goals: % - data representation accuracy % - data set topology representation accuracy % Here are 500 data points sampled from the unit cube: D = rand(500,3); subplot(1,3,1), plot3(D(:,1),D(:,2),D(:,3),'+r') view(3), axis on, rotate3d on title('Data') % The ROTATE3D command enables you to rotate the picture by % dragging the pointer above the picture with the leftmost mouse % button pressed down. pause % Strike any key to train the SOM... clc % DEFAULT TRAINING PROCEDURE % ========================== % Above, the initialization was done randomly and training was done % with sequential training function (SOM_SEQTRAIN). By default, the % initialization is linear, and batch training algorithm is % used. In addition, the training is done in two phases: first with % large neighborhood radius, and then finetuning with small radius. % The function SOM_MAKE can be used to both initialize and train % the map using default parameters: pause % Strike any key to use SOM_MAKE... sMap = som_make(D); % Here, the linear initialization is done again, so that % the results can be compared. sMap0 = som_lininit(D); subplot(1,3,2) som_grid(sMap0,'Coord',sMap0.codebook,... 'Markersize',2,'Linecolor','k','Surf',sMap0.codebook(:,3)) axis([0 1 0 1 0 1]), view(-120,-25), title('After initialization') subplot(1,3,3) som_grid(sMap,'Coord',sMap.codebook,... 'Markersize',2,'Linecolor','k','Surf',sMap.codebook(:,3)) axis([0 1 0 1 0 1]), view(3), title('After training'), hold on % Here you can see that the 2-dimensional map has folded into the % 3-dimensional space in order to be able to capture the whole data % space. pause % Strike any key to evaluate the quality of maps... clc % BEST-MATCHING UNITS (BMU) % ========================= % Before going to the quality, an important concept needs to be % introduced: the Best-Matching Unit (BMU). The BMU of a data % vector is the unit on the map whose model vector best resembles % the data vector. In practise the similarity is measured as the % minimum distance between data vector and each model vector on the % map. The BMUs can be calculated using function SOM_BMUS. This % function gives the index of the unit. % Here the BMU is searched for the origin point (from the % trained map): bmu = som_bmus(sMap,[0 0 0]); % Here the corresponding unit is shown in the figure. You can % rotate the figure to see better where the BMU is. co = sMap.codebook(bmu,:); text(co(1),co(2),co(3),'BMU','Fontsize',20) plot3([0 co(1)],[0 co(2)],[0 co(3)],'ro-') pause % Strike any key to analyze map quality... clc % SELF-ORGANIZING MAP QUALITY % =========================== % The maps have two primary quality properties: % - data representation accuracy % - data set topology representation accuracy % The former is usually measured using average quantization error % between data vectors and their BMUs on the map. For the latter % several measures have been proposed, e.g. the topographic error % measure: percentage of data vectors for which the first- and % second-BMUs are not adjacent units. % Both measures have been implemented in the SOM_QUALITY function. % Here are the quality measures for the trained map: [q,t] = som_quality(sMap,D) % And here for the initial map: [q0,t0] = som_quality(sMap0,D) % As can be seen, by folding the SOM has reduced the average % quantization error, but on the other hand the topology % representation capability has suffered. By using a larger final % neighborhood radius in the training, the map becomes stiffer and % preserves the topology of the data set better. echo off