%SOM_DEMO3 Self-organizing map visualization. % 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 080200 070600 clf reset; figure(gcf) echo on clc % ========================================================== % SOM_DEMO3 - VISUALIZATION % ========================================================== % som_show - Visualize map. % som_grid - Visualization with free coordinates. % % som_show_add - Add markers on som_show visualization. % som_show_clear - Remove markers from som_show visualization. % som_recolorbar - Refresh and rescale colorbars in som_show % visualization. % % som_cplane - Visualize component/color/U-matrix plane. % som_pieplane - Visualize prototype vectors as pie charts. % som_barplane - Visualize prototype vectors as bar charts. % som_plotplane - Visualize prototype vectors as line graphs. % % pcaproj - Projection to principal component space. % cca - Projection with Curvilinear Component Analysis. % sammon - Projection with Sammon's mapping. % som_umat - Calculate U-matrix. % som_colorcode - Color coding for the map. % som_normcolor - RGB values of indexed colors. % som_hits - Hit histograms for the map. % The basic functions for SOM visualization are SOM_SHOW and % SOM_GRID. The SOM_SHOW has three auxiliary functions: % SOM_SHOW_ADD, SOM_SHOW_CLEAR and SOM_RECOLORBAR which are used % to add and remove markers and to control the colorbars. % SOM_SHOW actually uses SOM_CPLANE to make the visualizations. % Also SOM_{PIE,BAR,PLOT}PLANE can be used to visualize SOMs. % The other functions listed above do not themselves visualize % anything, but their results are used in the visualizations. % There's an important limitation that visualization functions have: % while the SOM Toolbox otherwise supports N-dimensional map grids, % visualization only works for 1- and 2-dimensional map grids!!! pause % Strike any key to create demo data and map... clc % DEMO DATA AND MAP % ================= % The data set contructed for this demo consists of random vectors % in three gaussian kernels the centers of which are at [0, 0, 0], % [3 3 3] and [9 0 0]. The map is trained using default parameters. D1 = randn(100,3); D2 = randn(100,3) + 3; D3 = randn(100,3); D3(:,1) = D3(:,1) + 9; sD = som_data_struct([D1; D2; D3],'name','Demo3 data',... 'comp_names',{'X-coord','Y-coord','Z-coord'}); sM = som_make(sD); % Since the data (and thus the prototypes of the map) are % 3-dimensional, they can be directly plotted using PLOT3. % Below, the data is plotted using red 'o's and the map % prototype vectors with black '+'s. plot3(sD.data(:,1),sD.data(:,2),sD.data(:,3),'ro',... sM.codebook(:,1),sM.codebook(:,2),sM.codebook(:,3),'k+') rotate3d on % From the visualization it is pretty easy to see what the data is % like, and how the prototypes have been positioned. One can see % that there are three clusters, and that there are some prototype % vectors between the clusters, although there is actually no % data there. The map units corresponding to these prototypes % are called 'dead' or 'interpolative' map units. pause % Strike any key to continue... clc % VISUALIZATION OF MULTIDIMENSIONAL DATA % ====================================== % Usually visualization of data sets is not this straightforward, % since the dimensionality is much higher than three. In principle, % one can embed additional information to the visualization by % using properties other than position, for example color, size or % shape. % Here the data set and map prototypes are plotted again, but % information of the cluster is shown using color: red for the % first cluster, green for the second and blue for the last. plot3(sD.data(1:100,1),sD.data(1:100,2),sD.data(1:100,3),'ro',... sD.data(101:200,1),sD.data(101:200,2),sD.data(101:200,3),'go',... sD.data(201:300,1),sD.data(201:300,2),sD.data(201:300,3),'bo',... sM.codebook(:,1),sM.codebook(:,2),sM.codebook(:,3),'k+') rotate3d on % However, this works only for relatively small dimensionality, say % less than 10. When the information is added this way, the % visualization becomes harder and harder to understand. Also, not % all properties are equal: the human visual system perceives % colors differently from position, not to mention the complex % rules governing perception of shape. pause % Strike any key to learn about linking... clc % LINKING MULTIPLE VISUALIZATIONS % =============================== % The other option is to use *multiple visualizations*, so called % small multiples, instead of only one. The problem is then how to % link these visualizations together: one should be able to idetify % the same object from the different visualizations. % This could be done using, for example, color: each object has % the same color in each visualization. Another option is to use % similar position: each object has the same position in each % small multiple. % For example, here are four subplots, one for each component and % one for cluster information, where color denotes the value and % position is used for linking. The 2D-position is derived by % projecting the data into the space spanned by its two greatest % eigenvectors. [Pd,V,me] = pcaproj(sD.data,2); % project the data Pm = pcaproj(sM.codebook,V,me); % project the prototypes colormap(hot); % colormap used for values echo off for c=1:3, subplot(2,2,c), cla, hold on som_grid('rect',[300 1],'coord',Pd,'Line','none',... 'MarkerColor',som_normcolor(sD.data(:,c))); som_grid(sM,'Coord',Pm,'Line','none','marker','+'); hold off, title(sD.comp_names{c}), xlabel('PC 1'), ylabel('PC 2'); end subplot(2,2,4), cla plot(Pd(1:100,1),Pd(1:100,2),'ro',... Pd(101:200,1),Pd(101:200,2),'go',... Pd(201:300,1),Pd(201:300,2),'bo',... Pm(:,1),Pm(:,2),'k+') title('Cluster') echo on pause % Strike any key to use color for linking... % Here is another example, where color is used for linking. On the % top right triangle are the scatter plots of each variable without % color coding, and on the bottom left triangle with the color % coding. In the colored figures, each data sample can be % identified by a unique color. Well, almost identified: there are % quite a lot of samples with almost the same color. Color is not as % precise linking method as position. echo off Col = som_normcolor([1:300]',jet(300)); k=1; for i=1:3, for j=1:3, if i<j, i1=i; i2=j; else i1=j; i2=i; end if i<j, subplot(3,3,k); cla plot(sD.data(:,i1),sD.data(:,i2),'ko') xlabel(sD.comp_names{i1}), ylabel(sD.comp_names{i2}) elseif i>j, subplot(3,3,k); cla som_grid('rect',[300 1],'coord',sD.data(:,[i1 i2]),... 'Line','none','MarkerColor',Col); xlabel(sD.comp_names{i1}), ylabel(sD.comp_names{i2}) end k=k+1; end end echo on pause % Strike any key to learn about data visualization using SOM... clc % DATA VISUALIZATION USING SOM % ============================ % The basic visualization functions and their usage have already % been introduced in SOM_DEMO2. In this demo, a more structured % presentation is given. % Data visualization techniques using the SOM can be divided to % three categories based on their goal: % 1. visualization of clusters and shape of the data: % projections, U-matrices and other distance matrices % % 2. visualization of components / variables: % component planes, scatter plots % % 3. visualization of data projections: % hit histograms, response surfaces pause % Strike any key to visualize clusters with distance matrices... clf clc % 1. VISUALIZATION OF CLUSTERS: DISTANCE MATRICES % =============================================== % Distance matrices are typically used to show the cluster % structure of the SOM. They show distances between neighboring % units, and are thus closely related to single linkage clustering % techniques. The most widely used distance matrix technique is % the U-matrix. % Here, the U-matrix of the map is shown (using all three % components in the distance calculation): colormap(1-gray) som_show(sM,'umat','all'); pause % Strike any key to see more examples of distance matrices... % The function SOM_UMAT can be used to calculate U-matrix. The % resulting matrix holds distances between neighboring map units, % as well as the median distance from each map unit to its % neighbors. These median distances corresponding to each map unit % can be easily extracted. The result is a distance matrix using % median distance. U = som_umat(sM); Um = U(1:2:size(U,1),1:2:size(U,2)); % A related technique is to assign colors to the map units such % that similar map units get similar colors. % Here, four clustering figures are shown: % - U-matrix % - median distance matrix (with grayscale) % - median distance matrix (with map unit size) % - similarity coloring, made by spreading a colormap % on top of the principal component projection of the % prototype vectors subplot(2,2,1) h=som_cplane([sM.topol.lattice,'U'],sM.topol.msize, U(:)); set(h,'Edgecolor','none'); title('U-matrix') subplot(2,2,2) h=som_cplane(sM, Um(:)); set(h,'Edgecolor','none'); title('D-matrix (grayscale)') subplot(2,2,3) som_cplane(sM,'none',1-Um(:)/max(Um(:))) title('D-matrix (marker size)') subplot(2,2,4) C = som_colorcode(Pm); % Pm is the PC-projection calculated earlier som_cplane(sM,C) title('Similarity coloring') pause % Strike any key to visualize shape and clusters with projections... clf clc % 1. VISUALIZATION OF CLUSTERS AND SHAPE: PROJECTIONS % =================================================== % In vector projection, a set of high-dimensional data samples is % projected to a lower dimensional such that the distances between % data sample pairs are preserved as well as possible. Depending % on the technique, the projection may be either linear or % non-linear, and it may place special emphasis on preserving % local distances. % For example SOM is a projection technique, since the prototypes % have well-defined positions on the 2-dimensional map grid. SOM as % a projection is however a very crude one. Other projection % techniques include the principal component projection used % earlier, Sammon's mapping and Curvilinear Component Analysis % (to name a few). These have been implemented in functions % PCAPROJ, SAMMON and CCA. % Projecting the map prototype vectors and joining neighboring map % units with lines gives the SOM its characteristic net-like look. % The projection figures can be linked to the map planes using % color coding. % Here is the distance matrix, color coding, a projection without % coloring and a projection with one. In the last projection, % the size of interpolating map units has been set to zero. subplot(2,2,1) som_cplane(sM,Um(:)); title('Distance matrix') subplot(2,2,2) C = som_colorcode(sM,'rgb4'); som_cplane(sM,C); title('Color code') subplot(2,2,3) som_grid(sM,'Coord',Pm,'Linecolor','k'); title('PC-projection') subplot(2,2,4) h = som_hits(sM,sD); s=6*(h>0); som_grid(sM,'Coord',Pm,'MarkerColor',C,'Linecolor','k','MarkerSize',s); title('Colored PC-projection') pause % Strike any key to visualize component planes... clf clc % 2. VISUALIZATION OF COMPONENTS: COMPONENT PLANES % ================================================ % The component planes visualizations shows what kind of values the % prototype vectors of the map units have for different vector % components. % Here is the U-matrix and the three component planes of the map. som_show(sM) pause % Strike any key to continue... % Besides SOM_SHOW and SOM_CPLANE, there are three other % functions specifically designed for showing the values of the % component planes: SOM_PIEPLANE, SOM_BARPLANE, SOM_PLOTPLANE. % SOM_PIEPLANE shows a single pie chart for each map unit. Each % pie shows the relative proportion of each component of the sum of % all components in that map unit. The component values must be % positive. % SOM_BARPLANE shows a barchart in each map unit. The scaling of % bars can be made unit-wise or variable-wise. By default it is % determined variable-wise. % SOM_PLOTPLANE shows a linegraph in each map unit. M = som_normalize(sM.codebook,'range'); subplot(1,3,1) som_pieplane(sM, M); title('som\_pieplane') subplot(1,3,2) som_barplane(sM, M, '', 'unitwise'); title('som\_barplane') subplot(1,3,3) som_plotplane(sM, M, 'b'); title('som\_plotplane') pause % Strike any key to visualize cluster properties... clf clc % 2. VISUALIZATION OF COMPONENTS: CLUSTERS % ======================================== % An interesting question is of course how do the values of the % variables relate to the clusters: what are the values of the % components in the clusters, and which components are the ones % which *make* the clusters. som_show(sM) % From the U-matrix and component planes, one can easily see % what the typical values are in each cluster. pause % Strike any key to continue... % The significance of the components with respect to the clustering % is harder to visualize. One indication of importance is that on % the borders of the clusters, values of important variables change % very rabidly. % Here, the distance matrix is calculated with respect to each % variable. u1 = som_umat(sM,'mask',[1 0 0]'); u1=u1(1:2:size(u1,1),1:2:size(u1,2)); u2 = som_umat(sM,'mask',[0 1 0]'); u2=u2(1:2:size(u2,1),1:2:size(u2,2)); u3 = som_umat(sM,'mask',[0 0 1]'); u3=u3(1:2:size(u3,1),1:2:size(u3,2)); % Here, the distance matrices are shown, as well as a piechart % indicating the relative importance of each variable in each % map unit. The size of piecharts has been scaled by the % distance matrix calculated from all components. subplot(2,2,1) som_cplane(sM,u1(:)); title(sM.comp_names{1}) subplot(2,2,2) som_cplane(sM,u2(:)); title(sM.comp_names{2}) subplot(2,2,3) som_cplane(sM,u3(:)); title(sM.comp_names{3}) subplot(2,2,4) som_pieplane(sM, [u1(:), u2(:), u3(:)], hsv(3), Um(:)/max(Um(:))); title('Relative importance') % From the last subplot, one can see that in the area where the % bigger cluster border is, the 'X-coord' component (red color) % has biggest effect, and thus is the main factor in separating % that cluster from the rest. pause % Strike any key to learn about correlation hunting... clf clc % 2. VISUALIZATION OF COMPONENTS: CORRELATION HUNTING % =================================================== % Finally, the component planes are often used for correlation % hunting. When the number of variables is high, the component % plane visualization offers a convenient way to visualize all % components at once and hunt for correlations (as opposed to % N*(N-1)/2 scatterplots). % Hunting correlations this way is not very accurate. However, it % is easy to select interesting combinations for further % investigation. % Here, the first and third components are shown with scatter % plot. As with projections, a color coding is used to link the % visualization to the map plane. In the color coding, size shows % the distance matrix information. C = som_colorcode(sM); subplot(1,2,1) som_cplane(sM,C,1-Um(:)/max(Um(:))); title('Color coding + distance matrix') subplot(1,2,2) som_grid(sM,'Coord',sM.codebook(:,[1 3]),'MarkerColor',C); title('Scatter plot'); xlabel(sM.comp_names{1}); ylabel(sM.comp_names{3}) axis equal pause % Strike any key to visualize data responses... clf clc % 3. DATA ON MAP % ============== % The SOM is a map of the data manifold. An interesting question % then is where on the map a specific data sample is located, and % how accurate is that localization? One is interested in the % response of the map to the data sample. % The simplest answer is to find the BMU of the data sample. % However, this gives no indication of the accuracy of the % match. Is the data sample close to the BMU, or is it actually % equally close to the neighboring map units (or even approximately % as close to all map units)? Sometimes accuracy doesn't really % matter, but if it does, it should be visualized somehow. % Here are different kinds of response visualizations for two % vectors: [0 0 0] and [99 99 99]. % - BMUs indicated with labels % - BMUs indicated with markers, relative quantization errors % (in this case, proportion between distances to BMU and % Worst-MU) with vertical lines % - quantization error between the samples and all map units % - fuzzy response (a non-linear function of quantization % error) of all map units echo off [bm,qe] = som_bmus(sM,[0 0 0; 99 99 99],'all'); % distance to all map units [dummy,ind] = sort(bm(1,:)); d0 = qe(1,ind)'; [dummy,ind] = sort(bm(2,:)); d9 = qe(2,ind)'; bmu0 = bm(1,1); bmu9 = bm(2,1); % bmus h0 = zeros(prod(sM.topol.msize),1); h0(bmu0) = 1; % crisp hits h9 = zeros(prod(sM.topol.msize),1); h9(bmu9) = 1; lab = cell(prod(sM.topol.msize),1); lab{bmu0} = '[0,0,0]'; lab{bmu9} = '[99,99,99]'; hf0 = som_hits(sM,[0 0 0],'fuzzy'); % fuzzy response hf9 = som_hits(sM,[99 99 99],'fuzzy'); som_show(sM,'umat',{'all','BMU'},... 'color',{d0,'Qerror 0'},'color',{hf0,'Fuzzy response 0'},... 'empty','BMU+qerror',... 'color',{d9,'Qerror 99'},'color',{hf9,'Fuzzy response 99'}); som_show_add('label',lab,'Subplot',1,'Textcolor','r'); som_show_add('hit',[h0, h9],'Subplot',4,'MarkerColor','r'); hold on Co = som_vis_coords(sM.topol.lattice,sM.topol.msize); plot3(Co(bmu0,[1 1]),Co(bmu0,[2 2]),[0 10*qe(1,1)/qe(1,end)],'r-') plot3(Co(bmu9,[1 1]),Co(bmu9,[2 2]),[0 10*qe(2,1)/qe(2,end)],'r-') view(3), axis equal echo on % Here are the distances to BMU, 2-BMU and WMU: qe(1,[1,2,end]) % [0 0 0] qe(2,[1,2,end]) % [99 99 99] % One can see that for [0 0 0] the accuracy is pretty good as the % quantization error of the BMU is much lower than that of the % WMU. On the other hand [99 99 99] is very far from the map: % distance to BMU is almost equal to distance to WMU. pause % Strike any key to visualize responses of multiple samples... clc clf % 3. DATA ON MAP: HIT HISTOGRAMS % ============================== % One can also investigate whole data sets using the map. When the % BMUs of multiple data samples are aggregated, a hit histogram % results. Instead of BMUs, one can also aggregate for example % fuzzy responses. % The hit histograms (or aggregated responses) can then be compared % with each other. % Here are hit histograms of three data sets: one with 50 first % vectors of the data set, one with 150 samples from the data % set, and one with 50 randomly selected samples. In the last % subplot, the fuzzy response of the first data set. dlen = size(sD.data,1); Dsample1 = sD.data(1:50,:); h1 = som_hits(sM,Dsample1); Dsample2 = sD.data(1:150,:); h2 = som_hits(sM,Dsample2); Dsample3 = sD.data(ceil(rand(50,1)*dlen),:); h3 = som_hits(sM,Dsample3); hf = som_hits(sM,Dsample1,'fuzzy'); som_show(sM,'umat','all','umat','all','umat','all','color',{hf,'Fuzzy'}) som_show_add('hit',h1,'Subplot',1,'Markercolor','r') som_show_add('hit',h2,'Subplot',2,'Markercolor','r') som_show_add('hit',h3,'Subplot',3,'Markercolor','r') pause % Strike any key to visualize trajectories... clc clf % 3. DATA ON MAP: TRAJECTORIES % ============================ % A special data mapping technique is trajectory. If the samples % are ordered, forming a time-series for example, their response on % the map can be tracked. The function SOM_SHOW_ADD can be used to % show the trajectories in two different modes: 'traj' and 'comet'. % Here, a series of data points is formed which go from [8,0,0] % to [2,2,2]. The trajectory is plotted using the two modes. Dtraj = [linspace(9,2,20); linspace(0,2,20); linspace(0,2,20)]'; T = som_bmus(sM,Dtraj); som_show(sM,'comp',[1 1]); som_show_add('traj',T,'Markercolor','r','TrajColor','r','subplot',1); som_show_add('comet',T,'MarkerColor','r','subplot',2); % There's also a function SOM_TRAJECTORY which lauches a GUI % specifically designed for displaying trajectories (in 'comet' % mode). pause % Strike any key to learn about color handling... clc clf % COLOR HANDLING % ============== % Matlab offers flexibility in the colormaps. Using the COLORMAP % function, the colormap may be changed. There are several useful % colormaps readily available, for example 'hot' and 'jet'. The % default number of colors in the colormaps is 64. However, it is % often advantageous to use less colors in the colormap. This way % the components planes visualization become easier to interpret. % Here the three component planes are visualized using the 'hot' % colormap and only three colors. som_show(sM,'comp',[1 2 3]) colormap(hot(3)); som_recolorbar pause % Press any key to change the colorbar labels... % The function SOM_RECOLORBAR can be used to reconfigure % the labels beside the colorbar. % Here the colorbar of the first subplot is labeled using labels % 'small', 'medium' and 'big' at values 0, 1 and 2. For the % colorbar of the second subplot, values are calculated for the % borders between colors. som_recolorbar(1,{[0 4 9]},'',{{'small','medium','big'}}); som_recolorbar(2,'border',''); pause % Press any key to learn about SOM_NORMCOLOR... % Some SOM Toolbox functions do not use indexed colors if the % underlying Matlab function (e.g. PLOT) do not use indexed % colors. SOM_NORMCOLOR is a convenient function to simulate % indexed colors: it calculates fixed RGB colors that % are similar to indexed colors with the specified colormap. % Here, two SOM_GRID visualizations are created. One uses the % 'surf' mode to show the component colors in indexed color % mode, and the other uses SOM_NORMALIZE to do the same. clf colormap(jet(64)) subplot(1,2,1) som_grid(sM,'Surf',sM.codebook(:,3)); title('Surf mode') subplot(1,2,2) som_grid(sM,'Markercolor',som_normcolor(sM.codebook(:,3))); title('som\_normcolor') pause % Press any key to visualize different map shapes... clc clf % DIFFERENT MAP SHAPES % ==================== % There's no direct way to visualize cylinder or toroid maps. When % visualized, they are treated exactly as if they were sheet % shaped. However, if function SOM_UNIT_COORDS is used to provide % unit coordinates, then SOM_GRID can be used to visualize these % alternative map shapes. % Here the grids of the three possible map shapes (sheet, cylinder % and toroid) are visualized. The last subplot shows a component % plane visualization of the toroid map. Cor = som_unit_coords(sM.topol.msize,'hexa','sheet'); Coc = som_unit_coords(sM.topol.msize,'hexa','cyl'); Cot = som_unit_coords(sM.topol.msize,'hexa','toroid'); subplot(2,2,1) som_grid(sM,'Coord',Cor,'Markersize',3,'Linecolor','k'); title('sheet'), view(0,-90), axis tight, axis equal subplot(2,2,2) som_grid(sM,'Coord',Coc,'Markersize',3,'Linecolor','k'); title('cylinder'), view(5,1), axis tight, axis equal subplot(2,2,3) som_grid(sM,'Coord',Cot,'Markersize',3,'Linecolor','k'); title('toroid'), view(-100,0), axis tight, axis equal subplot(2,2,4) som_grid(sM,'Coord',Cot,'Surf',sM.codebook(:,3)); colormap(jet), colorbar title('toroid'), view(-100,0), axis tight, axis equal echo off

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