multidimensional scaling and kohonen's self-organizing maps
multidimensional scaling and kohonen's self-organizing maps multidimensional scaling and kohonen's self-organizing maps
D 2 (Y ) =nXi>j 2 ij ;kXl=1y (l)i! 22; y (l)jwhere y (l)iare components of Y i objects in the k-dimensional target space and thereduction in the number of the degrees of freedom going from N dimensions to kdimensions is taken into account by setting all components of Y 0 =0andk ; 1 componentsof Y 1 to zero, y (l)1=0l=1::k ; 1. For this measure we may obtain the bestrepresentation by solving a set of non-linear equations [11] instead of minimization:nXj6=iy (m)i 3 Xn ; y (m)j + y (m)ij6=i; y (m)j kXl6=my (l)i 2; y (l)j ;nXj6=i 2 ijy (m)i; y (m)j=0Unfortunately it is as hard to solve this system of nonlinear equations as it is tominimize the stress function.IV. COMPUTATIONAL RESULTSMinimization in MDS is usually done via gradient procedure. Since we are lookingfor a global minimum we have used simulated annealing method for minimization.We have applied SOM and MDS algorithms to a number of cases in which the qualityof maps could be assessed easily. Due to the lack ofspacewe will present onlytwocases: the data related to semantic maps about animals and a series of hypercubesin 3-5 dimensions, with cube corners represented in two-dimensional target space.Congurations of points obtained from SOM and MDS are compared in gures below.9HORSE876543DUCK21COWDOVELIONTIGERHENCATWOLFDOGFOXEAGLEGOOSE0OWL0 1 2 3 4 5 6 7 8 9GOOSEDUCKDOVEHAWK OWLHENEAGLECATFOXDOGTIGER LIONFIG. 1. The two-dimensional representations of the 13-dimensional semantic data obtainedby SOM (left) with a 10 x 10 neurons map, a training of 10000 cycles, with nalstress of 0.25, and the MDS (right) with nal stress of 0.20 after 10 iterations.COWHORSEZEBRAWOLF
56738874436511 22FIG. 2. The two-dimensional representations of the 8 points of the 3D cube obtained bySOM (left) with a 20 x 20 neuron map, a training of 10000 cycles, a nal quantization errorof 0.001 and stress 0.321, MDS (right) has the nal stress value 0.246 after 22 iterations.135 10 141613614129191 2154105124 31111328168 7 157 6FIG. 3. The two-dimensional representations of the 16 points of the 4D hypercube obtainedby SOM (left) with a 20 x 20 neurons map, a training of 10000 cycles, quantizationerror of 0.001, stress value 0.327, MDS (right) has the nal stress of 0.312 after 18 iterations.292526 3031 16 322826182231 2719233 7913101415 12112730101426151152118226 27 317 1 4 82419232029 25 17 211391 528321216242084FIG. 4. The two-dimensional representations of the 32 points of the 5D hypercube obtainedby SOM (left) with a 20 x 20 neurons map, a training of 10000 cycles, the stress valueof 0.353 and by MDS (right) with a nal stress of 0.333 after 18 iterations.In gures 2-4 all corners of the hypercube that are adjacent toeach other areconnected by lines. All these lines should be short but SOM tries to use all neurons
56738874436511 22FIG. 2. The two-dimensional representations of the 8 points of the 3D cube obtained bySOM (left) with a 20 x 20 neuron map, a training of 10000 cycles, a nal quantization errorof 0.001 <strong>and</strong> stress 0.321, MDS (right) has the nal stress value 0.246 after 22 iterations.135 10 141613614129191 2154105124 31111328168 7 157 6FIG. 3. The two-dimensional representations of the 16 points of the 4D hypercube obtainedby SOM (left) with a 20 x 20 neurons map, a training of 10000 cycles, quantizationerror of 0.001, stress value 0.327, MDS (right) has the nal stress of 0.312 after 18 iterations.292526 3031 16 322826182231 2719233 7913101415 12112730101426151152118226 27 317 1 4 82419232029 25 17 211391 528321216242084FIG. 4. The two-dimensional representations of the 32 points of the 5D hypercube obtainedby SOM (left) with a 20 x 20 neurons map, a training of 10000 cycles, the stress valueof 0.353 <strong>and</strong> by MDS (right) with a nal stress of 0.333 after 18 iterations.In gures 2-4 all corners of the hypercube that are adjacent toeach other areconnected by lines. All these lines should be short but SOM tries to use all neurons
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