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G. in C on the Unix workstations, we typically use drand48(). Donald Knuth has an algorithm which provides a means of creating a pseudo-Gaussian distribution from a uniform distribution. 029899776 double cStat::normalMean(double mean,double stdev) { int j; float r,x; float rsq; r=0; for(j=0;j<12;j++) r += drand48(); r = (r-6)/4; rsq = r*r; x = ((((A9*rsq+A7)*rsq + A5)*rsq + A3)*rsq+A1)*r; 52 CHAPTER 3. HEBBIAN LEARNING return mean+x*stdev; } This is a function which takes in two doubles (the mean and standard deviation of the distribution you wish to model) and returns a single value from that distribution.

The weights are not allowed to exceed w+ nor decrease beyond w− where w− = −w+ . e. ∆wij = axj yi − aE(x)yi − aE(y)(xj − E(x)) Both yi and E(y) are functions of w, but in neither case are we multiplying these by w itself. Therefore, as noted earlier, the weights will not tend to a multiple of the principal eigenvector but + − will saturate at the bounds ( wij or wij ) of their permissible values. Because the effects of the major eigenvectors will still be felt, there will not be a situation where a weight will tend to w− in a direction where the principal eigenvector has a positive correlation with the other weights.

The operations carried out at each neuron are identical. This is essential if we are to take full advantage of parallel processing. The major disadvantage of this algorithm is that it finds only the Principal Subspace of the eigenvectors not the actual eigenvectors themselves. 2 Oja’s Weighted Subspace Algorithm The final stage is the creation of algorithms which find the actual Principal Components of the input data. In 1992, Oja et al recognised the importance of introducing asymmetry into the weight decay process in order to force weights to converge to the Principal Components.