#### Abstract

Neural networks, trained with the backpropagation algorithm have: been applied to various classification problems. For linearly separable and nonseparahle problems, they have been shown to approximate the a posteriori probability of an input vector X belonging to a specific class C. In order to achieve high accuracy, large training data sets have to be used. For a small number of input dimensions, the accuracy of estimation was inferior to estimates using the Parzen density estimation. In this thesis, we propose two new techniques, lowering the mean square estimation error drastically and achieving better classification. In the past, t:he desired output patterns used for training have been of binary nature, using one for the class C the vector belongs to, and zero for the other classes. This work will show that by training against the columns of a Hadamard matrix, and then taking the inverse Hadamard transform of the network output, we can obtain more accurate estimates. The second change proposed in comparison with standard backpropagation networks will be the use of redundant output nodes. In standard backpropagat:ion the number of output nodes equals the number of different classes. In this thesis, it is shown that adding redundant output nodes enables us to decrease the mean square error at the output further, reaching better classification and lower mean square error rates than the Parzen density estimator. Comparisons between the statistical methods, the Parzen density estimation and histogramming, the conventional neural network and the Hadamard transformed neural network with redundant output nodes are given. Further, the effects of the proposed changes to the backpropagation algorithm on the convergence speed and the risk of getting stuck in a local minimum are: studied.

#### Date of this Version

August 1997