On a Parameter Estimation Method for Gibbs-Markov Random Fields

Fig. 2. space by Patrick-Fisher's algorithm (solid line) and E (dotted line). Bayes error estimates for SONAR data transformed to IO-dimensional high-dimensional data these results might be more in favor of E.) This is a result of the fact that each iteration of simplex requires that the samples be transformed to the low-dimensional space, and then the Bayes error estimated in that space, which in turn requires computation of distances, determination of optimal thresholds, and classification. The iterative process is repeated until the convergence is achieved. In contrast, FLD and KL are noniterative techniques, which merely compute eigenvalues and eigenvectors of certain matrices. For PF, which is also an iterative algorithm, the difference in CPU timing is about an order of magnitude for SONAR data. The aim of this correspondence was to investigate the possibility of constructing such linear transformation of labeled multidimensional vectors that would hopefully ensure the maximum attainable classification accuracy in the transformed space. In other words, the goal was to come as close as possible to computing a transformation matrix that would minimize the Bayes error in the low-dimensional space, and to devise a practical algorithm for such purpose. The proposed algorithm, called E, consists in finding such matrix that minimizes the estimate of the Bayes error, computed on the training data set projected to the low-dimensional space. The most reliable technique for Bayes error estimation available was used, and the simplex algorithm played the role of the optimization algorithm. In all examples, E demonstrated superior performance in comparison with standard algorithms, coming close to the theoretical limits on classification accuracy. This is payed through significant increase of the computational load. Still we managed to keep its complexity within realistic bounds, thus realizing designated goals. A simplex method for function minimization , " Comput. the probability of misclassification for linear feature selection, " Annals Statist. On the optimal number of features in the classification of multivariate Gaussian data, " Pattern Recogn., vol. Analysis of hidden units in a layered network trained to classify sonar targets, " On the minimal dimension of sufficient statistics, " IEEE Trans. Abstract-This correspondence is about a Gibbs-Markov random field (GMRF) parameter estimation technique proposed by Derin and Elliott. We will refer to this technique as the histogramming (H) method. First, the relation of the H method to the (conditional) maximum likelihood (ML) method is considered. Second, a bias-reduction based modification of …