Maximum likelihood training of probabilistic neural networks with rotationally related covariance matrices

Maximum likelihood algorithms are available for training two fundamental kinds of Gaussian probabilistic neural networks (PNNs), called herein homoscedastic ("same scatter") and heteroscedastic ("different scatter") PNNs. These are the only PNNs in the literature having readily derived maximum likelihood training algorithms. A new kind of PNN is defined in this paper, and a maximum likelihood training algorithm is derived. This new PNN is called a strophoscedastic ("twisted scatter") PNN to reflect the statistical character of its representation (as yet unnamed in the statistical literature). Structurally, in a sense made precise below, strophoscedastic PNNs fall between homoscedastic and heteroscedastic PNNs. Strophoscedastic PNNs are significant because they have a representational power similar to heteroscedastic PNNs and a parametric parsimony (and, hence, an inherent numerical stability) similar to homoscedastic PNNs.