In this paper a technique is developed in order to numerically calculate the hyper-volume under a multidimensional Gaussian function over a region of the space defined by an arbitrary hyper-quadratic boundary. The technique is a modified version of the technique developed by Fukunaga and rile. The latter technique can be used only if the hyper-quadratic boundary results from the intersection of the Gaussian function, under which the hyper-volume is being calculated, with some other Gaussian function as opposed to an arbitrary hyper-quadratic boundary.
A practical problem in which the hyper-volume calculation mentioned above is of interest arises in statistical pattern classification involving Gaussian classes. In this situation the Gaussian function under consideration is actually a probability density function and the arbitrary hyper-quadratic boundary results from the intersection between two estimated distributions which partitions the feature space into two disjoint decision regions. For this case the hyper-volume under the probability density function of any class in the region for which patterns are classified into that class, is actually the probability of correctly classifying vectors from the class.
The proposed technique has been successfully implemented and it has proven to be quite efficient and reasonably simple. Real data have been used to demonstrate the applicability and efficiency of the technique and to study the effect of estimation on the value of the probability of correct classification.
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