Comparison of kernel based PDF estimation methods

There are a number of challenging estimation, tracking, and decision theoretic problems that require the estimation of Probability Density Functions (PDFs). When using a traditional parametric approach, the functional model of the PDF is assumed to be known. However, these models often do not capture the complexity of the underlying distribution. Furthermore, the problems of validating the model and estimating its parameters are often complicated by the sparsity of prior examples. The need for exemplars grows exponentially with the dimension of the feature space. These methods may yield PDFs that do not generalize well to unseen data because these tend to overfit or underfit the training exemplars. We investigate and compare alternate approaches for estimating a PDF and consider instead kernel based estimation methods which generalize the Parzen estimator and use a Linear Mixture of Kernels (LMK) model. The methods reported here are derived from machine learning methods such as the Support Vector Machines and the Relevance Vector Machines. These PDF estimators provide the following benefits: (a) they are data driven; (b) they do not overfit the data and consequently have good generalization properties; (c) they can accommodate highly irregular and multi-modal data distributions; (d) they provide a sparse and succinct description of the underlying data which leads to efficient computation and communication. Comparative experimental results are provided illustrating these properties using simulated Mixture of Gaussian-distributed data.