Gaussian kernels for incomplete data

Diego P. P. Mesquita, João P. P. Gomes, Francesco Corona, Amauri Holanda de Souza Júnior, Juvêncio S. Nobre

Abstract: Highlights • We propose a method to estimate the expected value of a Gaussian kernel in the presence of incomplete data. • Our method deals with missing data directly and do not rely on any imputation method. • The proposed formulation models the distribution of the data using a Gaussian Mixture Model (GMM). • The validity of our method is empirically assessed under a range of conditions on simulated and real problems. • Results showed that our method was able to better estimate Gaussian kernels when compared to other missing data treatments. Abstract This paper discusses a method to estimate the expected value of the Gaussian kernel in the presence of incomplete data. We show how, under the general assumption of a missing-at-random mechanism, the expected value of the Gaussian kernel function has a simple closed-form solution. Such a solution depends only on the parameters of the Gamma distribution which is assumed to represent squared distances. Furthermore, we show how the parameters governing the Gamma distribution depend only on the non-central moments of the kernel arguments, via the second-order moments of their squared distance, and can be estimated by making use of any parametric density estimation model of the data distribution. We approximate the data distribution with the maximum likelihood estimate of a Gaussian mixture distribution. The validity of the method is empirically assessed, under a range of conditions, on synthetic and real problems and the results compared to existing methods. For comparison, we consider methods that indirectly estimate a Gaussian kernel function by either estimating squared distances or by imputing missing values and then computing distances. Based on the experimental results, the proposed method consistently proves itself an accurate technique that further extends the use of Gaussian kernels with incomplete data.