Matérn Kernel for incomplete data

Abstract

Machine learning problems with incomplete data are constantly addressed in various real-world domains. Statistical methods dealing with missing attributes are characterized by assumptions about data distribution through a density function. In this context, approaches that utilize similarity-based methods, become very promising research objects since these methods generally assume that data is fully observed and are not naturally equipped to handle incomplete. In this work, methods will be proposed to estimate the expected value of the Matérn Kernel in the presence of incomplete data vectors without any preprocessing steps. The EMK-MC and EMK-UT methods demonstrate the capability to address the kernel estimation problem directly, meaning they estimate the transformation of interest instead of embedding it within a preprocessing framework. To obtain such estimates, incomplete vectors are treated as continuous random variables, and based on the assumption that the Euclidean distance between points of interest follows a Nakagami distribution, sampling methods are used to generate points that depend only on the distribution of interest. Through a Gaussian mixture model, the data distribution is approximated by maximum likelihood estimation via the Expectation-Maximization algorithm, while simultaneously iteratively estimating the missing values. This allows the model to be fitted to the observed data, considering the uncertainty of the missing values and the relationships between variables. The performances of the proposed methods are compared to three methods on real and synthetic datasets. Considering the root mean square error obtained by computing the difference between the estimated kernel value and the true value, the consistency of the achieved performance remains evident in the majority of the scenarios evaluated for real-world datasets. The proposed methods, EMK-MC and EMK-UT, are superior in approximately 43% and 38% of the evaluated scenarios, respectively. As for the scenarios evaluated in synthetic datasets, the proposed approaches outperform all evaluated scenarios.