A Flexible Family of Compactly Supported Covariance Functions Based on Cutoff Polynomials

Abstract

AbstractIn time series analysis, signal covariance modeling is an essential part of stochastic methods like prediction or filtering. In geodetic applications, covariance functions are rarely treated as true compactly supported functions although large amounts of data would approve such. Covariance models for complex correlation shapes are also rare. Ideally, general families of covariance functions with a large flexibility are desirable to model complex correlations structures like negative correlations. In this paper, we derive isotropic finite covariance functions that are parametrized in a way that positive definiteness is guaranteed. These are based on cutoff polynomials which are derived from operations such as autoconvolution and autocorrelation. Next to the compact support, the resulting autocovariance models share the advantages of (a) positive definiteness by design, (b) extensibility to arbitrary orders and (c) extensive flexibility by employing multiple tunable shape parameters. All these realize various correlation shapes such as negative correlations (the so-called hole effect) and several oscillations. The methodological concepts are derived for homogeneous and isotropic random fields in $$\mathbb {R}^d$$ ℝ d . The family of covariance functions is then derived for one-dimensional applications. A data example demonstrates the covariance modeling approach using stationary time-series data.