Compactly supported radial basis functions as probability density functions

2026-07-29

AI Summary

This work addresses the underexplored use of compactly supported radial basis functions (RBFs) in probabilistic density modeling by systematically formulating them as parametric probability density functions. Specifically, it introduces a mixture density model based on the Wendland 𝒞² kernel and derives closed-form expressions for key statistical quantities—including moments and cumulative distribution functions—under both truncated and untruncated support. An incremental learning algorithm is proposed, combining k-means for centroid initialization with stochastic gradient descent for parameter optimization. Experimental results demonstrate that the proposed method achieves likelihood performance and model complexity comparable to Gaussian mixture models on both synthetic and real-world datasets, while enabling exact computation of distributional properties owing to its compact support and analytical tractability.

Abstract

Compactly Supported Radial Basis Functions (CS-RBFs) are a fundamental tool in multivariate approximation theory. However, their use in statistics and probability modeling remains underexplored, having been used mainly to express covariance functions in Gaussian processes or as kernel functions. This work explores CS-RBFs as a novel parametric family of probability density functions, focusing in particular on Wendland $\mathscr{C}^2$ kernels. The primary contribution of this work is the derivation of analytical expressions for various statistical properties, such as moments and the cumulative distribution function, of CS-RBFs as univariate and conditional densities. The approach comprises two alternative scenarios: when the CS-RBF support lies entirely within the variable's domain (untruncated support) and when part of it is outside (truncated support). Mixture models employing CS-RBFs are also analyzed, and their main properties are detailed. Furthermore, we introduce an incremental learning algorithm for density estimation with CS-RBF mixture models, in which centers are determined using k-means and weights and shape parameters are optimized by stochastic gradient descent. Experiments on synthetic and real-world datasets show that CS-RBF densities provide competitive results in terms of likelihood and model complexity in comparison with Gaussian mixture models. In addition, these CS-RBF densities allow the exact computation of key distributional properties in univariate and conditional settings.

Problem

• Compactly Supported Radial Basis Functions
• probability density functions
• statistical properties
• mixture models
• density estimation

Innovation

• Compactly Supported Radial Basis Functions
• Probability Density Functions
• Analytical Moments
• Mixture Models
• Incremental Learning