紧支撑径向基函数作为概率密度函数
Compactly supported radial basis functions as probability density functions
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中文总结 AI 辅助
本研究将紧支撑径向基函数（CS-RBFs）作为新型概率密度函数族，推导其统计特性解析式，提出基于CS-RBF混合模型的增量学习算法，实验显示其在密度估计中与高斯混合模型表现相当。
中文摘要 AI 辅助
紧支撑径向基函数（CS-RBFs）是多元逼近理论的基础工具，但其在统计与概率建模中的应用仍未被充分探索，目前主要用于表达高斯过程中的协方差函数或作为核函数。本研究将CS-RBFs作为一种新型参数化概率密度函数族展开研究，尤其聚焦于Wendland $\boldsymbol{\fancyscript{C}}^2$核。核心贡献在于推导了CS-RBFs作为单变量及条件密度时，各类统计特性（如矩、累积分布函数）的解析表达式，涵盖两种情形：CS-RBF支撑完全位于变量定义域内（未截断支撑）与部分位于定义域外（截断支撑）。本研究还分析了采用CS-RBFs的混合模型，详述其主要特性，并引入一种基于CS-RBF混合模型的密度估计增量学习算法，其中中心通过k-means确定，权重与形状参数采用随机梯度下降优化。在合成数据集与真实世界数据集上的实验表明，与高斯混合模型相比，CS-RBF密度在似然值与模型复杂度方面表现具有竞争力，且该类CS-RBF密度可在单变量及条件场景下精确计算关键分布特性。
英文摘要
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.
