矩估计方法用于高斯混合模型的理论分析与遥感实证

Theoretical Analysis and Remote Sensing Empirical Study of the Method of Moments for Gaussian Mixture Models

摘要: 本文系统介绍并实现了基于矩估计(Method of Moments, MoM)用于一维三成分高斯混合模型(Gaussian Mixture Model, GMM)参数估计的方法，推导单分量原点矩的闭式表达，利用1阶到8阶的原点矩和中心距关系，构造了8个关于模型参数的非线性方程，用矩阵最小二乘法得到混合模型的权重的初值，采用Newton-Raphson迭代法求出非线性方程组的解。最后使用中国西部高原纳木错部分区域冰、水、雪、陆地多种物质的光谱进行了验证，针对估计出的每种物质的混合模型概率密度函数，结合距离函数进行了初步分类，分类结果显示了该方法的有效性。

Abstract: This paper systematically introduces and implements the method of moments (MoM) for parameter estimation of a one-dimensional three-component Gaussian mixture model (GMM). We derive the closed-form expressions of raw moments for single Gaussian components, and by leveraging the relationship between the first to eighth raw moments and central moments, we construct eight nonlinear equations with respect to the model parameters. The initial values of the mixture weights are obtained using a matrix least squares method, and the solutions to the nonlinear system are then found via the Newton-Raphson iterative method. Finally, we validate the approach using spectral data of ice, water, snow, and land from the Nam Co region on the Qinghai-Xizang Plateau in western China. For each estimated material’s mixture probability density function, preliminary classification is performed in combination with a distance function. The classification results demonstrate the effectiveness of the proposed method.