基于约束B样条光滑方法的协方差函数估计

The covariance function estimation based on the constrained B-spline smoothing method

协方差函数在描述信号时域特征和空间相关结构等方面发挥着关键作用.近年来,许多文献研究了协方差函数的非参数估计问题.然而大部分的非参数协方差函数估计量常常无法保证正定性.为克服上述困难,本文在平稳随机过程背景下,首先研究无约束条件的B样条协方差函数估计量的渐近性质;在此基础上,基于完全单调函数与B样条基函数的关系,构造带有约束条件的B样条协方差函数估计量,此估计量满足正定性.然后,通过研究带约束条件的B样条与无约束条件的B样条协方差函数估计量的关系,建立带约束条件的B样条协方差函数估计量的渐近性质.通过数据仿真模拟实验发现,本文所提出的估计量具有优异的正定性和平均均方误差(average mean squared error,AMSE)表现.最后,本文给出加拿大气温数据和澳大利亚电力需求数据中的正定协方差函数估计量.

Covariance functions play a key role in describing signal time-domain characteristics and spatial correlation structures.In recent years,the nonparametric estimation of covariance functions has been studied extensively.However,the positive definiteness cannot be guaranteed for most nonparametric covariance function estimations.In order to overcome the difficulties,in the context of stationary random processes,this paper first studies the asymptotic properties of the unconstrained B-spline covariance function estimator,based on which,the asymptotic property of the constrained B-spline covariance estimator is established.Numerical results of the simulation experiments indicate that the proposed constrained B-spline estimator has an excellent performance of the positive definiteness and AMSE(average mean squared error).Finally,the proposed positive definite covariance estimator is computed for the Canadian temperature data and the Australian electricity demands data.