基于变分法的一元非线性回归分析

Analysis of Nonlinear Regression Based on the Variational Method

研究具有统计相关关系的二维连续型随机变量(ξ,η)的非线性回归分析问题:η=f(ξ)+ε,ξ的分布是已知的,ξ与ε独立,ε～N(0,σ2).首先推得(ξ,η)的联合密度φ(x,y),通过对φ(x,y)统计学性质、几何性质研究,将非线性回归问题归结为泛函极值问题,应用变分法,对于ξ服从不同的分布及η满足不同的约束条件,得出了依据(ξ,η)的一组样本值确定回归曲线f(x)的解析解的方法.

The paper studies the analysis of two dimensional continuous random variable nonlinear regression with statistical correlation, that is, η = f（ξ） ＋ ε , the distribution ξ is known and ξ is independent of ε, ε N（0, σ2 ） . The joint density （ξ,η） is φ（x,y） , and based on the study of the statistical properties φ（x,y） , geometrical properties, the nonlinear regression problem is considered to be attributed to functional extremum problem. By the variational method, for the different distxibutions of ξ and the dif- ferent constraint conditions of η , according to a group of sample values of （ξ, η） , the analytical method to determine the regression curvesf（x） is obtained.