可线性化的非线性回归的有关问题与几种回归方法的比较

Issues in Linearization Nonlinear Regression and Comparison of Several Regression Methods

对于幂函数、指数函数等类型的非线性回归,只当采用乘积随机误差时才能够线性化.导出了采用乘积随机误差时幂函数型因变量的数学期望的表达式,表明因变量的估计值并非是其数学期望的估值;导出了非线性回归与其线性化回归二者的残差平方和之间的关系式,表明当对非线性回归的因变量做了变换时,传统方法所求非线性回归系数不满足该因变量的残差平方和为最小.故幂函数、指数函数等类型的回归计算,应采用非线性回归方法求解.实例进一步表明,非线性回归方法高斯-牛顿法和麦夸尔特法均显著优于传统方法,且借助MATLAB软件易于实现.

For the non-linear regression of power function,exponential function and so on,can be linearized only when using the product random error.This paper,the formula of mathematical expectation of dependent variable of power function type was deduced when using the product random error.The formula indicates estimated value of the dependent variable is not its estimated value of mathematical expectation.The relationship between the two residual sum of squares nonlinear regression and its linear regression was deduced.The formula indicates that,the calculated nonlinear regression coefficient using traditional method does not meet residual sum of squares of the dependent variable to be minimum when dependent variable of nonlinear regression was changed.Therefore,regression of power function,exponential function and so on,should be solved by nonlinear regression method.The example proceed to show that Nonlinear regression method Guass-Newton method and Marquardt method is notably better than the traditional method,and the methods are easy to achieve using MATLAB software.