人口与资源预测中Logistic模型承载量参数的自回归估计  

Carrying Capacity Estimation of Logistic Model in Population and Resources Prediction by Nonlinear Autoregression

作　　者：陈彦光[1] 

机构地区：[1]北京大学城市与环境学院,北京100871

出　　处：《自然资源学报》2009年第6期1105-1114,共10页Journal of Natural Resources

基　　金：国家自然科学基金资助项目(40771061)

摘　　要：在人口、资源、环境和生态诸多系统的预测中,Logistic函数都是一个非常重要的模型。但是,确定Logistic模型的参数又是一件困难的事情。论文基于最小二乘算法和多元回归分析发展一种参数估值方法:首先对模型求导,化为微分方程;然后将微分方程离散化,得到差分方程;将差分方程转换为二元线性回归模型,利用自回归分析确定模型参数;最后将回归系数转换为Logistic模型参数。根据对称性思想提出了Logistic模型参数估计效果检测的对称性指数。大量的试验表明,只要研究对象具有S形曲线的增长特征,这种方法就行之有效。借助城市人口和城市化水平预测,给出了两个计算实例,由此说明上述方法的具体应用步骤。Logistic model is very important for the prediction in population, resources, environment, and ecology. However, it is difficult for us to evaluate the parameter indicative of carrying capacity. In this paper, a new method is advanced to estimate the values of the parameters of Logistic model by using bivariate nonlinear autoregression （BNA）. The principal purpose is to solve the problem of evaluating the carrying capacity and related parameters through a simple approach. Suppose that a Logistic function is expressed as y = y^＊/（ 1 ＋ aexp（-bt） ）. The main steps of the computation method can be summarized as follows. （ 1 ） Taking derivative of the Logistic function with respect to time give a differential equation in the form dy/dt = by （ 1 - y/y^＊ ）. （2） Transforming the differential equation into a difference equation △t/△t = byt-1 （ 1 - yt-1 /y^＊, which can be rewritten as △yt =b△tyt-1 -b△ty^2t-1/y^＊ or yt = （1 ＋b△t）yt-1 -b△ty^2t-1/y^＊. Generally speaking, △t = 1, then we have △yt = byt-1 - by^2t-1/y^＊ or yt = （ 1 ＋ b） yt-1-by^2t-1/y^＊.（3） Taking yt-1 and y^2t- 1 as two independent variables, while taking Ay, or y, as a dependent variable, we can make a multivariable linear regression analysis, which is in essence a bivariate nonlinear autoregression process. （4） A least squares calculation can give two regression coefficients such as b△t and b△t/y^＊ , or 1 ＋ b△t and b△t/y^＊ , from which b and y^＊ can be evaluated easily. （5） Substituting y^＊ value into the Logistic function, we can estimate a and b values by regression analysis based on the least squares computation. In this fashion, a Logistic model can be built step by step. The method is applied to the population based on household register in the city of Wuhan （1986 -2002） , illustrating how to use BNA. The carrying capacity parameter value of Wuhan＇ s population is estimated as 9390532, and the Logistic model is built in the form Pt = 9390532/[ 1 �

关 键 词：LOGISTIC模型 承载量 人口预测 城市化水平 自回归 对称 

分 类 号：N945.12[自然科学总论—系统科学] C921[社会学—人口学]