最小二乘直线拟合方法及其在同位素地质年代学中的应用研究

Methods for Least Squares Fitting of a Straight Line and Their Application in Geochronology

最小二乘法(least squares method)是直线拟合常用的方法,在自然科学和社会科学内被广泛应用,尤其在同位素地质年代学领域更是必不可少。仅考虑x或y误差的普通最小二乘法(normal/ordinary least squares)广为人知,但事实上最小二乘法并非如此简单,尤其是在同时考虑x、y误差(乃至误差相关)并采用加权处理时,其数学处理方法会变得十分复杂,而此时普通最小二乘法显得极不合理。本文在前人研究的基础上,结合同位素地质年龄计算的需要,对直线拟合的最小二乘法进行了系统地总结研究,详细介绍了普通最小二乘法、加权普通最小二乘法(normal/ordinary weighting least squares)、标准加权最小二乘法(standard weighting model)、标准独立加权最小二乘法(standard independent weighting model)、独立加权最小二乘法(independent weighting model)及误差相关最小二乘法(error correlated independent weighting model)的数学原理及相关变量的计算过程。在此基础上,进一步阐述了MSWD(mean squared weighted deviation)这个同位素地质年代学中经常使用的参数的数学意义,以及MSWD对计算结果的评判意义。准确理解这些数学方法,对于我们合理选择同位素地质年龄相关参数的计算方法,科学评价直线拟合结果并探讨其地质意义至关重要。我们的研究同时有助于拓展和完善数学领域最小二乘法的基本理论,并可用于其他领域相似的研究之中。

The methods for least squares fitting of a straight line are applied widely in natural sciences and social sciences, and geology is no exception, especially in geochronology. It is the normal/ordinary least squares that is well known by much of the population, where errors in only one coordinate are considered. In fact, The least square linear regressions is much more complex, especially when errors in both coordinates even the error correlations are considered. Based on the previous research, an overview of the methods for least squares fitting and its application in geochronology are provided here. These methods includes normal/ordinary least squares, normal/ ordinary weighting least squares, standard weighting model, standard independent weighting model, independent weighting model, and error correlated independent weighting model. What＇ s more, we give a brief study of the parameter of MSWD, which is commonly used but makes many people confused in geochronological study. It is very important for us to know all of these algorithm and their underlying assumption exactly, which will of course benefit us a lot in choosing the best approach to calculate the parameters we need and assessing the results in geochronology. Our study will also help improve the basic methods for least squares fitting in mathematics and could be used in other areas.