一种新的线性回归模型及其应用示例

回归分析是统计分析中常用的方法之一。传统的回归模型不具备全域分析能力，而变量场之间的关系多采用SVD（Singular Value Decomposition）进行分析，与传统的回归分析有所脱节。更为广义的线性回归模型是传统线性回归模型的延拓，在标量情况下，该模型可转化为传统线性回归模型。该模型的基本特征包含乘法不可互易性、等价于传统线性回归（因子项为标量时）、可分析性、延拓性、降维特征及容错性等。该模型解决了传统的线性回归模型不具备全域分析能力及模型表达能力受限于模型维数的现实问题。本文采用了NCEP（National Centers for Environmental Prediction）降水、高度场、风场月平均资料及国家气候中心西太平洋副热带高压指数资料，利用该模型和传统回归方案进行对比分析，分析结果表明，该模型具有一定的实用参考价值。

A New Linear Regression Model and Its Application

Regression analysis is one of the commonly used methods in statistical analysis. However, traditional regression models have less ability for global analysis, and the relationship between variables is often analyzed by methods like the SVD (Singular Value Decomposition), which lack connections with traditional regression analysis. A MGLRM (more generalized linear regression model) is a continuation of traditional linear regression model. In the case that both the predictand and the predictors are scalars, the MGLRM can be transformed into the traditional linear regression model. The MGLRM''s basic features include non-commutative multiplication, equivalence to traditional linear regression as predictors in the model are scalars, analysis, extension, dimension-reduction, and robustness, etc. The MGLRM solves problems in traditional linear regression models that have less ability for global analysis and limited expressive ability due to the dimensions of the regression equation. In this paper, the MGLRM and the traditional regression model are applied for statistical analysis of monthly average data of precipitation, height, and wind fields from the NCEP (National Centers for Environmental Prediction) and the western Pacific subtropical high index data from the National Climate Center. Comparison of the results show that the MGLRM has practical implications.