气候变化研究中复Hermite 矩阵解的统计和物理意义

复经验正交分析在气象学和海洋学研究领域有着广泛的应用,其核心部分就是求解复Hermite矩阵的特征值、复特征向量和复主成分.但是,在以往的研究中,并没有讨论复Hermite矩阵解的统计和物理意义.研究证明,复Hermite矩阵的特征值反映了方差贡献、异常能量,复特征向量本身并没有明确的物理意义,而有明确的统计意义.但是复主成分有清晰的物理意义.并且其实部和虚部不是独立的.因此,在研究二维矢量场变化的优势模态时,只能使用一维线性回归分析方法.

Statistical and Physical Significances of Resolution of Complex Hermite Matrix in Climate Change Studies

The complex Empirical Orthogonal Function (CEOF) analysis has been extensively used in the meteorological and oceanic fields, and the key part of this method is to extract eigenvalues, eigenvectors, and complex principal components from a Hermite matrix. In previous studies, however, no one explores statistical and physical significances of the resolution of a Hermite Matrix. It is demonstrated that the eigenvalues of a complex Hermite Matrix represent variance contribution or anomalous energy, and eigenvectors have clear statistical significances and no apparent physical significances. In contrast, a complex principal component has a clear physical significance, and their real and imaginary parts are related to each other. Thus, the dimensional linear regression method can be used to extract predominant modes of vector variability.