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Proposal of the Concept of Endoharmony in the Method of Least Squares and Theoretical/Practical Study of the Concept

最小二乘法中内和谐概念的提出及其理论与实践研究

基本信息

批准号：
13650069
负责人：
IRI Masao
金额：
$ 0.32万
依托单位：
CHUO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13650069/
关键词：
Least Squares Polynomial Regression Differentiation with respect to the Point of Interest Endoharmony Hermite Polynomials Collocation Orthogouality Hemite多項式

项目摘要

"Endoharmony" (the Greek standing for "internal consistency") is an important but unno-ticed new concept in the polynomial regression of observational data with the method of least squares. When the data-size is large and we are interested in the behaviour of the data around a point of interest, we usually introduce the weight or window function. Then, as the point of interest moves, the coefficients of polynomials change accordingly. The endoharmony is the property of the method itself (i.e., independent of the data) which ensures that the differentiation of a coefficient with respect to the point of interest should coincide with the coefficient of the degree higher by one. Although this property is naturally expected to hold intuitively.It has not been studied under what circumstances it holds valid. This research clarifies that it is possible to choose the weight function and the distribution of observational points in such away that the resulting least-square method may enjoy the property of endoharmony. In brief, the sufficient condtions for endoharmony has been theoretically established, namely, endoharmony holds at some specific locations of the point of interest if we take a weight function of the Gaussian type and if the observational points are arranged regularly (i.e., equally spaced in each coordinate direction).It has also been shown that, under those sufficient conditions, endoharmony practically-numerically holds valid almost everywhere (except the region near the boundary). The theoretical results are backed up by a number of numerical examples.

在用最小二乘法进行观测数据的多项式回归中，“内协调”(希腊语中“内部一致性”的意思)是一个重要而又不确定的新概念。当数据量很大，并且我们对兴趣点附近的数据行为感兴趣时，我们通常引入权重或窗口函数。然后，随着兴趣点的移动，多项式的系数也随之改变。不协调是方法本身的属性(即，独立于数据)，它确保相对于兴趣点的系数的差值应与较高程度的系数一致一。虽然这一性质自然被认为是直观的，但在什么情况下它是有效的还没有被研究过。这项研究表明，可以在一定距离内选择权函数和观测点的分布，从而使所得到的最小二乘法具有不协调性。简而言之，从理论上建立了不协调的充分条件，即如果我们取高斯型的权函数，并且观测点按规则排列(即在每个坐标方向上均匀分布)，则在兴趣点的某些特定位置上不协调成立。在这些充分条件下，几乎所有地方(除了边界附近的区域)上的不协调在实践-数值上都成立。理论结果得到了大量数值算例的支持。