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

在最小二乘法对观测数据进行多项式回归的过程中，“内和谐”是一个重要但未被注意到的新概念。当数据量很大，并且我们对数据在某个兴趣点附近的行为感兴趣时，我们通常引入权值或窗口函数。然后，随着兴趣点的移动，多项式的系数也随之变化。内调和性是方法本身的性质（即，独立于数据），它确保系数相对于感兴趣点的微分应该与高1度的系数一致。虽然这个性质很自然地被直观地认为是成立的。还没有研究在什么情况下它是有效的。本研究表明，可以在一定距离内选择权函数和观测点的分布，从而使最小二乘法具有内调和性。简而言之，从理论上建立了内调和的充分条件，即，如果我们取高斯型的权函数，并且观测点有规则地排列（即在每个坐标方向上间隔相等），则内调和在感兴趣点的某些特定位置成立。还证明，在这些充分条件下，内调和几乎在任何地方（边界附近区域除外）在实际数值上都是有效的。理论结果得到了若干数值算例的支持。

"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.