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

“内调和”（希腊语，意为“内部一致性”）是用最小二乘法对观测数据进行多项式回归时一个重要但未引起注意的新概念。当数据量很大并且我们对感兴趣点周围的数据行为感兴趣时，我们通常会引入权重或窗口函数。然后，随着兴趣点的移动，多项式的系数相应地改变。内调和是方法本身的属性（即，独立于数据），这确保了系数相对于感兴趣点的微分应该与高1的次数的系数一致。虽然人们自然地期望这一性质在直觉上成立，但还没有研究它在什么情况下成立。本研究阐明了权函数和观测点分布的选择是可能的，从而使最小二乘法具有内调和性质。简而言之，理论上已经建立了内调和的充分条件，即如果我们采用高斯型的权重函数并且如果观测点是规则排列的（即，它还表明，在这些充分条件下，内调和实际上-数值上几乎在任何地方都有效（除了边界附近的区域）。理论结果得到了一些数值例子的支持。

"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.