| Research Abstract |
"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.
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