VII. Reduction of error by linear compounding
| Autor: | W. F. Sheppard |
|---|---|
| Rok vydání: | 1921 |
| Předmět: | |
| Zdroj: | Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character. 221:199-237 |
| ISSN: | 2053-9258 0264-3952 |
| DOI: | 10.1098/rsta.1921.0007 |
| Popis: | 1. Introductory .—This paper is a development of two earlier papers, which for brevity I call “Reduction” and “Fitting” respectively. The paper immediately preceding “Fitting” is referred to as “Factorial Moments.” These earlier papers deal with two problems, which are closely connected and have the same solution. For both of them, the data are a set of quantities , u 0 , u 1 , u 2 , ... of the same kind, which we regard as representing certain true values U 0 , U 1 , U 2 , ..., with errors e 0 , e 1 , e 2 , ..., so that u r = U r + e r . These errors may be independent or may be correlated in any way. The first problem is based on the assumption (which defines the class of cases we are dealing with) that the sequence of U 's is fairly regular, so that differences after those of a certain order, which we will call j , are negligible. This being so, we may alter any u , or any linear compound of the u 's, such as an interpolation-formula, by adding to it any linear compound of the negligible differences. (I use the term “linear compound” in preference to “linear function,” since there is no consideration of functionality.) The problem is to find the value of the resulting sum when, by suitable choice of the coefficients in the added portion, the mean square of error of the sum is a minimum. This is the problem of “reduction of error.” For the second problem it is assumed that U r is a polynomial in r of degree j , and the problem is to find the coefficients in this polynomial by the method of least squares. This is the problem of “fitting.” |
| Databáze: | OpenAIRE |
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