fitting(redirected from fittings)
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in statistics, a method by means of which we obtain an analytic and graphic expression of the statistical regularity on which a given empirical series of statistical data is based. By means of fitting, the broken line of the steps of an empirical series is replaced with a smooth, “fitted” curve (in particular cases, with a straight line) and the equation of this curve is computed. During fitting, the following three problems are solved in sequence: the type of equation (shape of the smooth curve) is selected, the parameters (coefficients) of this equation are computed, and the levels (ordinates) of the “theoretical” statistical series obtained are computed (on the basis of the equation) or measured (by the graph of the curve). The type of equation and, correspondingly, the shape of the smooth curve are selected on the basis of general in-formation about (or frequently from practical experience with) the essence of the phenomenon, the regularities of its structure and development, the relationships among its attributes, and so forth (so-called analytic fitting). Where such advance information is not available, the type of equation (shape of the curve) can often be suggested by the graphic shape of the broken line that expresses the given empirical series.
In socioeconomic statistics, fitting is used in three typical cases: (1) fitting a series of distributions, (2) fitting broken lines of regression, and (3) fitting series in a dynamic process.
The purpose of fitting series of distributions is to give quantitative and graphic expressions for the nature of the regularity of the distribution of units of the aggregate on the basis of an assigned attribute (for example, their normal distribution and distribution according to Poisson’s law). In this we preserve the equality of certain primary numerical characteristics of the given empirical series and the theoretical series obtained: the average magnitude of the attribute, the mean quadratic deviation, and the total number of units in the aggregate. We use a particular goodness-of-fit test to establish the degree to which the levels (ordinates) of the theoretical series obtained correspond in aggregate to the empirical steps. In some special cases, for example when fitting population distribution by age given in the census, specially developed procedures and formulas are used to eliminate the well-known “accumulation of ages” ending in 0 and 5. Fitting distributions always assumes the availability of a sufficiently numerous empirical series of data.
Fitting broken lines of regression is done when studying the relationships of attributes in order to obtain a smooth line of regression and a regression (correlation) equation that ex-presses the dependence of the average values of one attribute on values of others; for example, yx = a + bx; yx,z = a + bx + cz.
We resort to fitting time series in dynamic processes to obtain an equation (and a smooth line) expressing the developmental trend of a process in time t; for example, y = a + bt, y = a + bt + ct2. In the last two cases of fitting, the coefficients a, b, c, … of the unknown equation are usually computed by the method of least squares. The fitting of statistical time series should not be confused with smoothing statistical series.
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Ezhov, A. I. Vyravnivanie i vychislenie riadov raspredelenii. Moscow, 1961.
Khotimskii, V. I. Vyravnivanie statisticheskikh riadov po metodu naimen’shikh kvadratov (sposob Chebysheva). Moscow-Leningrad, 1925. Second ed., Moscow, 1959.
Chetverikov, N. S. “O tekhnike vychisleniia parabolicheskikh krivykh.” In Voprosy kon“iunktury, vol. 2. Moscow, 1926. Reprinted in Chetverikov’s book Statisticheskie i stokhasticheskie issledovaniia. Moscow, 1963. Pages 190-210.
Iastremskii, B. S. Nekotorye voprosy matematicheskoi statistiki. Moscow, 1961. Chapter 2.
Obukhov, V. M. “K voprosu o nakhozhdenii uravneniia regressii, udoletvoriaiushchego dannomu empiricheskomu riadu.” Trudy TsSU, vol. 16, issue II. Moscow, 1923.
F. D. LIVSHITS