Regression analysis. The least squares smoothing One of often meeting problems at processing results of experiment is selection of the formula for an establishment of functional dependence between experimental data, the so-called regression equation . Before to start selection of the formula, it is expedient to put empirical data on the graph and approximately, by hand to lead through the received points the most plausible curve. Very often the general view of a curve happens is known from other reasons that simplifies a problem, reducing it to search of numerical coefficients of known functional dependence of a general view (for example, linear, square-law, logarithmic, etc.). Thus those empirical data that most likely contain the greatest errors become often visible. Except for the received experimental points the essential moment at carrying out of a curve are reasons of the general character: as the curve near to zero behaves, whether it crosses coordinate axes, whether touch es them, whether has asymptotes, etc. After this preliminary work is done, selection of the formula – the equations of regress begins actually. In solution of the problems connected with search of the equations of regress is engaged regression analysis , and one of it of algorithms most widely put into practice is the method of the least squares [3, 5, 6, 7]. Generally the method of the least squares problem is formulated as follows. Let required functional dependence of value y from value х is expressed by the formula:
where
It is supposed, that values of argument
х
are established precisely, and corresponding values of function
y
are defined in experiment with some error. If measurements were made without mistakes for definition of parameters
So, for receiving of estimations of parameters
The method of the least squares
consists that
estimations of parameters
reaches the least value.
It is necessary for definition of these estimations to differentiate (2) by all estimations
The system of the equations (3) refers to as system of Gauss’ normal equations . Here for brevity records the following designations of the sums are accepted:
It is necessary to note, that the system of the equations (3) sometimes appears ill-conditioned , i.e. its solutions are rather sensitive to the slightest changes in results of measurements. Special methods now are developed for ill-conditioned systems, for example, regularization methods.
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