The Effect of a Transformation Group on the Quality Indicator of a Linear Regression Model
УДК 514.172; 519.654
Abstract
The construction of functional dependencies between the observed phenomena is an important area of modern applied mathematics. The basis of such constructions is often a statistical data array. The adequacy of the models obtained directly depends on the quality of these data. In general, one has to choose one of the possible models, based on a certain indicator. However, the resulting samples may be in some sense identical, but the models constructed will be different.
This paper discusses one of the methods for constructing linear regression — the method of least squares. The problem of changing the quality functional of a regression model under the orthogonal transformation of the initial data set is studied. A geometric interpretation of the regression model itself and its functional quality, as well as the statistical indicator of the relationship between variables — the correlation coefficient, is given. Formulas are shown explicitly showing the relationship between the functionals of quality during rotation of a set relative to one of the axes of coordinates in two- and threedimensional spaces. Based on the formulas obtained, an algorithm is presented that allows one to obtain the value of the quality functional with any proper movement of n-dimensional space.
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References
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