Mathematical Modeling Using Multiple and Fully-Connected Linear Regressions

Abstract

This article is devoted to the synthesis of traditional multiple linear regression models with fully connected linear regression models. It has been noted that these models, in a certain sense, complement each other — the disadvantages of multiple regression are compensated by the fully connected one, and the disadvantages of the fully connected one are compensated by the multiple one. Multiple regression estimates with partial multicollinearity of factors are unstable, and with complete multicollinearity they do not exist at all. There are no obstacles to the use of fully connected regression in such conditions, and its estimates, on the contrary, do not exist in the complete absence of linear dependence between factors. The estimated fully connected regression is an equation of a line in space, as opposed to a multiple regression, which is a hyperplane in space, so you can find estimates of a fully connected regression with only two observations at your disposal. An algorithm for estimating fully connected regressions using the maximum likelihood method is considered. Based on a sample size of 21, a multiple and fully connected linear regression of passenger rail traffic in the Irkutsk region was constructed, containing 23 input variables. During the construction process, it was possible to cope with multicollinearity and ensure protection of all input variables in the model. The constructed multiple and fully connected regression is adequate and fully satisfies the substantive meaning of the problem being solved, therefore its interpretation is given. Based on the results of the study, we can conclude that a combination of multiple and fully connected regressions can be very effective in solving data analysis problems.