Multi-criteria method of least absolute deviations in regression analysis

Abstract

The paper provides a brief review of publications on methods for constructing regression models using vector loss functions. In particular, approaches with partial loss functions corresponding to the methods of least squares, absolute deviations (МLAD), and anti-robust estimation are considered. The problem of identifying the parameters of a linear regression model is solved using the multicriteria method of least modules, in which the loss function is a vector, each component of which (i.e., a partial loss function) is specified on a pre-fixed subsample of the initial data sample. A computational algorithm for searching for parameter vectors corresponding to the Pareto vertices of a polyhedron (simplex) given by corresponding MIS constraints with a vector objective function and consisting of solving a series of linear programming problems is proposed. In this case, the entire set of Pareto parameter estimates can be formed as a union of the Pareto faces of a polyhedron, which are convex combinations of its Pareto vertices. The proposed approach is based on the fundamental work of L. Yu. and M. Zelenу “The set of all non-dominated solutions in the linear case and the multicriteria simplex method”. Multi-criteria method of least absolute deviations was applied to build a linear regression model of the volume of loading of the main types of cargo by rail. The volumes of transportation by road, sea, pipeline and inland waterways were used as independent variables. The original sample was divided into two non-overlapping sub-samples. A total of 19 Pareto vectors of parameter estimates were obtained. The change in the values of all parameters lacks not only monotonicity, but also any other pronounced regularity, while only one parameter does not change its positive sign. The scatter of all parameters reaches high values. The presented multi-criteria method for estimating the parameters of a linear regression model makes it possible to reflect the trends that appear in certain sections of the processed data sample and increases variability in the study of the patterns of functioning of complex objects using regression analysis methods.