Román Salmerón

Collinearity: revisiting the variance inflation factor in ridge regression

Ridge regression has been widely applied to estimate under collinearity by defining a class of estimators that are dependent on the parameter k. The variance inflation factor (VIF) is applied to detect the presence of collinearity and also as an objective method to obtain the value of k in ridge regression. Contrarily to the definition of the VIF, the expressions traditionally applied in ridge regression do not necessarily lead to values of VIFs equal to or greater than 1. This work presents an alternative expression to calculate the VIF in ridge regression that satisfies the aforementioned condition and also presents other interesting properties.

Locating hyperplanes to fitting set of points: A general framework

This paper presents a family of new methods for locating/fitting hyperplanes with respect to a given set of points. We introduce a general framework for a family of aggregation criteria of different distance-based errors. The most popular methods found in the specialized literature can be cast within this family as particular choices of the errors and the aggregation criteria. Mathematical programming formulations for these methods are stated and some interesting cases are analyzed. It is also proposed a new goodness of fitting index which extends the classical coefficient of determination. A series of illustrative examples and extensive computational experiments implemented in R are provided to show the performances of some of the proposed methods.Abstract This paper presents a family of methods for locating/fitting hyperplanes with respect to a given set of points. We introduce a general framework for a family of aggregation criteria, based on ordered weighted operators, of different distance-based errors. The most popular methods found in the specialized literature, namely least sum of squares, least absolute deviation, least quantile of squares or least trimmed sum of squares among many others, can be cast within this family as particular choices of the errors and the aggregation criteria. Unified mathematical programming formulations for these methods are provided and some interesting cases are analyzed. The most general setting give rise to mixed integer nonlinear programming problems. For those situations we present inner and outer linear approximations to assess tractable solution procedures. It is also proposed a new goodness of fitting index which extends the classical coefficient of determination and allows one to compare different fitting hyperplanes. A series of illustrative examples and extensive computational experiments implemented in R are provided to show the applicability of the proposed methods.

The raise estimator estimation, inference, and properties

ABSTRACT Several methods using different approaches have been developed to remedy the consequences of collinearity. To the best of our knowledge, only the raise estimator proposed by García et al. (2010) deals with this problem from a geometric perspective. This article fully develops the raise estimator for a model with two standardized explanatory variables. Inference in the raise estimator is examined, showing that it can be obtained from ordinary least squares methodology. In addition, contrary to what happens in ridge regression, the raise estimator maintains the coefficient of determination value constant. The expression of the variance inflation factor for the raise estimator is also presented. Finally, a comparative study of the raise and ridge estimators is carried out using an example.

Variance Inflation Factor and Condition Number in multiple linear regression

ABSTRACT The Variance Inflation Factor and the Condition Number are measures traditionally applied to detect the presence of collinearity in a multiple linear model. This paper presents the relation and the difference between both measures from theoretical and empirical perspectives by using Monte Carlo simulations and taking special interest in the computational techniques.

A Multicriteria Selection System Based on Player Performance: Case Study—The Spanish ACB Basketball League

In this paper, we describe an approach to rank sport players based on their efficiency. Although is extremely useful to analyze the performance of team games there is no unanimity on the use of a single index to perform such a ranking. We propose a method to summarize the huge amount of information collected at different aspects of a sport team which is almost daily publicly available. The tool will allow agents involved in a player’s negotiation to show the strengths (and weaknesses) of the player with respect to other players. The approach is based on applying a multicriteria outranking methodology using as alternatives the potential players and as criteria different efficiency indices. A novel automatic parameter tuning approach is detailed that will allow coaches and sports managers to design templates and sports strategies that improve the efficiency of their teams. We report the results performed over the available information on the ACB Basketball League, and we show how it can be easily implemented and interpreted in practice by decision-makers non familiar with the mathematical side of the methodology.