Premium
Geometry and Degrees of Freedom of Linearly Constrained Generalized Lasso
Author(s) -
Zeng Peng,
Hu Qinqin,
Li Xiaoyu
Publication year - 2017
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12288
Subject(s) - mathematics , degrees of freedom (physics and chemistry) , rank (graph theory) , lasso (programming language) , design matrix , projection (relational algebra) , matrix (chemical analysis) , polytope , least squares function approximation , rotation (mathematics) , selection (genetic algorithm) , linear model , combinatorics , algorithm , statistics , geometry , estimator , artificial intelligence , computer science , physics , materials science , quantum mechanics , world wide web , composite material
Abstract The least squares fit in a linear regression is always unique even when the design matrix has rank deficiency. In this paper, we extend this classic result to linearly constrained generalized lasso. It is shown that under a mild condition, the fit can be represented as a projection onto a polytope and, hence, is unique no matter whether design matrix X has full column rank or not. Furthermore, a formula for the degrees of freedom is derived to characterize the effective number of parameters. It directly yields an unbiased estimate of degrees of freedom, which can be incorporated in an information criterion for model selection.