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Monotone Splines Lasso

We consider the problems of variable selection and estimation in nonparametric additive regression models for high-dimensional data. In recent years, several methods have been proposed to model nonlinear relationships when the number of covariates exceeds the number of observations by using spline basis functions and group penalties. Nonlinear {\it monotone} effects on the response play a central role in many situations, in particular in medicine and biology. We construct the monotone splines lasso (MS-lasso) to select variables and estimate effects using monotone spline (I-splines). The additive components in the model are represented by their I-spline basis function expansion and the component selection becomes that of selecting the groups of coefficients in the I-spline basis function expansion. We use a recent procedure, called cooperative lasso, to select sign-coherent groups, that is selecting the groups with either exclusively non-negative or non-positive coefficients. This leads to the selection of important covariates that have nonlinear monotone increasing or decreasing effect on the response. We also introduce an adaptive version of the MS-lasso which reduces both the bi