Adaptive basis expansion via \ell _1 trend filtering

We propose a new approach for nonlinear regression modeling by employing basis expansion for the case where the underlying regression function has inhomogeneous smoothness. In this case, conventional nonlinear regression models tend to be over- or underfitting, where the function is more or less smoother, respectively. First, the underlying regression function is roughly approximated with a locally linear function using an \(\ell _1\) penalized method, where this procedure is executed by extending an algorithm for the fused lasso signal approximator. We then extend the fused lasso signal approximator and develop an algorithm. Next, the residuals between the locally linear function and the data are used to adaptively prepare the basis functions. Finally, we construct a nonlinear regression model with these basis functions along with the technique of a regularization method. To select the optimal values of the tuning parameters for the regularization method, we provide an explicit form of the generalized information criterion. The validity of our proposed method is then demonstrated through several numerical examples.