Abstract
We consider a function estimation method with change point detection using truncated power spline basis and elastic-net-type L1-norm penalty. The L1-norm penalty controls the jump detection and smoothness depending on the value of the parameter. In terms of the proposed estimators, we introduce two computational algorithms for the Lagrangian dual problem (coordinate descent algorithm) and constrained convex optimization problem (an algorithm based on quadratic programming). Subsequently, we investigate the relationship between the two algorithms and compare them. Using both simulation and real data analysis, numerical studies are conducted to validate the performance of the proposed method.
Highlights
A nonparametric function estimation is useful for estimating the actual relationship of data exhibiting nonlinear relationships
The Akaike information criterion (AIC) and cross validation (CV) involve the selection of a model with a relatively large variance and small bias compared with the Bayesian information criterion (BIC)
We develop a nonparametric regression function estimation method with change point detection
Summary
A nonparametric function estimation is useful for estimating the actual relationship of data exhibiting nonlinear relationships. The selection of knots and the detection of change points in regression splines significantly affect the performance of the model. The proposed function estimator is defined by a linear combination of multidegree-based splines to simultaneously provide a change point detection and smoothing. Two computational algorithms are introduced to address the constrained convex optimization problem (algorithm based on QP) and Lagrangian dual problem (CDA); the relationship between them is investigated Numerical studies using both simulated and real datasets are provided to demonstrate the performance of the proposed method. We express the estimator as a linear combination of a polynomial and a truncated power-spline basis for zero and positive integer degrees, respectively. Because the goal is to achieve a smooth function estimation that can detect change point(s), the function space to which f belongs can be specified as follows.
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