Change-Point estimation using shape-restricted regression splines - Liao, Xiyue, author; Meyer, Mary C., advisor; Breidt, F. Jay, committee member; Homrighausen, Darren, committee member; Belfiori, Elisa, committee member (Colorado State University. Libraries, 2016)
Change-Point estimation is in need in fields like climate change, signal processing, economics, dose-response analysis etc, but it has not yet been fully discussed. We consider estimating a regression function ƒm and a change-point m, where m is a mode, an inflection point, or a jump point. Linear inequality constraints are used with spline regression functions to estimate m and ƒm simultaneously using profile methods. For a given m, the maximum-likelihood estimate of ƒm is found using constrained regression methods, then the set of possible change-points is searched to find the ˆm that maximizes the likelihood. Convergence rates are obtained for each type of change-point estimator, and we show an oracle property, that the convergence rate of the regression function estimator is as if m were known. Parametrically modeled covariates are easily incorporated in the model. Simulations show that for small and moderate sample sizes, these methods compare well to existing methods. The scenario when the random error is from a stationary autoregressive process is also presented. Under such a scenario, the change-point and parameters of the stationary autoregressive process, such as autoregressive coefficients and the model variance, are estimated together via Cochran-Orcutt-type iterations. Simulations are conducted and it is shown that the change-point estimator performs well in terms of choosing the right order of the autoregressive process. Penalized spline-based regression is also discussed as an extension. Given a large number of knots and a penalty parameter which controls the effective degrees of freedom of a shape-restricted model, penalized methods give smoother fits while balance between under- and over-fitting. A bootstrap confidence interval for a change-point is established. By generating random change-points from a curve on the unit interval, we compute the coverage rate of the bootstrap confidence interval using penalized estimators, which shows advantages such as robustness over competitors. The methods are available in the R package ShapeChange on the Comprehensive R Archival Network (CRAN). Moreover, we discuss the shape selection problem when there are more than one possible shapes for a given data set. A project with the Forest Inventory & Analysis (FIA) scientists is included as an example. In this project, we apply shape-restricted spline-based estimators, among which the one-jump and double-jump estimators are emphasized, to time-series Landsat imagery for the purpose of modeling, mapping, and monitoring annual forest disturbance dynamics. For each pixel and spectral band or index of choice in temporal Landsat data, our method delivers a smoothed rendition of the trajectory constrained to behave in an ecologically sensible manner, reflecting one of seven possible “shapes”. Routines to realize the methodology are built in the R package ShapeSelectForest on CRAN, and techniques in this package are being applied for forest disturbance and attribute mapping across the conterminous U.S.. The Landsat community will implement techniques in this package on the Google Earth Engine in 2016. Finally, we consider the change-point estimation with generalized linear models. Such work can be applied to dose-response analysis, when the effect of a drug increases as the dose increases to a saturation point, after which the effect starts decreasing.