Korkas, Karolos
(2014)
Randomised and L1-penalty approaches to segmentation in time series and regression models.
PhD thesis, London School of Economics and Political Science.
Abstract
It is a common approach in statistics to assume that the parameters of a stochastic model change. The simplest model involves parameters than can be exactly or approximately piecewise constant. In such a model, the aim is the posteriori detection of the number and location in time of the changes in the parameters. This thesis develops segmentation methods for non-stationary time series and regression models using randomised methods or methods that involve L1 penalties which force the coefficients in a regression model to be exactly zero. Randomised techniques are not commonly found in nonparametric statistics, whereas L1 methods draw heavily from the variable selection literature. Considering these two categories together, apart from other contributions, enables a comparison between them by pointing out strengths and weaknesses. This is achieved by organising the thesis into three main parts.
First, we propose a new technique for detecting the number and locations of the change-points in the second-order structure of a time series. The core of the segmentation procedure is the Wild Binary Segmentation method (WBS) of Fryzlewicz (2014), a technique which involves a certain randomised mechanism. The advantage of WBS over the standard Binary Segmentation lies in its localisation feature, thanks to which it works in cases where the spacings between change-points are short. Our main change-point detection statistic is the wavelet periodogram which allows a rigorous estimation of the local autocovariance of a piecewise-stationary process. We provide a proof of consistency and examine the performance of the method on simulated and real data sets.
Second, we study the fused lasso estimator which, in its simplest form, deals with the estimation of a piecewise constant function contaminated with Gaussian noise (Friedman et al. (2007)). We show a fast way of implementing the solution path algorithm of Tibshirani and Taylor (2011) and we make a connection between their algorithm and the taut-string method of Davies and Kovac (2001). In addition, a theoretical result and a simulation study indicate that the fused lasso estimator is suboptimal in detecting the location of a change-point.
Finally, we propose a method to estimate regression models in which the coefficients vary with respect to some covariate such as time. In particular, we present a path algorithm based on Tibshirani and Taylor (2011) and the fused lasso method of Tibshirani et al. (2005). Thanks to the adaptability of the fused lasso penalty, our proposed method goes beyond the estimation of piecewise constant models to models where the underlying coefficient function can be piecewise linear, quadratic or cubic. Our simulation studies show that in most cases the method outperforms smoothing splines, a common approach in estimating this class of models.
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