Penalized Regression Methods in Time Series and Functional Data Analysis

  • Author / Creator
    Zhang, Li
  • In this thesis, we study penalized methods in time series and functional data analysis. In the first part, we introduce regularized periodograms for spectral analysis of unevenly spaced time series. The regularized periodograms, called regularized least squares periodogram and regularized quantile periodogram, are derived from trigonometric least squares and quantile regression with Group Lasso penalty. A simple model provides a theoretical justification for the use of regularized least squares periodogram as a tool for detecting hidden frequency in the time series. We give a data-dependent procedure for selection of the regularization parameter. An extensive simulation studies are conducted to examine whether our periodogram functions have the power to detect frequencies from the unevenly spaced time series with big gaps and outliers. In the second part, we propose a penalized likelihood approach for the estimation of the spectral density of a stationary time series. The approach involves $L_1$ penalties, which were shown to be an attractive regularization device for nonparametric regression, image reconstruction, and model selection. We explore the use of penalties based on the total variation of the estimated derivatives of spectral density. An asymptotic analysis of the integrated absolute error between the estimator and the true spectral density is presented and gives a consistency result under certain regularity conditions. We also investigate the convergence of the total variation penalized Whittle likelihood estimator to the true spectral density via simulations. In the third part, we treat discrete time series data have as functional covariates in functional regression models with a scalar response. We develop an efficient wavelet-based regularized linear quantile regression framework for coefficient estimation in the functional linear model. The coefficient functions are sparsely represented in the wavelet domain, and we suppose that only a few of them are linked to the response. Subsequently, we derive an estimator for the slope functions through composite quantile regression and sparse Group Lasso penalty. We also establish the rate of convergence of the estimator under mild conditions, showing that this rate is dependent on both the sample size and the number of observed discrete points for the predictive functions. Finally, we conduct a simulation study to figure whether our method can identify relevant functional variables. We illustrate the empirical performance of all the proposed methods on several real data examples.

  • Subjects / Keywords
  • Graduation date
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
  • Language
  • Institution
    University of Alberta
  • Degree level
  • Department
    • Department of Mathematical and Statistical Sciences
  • Specialization
    • Statistics
  • Supervisor / co-supervisor and their department(s)
    • Mizera, Ivan (Department of Mathematical and Statistical Sciences)
  • Examining committee members and their departments
    • Kong, Linglong (Department of Mathematical and Statistical Sciences)
    • Wiens, Douglas (Department of Mathematical and Statistical Sciences)
    • Anton, Cristina (Mathematics and Statistics, MacEwan University )
    • Li, Guodong (Department of Statistics and Actuarial Science, University of Hong Kong )
    • Mizera, Ivan (Department of Mathematical and Statistical Sciences)