Non Asymptotic Analysis Of Learning Long Range Autoregressive Generalized Linear Models For Discrete High Dimensional Data

Author : Hanh Nguyen
Publisher :
ISBN 13 :
Total Pages : 115 pages
Book Rating : 4.:/5 (113 download)

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Download or read book Lasso for Autoregressive and Moving Average Coeffi[ci]ents Via Residuals of Unobservable Time Series written by Hanh Nguyen and published by . This book was released on 2018 with total page 115 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation contains four topics in time series data analysis. First, we propose the oracle model selection for autoregressive time series when the observations are contaminated with trend. An adaptive least absolute shrinkage and selection operator (LASSO) type model selection method is used after the trend is estimated by non-parametric B-splines method. The first step is to estimate the trend by B-splines method and then calculate the detrended residuals. The second step is to use the residuals as if they were observations to optimize an adaptive LASSO type objective function. The oracle properties of such an Adaptive Lasso model selection procedure are established; that is, the proposed method can identify the true model with probability approaching one as the sample size increases, and the asymptotic properties of estimators are not affected by the replacement of observations with detrended residuals. The extensive simulation studies of several constrained and unconstrained autoregressive models also confirm the theoretical results. The method is illustrated by two time series data sets, the annual U.S. tobacco production and annual tree ring width measurements. Second, we generalize our first topic to a more general class of time series using the autoregressive and moving-average (ARMA) model. The ARMA model class is the building block for stationary time series analysis. We adopt the two-step method non-parametric trend estimation with B-spline and model selection and model estimation with the adaptive LASSO. We prove that such model selection and model estimation procedure possesses the oracle properties. Another important objective of this topic is forecasting time series with trend. We approach the forecasting problem by two methods: the empirical method by using the one-step ahead prediction in time series and the bagging method. Our simulation studies show that both methods are efficient with the decreased mean square error when the sample size increases. Simulation studies are adopted to illustrate the asymptotic result of our proposed method for model selection and model estimation with twelve ARMA(p, q) models, in which p an q are in the range from 1 to 15. The method is also illustrated by two time series data sets from the New York State Energy Research and Development Authority (NYSERDA), a public benefit corporation which offers data and analysis to help New Yorkers increase energy efficiency. Third, we propose a new model class, which is motivated by lag effects of covariates on the dependent variable. Our paper aims at providing more accurate statistical analysis for the relationship, for example, between the outcome of an event that occurs once every several years and the covariates that have observations every year. Lag effects have received a great deal of attention since Almon (1965) proposed linear distributed lag models to model the dependence of time series on several regressors from a correlated sequence. Motivated by the linear distributed lag model, we propose distributed generalized linear models as well as the estimation procedure for the model coefficients. The estimators from our proposed procedure are shown to be oracle or asymptotically efficient. Simulation studies confirm the asymptotic properties of the estimators and present consequences of model misspecification as well as better model prediction accuracy. The application is illustrated by analysis of the presidential election data in 2016. Fourth, we aim to provide an easy-to-use data analysis procedure for linear regression with non-independent errors. In practice, errors in regression model may be non-independent. In such situation, it is usually suitable to assume that the error terms for the model follow a time series structure. In fact, this type of model structure (reffered as RegARMA) has received great interests from researchers. Pierce (1971) discussed a nonlinear least squares estimation of RegARMA; Greenhouse et al. (1987) studied biological rhythm data by using the RegARMA model. Recently, Wu and Wang (2012) used the shrinkage estimation procedure to analyze data using RegARMA. However, in the literature the trend factor of the time series has not been considered. We will use the same idea of the two step-procedure as in the first project and the second project for our approach. We first estimate the trend of the time series by using a non-parametric method such as B-spline or linear Kernel. We then use the adaptive LASSO method for model selection and model estimation of the linear part and the time series error part. Simulation results show that our approach works quite well. However, it would be very interesting and challenging to improve the estimations and extend the estimation method to more complicated models, which will be the focus of the future research.