Time Series with Mixed Spectra Theory and Methods

Time series with mixed spectra are characterized by hidden periodic components buried in random noise. Despite strong interest in the statistical and signal processing communities, no book offers a comprehensive and up-to-date treatment of the subject. Filling this void, Time Series with Mixed Spectra focuses on the methods and theory for the statistical analysis of time series with mixed spectra. It presents detailed theoretical and empirical analyses of important methods and algorithms. Using both simulated and real-world data to illustrate the analyses, the book discusses periodogram analysis, autoregression, maximum likelihood, and covariance analysis. It considers real- and complex-valued time series, with and without the Gaussian assumption. The author also includes the most recent results on the Laplace and quantile periodograms as extensions of the traditional periodogram. Complete in breadth and depth, this book explains how to perform the spectral analysis of time series data to detect and estimate the hidden periodicities represented by the sinusoidal functions. The book not only extends results from the existing literature but also contains original material, including the asymptotic theory for closely spaced frequencies and the proof of asymptotic normality of the nonlinear least-absolute-deviations frequency estimator.

Introduction Periodicity and Sinusoidal Functions Sampling and Aliasing Time Series with Mixed Spectra Complex Time Series with Mixed Spectra Basic Concepts Parameterization of Sinusoids Spectral Analysis of Stationary Processes Gaussian Processes and White Noise Linear Prediction Theory .Asymptotic Statistical Theory Cramér-Rao Lower Bound Cramér-Rao Inequality CRLB for Sinusoids in Gaussian Noise Asymptotic CRLB for Sinusoids in Gaussian Noise CRLB for Sinusoids in NonGaussian White Noise Autocovariance Function Autocovariances and Autocorrelation Coefficients Consistency and Asymptotic Unbiasedness Covariances and Asymptotic Normality Autocovariances of Filtered Time Series Linear Regression AnalysisLeast Squares Estimation Sensitivity to Frequency Offset Frequency Identification Frequency SelectionLeast Absolute Deviations Estimation Fourier Analysis Approach Periodogram Analysis Detection of Hidden Sinusoids Extension of the PeriodogramContinuous Periodogram Time-Frequency Analysis Estimation of Noise Spectrum Estimation in the Absence of SinusoidsEstimation in the Presence of Sinusoids Detection of Hidden Sinusoids in Colored Noise Maximum Likelihood Approach Maximum Likelihood Estimation Maximum Likelihood under Gaussian White Noise The Case of Laplace White Noise The Case of Gaussian Colored Noise Determining the Number of Sinusoids Autoregressive Approach Linear Prediction Method Autoregressive Reparameterization Extended Yule-Walker Method Iterative Filtering Method Iterative Quasi Gaussian Maximum Likelihood Method Covariance Analysis Approach Eigenvalue Decomposition of Covariance Matrix Principal Component Analysis Method Subspace Projection Method Subspace Rotation Method Estimating the Number of Sinusoids Sensitivity to Colored Noise Further Topics Single Complex Sinusoid Tracking Time-Varying Frequencies Periodic Functions in Noise Beyond Single Time Series Quantile Periodogram Appendix Trigonometric Series Probability Theory Numerical Analysis Matrix Theory Asymptotic Theory Bibliography Proofs of Theorems appear at the end of most chapters.

Ta-Hsin Li is a research statistician at the IBM Watson Research Center. He was previously a faculty member at Texas A&M University and the University of California, Santa Barbara. Dr. Li is a fellow of the American Statistical Association and an elected senior member of the Institute of Electrical and Electronic Engineers. He is an associate editor for the EURASIP Journal on Advances in Signal Processing, the Journal of Statistical Theory and Practice, and Technometrics. He received a Ph.D. in applied mathematics from the University of Maryland.