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 Analysis
Least Squares Estimation
Sensitivity to Frequency Offset
Frequency Identification
Frequency Selection
Least Absolute Deviations Estimation
Fourier Analysis Approach
Periodogram Analysis
Detection of Hidden Sinusoids
Extension of the Periodogram
Continuous Periodogram
Time-Frequency Analysis
Estimation of Noise Spectrum
Estimation in the Absence of Sinusoids
Estimation 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.
Biography
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.
"It masterfully integrates the most significant advances in the literature."
—Journal of the American Statistical Association"… an excellent introduction and overview of the literature dealing with statistical inference on time-series involving sinusoids. It will be an indispensable reference that research workers and graduate students of allied fields will rely on in the future."
—Mathematical Reviews, January 2015"It is extremely thorough in its approach. Every term is carefully defined, and many proofs are given in elaborate detail. … The range of problems and methods considered in the book is extensive."
—Journal of Time Series Analysis, 2015
