1st Edition

Time Series with Mixed Spectra

By Ta-Hsin Li Copyright 2014
    680 Pages 105 B/W Illustrations
    by Chapman & Hall

    680 Pages 105 B/W Illustrations
    by Chapman & Hall

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    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.

    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

    Trigonometric Series
    Probability Theory
    Numerical Analysis
    Matrix Theory
    Asymptotic Theory


    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.

    "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