Time Series with Mixed Spectra: 1st Edition (Paperback) book cover

Time Series with Mixed Spectra

1st Edition

By Ta-Hsin Li

Chapman and Hall/CRC

680 pages | 105 B/W Illus.

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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.


"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

Table of Contents


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.

About the Author

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.

Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General
MATHEMATICS / Probability & Statistics / Bayesian Analysis