Lecture

Spectral Estimation in Time Series

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In course

MATH-342: Time series

A first course in statistical time series analysis and applications. ...

Description

This lecture covers the concepts of spectral estimation in time series analysis, including topics such as imaging kernels, tapering methods, parametric estimation, and multitapering. The instructor explains how to estimate the spectrum of a process, reduce bias in spectral estimators, and analyze the performance of periodograms. The lecture also delves into AR models, Yule-Walker estimation, and the importance of AR processes in approximating continuous spectra. Various algorithms and methods for spectral estimation are discussed, providing insights into the analysis of time series data.

Instructor

Sofia Charlotta Olhede

I am a professor of Statistics at EPFL. I did my PhD in Mathematics at Imperial College London, and graduated in 2003. I was an assistant professor (lecturer) at Imperial College London 2002-2006, and an associate professor (senior lecturer) 2006-2007. In 2007 I moved to University College London and became a full professor. I moved to EPFL in 2019, and in 2018 I became an Institute of Mathematical Statistics Fellow with citation “For seminal contributions to the theory and application of large and heteroge

Official source

Related concepts (32)

Spectral density estimation

In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signal. Intuitively speaking, the spectral density characterizes the frequency content of the signal. One purpose of estimating the spectral density is to detect any periodicities in the data, by observing peaks at the frequencies corresponding to these periodicities.

Least-squares spectral analysis

Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum based on a least-squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in the long and gapped records; LSSA mitigates such problems. Unlike in Fourier analysis, data need not be equally spaced to use LSSA.

Autoregressive model

In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregressive model specifies that the output variable depends linearly on its own previous values and on a stochastic term (an imperfectly predictable term); thus the model is in the form of a stochastic difference equation (or recurrence relation which should not be confused with differential equation).

Periodogram

In signal processing, a periodogram is an estimate of the spectral density of a signal. The term was coined by Arthur Schuster in 1898. Today, the periodogram is a component of more sophisticated methods (see spectral estimation). It is the most common tool for examining the amplitude vs frequency characteristics of FIR filters and window functions. FFT spectrum analyzers are also implemented as a time-sequence of periodograms. There are at least two different definitions in use today.

Spectrum (physical sciences)

In the physical sciences, the term spectrum was introduced first into optics by Isaac Newton in the 17th century, referring to the range of colors observed when white light was dispersed through a prism. Soon the term referred to a plot of light intensity or power as a function of frequency or wavelength, also known as a spectral density plot. Later it expanded to apply to other waves, such as sound waves and sea waves that could also be measured as a function of frequency (e.g., noise spectrum, sea wave spectrum).

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