Lecture

Spectral Estimation: Periodogram and Tapering

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In course

MATH-342: Time series

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

Description

This lecture covers p-variate spectral representations, ACVS estimation, and spectral estimation in time series analysis. It delves into the properties of spectral density functions, cross-spectral densities, and the estimation of auto-covariance sequences. The instructor explains the use of spectral estimators, the periodogram, and the impact of tapers on reducing bias in spectral estimation.

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.

Spectral density

The power spectrum of a time series describes the distribution of power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum.

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.

Maximum entropy spectral estimation

Maximum entropy spectral estimation is a method of spectral density estimation. The goal is to improve the spectral quality based on the principle of maximum entropy. The method is based on choosing the spectrum which corresponds to the most random or the most unpredictable time series whose autocorrelation function agrees with the known values. This assumption, which corresponds to the concept of maximum entropy as used in both statistical mechanics and information theory, is maximally non-committal with regard to the unknown values of the autocorrelation function of the time series.

Time series

In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart).

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