Lecture 8

Spectral estimation

Approaches

Periodogram the basic modulus-squared of the Fourier transform Bartlett's method is a periodogram spectral estimate formed by averaging the discrete Fourier transform of multiple segments of the signal to reduce variance of the spectral density estimate Welch's method a windowed version of Bartlett's method that uses overlapping segments Blackman-Tukey is a periodogram smoothing technique to reduce variance of the spectral density estimate Autoregressive moving average estimation, based on fitting to an ARMA model to the time series of signal samples; in addition to ARMA, there are separate moving average and autoregressive methods Multitaper is a periodogram-based method that uses multiple tapers, or windows, to form independent estimates of the spectral density to reduce variance of the spectral density estimate Maximum entropy spectral estimation is an all-poles method useful for SDE when singular spectral features, such as sharp peaks, are expected. Least-squares spectral analysis, based on least squares fitting to known frequencies Non-uniform discrete Fourier transform is used when the signal samples are unevenly spaced in time Singular spectrum analysis is a nonparametric method that uses a singular value decomposition of the covariance matrix to estimate the spectral density Short-time Fourier transform

FT based approaches AR based approaches

Methods we have looked at already

DFT / FFT Non-parametric

AR (ARMA) models parametric

Wavelet models

Singular spectrum analysis

We have already considered this – it is the method of embedding followed by singular value decomposition

The significance of the eigen values (singular values) provides us with a nice way of judging the number of non-stationary oscillators present in the data

This can then be used to help smooth, filter etc the original timeseries

SSA: sunspot data

SSA: sunspot data

Note how the core resonances in the data are picked up – particularly the dominant 11 year cycle The 'music' algorithm Uses the SSA basis to remove the noise components and so produce a better spectral estimate

Looking for 1,2,3, …, 7,8 harmonic components using music Least squares – fitting to known frequencies This is just Fourier Analysis!

But, in FT we consider the least squares fitting of a wide range of frequencies

If we are only interested in one – then we can encode this directly

Factor weights

prob(spectral peak)

In effect we create a filter bank Comparison

Fourier (DFT)

Welch (smoothed DFT)

AR(11) model

Music(11) STFT, wavelet