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.