# Cepstral transforms and harmonic identification

Just as you can generalise linear models for i.i.d observations you can do it with time series. You can also do it for the power-spectral representation of the time series, which includes as a special case the cepstral representation of the series.

I haven’t actually read the foundational literature here, just used some algorithms; but it seems to be mostly a hack for rapid identification of correlation lags where said lags are long.

• Proietti, T., & Luati, A. (2013). Generalised Linear Cepstral Models for the Spectrum of a Time Series.

In this chapter we consider a class of parametric spectrum estimators based on a generalized linear model for exponential random variables with power link. The power transformation of the spectrum of a stationary process can be expanded in a Fourier series, with the coefficients representing generalised autocovariances. Direct Whittle estimation of the coefficients is generally unfeasible, as they are subject to constraints (the autocovariances need to be a positive semidefinite sequence). The problem can be overcome by using an ARMA representation for the power transformation of the spectrum. Estimation is carried out by maximising the Whittle likelihood, whereas the selection of a spectral model, as a function of the power transformation parameter and the ARMA orders, can be carried out by information criteria. The proposed methods are applied to the estimation of the inverse autocorrelation function and the related problem of selecting the optimal interpolator, and for the identification of spectral peaks. More generally, they can be applied to spectral estimation with possibly misspecified models.

## Harmonic regression

A random thing I saw mentioned - I wonder if this is just another smoother for regressions?

Estimating the magnitude of individual cyclic components in a signal, e.g.

Rather than count peaks to guess the period or frequency […] fit regressions at many frequencies to find hidden sinusoids. Use the estimated amplitude at these frequencies to locate hidden periodic components. It is particularly easy to estimate the amplitude at a grid of evenly spaced frequencies from 0 to 1/2.