Probabilistic spectral analysis

Graphical introduction to nonstationary modelling of audio data. The input (bottom) is a sound recording of female speech. We seek to decompose the signal into Gaussian process carrier waveforms (blue block) multiplied by a spectrogram (green block). The spectrogram is learned from the data as a nonnegative matrix of weights times positive modulators (top).(W. J. Wilkinson, Andersen, et al. 2019a)

I am interested in probabilistic analogues of time frequency analysis and, what is nearly the same thing, autocorrelation analysis. I am especially interested in this for audio signals, which can be very very large, but have certain simplicities - i.e. being scalar functions of a univariate time index, usually regularly sampled.

In signal processing we frequently use Fourier transforms as a notionally nonparametric model for such a system, or a source of features for analysis.

That is classic stuff but it is (for me) always unsatisfying just taking the Fourier transform of something and hoping to have learned stuff about the system. There are a lot of arbitrary tuning parameters and awkward assumptions about, e.g. local stationarity and arbitrary ways of introducing non-local correlation. The same holds for the deterministic autocorrelogram, on which I have recently published a paper. I got good results, but I had no principled way to select the regularisation and interpretation of the methods. Unsatisfying.

I think we can do better by looking at the probabilistic behaviour of Fourier transforms and treating these as Bayesian nonparametric problems. This could solve a few problems at once.

This is now an active area, with a couple of approahces.

The central tool here in practice is Bochner’s theorem, which states that the Fourier transform of some spectral measure is a valid covariance kernel:

\[\kappa(\Delta t)=\mathcal{F}_{\Delta t}.\]

Taking this insight and running with it you can do lots of fun stuff. Turner and Sahani (2014) is sometimes mentioned as ground-zero of this kind of research, although the connections are certainly much older, e.g. Curtain (1975). Wiener and Khintchine approaches were not far from this (Wiener and Masani 1958, 1957) and it is implicit in Kalman-Bucy filtering (Kalman 1959, 1960; Kailath 1971) In modern times we have related but more specialised techniques such as the probabilistic phase vocoder (Godsill and Cemgil 2005). See also the connections to time series state models of Hartikainen and Särkkä (2010), Lindgren, Rue, and Lindström (2011), Reece and Roberts (2010) and Liutkus, Badeau, and Richard (2011).

There is a parallel body of literature here in the AdaptSpect family of methods (Bertolacci et al. 2020; Rosen, Wood, and Stoffer 2012). I just enjoyed, a seminar by Michael Bertolacci on this theme.

There are nice introductions in some papers (Solin 2016; Alvarado, Alvarez, and Stowell 2019; W. J. Wilkinson, Andersen, et al. 2019a), which unite various pieces I was discussing above with actual applications. I will work through these methods here for my own edification.


The basic setting here is the same as for typical audio signal analysis; we begin with a (random) signal \(f:\mathbb{R}\to\mathbb{R}\), where the argument is a continuous time index. We do not know this signal, but will infer its properties will have some countable number of discrete observations, \(\mathbf{f}:=\{f(t_k);k=1,2,\dots,K\}.\)

We imagine observations from this signal are modelled by a Gaussian process, giving us the same setup as Gaussian process regression. We introduce the additional assumption here that the scalar index \(\mathcal{I}:=\mathbb{R}\). ) represents time.

I suppose what we are doing here is requiring that there be some model for sampling error and that it may as well be the most convenient possible model to work with, which is additive Gaussian. More general noise models are indeed possible, and if we allow other Gaussian processes as additive noise models then we are on the way to constructing a source separation model. That is is indeed what (Liutkus, Badeau, and Richard 2011) do.1

Anyway, with these choices, this becomes absolutely the classic Gaussian process regression with some specialisation. (univariate index, mean-0)

It is also not far from the classic time frequency spectral analysis setup, where we take Fourier transforms over fixed size windows to estimate a kind of deterministic approximation to \(\kappa\) (thanks Wiener-Khintchine theorem); in that context we are effectively assuming that for each window we have an independent estimation problem, and a periodic kernel. I should make that relationship precise. 🏗

There is clearly a lot wrapped up in the kernel, \(\kappa(t, t';\mathbf{\theta}).\) We will come back to that.

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  1. We might more generally consider a sampling problem where we observe the signal through inner products with some sampling kernel, possibly even a stochastic one, but that sounds complicated.↩︎