Probabilistic spectral analysis

November 13, 2019 — November 25, 2020

Figure 1

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 et al. 2019a)

I am interested in probabilistic analogues of time frequency analysis and, what is nearly the same thing, autocorrelation analysis, but for non-stationary signals. This is natural with Gaussian processes.

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 an active area, with a several approaches.

1 Classic: stochastic processes studied via correlation function

I’ve discussed basic stationary signal analysis everywhere, but why not check out some backgrounders in (Wiener and Masani 1957; Yaglom 1987)?

2 Non-stationary spectral kernel

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 (R. Kalman 1959; R. E. Kalman 1960; Kailath 1971). There are natural extensions of classic results, e.g. a Shannon-Nyquist theorem Tobar (2019). 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 are nice introductions in some papers (Solin 2016; Alvarado, Alvarez, and Stowell 2019; W. J. Wilkinson 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 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 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, \(\kappa(t, t';\mathbf{\theta}).\)

Typically this is some kind of spectral mixture kernel (W. J. Wilkinson et al. 2019b; Wilson and Adams 2013). W. J. Wilkinson et al. (2019a) summarizes these as:

\[ \begin{aligned} \kappa_{\mathrm{sm}}\left(t, t^{\prime}\right) &=\sum_{d=1}^{D} \kappa_{z}^{(d)}\left(t, t^{\prime}\right) \\ \kappa_{z}^{(d)}\left(t, t^{\prime}\right) &=\sigma_{d}^{2} \cos \left(\omega_{d}\left(t-t^{\prime}\right)\right) \kappa_{d}\left(t, t^{\prime}\right) \end{aligned} \]

\(\kappa_{d}\) is free to be chosen, but is typically from the Matérn class of kernel functions. Parameters \(\omega_{d}\) determine the periodicity of the kernel components, which can be interpreted as the centre frequencies of the filters in a probabilistic filter bank. By choosing the exponential kernel \(\kappa_{d}\left(t, t^{\prime}\right)=\exp \left(\left|t-t^{\prime}\right| / \ell_{d}\right)\) we recover exactly the probabilistic phase vocoder (Cemgil & Godsill, 2005 ), and the lengthscales \(\ell_{d}\) control the filter bandwidths.

More generally we would like this to be a non-stationary kernel, which requires a model for the density of these kernels. W. J. Wilkinson et al. (2019a) uses a NMF model with a GP prior on some matrix rows and applying a softmax link. (Remes, Heinonen, and Kaski (2018) seems to get a similar structure?)

3 Locally stationary

Connection to the short time Fourier transform, where signals are assumed stationary. Change point detection version TBD.

There is an alternative approach which looks at switching between covariance kernels/spectrogram. One strand is the AdaptSpect family of methods (Bertolacci et al. 2020; Rosen, Wood, and Stoffer 2012), whgcih develop fast MCMC samplers by using Whittle likelihood approaches over randomised change points. Disclosure of bias: I just enjoyed a seminar by my colleague Michael Bertolacci on this theme, and Sally Cripps née Wood works 20m from me and was co-author these.

Russell Tsuchida has made me aware of a parallel body o -g mouse off f work (Adams and MacKay 2007; Edwards, Meyer, and Christensen 2019; Osborne 2010; Roberts et al. 2013; Saatçi, Turner, and Rasmussen 2010) which keeps the spectrogram implicit and changes the covariance kernel. This is still reasonably fast thanks to Lattice GP tricks.

TODO: compare and contrast these methods. ls -al /dev/disk/by-uuid/I suspect a major difference is that the former targets statisticians and the latter ML people but they can probably be combined, or at least a neat cherry-picked method leveraging both should be feasible.

4 Non-Gaussian approaches

For now, see sparse stochastic processes.

5 References

Adams, and MacKay. 2007. Bayesian Online Changepoint Detection.” arXiv:0710.3742 [Stat].
Alvarado, Alvarez, and Stowell. 2019. Sparse Gaussian Process Audio Source Separation Using Spectrum Priors in the Time-Domain.” In ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
Alvarado, and Stowell. 2018. Efficient Learning of Harmonic Priors for Pitch Detection in Polyphonic Music.” arXiv:1705.07104 [Cs, Stat].
Bertolacci. 2019. “Hierarchical Bayesian Mixture Models for Spatiotemporal Data with Nonstandard Features.”
Bertolacci, Cripps, Rosen, et al. 2019. Climate Inference on Daily Rainfall Across the Australian Continent, 1876–2015.” Annals of Applied Statistics.
Bertolacci, Rosen, Cripps, et al. 2020. AdaptSPEC-X: Covariate Dependent Spectral Modeling of Multiple Nonstationary Time Series.” arXiv:1908.06622 [Stat].
Bruinsma, and Turner. 2018. Learning Causally-Generated Stationary Time Series.” arXiv:1802.08167 [Stat].
Cemgil. 2009. Bayesian Inference for Nonnegative Matrix Factorisation Models.” Computational Intelligence and Neuroscience.
Cheng, Sa-Ngasoongsong, Beyca, et al. 2015. Time Series Forecasting for Nonlinear and Non-Stationary Processes: A Review and Comparative Study.” IIE Transactions.
Choudhuri, Ghosal, and Roy. 2004a. Contiguity of the Whittle Measure for a Gaussian Time Series.” Biometrika.
———. 2004b. Bayesian Estimation of the Spectral Density of a Time Series.” Journal of the American Statistical Association.
Cunningham, Shenoy, and Sahani. 2008. Fast Gaussian Process Methods for Point Process Intensity Estimation.” In Proceedings of the 25th International Conference on Machine Learning. ICML ’08.
Curtain. 1975. Infinite-Dimensional Filtering.” SIAM Journal on Control.
Duvenaud, Nickisch, and Rasmussen. 2011. Additive Gaussian Processes.” In Advances in Neural Information Processing Systems.
Dym, and McKean. 2008. Gaussian Processes, Function Theory, and the Inverse Spectral Problem. Dover Books on Mathematics.
Edwards, Meyer, and Christensen. 2015. Bayesian Semiparametric Power Spectral Density Estimation with Applications in Gravitational Wave Data Analysis.”
———. 2019. Bayesian Nonparametric Spectral Density Estimation Using B-Spline Priors.” Statistics and Computing.
Févotte, Bertin, and Durrieu. 2008. Nonnegative Matrix Factorization with the Itakura-Saito Divergence: With Application to Music Analysis.” Neural Computation.
Girolami, and Rogers. 2005. Hierarchic Bayesian Models for Kernel Learning.” In Proceedings of the 22nd International Conference on Machine Learning - ICML ’05.
Godsill, and Cemgil. 2005. Probabilistic Phase Vocoder and Its Application to Interpolation of Missing Values in Audio Signals.” In 2005 13th European Signal Processing Conference.
Hartikainen, and Särkkä. 2010. Kalman Filtering and Smoothing Solutions to Temporal Gaussian Process Regression Models.” In 2010 IEEE International Workshop on Machine Learning for Signal Processing.
Hensman, Durrande, and Solin. 2018. Variational Fourier Features for Gaussian Processes.” Journal of Machine Learning Research.
Hermansen. 2008. Bayesian nonparametric modelling of covariance functions, with application to time series and spatial statistics.”
Jesus, and Chandler. 2017. Inference with the Whittle Likelihood: A Tractable Approach Using Estimating Functions.” Journal of Time Series Analysis.
Kailath. 1971. “The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures.” The Annals of Mathematical Statistics.
Kalman, R. 1959. On the General Theory of Control Systems.” IRE Transactions on Automatic Control.
Kalman, R. E. 1960. A New Approach to Linear Filtering and Prediction Problems.” Journal of Basic Engineering.
Karvonen, and Särkkä. 2016. Approximate State-Space Gaussian Processes via Spectral Transformation.” In 2016 IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP).
Kirch, Edwards, Meier, et al. 2019. Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis.” Bayesian Analysis.
Lindgren, Rue, and Lindström. 2011. An Explicit Link Between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Liutkus, Badeau, and Richard. 2011. Gaussian Processes for Underdetermined Source Separation.” IEEE Transactions on Signal Processing.
Liutkus, Rafii, Pardo, et al. 2014. Kernel Spectrogram Models for Source Separation.” In.
Macaro, and Prado. 2014. Spectral Decompositions of Multiple Time Series: A Bayesian Non-Parametric Approach.” Psychometrika.
Meier, Kirch, and Meyer. 2020. Bayesian Nonparametric Analysis of Multivariate Time Series: A Matrix Gamma Process Approach.” Journal of Multivariate Analysis.
Meyer, Edwards, Maturana-Russel, et al. 2020. Computational Techniques for Parameter Estimation of Gravitational Wave Signals.” WIREs Computational Statistics.
Nickisch, Solin, and Grigorevskiy. 2018. State Space Gaussian Processes with Non-Gaussian Likelihood.” In International Conference on Machine Learning.
Osborne. 2010. Bayesian Gaussian Processes for Sequential Prediction, Optimisation and Quadrature.”
Rasmussen, and Nickisch. 2010. Gaussian Processes for Machine Learning (GPML) Toolbox.” Journal of Machine Learning Research.
Reece, and Roberts. 2010. An Introduction to Gaussian Processes for the Kalman Filter Expert.” In 2010 13th International Conference on Information Fusion.
Remes, Heinonen, and Kaski. 2017. Non-Stationary Spectral Kernels.” In Advances in Neural Information Processing Systems 30.
———. 2018. Neural Non-Stationary Spectral Kernel.” arXiv:1811.10978 [Cs, Stat].
Roberts, Osborne, Ebden, et al. 2013. Gaussian Processes for Time-Series Modelling.” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.
Rosen, Stoffer, and Wood. 2009. Local Spectral Analysis via a Bayesian Mixture of Smoothing Splines.” Journal of the American Statistical Association.
Rosen, Wood, and Stoffer. 2012. AdaptSPEC: Adaptive Spectral Estimation for Nonstationary Time Series.” Journal of the American Statistical Association.
Saatçi, Turner, and Rasmussen. 2010. Gaussian Process Change Point Models.” In Proceedings of the 27th International Conference on International Conference on Machine Learning. ICML’10.
Särkkä, Simo. 2007. On Unscented Kalman Filtering for State Estimation of Continuous-Time Nonlinear Systems.” IEEE Transactions on Automatic Control.
Särkkä, S., and Hartikainen. 2013. Non-Linear Noise Adaptive Kalman Filtering via Variational Bayes.” In 2013 IEEE International Workshop on Machine Learning for Signal Processing (MLSP).
Särkkä, Simo, and Nummenmaa. 2009. Recursive Noise Adaptive Kalman Filtering by Variational Bayesian Approximations.” IEEE Transactions on Automatic Control.
Särkkä, Simo, and Solin. 2019. Applied Stochastic Differential Equations. Institute of Mathematical Statistics Textbooks 10.
Särkkä, Simo, Solin, and Hartikainen. 2013. Spatiotemporal Learning via Infinite-Dimensional Bayesian Filtering and Smoothing: A Look at Gaussian Process Regression Through Kalman Filtering.” IEEE Signal Processing Magazine.
Solin. 2016. Stochastic Differential Equation Methods for Spatio-Temporal Gaussian Process Regression.”
Solin, and Särkkä. 2013. Infinite-Dimensional Bayesian Filtering for Detection of Quasiperiodic Phenomena in Spatiotemporal Data.” Physical Review E.
———. 2014. Explicit Link Between Periodic Covariance Functions and State Space Models.” In Artificial Intelligence and Statistics.
Sykulski, Olhede, Guillaumin, et al. 2019. The Debiased Whittle Likelihood.” Biometrika.
Tobar. 2019. Band-Limited Gaussian Processes: The Sinc Kernel.” Advances in Neural Information Processing Systems.
Tobar, Araya-Hernández, Huijse, et al. 2020. Bayesian Reconstruction of Fourier Pairs.” arXiv:2011.04585 [Eess, Stat].
Tuft. 2020. Statistical Learning for the Spectral Analysis of Time Series Data.”
Turner, and Sahani. 2014. Time-Frequency Analysis as Probabilistic Inference.” IEEE Transactions on Signal Processing.
Valenzuela, and Tobar. 2019. Low-Pass Filtering as Bayesian Inference.” In ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
Virtanen, Cemgil, and Godsill. 2008. Bayesian Extensions to Non-Negative Matrix Factorisation for Audio Signal Modelling.” In 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.
Wiener, and Masani. 1957. The Prediction Theory of Multivariate Stochastic Processes.” Acta Mathematica.
———. 1958. The Prediction Theory of Multivariate Stochastic Processes, II.” Acta Mathematica.
Wilkinson, W. 2019. Gaussian Process Modelling for Audio Signals.”
Wilkinson, William J., Andersen, Reiss, et al. 2019a. End-to-End Probabilistic Inference for Nonstationary Audio Analysis.” arXiv:1901.11436 [Cs, Eess, Stat].
Wilkinson, William J., Andersen, Reiss, et al. 2019b. Unifying Probabilistic Models for Time-Frequency Analysis.” In ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).
Wilson, and Adams. 2013. Gaussian Process Kernels for Pattern Discovery and Extrapolation.” In International Conference on Machine Learning.
Yaglom. 1987. Correlation Theory of Stationary and Related Random Functions. Volume II: Supplementary Notes and References. Springer Series in Statistics.
Zheng, Zhu, and Roy. 2010. Nonparametric Bayesian Inference for the Spectral Density Function of a Random Field.” Biometrika.

Footnotes

  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.↩︎