Modelling of complex signals using gaussian processes

Tobar, Felipe; Turner, Richard E.

doi:10.1109/icassp.2015.7178363

Cited by 11 publications

(11 citation statements)

References 12 publications

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“…(5) Furthermore, in order to avoid the treatment of complex-valued random variables [29], [30], we identify the real and imaginary parts of the Fourier matrix respectively by,…”

Section: B the Induced Generative Model In Time And Frequencymentioning

confidence: 99%

Bayesian Reconstruction of Fourier Pairs

Tobar

Araya-Hernandez

Huijse

et al. 2021

IEEE Trans. Signal Process.

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In a number of data-driven applications such as detection of arrhythmia, interferometry or audio compression, observations are acquired indistinctly in the time or frequency domains: temporal observations allow us to study the spectral content of signals (e.g., audio), while frequency-domain observations are used to reconstruct temporal/spatial data (e.g., MRI). Classical approaches for spectral analysis rely either on i) a discretisation of the time and frequency domains, where the fast Fourier transform stands out as the de facto off-the-shelf resource, or ii) stringent parametric models with closed-form spectra. However, the general literature fails to cater for missing observations and noise-corrupted data. Our aim is to address the lack of a principled treatment of data acquired indistinctly in the temporal and frequency domains in a way that is robust to missing or noisy observations, and that at the same time models uncertainty effectively. To achieve this aim, we first define a joint probabilistic model for the temporal and spectral representations of signals, to then perform a Bayesian model update in the light of observations, thus jointly reconstructing the complete (latent) time and frequency representations. The proposed model is analysed from a classical spectral analysis perspective, and its implementation is illustrated through intuitive examples. Lastly, we show that the proposed model is able to perform joint time and frequency reconstruction of real-world audio, healthcare and astronomy signals, while successfully dealing with missing data and handling uncertainty (noise) naturally against both classical and modern approaches for spectral estimation.

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“…(5) Furthermore, in order to avoid the treatment of complex-valued random variables [29], [30], we identify the real and imaginary parts of the Fourier matrix respectively by,…”

Section: B the Induced Generative Model In Time And Frequencymentioning

confidence: 99%

Bayesian Reconstruction of Fourier Pairs

Tobar

Araya-Hernandez

Huijse

et al. 2021

IEEE Trans. Signal Process.

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“…GPs are flexible nonparametric models for functions, in particular, for latent signals involved in SE settings. Besides their strength as a generative model, there are two key properties that position GPs as a sound prior within SE: first, as the Fourier transform is a linear operator, the Fourier transform of a GP (if it exists) is also a (complex-valued) GP [30,31] and, critically, the signal and its spectrum are jointly Gaussian. Second, Gaussian random variables are closed under conditioning and marginalisation, meaning that the exact posterior distribution of the spectrum conditional to a set of partial observations of the signal is also Gaussian.…”

Section: Gaussian Process Priors Over Functionsmentioning

confidence: 99%

“…As a complex-valued linear transformation of f (t) ∼ GP, the local spectrum F c (ξ) is a complex-GP [31,30] and thus completely determined by its covariance and pseudocovariance [33] given by…”

Section: The Local-spectrum Gaussian Processmentioning

confidence: 99%

Bayesian Nonparametric Spectral Estimation

Tobar¹

2018

Preprint

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Spectral estimation (SE) aims to identify how the energy of a signal (e.g., a time series) is distributed across different frequencies. This can become particularly challenging when only partial and noisy observations of the signal are available, where current methods fail to handle uncertainty appropriately. In this context, we propose a joint probabilistic model for signals, observations and spectra, where SE is addressed as an exact inference problem. Assuming a Gaussian process prior over the signal, we apply Bayes' rule to find the analytic posterior distribution of the spectrum given a set of observations. Besides its expressiveness and natural account of spectral uncertainty, the proposed model also provides a functional-form representation of the power spectral density, which can be optimised efficiently. Comparison with previous approaches, in particular against Lomb-Scargle, is addressed theoretically and also experimentally in three different scenarios. Code and demo available at github.com/GAMES-UChile.

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“…The power spectral density in eq. ( 9) corresponds to a complex-valued kernel and therefore to a complex-valued GP [14,15] . In order to restrict this generative model only to real-valued GPs, the proposed power spectral density has to be symmetric with respect to ω [16], we then make S ij (ω) symmetric simply by reassigning S ij (ω) → 1 2 (S ij (ω) + S ij (−ω)), this is equivalent to choosing R i (ω) to be a vector of two mirrored complex SE functions.…”

Section: The Multi-output Spectral Mixture Kernelmentioning

confidence: 99%

Spectral Mixture Kernels for Multi-Output Gaussian Processes

Parra,

Tobar

2017

Preprint

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Early approaches to multiple-output Gaussian processes (MOGPs) relied on linear combinations of independent, latent, single-output Gaussian processes (GPs). This resulted in cross-covariance functions with limited parametric interpretation, thus conflicting with the ability of single-output GPs to understand lengthscales, frequencies and magnitudes to name a few. On the contrary, current approaches to MOGP are able to better interpret the relationship between different channels by directly modelling the cross-covariances as a spectral mixture kernel with a phase shift. We extend this rationale and propose a parametric family of complex-valued cross-spectral densities and then build on Cramér's Theorem (the multivariate version of Bochner's Theorem) to provide a principled approach to design multivariate covariance functions. The so-constructed kernels are able to model delays among channels in addition to phase differences and are thus more expressive than previous methods, while also providing full parametric interpretation of the relationship across channels. The proposed method is first validated on synthetic data and then compared to existing MOGP methods on two real-world examples.

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Modelling of complex signals using gaussian processes

Cited by 11 publications

References 12 publications

Bayesian Reconstruction of Fourier Pairs

Bayesian Reconstruction of Fourier Pairs

Bayesian Nonparametric Spectral Estimation

Spectral Mixture Kernels for Multi-Output Gaussian Processes

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