Compositionally-warped Gaussian processes

Rios, Gonzalo; Tobar, Felipe

doi:10.1016/j.neunet.2019.06.012

Cited by 35 publications

(28 citation statements)

References 26 publications

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“…and the inclusion of the Jacobian term that takes the change of volume induced by the warping into account, thereby ensuring a valid probability measure, for details see [28]:…”

Section: Likelihood Warpingmentioning

confidence: 99%

Warped Bayesian Linear Regression for Normative Modelling of Big Data

Fraza

Dinga²,

Beckmann

et al. 2021

Preprint

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Normative modelling is becoming more popular in neuroimaging due to its ability to make predictions of deviation from a normal trajectory at the level of individual participants. It allows the user to model the distribution of several neuroimaging modalities, giving an estimation for the mean and centiles of variation. With the increase in the availability of big data in neuroimaging, there is a need to scale normative modelling to big data sets. However, the scaling of normative models has come with several challenges. So far, most normative modelling approaches used Gaussian process regression, and although suitable for smaller datasets (up to a few thousand participants) it does not scale well to the large cohorts currently available and being acquired. Furthermore, most neuroimaging modelling methods that are available assume the predictive distribution to be Gaussian in shape. However, deviations from Gaussianity can be frequently found, which may lead to incorrect inferences, particularly in the outer centiles of the distribution. In normative modelling, we use the centiles to give an estimation of the deviation of a particular participant from the 'normal' trend. Therefore, especially in normative modelling, the correct estimation of the outer centiles is of utmost importance, which is also where data are sparsest. Here, we present a novel framework based on Bayesian Linear Regression with likelihood warping that allows us to address these problems, that is, to scale normative modelling elegantly to big data cohorts and to correctly model non-Gaussian predictive distributions. In addition, this method provides also likelihood-based statistics, which are useful for model selection. To evaluate this framework, we use a range of neuroimaging-derived measures from the UK Biobank study, including image-derived phenotypes (IDPs) and whole-brain voxel-wise measures derived from diffusion tensor imaging. We show good computational scaling and improved accuracy of the warped BLR for certain IDPs and voxels if there was a deviation from normality of these parameters in their residuals. The present results indicate the advantage of a warped BLR in terms of; computational scalability and the flexibility to incorporate non-linearity and non-Gaussianity of the data, giving a wider range of neuroimaging datasets that can be correctly modelled.

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“…and the inclusion of the Jacobian term that takes the change of volume induced by the warping into account, thereby ensuring a valid probability measure, for details see [28]:…”

Section: Likelihood Warpingmentioning

confidence: 99%

Warped Bayesian Linear Regression for Normative Modelling of Big Data

Fraza

Dinga²,

Beckmann

et al. 2021

Preprint

View full text Add to dashboard Cite

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“…The main research direction, however, should be towards relaxing the Gaussian assumption on the time series, though this is challenging due to the conjugacy between the Gaussian distribution and the Fourier operator. One way of lifting this restriction would be to consider a latent representation of the data, which could be Gaussian as in [25], so that the observed data are a (nonlinear) transformation of such a hidden representation. This is equivalent to considering a nonlinear likelihood H in the proposed model.…”

Section: Discussionmentioning

confidence: 99%

“…There is a vast literature on Gaussian models with non-Gaussian likelihoods that are used for classification [17,Ch. 3] and regression [24], which can be motivated from neural networks [25], parametric non-linear functions [26] and even Gaussian processes [27] or Gaussian mixtures. Therefore, as we focus on modelling the temporal dependence of signals and exploiting it in the spectral analysis setting, we argue that the Gaussian assumption is pertinent as it can be complemented with a non-Gaussian likelihood if required; however, such extensions are beyond the scope of this work.…”

Section: A Multivariate Normal Prior For Temporal Signalsmentioning

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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“…Future work could include exploring financial data sets with non-Gaussian likelihoods by warping GPs as proposed by [38,39], or by using Student's t-distribution likelihoods to better identify heteroskedasticity as used by GARCH and other financial models. Furthermore, better initialisation of hyperparameters and training can also greatly improve the results of the models which should remain an active area of research.…”

Section: Discussionmentioning

confidence: 99%

Gaussian Process Imputation of Multiple Financial Series

Wolff

Cuevas

Tobar

2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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In Financial Signal Processing, multiple time series such as financial indicators, stock prices and exchange rates are strongly coupled due to their dependence on the latent state of the market and therefore they are required to be jointly analysed. We focus on learning the relationships among financial time series by modelling them through a multi-output Gaussian process (MOGP) with expressive covariance functions. Learning these market dependencies among financial series is crucial for the imputation and prediction of financial observations. The proposed model is validated experimentally on two realworld financial datasets for which their correlations across channels are analysed. We compare our model against other MOGPs and the independent Gaussian process on real financial data.

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Compositionally-warped Gaussian processes

Cited by 35 publications

References 26 publications

Warped Bayesian Linear Regression for Normative Modelling of Big Data

Warped Bayesian Linear Regression for Normative Modelling of Big Data

Bayesian Reconstruction of Fourier Pairs

Gaussian Process Imputation of Multiple Financial Series

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