Gaussian processes for time-series modelling by S. Roberts, M. Osborne, M. Ebden, S. Reece, N. Gibson, and S. Aigrain Philosophical Transactions A Volume.

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Presentation transcript:

Gaussian processes for time-series modelling by S. Roberts, M. Osborne, M. Ebden, S. Reece, N. Gibson, and S. Aigrain Philosophical Transactions A Volume 371(1984): February 13, 2013 ©2013 by The Royal Society

A simple example of curve fitting. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

The conceptual basis of GPs starts with an appeal to simple multi-variate Gaussian distributions. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

The change in distributions on x1 and x2 is here presented in a form more familiar to time-series analysis. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

(a) The posterior distribution (the black line showing the mean and the grey shading ±σ) for a 10 day example, with observations made at locations 2, 6 and 8. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

(a–c) Functions drawn from a GP with a squared exponential covariance function with output scale h=1 and length scales λ=0.1 (a), 1 (b), 10 (c). S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

(a) Given six noisy data points (error bars are indicated with vertical lines), we are interested in estimating a seventh at x*=0.2. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

(a) Estimation of y* (solid line) and ±2σ posterior deviance for a function with short-term and long-term dynamics, and (b) long-term dynamics and a periodic component. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

A simple example of a GP applied sequentially. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

The GP is run sequentially making forecasts until a new datum is observed. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Random draws from GPs with different kernels. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Example covariance functions (a) for the modelling of data with changepoints and associated draws (b) from the resultant GPs, indicating what kind of data that they might be appropriate for. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

The effect of including a simple mean function. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Samples (dots) obtained by optimizing the log-likelihood (grey curve) using a global optimizer, and the maximum-likelihood approximation (vertical line) of the likelihood surface. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Samples obtained by taking draws from the posterior using a Markov chain Monte Carlo method. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

A set of samples that would lead to unsatisfactory behaviour from simple Monte Carlo. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Bayesian quadrature fits a GP to the integrand, and thereby performs inference about the integral. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Prediction and regression of tide height data for (a) independent and (b) multi-output GPs. (Online version in colour.). S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

(a) Comparison of active sampling of tide data using independent and (b) multi-output GPs. Note that, in the case of multi-output GPs, one sensor reading (Sotonmet) slightly leads the other readings and is hence sampled much more frequently. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Prediction for the Nile dataset using input-scale changepoint covariance (a) and the corresponding posterior distribution for time since the changepoint (b). S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Online (sequential) one-step predictions (a) and posterior for the location of changepoint for the Dow–Jones industrial average data using an output-scale changepoint covariance (b). S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

Predictive distribution for a quasi-periodic GP model using a mixed SE and RQ kernel, trained and conditioned on observations made with the 0.8 m Automated Patrol Telescope [24] using the Strömgren b and y filters. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society

As an example of a complex mean function, we here model data from an exoplanet transit light curve. S. Roberts et al. Phil. Trans. R. Soc. A 2013;371: ©2013 by The Royal Society