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gp-code-osborne

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MATLAB code for Gaussian processes

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- - AT_cov_fn.m
  - faulty_2term_cov_fn.m
  - heterosked_noise_fn.m
  - hom_cov_fn.m
  - iid_noise_fn.m
  - log_cov_fn.m
  - MIDAS_cov_fn.m
  - MIDAS_indep_cov_fn.m
  - ndimsqdexp_cov_fn.m
  - ndimsqdexp_cov_fn_withderivs.m
  - ndimsqdexp_isotropic_cov_fn.m
  - ndimsqdexp_isotropic_cov_fn_withderivs.m
  - ndimsqdexpperiodic_cov_fn.m
  - ndimsqdexpperiodic_cov_fn_withderivs.m
  - ndimsqdexpperiodic_isotropic_cov_fn.m
  - ndimsqdexpperiodic_isotropic_cov_fn_withderivs.m
  - oned_cov_fn.m
  - poly_cov_fn.m
  - prod2term_cov_fn.m
  - sensor2term_cov_fn.m
  - sensor_cov_fn.m
  - sets_cov_fn.m
  - shipping_cov_fn.m
  - simple2term_cov_fn.m
  - Tides_cov_fn.m
  - versatile_cov_fn.m
  - wderiv_cov_fn.m
  - WS_cov_fn.m
- - affine_mean_fn.m
  - constant_mean_fn.m
  - ndimssqdexp_mean_fn.m
  - planar_mean_fn.m
  - quadratic_mean_fn.m
  - sensors_mean_fn.m
  - wderiv_mean_fn.m
- - ackley.m
  - ackley10.mat
  - ackley2.mat
  - ackley5.mat
  - banana.m
  - bounds.m
  - bra.m
  - branin.m
  - branin.mat
  - cam.m
  - camelback.m
  - camelback.mat
  - gold.m
  - goldsteinprice.mat
  - gpr2.m
  - griewank10.mat
  - griewank2.mat
  - griewank5.mat
  - har3.m
  - har6.m
  - hartman3.mat
  - hartman6.mat
  - hm3.m
  - hm6.m
  - mich.m
  - rast.m
  - rast.mat
  - s10.m
  - sch.m
  - sgr.m
  - sh5.m
  - sh7.m
  - shekel.m
  - shekel10.m
  - shekel10.mat
  - shekel5.m
  - shekel5.mat
  - shekel7.m
  - shekel7.mat
  - shu.m
  - shubert.mat
  - Tom.m
- activedataseln.m
- activedataseln_old.m
- adjust_w_max.m
- basicregression.m
- best_hyperparams.m
- big_chi_mat.m
- big_ups_mat.m
- bmcparams.m
- bmcparams_ahs.m
- bmcparams_ML.m
- bmcparams_old.m
- bq_ais.m
- bq_params.m
- bqgi.m
- brutish_fault_bucket.m
- bz_quad.m
- bz_quad_hps.m
- calculate_hyper2sample_likelihoods.m
- calculate_hyper2sample_likelihoods_ML.m
- calculate_sample_uncertainty.m
- candidate_combs_cell.mat
- correlation_plot.m
- corrpriors.m
- covfn.m
- covfnper.m
- create_lhs_hypersamples.m
- D3cov.m
- d_log_scales_gaussian.m
- dblderivcovfn.m
- DDKwderivrot.m
- del_hyperparams.m
- derivativise.m
- derivcovfn.m
- determine_candidates.m
- determine_starting_points.m
- disp_gp_hps.m
- disp_hyperparams.m
- disp_importances.m
- disp_spgp_hps.m
- DKwderivrot.m
- DTheta_consts.m
- DTKwderivrot.m
- evidence.m
- evidence_ahs.m
- EXAMPLE.m
- expected_uncertainty_evidence.m
- fast_exp_loss_min.m
- fault_bucket.m
- fcov.m
- fit_hypers_multiple_restart.m
- gaussian_mat.m
- gcov.m
- get_diag_noise.m
- get_diag_sqd_noise.m
- get_hyper2_samples_to_move.m
- get_hyper_samples_to_move.m
- get_hyper_scale.m
- get_mu.m
- get_noise.m
- get_outta_here.m
- GP_EXAMPLE.m
- gp_plot.m
- gpgo.m
- GPGO_EXAMPLE.m
- GPGO_EXAMPLE_wderivs.m
- GPGO_EXAMPLE_woderivs.m
- gpmeancov.m
- gpparams.m
- gTcov.m
- gTgcov.m
- gTHcov.m
- Hcov.m
- hess_log_scales_lik_gaussian.m
- hp_heuristics.m
- hyper2params.m
- hyperparams.m
- hypersample_weights.m
- hyperweights.m
- import_likelihood_vector.m
- improve_bmc_conditioning.m
- input_importance.m
- input_importance_wm.m
- integrate_ahs.m
- integrate_gaussians.m
- integrate_HMC.m
- integrate_ML.m
- jitter_correction.m
- kersting_fault_bucket.m
- Kwderiv.m
- Kwderivrot.m
- laplot_nice.m
- log_ev_plots.m
- log_evidence.m
- log_gaussian_mat.m
- log_transform.m
- logmvnpdf.m
- lw_train_gp.m
- Maike_script.m
- make_gp_plot.m
- manage_hyper2_samples.m
- manage_hyper_samples.m
- manage_hyper_samples_HMC.m
- manage_hyper_samples_ML.m
- manage_hyper_samples_old.m
- min_zoom.m
- multdim_tracking.m
- negval.m
- pinkmap.mat
- plot_hp_posteriors.m
- plot_nice.m
- plot_predictive_posterior.m
- plot_regression.m
- plot_sep_ts.m
- plot_thesis.m
- plot_ts.m
- posterior_gp.m
- posterior_hp.m
- posterior_hp_ahs.m
- posterior_hp_params.m
- posterior_spgp.m
- postmean_hp.m
- predict.m
- predict_ahs.m
- predict_BMC.m
- predict_bq.m
- predict_exact.m
- predict_gp.m
- predict_HMC.m
- predict_HMC_seq.m
- predict_MC.m
- predict_ML.m
- predict_ML_GA.m
- predict_spgp.m
- process_flag.m
- regression.m
- revise_bz_quad_matrix.m
- revise_gp.m
- revise_spgp.m
- robust_gpparams.m
- sample_candidate_likelihoods.m
- sample_hyperparameters.m
- sbq.m
- SBQ_EXAMPLE.m
- sbq_gpml.m
- sbq_gpml_ais.m
- set_affine_mean.m
- set_candidates.m
- set_constant_mean.m
- set_covvy.m
- set_gp.m
- set_gp_data.m
- set_hps_struct.m
- set_planar_mean.m
- set_quadratic_mean.m
- set_spgp.m
- shaded_sd.m
- simple_bmc_integral.m
- simple_bz_quad.m
- simple_gpmean.m
- simple_gpmeancov.m
- simple_zoom_pt.m
- small_chi_const.m
- small_ups2_vec.m
- small_ups_vec.m
- spgp_centres.m
- SPGP_EXAMPLE.m
- spgp_plot.m
- spgpgo.m
- SPGPGO_EXAMPLE.m
- spnegval.m
- sqdexp2gaussian.m
- suggest_input_scale_samples.m
- test_ahs_integration.m
- track.m
- track_ahs.m
- track_ahs_test.m
- track_BQ.m
- track_BQML.m
- track_gp.m
- track_HMC.m
- track_likelihood_fullfn.m
- track_ML.m
- track_q_fullfn.m
- track_sbq.m
- tracking.m
- tracking_script.m
- train_gp.m
- train_gpgo.m
- train_spgp.m
- update_gaussian_mat.m
- update_likelihood_gp.m
- weighted_gpmeancov.m
- weightednegval.m
- weights.m
- weights_ahs.m
- weights_ML.m
- window_gp.m
- zoom_hyper_samples.m
- axisepmgp.m
- draw_from_linearisation.m
- ellipsoid_volume_test.m
- epmgp.m
- genzmgp.m
- mvncdf_bq.m
- mvncdf_bq_roman_test.m
- notes20120501forMikeOsborne.m
- plot_bounding_ellipse.m
- plot_bqmvncdf_covariance_comparison.m
- plot_covariance_comparison.m
- plot_covariance_ellipses.m
- plot_linearisation.m
- plot_point_slice_convolution_observations.m
- qsclatmvnv.m
- rotating_MV_gaussian.m
- test_mvncdf_bq.m
- truncNormMoments.m
- - bmc.m
  - bmc_integrate.m
  - bmc_marginal_mean.m
  - bmc_marginal_variance.m
  - log_bmc.m
  - online_bmc.m
  - online_bq_ais.m
  - online_bq_gpml_ais.m
  - online_log_bmc.m
- - bmc_intro.m
  - bmc_intro_v2.m
  - bmc_intro_v3.m
  - bmc_intro_v4.m
  - bmc_intro_v6.m
  - custom_gpml_plot.m
  - draw_sbq_plots.m
  - gen_eue_plot.m
  - integrate_hypers_figure.m
  - integrate_hypers_figure_v2.m
  - log_gp_figure.m
  - log_prior_demo.m
  - log_prior_demo_show_bad.m
  - log_prior_demo_v2.m
  - log_prior_demo_v3.m
  - log_prior_demo_v4.m
  - marginals_demo.m
  - min_in_box_plots.m
  - new_lin_demo.m
  - plot_1d_problems.m
  - plot_sample_path_3d.m
  - print_problem_descriptions.m
  - saveeps.m
  - savepng.m
  - untransformed_vs_transformed.m
- - define_integration_problems.m
  - estimate_truth_via_mcmc.m
  - friedman_data.mat
  - gen_friedman_data.m
  - gp_log_likelihood.m
- - ais_mh.m
  - make_online_slow.m
  - online_ais_mh.m
  - online_smc.m
  - simple_monte_carlo.m
- - run_one_job.sh
  - submit_all.sh
  - submit_all_ubuntu.sh
- - bests.mat
  - check_required_options.m
  - concordance
  - create_spectrum_likelihood_handle.m
  - create_spectrum_likelihood_handle_dla.m
  - Gamma.eps
  - Gaussian.eps
  - meanVoightProfile.m
  - okaythismightnotwork.zip
  - priors.mat
  - read_fits_data.m
  - RQ.eps
  - test_thing.m
- - auto_correlation.m
  - call_python_numeric.m
  - colorbrew.m
  - crosshatch_poly.m
  - draw_lengthscale.m
  - estimate_truth_fast.m
  - exp10.m
  - expnormal.m
  - final_results_table.m
  - get_likelihood_hessian.m
  - gpml_lengthscale_likelihood.m
  - gpml_plot.m
  - gridLegend.m
  - jbfill.m
  - latex_table.m
  - likelihood_laplace.m
  - log_lognpdf.m
  - log_of_normal_to_log_normal.m
  - log_volume_between_two_gaussians.m
  - logmvnpdf.m
  - logsumexp.m
  - matlab2tikz.m
  - matlab2tikzInputParser.m
  - matlabfrag.m
  - parseArgs.m
  - print_table.m
  - python.m
  - remove_duplicate_samples.m
  - save2pdf.m
  - savepng.m
  - set_fig_units_cm.m
  - subaxis.m
- call_one_experiment.m
- compile_all_results.m
- compile_cosmo_results.m
- compile_real_results.m
- debug_sbq.m
- define_integration_methods.m
- define_sample_sizes.m
- exp_loss_movies.m
- run_all_experiments.m
- run_one_experiment.m
- run_real_experiments.m
- BQR_choosing_candidates_for_SBQ2.m
- BQR_oned_plot.m
- BQR_plot_logGP_vs_GP_2.m
- censored_integrals2.m
- log_gp_lik.m
- oned_BQR_plot.m
- posterior_as_fn_of_input_scale.m
- prob_bqr_incr_samples.m
- problem_bbq_predict_bq.m
- quick_test_bqr_incr_samples_slice.m
- rotating_MV_gaussian.m
- simple_test_sbq.m
- simple_test_sbq_gpml.m
- spline.m
- test_bbq_predict_bq.m
- test_bqr_incr_samples_hmc.m
- test_bqr_incr_samples_hmc_incl_ML.m
- test_bqr_incr_samples_slice.m
- test_bqr_incr_samples_slice_incl_ML.m
- test_bqr_on_analytic_integrals.m
- test_bqr_on_flux_prediction.m
- test_bqr_on_GP_prediction.m
- test_bqr_on_GP_prediction_ML.m
- test_sbq_on_GP_prediction.m
- test_simple_zoom.m
- - derivest.m
  - directionaldiff.m
  - gradest.m
  - hessdiag.m
  - hessian.m
  - jacobianest.m
- - build_emd.m
  - demo_emd.m
  - emd.c
  - emd.h
  - EMD_example.mat
  - EMD_example_2.mat
  - EMD_example_3.mat
  - EMD_good_example.mat
  - emd_mex.c
  - emd_mex.dll
  - emd_mex.m
  - emd_mex.mexglx
  - example4a.png
  - example4b.png
- - license.txt
- accelerate.m
- allcombs.m
- argmin.m
- bound.m
- build_gaussian_filter.m
- cell2mat2d.c
- cell2mat2d.m
- colorbrew.m
- diag_inds.m
- Direct.m
- downdatechol.m
- dprod3.m
- error_ellipse.m
- expandScalarInputs.m
- extract_likelihood_samples.m
- ezmiller.m
- fastsmooth.m
- find_candidates.m
- find_farthest.m
- find_likelihood_samples.m
- find_likelihood_samples_lhs.m
- find_list.m
- find_needle.m
- gradient_ascent.m
- hellinger_distance.m
- improve_covariance_conditioning.m
- intlogspace.m
- inv_logistic.m
- K_wrapper.m
- kfCovEstMissDataRecurse.m
- kron2d.c
- kron2d.m
- kron3d.m
- kron_solve_chol.m
- kronmldivide.m
- kronmult.m
- libsvm-3.0.zip
- libsvm-mat-2.86.zip
- libsvm-mat-3.0-1.zip
- linsolve3.m
- linspacey.m
- logdet.m
- logistic.m
- lognormpdf.m
- make.m
- mat2cell2d.c
- mat2cell2d.m
- mat2cell3d.c
- mat2cell4d.c
- matlabfrag.m
- matrify.m
- meshgrid.c
- meshgrid2d.c
- meshgrid2d.m
- metrickcentres.m
- mf_legend.m
- min_around_points.m
- min_in_box.m
- minFunc.zip
- mvncdf_mosb.m
- mvncdfN.m
- mvtcdfqmc.m
- myaa.m
- notnear.m
- notnear2.m
- perform_convolution.m
- performance.m
- perturbchol.m
- plot_1d_minimize.m
- plot_hessian_approx.m
- process_cov_inputs.m
- prod3.m
- qsimvnef.m
- qsimvnef_mosb.m
- remove_close_data.m
- repmat.c
- revisechol.m
- revisedatahalf.m
- rot_matrix.m
- roundto.m
- sepn.m
- sepn2.m
- set_defaults.m
- setdist.m
- solve_chol.c
- solve_chol.m
- solve_chol.mexa64
- solve_chol3.m
- solve_tz.m
- sq_dist.m
- sqd_dist_stack.m
- squared_distance.m
- suplabel.m
- tensify.m
- toepsolve.c
- toepsolve.m
- toepsolve.mexa64
- tr.m
- trace3.m
- tri2.m
- triu_inds.m
- updatechol.m
- updatedatahalf.m
- updatelogL.m
- wienerDriver.m
- wienerFilter.m
LICENSE
README.md

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MATLAB code for Gaussian process regression, sequential prediction, global optimisation and Bayesian quadrature, written by Maike Osborne during and after a DPhil at Oxford, with contributions from Roman Garnett and David Duvenaud. The last substantive change was made in 2016.

This repository is an archive. It is not maintained, has no automated tests, and targets MATLAB releases of around R2015b. It is kept so that the methods in the papers listed below remain inspectable and, with some effort, reproducible. Anyone wanting a Gaussian process library for new work should look elsewhere.

## What is here

The distinctive feature of this code, relative to toolboxes such as GPML, is that hyperparameters are marginalised rather than optimised. A GP is represented by a `gp` structure holding a set of hyperparameter samples (`gp.hypersamples`), each with its own cached Cholesky factor and log-likelihood. Predictions are combined across samples using Bayesian quadrature weights, and the sample set is managed adaptively by moving samples towards regions of high likelihood.

`CoreGP/` is the library proper. The functions to start with are:

- `train_gp`, `lw_train_gp`: fit a GP to data, optimising the likelihood locally around each of a number of hyperparameter samples. `lw_train_gp` is the lighter-weight version used in the examples.
- `predict_gp`: predictive mean and standard deviation, either from the best hypersample or marginalised over all of them.
- `revise_gp`: the workhorse that conditions a GP on data, with rank-one Cholesky updates and downdates so that data can be added and removed incrementally.
- `posterior_gp`: posterior mean, covariance and their input derivatives for a single hypersample.
- `gpgo`, `spgpgo`: Gaussian process global optimisation, with optional derivative observations. `spgpgo` uses a sparse GP.
- `sbq`, `sbq_gpml`: sequential Bayesian quadrature for estimating a model evidence integral.
- `track_gp`, `fault_bucket`: sequential prediction over time series, including a changepoint and fault model.
- `train_spgp`, `predict_spgp`, `revise_spgp`: sparse pseudo-input GP variants of the above.

Covariance functions live in `CoreGP/covariance_functions/` and are named by string in `set_gp` (`'sqdexp'`, `'matern'`, `'ratquad'`, `'poly'`, `'prodcompact'`). Mean functions are in `CoreGP/mean_functions/` (`'constant'`, `'affine'`, `'planar'`, `'quadratic'`). Standard optimisation test functions are in `CoreGP/testfunctions/`.

The other directories are:

- `Utils/`: numerical helpers, including the Cholesky update and downdate routines, Kronecker solvers, and some vendored third-party code (see below).
- `Gaussian_integral_code/`: Bayesian quadrature for multivariate Gaussian probabilities (`mvncdf_bq`), unpublished work from 2012, together with the expectation propagation (`epmgp`) and lattice (`qsclatmvnv`) methods it was compared against. Some scripts here depend on an external `convex_programming` directory that is not included.
- `sbq_experiments/`: the experimental harness for the NIPS 2012 paper on sequential Bayesian quadrature, largely written by David Duvenaud. Includes the competing Monte Carlo methods, the integration problems and the SDSS spectra used as real data.
- `Scripts/`: one-off experiment and plotting scripts, mostly for Bayesian quadrature for ratios.

## Getting started

Add the repository to the MATLAB path and run one of the example scripts in `CoreGP/`:

```matlab
addpath(genpath('/path/to/gp-code-osborne'))
GP_EXAMPLE        % regression on a one-dimensional test function
GPGO_EXAMPLE      % global optimisation of the Branin function
SBQ_EXAMPLE       % evidence estimation for a Gaussian likelihood
SPGP_EXAMPLE      % sparse GP regression
```

The minimal regression call is

```matlab
opt.optim_time = 60;
gp = lw_train_gp('sqdexp', 'constant', [], X_data, y_data, opt);
[mean, sd] = predict_gp(X_star, gp);
```

where `X_data` is N by D, `y_data` is N by 1, and `optim_time` is the training budget in seconds.

Some routines in `Utils/` have MEX implementations (`solve_chol`, `toepsolve`, `emd_mex`). Prebuilt binaries are included only for 64-bit Linux and, for `emd_mex`, 32-bit Linux and Windows. `solve_chol` and `toepsolve` fall back to pure MATLAB if no binary is found for your platform; `emd_mex` does not, and `Utils/emd/build_emd.m` must be run first. `Utils/make.m` rebuilds `solve_chol` under MATLAB or Octave. The Parallel Computing Toolbox is used where `opt.parallel` is true but is not required.

## Papers

The code was written for the following, listed under the name the papers were published under.

- M. A. Osborne. *Bayesian Gaussian Processes for Sequential Prediction, Optimisation and Quadrature.* DPhil thesis, University of Oxford, 2010. Most of `CoreGP/`.
- M. A. Osborne, R. Garnett and S. J. Roberts. Gaussian processes for global optimization. *LION 3*, 2009. `gpgo`.
- M. A. Osborne, S. J. Roberts, A. Rogers and N. R. Jennings. Real-time information processing of environmental sensor network data using Bayesian Gaussian processes. *ACM Transactions on Sensor Networks*, 2012. `track_gp` and the sensor covariance functions.
- R. Garnett, M. A. Osborne, S. Reece, A. Rogers and S. J. Roberts. Sequential Bayesian prediction in the presence of changepoints and faults. *The Computer Journal*, 2010. `fault_bucket`.
- M. A. Osborne, R. Garnett, S. J. Roberts, C. Hart, S. Aigrain and N. Gibson. Bayesian quadrature for ratios. *AISTATS*, 2012. `Scripts/`.
- M. A. Osborne, D. Duvenaud, R. Garnett, C. E. Rasmussen, S. J. Roberts and Z. Ghahramani. Active learning of model evidence using Bayesian quadrature. *NIPS*, 2012. `sbq` and `sbq_experiments/`.

## Known issues

- Conditioning failures are handled heuristically. `revise_gp` retries with added jitter and, if the Cholesky decomposition still fails, falls back to a diagonal approximation of the covariance with a warning. Predictions after that warning ignore correlations between data points and should not be trusted.
- `improve_covariance_conditioning` returns early with a fixed jitter; the importance-weighted heuristic that follows is dead code.
- Several `_old` files sit alongside their replacements and are not used by the examples.
- There are no tests. The example scripts are the only check that anything runs.

## Licence

The code written by Maike Osborne in `CoreGP/`, `Utils/`, `Gaussian_integral_code/` and `Scripts/` is released under the MIT licence; see `LICENSE`. Any file in those directories that carries its own copyright or licence notice is governed by that notice instead. In particular, three files in `CoreGP/` are copyright Roman Garnett under the ISC licence stated in their headers, and the third-party components listed below carry their own licences.

`sbq_experiments/` is of mixed authorship, much of it by David Duvenaud, and is not covered by the MIT grant. It is retained so that the NIPS 2012 experiments can be reproduced; rights remain with the respective authors.

## Third-party code

- `sbq_experiments/bmc/gpml/`: GPML toolbox version 3.1, Rasmussen and Nickisch.
- `Utils/minFunc.zip`: minFunc, Mark Schmidt.
- `Utils/libsvm-*.zip`: LIBSVM, Chang and Lin.
- `Utils/derivest/`: DERIVEST, John D'Errico.
- `Utils/kdtree/`: kd-tree implementation, Pramod Vemulapalli, BSD licence enclosed.
- `Utils/emd/`: Earth mover's distance, Rubner's C code with a MEX wrapper.
- `sbq_experiments/util/matlab2tikz.m`: matlab2tikz, Nico Schlömer.
- `Gaussian_integral_code/epmgp.m`, `axisepmgp.m`, `truncNormMoments.m`: EPMGP, John P. Cunningham.
- `Gaussian_integral_code/qsclatmvnv.m`: lattice rule for multivariate normal probabilities, Alan Genz.

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