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GPCM

Implementation of the GPCM and variations

Citation:

@inproceedings{Bruinsma:2022:Modelling_Non-Smooth_Signals_With_Complex,
    title = {Modelling Non-Smooth Signals With Complex Spectral Structure},
    year = {2022},
    author = {Wessel P. Bruinsma and Martin Tegn{\' e}r and Richard E. Turner},
    booktitle = {Proceedings of the 25th International Conference on Artificial Intelligence and Statistics},
    series = {Proceedings of Machine Learning Research},
    publisher = {PMLR},
    eprint = {https://arxiv.org/abs/2203.06997},
}

Contents:

• Installation
• Example
• Available Models and Inference Schemes
• Making Predictions
• Sample Experiments
• Reproduce Experiments From the Paper

Installation

See the instructions here. Then simply

pip install gpcm

If you have a GPU available, it is recommended that you use a GPU-accelerated version of JAX.

Example

import numpy as np

from gpcm import RGPCM

model = RGPCM(window=2, scale=1, n_u=30, t=(0, 10))

# Sample from the prior.
t = np.linspace(0, 10, 100)
K, y = model.sample(t)  # Sampled kernel matrix and sampled noisy function

# Fit model to the sample.
model.fit(t, y)

# Compute the ELBO.
print(model.elbo(t, y))

# Make predictions.
posterior = model.condition(t, y)
mean, var = posterior.predict(t)

Available Models and Approximation Schemes

The following models are available:

The simplest way of constructing a model is to set the following keywords:

Example:

from gpcm import RGPCM

model = RGPCM(window=4, scale=0.5, t=(0, 10))

Please see the API for a detailed description of the keyword arguments which you can set. Amongst these keyword arguments, we highlight the following few which are important:

The constructors of these models also take in a keyword scheme, which can be set to one of the following values:

Example:

from gpcm import RGPCM

model = RGPCM(scheme="mean-field-ca", window=4, scale=0.5, t=(0, 10))

Making Predictions With a Model

The implemented models follow the interface from ProbMods.

To begin with, construct a model:

from gpcm import GPCM

model = GPCM(window=4, scale=0.5, t=(0, 10))

Sample From the Paper

Sampling gives back the sampled kernel matrix and the noisy outputs.

K, y = model.sample(t)

Fit the Model to Data

It is recommended that you normalise the data before fitting. It is also recommended that you do not fit the more to more than 1000 data points.

model.fit(t, y)

The function fit takes in the keyword argument iters. The rule of thumb which you can use is as follows:

The function fit also takes in the keyword argument rate. The rule of thumb which you can use here is as follows:

Compute the ELBO

It is recommended that you normalise the data before computing the ELBO. It is also recommended that you do not compute the ELBO for more than 1000 data points.

elbo = model.elbo(t, y)

Condition the Model on Data

It is recommended that you normalise the data before conditioning and unnormalise the predictions. It is also recommended that you do not condition on more than 1000 data points.

posterior_model = model.condition(t, y)

Make Predictions

Predictions for new inputs:

mean, var = posterior_model.predict(t_new)

Predictions for the kernel:

pred = posterior_model.predict_kernel()
x, mean, var = pred.x, pred.mean, pred.var

Predictions for the PSD:

pred = posterior_model.predict_psd()
x, mean, var = pred.x, pred.mean, pred.var

Sample Experiments

Learn a GP With a Known Kernel

python experiments/eq.py

python experiments/smk.py

Learn the Mauna Loa CO2 Data Set

python experiments/mauna_loa.py

Reproduce Experiments From the Paper

See here.