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Multi-group Gaussian process (MGGP)

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# [Multi-group Gaussian process (MGGP)](https://arxiv.org/abs/2110.08411)
![Build Status](https://github.com/andrewcharlesjones/multi-group-GP/actions/workflows/main.yml/badge.svg)

The multi-group Gaussian process (MGGP) is a generalization of a traditional GP to the setting in which observations are partitioned into a set of known subgroups.

The preprint can be found here: https://arxiv.org/abs/2110.08411.

The multigroupGP package is primarily written using [JAX](https://github.com/google/jax).

## Installation

The Python package can be installed with `pip`:

`pip install multigroupGP`

You may also need to install the dependencies in `requirements.txt`.

## Usage

The multigroupGP package supports both standard GPs and multi-group GP. Crucially, the primary difference between multi- and single-group GPs is the choice of covariance function. In this README, we first show typical usage for standard GPs (since this is likely more familiar to users), and then show how to use the package in the multi-group setting.

### Covariance functions

Currently, there are four covariance implemented in the `multigroupGP` package, each as a subclass inheriting from an abstract `Kernel` class:

- `RBF` (radial basis function kernel, also known as the exponential quadratic)
- `MultiGroupRBF` (multi-group extension of the RBF kernel)
- `Matern12` (Matérn 1/2 kernel)
- `MultiGroupMatern12` (multi-group extension of the Matérn 1/2 kernel)

Each covariance function can be initialized and then called to evaluate the function on all pairs of samples (all classes assume the samples are on the rows). For example, if we have two data matrices `X1` and `X2`, the we could do:

```python
from multigroupGP import RBF
rbf = RBF(amplitude=1., lengthscale=1.)
K = rbf(X1, X2, log_params=False)
```

Here, `log_params` is a boolean argument specifying whether to exponentiate the parameters (this is useful for ensuring positivity).

### Estimation and inference

The `multigroupGP` package offers both maximum likelihood estimation and full Bayesian inference.

**Maximum likelihood**: To fit the model and covariance function parameters via MLE, simply call the `fit` function on a GP object. This uses the Adam optimizer by default.

**Full Bayes**: A PyStan implementation of the MGGP with the multi-group RBF kernel is available in `multigroupGP/models/mggp.stan`. Currently, this only supports the two-group setting, but this will be generalized in the future. Please see `experiments/simulations/posterior_inference.py` for an example.

### Standard GP

Given an `n x p` matrix `X` of explanatory variables and an `n`-vector `y` containing responses, a standard GP can be fit as follows:

```python
from multigroupGP import GP, RBF
gp = GP(kernel=RBF())
gp.fit(X, y)
```

After fitting, predictions can be made for a test matrix `Xtest` as follows:

```python
gp.predict(Xtest)
```

### Multi-group GP

If our data contains multiple known subgroups of samples, we can account for these with an MGGP. Suppose we again have an explanatory matrix `X` and response vector `y`, and we also have an `n`-vector of integer group labels, `Xgroups`. To initialize an MGGP, we have to make two changes:

1. Switch the `is_mggp` argument to `True`,
2. Use an appropriate multi-group covariance (kernel) function.

We can then fit an MGGP as follows:

```python
from multigroupGP import GP, MultiGroupRBF
mggp = GP(kernel=MultiGroupRBF(), is_mggp=True)
mggp.fit(X, y, groups=Xgroups)
```

By default, the MGGP assumes that groups are equally similar to one another a priori. However, if we have prior knowledge about group similarities, we can encode this in the `group_distances` argument of the `fit` function. If we have `k` total groups, `group_distances` argument should be a `k x k` numpy array, where element `ij` contains the "distance" between group `i` and group `j` (lower values indicate that the groups are more similar). Note that the `group_distances` argument is only relevant when `k>2` because the distances will be arbitrarily rescaled during fitting in the two-group case. We can then fit the MGGP as follows:

```python
from multigroupGP import GP, MultiGroupRBF
mggp = GP(kernel=MultiGroupRBF(), is_mggp=True)
mggp.fit(X, y, groups=Xgroups, group_distances=Xgroup_distances)
```

After fitting, predictions can be made for a test matrix `Xtest` as follows:

```python
mggp.predict(Xtest, groups_test=Xtest_groups)
```

## Interpretation of the `a` parameter

The multi-group covariance functions here each have a parameter `a` that controls the strength of similarity between groups. Heuristically, lower values of `a` correspond to MGGPs whose function draws are more similar across groups, and higher values of `a` correspond to MGGPs whose function draws are more dissimilar across groups. We visualize this below by sampling a function for each of two groups using the multi-group RBF kernel:

## Example

Below is a full example for performing multi-group regression with the MGGP. See the `examples/` folder for more.

```python
import jax.numpy as jnp
import jax.random as random
import matplotlib.pyplot as plt
from multigroupGP import GP, MultiGroupRBF
import numpy as onp

key = random.PRNGKey(1)

## Generate data
noise_variance = 0.05
n0, n1 = 100, 100
p = 1
n = n0 + n1
X0 = random.uniform(key, minval=-10, maxval=10, shape=(n0, 1))
X1 = random.uniform(key, minval=-10, maxval=10, shape=(n1, 1))
X = jnp.concatenate([X0, X1], axis=0)

group_dist_mat = onp.ones((2, 2)) - onp.eye(2)

X_groups = onp.concatenate([onp.zeros(n0), onp.ones(n1)]).astype(int)

kernel_true = MultiGroupRBF(amplitude=1.0, lengthscale=1.0, group_diff_param=1.0)
K_XX = (
    kernel_true(
        x1=X,
        groups1=X_groups.astype(float),
        group_distances=group_dist_mat,
    )
    + noise_variance * jnp.eye(n)
)
Y = random.multivariate_normal(random.PRNGKey(12), jnp.zeros(n), K_XX)

## Set up GP
kernel = MultiGroupRBF()
mggp = GP(kernel=kernel, key=key, is_mggp=True)
mggp.fit(
    X,
    Y,
    groups=X_groups,
    group_distances=group_dist_mat,
    group_specific_noise_terms=True,
)

## Predict
ntest = 200
Xtest_onegroup = jnp.linspace(-10, 10, ntest)[:, None]
Xtest = jnp.concatenate([Xtest_onegroup, Xtest_onegroup], axis=0)
Xtest_groups = onp.concatenate([onp.zeros(ntest), onp.ones(ntest)]).astype(int)
preds_mean = mggp.predict(Xtest, groups_test=Xtest_groups)

## Plot
plt.figure(figsize=(10, 5))
plt.scatter(X[:n0], Y[:n0], color="black", label="Data, group 1")
plt.scatter(X[n0:], Y[n0:], color="gray", label="Data, group 2")
plt.plot(Xtest[:ntest], preds_mean[:ntest], color="red", label="Predictions, group 1")
plt.plot(
    Xtest[ntest:], preds_mean[ntest:], color="orange", label="Predictions, group 2"
)
plt.xlabel(r"$X$")
plt.ylabel(r"$y$")
plt.legend()
plt.tight_layout()
plt.show()
```

## Experiments

Code for reproducing the experiments in the paper can be found in the `experiments/` folder.

## Bugs

Please open a GitHub issue to report bugs.

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