# A simple demonstration of coregionalization¶

This notebook shows how to construct a multi-output GP model using GPflow. We will consider a regression problem for functions $$f: \mathbb{R}^D \rightarrow \mathbb{R}^P$$. We assume that the dataset is of the form $$(X_1, f_1), \dots, (X_P, f_P)$$, that is, we do not necessarily observe all the outputs for a particular input location (for cases where there are fully observed outputs for each input, see Multi-output Gaussian processes in GPflow for a more efficient implementation). We allow each $$f_i$$ to have a different noise distribution by assigning a different likelihood to each.

For this problem, we model $$f$$ as a coregionalized Gaussian process, which assumes a kernel of the form:

\begin{equation} \textrm{cov}(f_i(X), f_j(X^\prime)) = k(X, X^\prime) \cdot B[i, j]. \end{equation}

The covariance of the $$i$$th function at $$X$$ and the $$j$$th function at $$X^\prime$$ is a kernel applied at $$X$$ and $$X^\prime$$, times the $$(i, j)$$th entry of a positive definite $$P \times P$$ matrix $$B$$. This is known as the intrinsic model of coregionalization (ICM) (Bonilla and Williams, 2008).

To make sure that B is positive-definite, we parameterize it as:

\begin{equation} B = W W^\top + \textrm{diag}(\kappa). \end{equation}

To build such a model in GPflow, we need to perform the following two steps:

• Create the kernel function defined previously, using the Coregion kernel class.

• Augment the training data X with an extra column that contains an integer index to indicate which output an observation is associated with. This is essential to make the data work with the Coregion kernel.

• Create a likelihood for each output using the SwitchedLikelihood class, which is a container for other likelihoods.

• Augment the training data Y with an extra column that contains an integer index to indicate which likelihood an observation is associated with.

:

import gpflow
import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt

%matplotlib inline

from gpflow.ci_utils import ci_niter

plt.rcParams["figure.figsize"] = (12, 6)
np.random.seed(123)


## Data preparation¶

We start by generating some training data to fit the model with. For this example, we choose the following two correlated functions for our outputs:

\begin{align} y_1 &= \sin(6x) + \epsilon_1, \qquad \epsilon_1 \sim \mathcal{N}(0, 0.009) \\ y_2 &= \sin(6x + 0.7) + \epsilon_2, \qquad \epsilon_2 \sim \mathcal{N}(0, 0.01) \end{align}
:

# make a dataset with two outputs, correlated, heavy-tail noise. One has more noise than the other.
X1 = np.random.rand(100, 1)  # Observed locations for first output
X2 = np.random.rand(50, 1) * 0.5  # Observed locations for second output

Y1 = np.sin(6 * X1) + np.random.randn(*X1.shape) * 0.03
Y2 = np.sin(6 * X2 + 0.7) + np.random.randn(*X2.shape) * 0.1

plt.figure(figsize=(8, 4))
plt.plot(X1, Y1, "x", mew=2)
_ = plt.plot(X2, Y2, "x", mew=2) ## Data formatting for the coregionalized model¶

We add an extra column to our training dataset that contains an index that specifies which output is observed.

:

# Augment the input with ones or zeros to indicate the required output dimension
X_augmented = np.vstack((np.hstack((X1, np.zeros_like(X1))), np.hstack((X2, np.ones_like(X2)))))

# Augment the Y data with ones or zeros that specify a likelihood from the list of likelihoods
Y_augmented = np.vstack((np.hstack((Y1, np.zeros_like(Y1))), np.hstack((Y2, np.ones_like(Y2)))))


## Building the coregionalization kernel¶

We build a coregionalization kernel with the Matern 3/2 kernel as the base kernel. This acts on the leading () data dimension of the augmented X values. The Coregion kernel indexes the outputs, and acts on the last () data dimension (indices) of the augmented X values. To specify these dimensions, we use the built-in active_dims argument in the kernel constructor. To construct the full multi-output kernel, we take the product of the two kernels (for a more in-depth tutorial on kernel combination, see Manipulating kernels).

:

output_dim = 2  # Number of outputs
rank = 1  # Rank of W

# Base kernel
k = gpflow.kernels.Matern32(active_dims=)

# Coregion kernel
coreg = gpflow.kernels.Coregion(output_dim=output_dim, rank=rank, active_dims=)

kern = k * coreg


Note: W = 0 is a saddle point in the objective, which would result in the value of W not being optimized to fit the data. Hence, by default, the W matrix is initialized with 0.1. Alternatively, you could re-initialize the matrix to random entries.

## Constructing the model¶

The final element in building the model is to specify the likelihood for each output dimension. To do this, use the SwitchedLikelihood object in GPflow.

:

# This likelihood switches between Gaussian noise with different variances for each f_i:
lik = gpflow.likelihoods.SwitchedLikelihood(
[gpflow.likelihoods.Gaussian(), gpflow.likelihoods.Gaussian()]
)

# now build the GP model as normal
m = gpflow.models.VGP((X_augmented, Y_augmented), kernel=kern, likelihood=lik)

# fit the covariance function parameters
maxiter = ci_niter(10000)
gpflow.optimizers.Scipy().minimize(
m.training_loss, m.trainable_variables, options=dict(maxiter=maxiter), method="L-BFGS-B",
)

:

      fun: -225.66185790128998
hess_inv: <11483x11483 LbfgsInvHessProduct with dtype=float64>
jac: array([ 9.80972157e-03,  2.32377024e-03,  1.01201782e-02, ...,
-1.20376378e-10,  4.51611141e-11,  3.58283369e-03])
message: b'CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH'
nfev: 1443
nit: 1317
status: 0
success: True
x: array([ 7.68640462e-02, -6.25721849e-01, -1.28675184e+00, ...,
-3.86079029e-12, -1.01535023e-12,  4.49399991e-01])


That’s it: the model is trained. Let’s plot the model fit to see what’s happened.

:

def plot_gp(x, mu, var, color, label):
plt.plot(x, mu, color=color, lw=2, label=label)
plt.fill_between(
x[:, 0],
(mu - 2 * np.sqrt(var))[:, 0],
(mu + 2 * np.sqrt(var))[:, 0],
color=color,
alpha=0.4,
)

def plot(m):
plt.figure(figsize=(8, 4))
Xtest = np.linspace(0, 1, 100)[:, None]
(line,) = plt.plot(X1, Y1, "x", mew=2)
mu, var = m.predict_f(np.hstack((Xtest, np.zeros_like(Xtest))))
plot_gp(Xtest, mu, var, line.get_color(), "Y1")

(line,) = plt.plot(X2, Y2, "x", mew=2)
mu, var = m.predict_f(np.hstack((Xtest, np.ones_like(Xtest))))
plot_gp(Xtest, mu, var, line.get_color(), "Y2")

plt.legend()

plot(m) From the plots, we can see:

• The first function (blue) has low posterior variance everywhere because there are so many observations, and the noise variance is small.

• The second function (orange) has higher posterior variance near the data, because the data are more noisy, and very high posterior variance where there are no observations (x > 0.5).

• The model has done a reasonable job of estimating the noise variance and lengthscale.

• The model recognises the correlation between the two functions and is able to suggest (with uncertainty) that because x > 0.5 the orange curve should follow the blue curve (which we know to be the truth from the data-generating procedure).

The covariance matrix between outputs is as follows:

:

B = coreg.output_covariance().numpy()
print("B =", B)
_ = plt.imshow(B)

B = [[2.87939352 2.67437141]
[2.67437141 4.47325892]] 