# 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](./multioutput.ipynb) 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.


```python
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}



```python
# 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.


```python
# 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 ([0]) data dimension of the augmented X values. The `Coregion` kernel indexes the outputs, and acts on the last ([1]) 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](./kernels.ipynb)).


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

# Base kernel
k = gpflow.kernels.Matern32(active_dims=[0])

# Coregion kernel
coreg = gpflow.kernels.Coregion(output_dim=output_dim, rank=rank, active_dims=[1])

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.


```python
# 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.


```python
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:


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

## References

Bonilla, Edwin V., Kian M. Chai, and Christopher Williams. "Multi-task Gaussian process prediction." _Advances in neural information processing systems_. 2008.

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