Kronecker Structured Covariances

PyMC3 contains implementations for models that have Kronecker structured covariances. This patterned structure enables Gaussian process models to work on much larger datasets. Kronecker structure can be exploited when - The dimension of the input data is two or greater (\(\mathbf{x} \in \mathbb{R}^{d}\,, d \ge 2\)) - The influence of the process across each dimension or set of dimensions is separable - The kernel can be written as a product over dimension, without cross terms:

\[k(\mathbf{x}, \mathbf{x'}) = \prod_{i = 1}^{d} k(\mathbf{x}_{i}, \mathbf{x'}_i) \,.\]

The covariance matrix that corresponds to the covariance function above can be written with a Kronecker product

\[\mathbf{K} = \mathbf{K}_2 \otimes \mathbf{K}_2 \otimes \cdots \otimes \mathbf{K}_d \,.\]

These implementations support the following property of Kronecker products to speed up calculations, \((\mathbf{K}_1 \otimes \mathbf{K}_2)^{-1} = \mathbf{K}_{1}^{-1} \otimes \mathbf{K}_{2}^{-1}\), the inverse of the sum is the sum of the inverses. If \(K_1\) is \(n \times n\) and \(K_2\) is \(m \times m\), then \(\mathbf{K}_1 \otimes \mathbf{K}_2\) is \(mn \times mn\). For \(m\) and \(n\) of even modest size, this inverse becomes impossible to do efficiently. Inverting two matrices, one \(n \times n\) and another \(m \times m\) is much easier.

This structure is common in spatiotemporal data. Given that there is Kronecker structure in the covariance matrix, this implementation is exact – not an approximation to the full Gaussian process. PyMC3 contains two implementations that follow the same pattern as gp.Marginal and gp.Latent. For Kronecker structured covariances where the data likelihood is Gaussian, use gp.MarginalKron. For Kronecker structured covariances where the data likelihood is non-Gaussian, use gp.LatentKron.

Our implementations follow Saatchi’s Thesis. MarginalKron follows “Algorithm 16” using the Eigendecomposition, and LatentKron follows “Algorithm 14”, and uses the Cholesky decomposition.

Using MarginalKron for a 2D spatial problem

The following is a canonical example of the usage of MarginalKron. Like Marginal, this model assumes that the underlying GP is unobserved, but the sum of the GP and normally distributed noise are observed.

For the simulated data set, we draw one sample from a Gaussian process with inputs in two dimensions whose covariance is Kronecker structured. Then we use MarginalKron to recover the unknown Gaussian process hyperparameters \(\theta\) that were used to simulate the data.

Example

We’ll simulate a two dimensional data set and display it as a scatter plot whose points are colored by magnitude. The two dimensions are labeled x1 and x2. This could be a spatial dataset, for instance. The covariance will have a Kronecker structure since the points lie on a two dimensional grid.

import arviz as az
import matplotlib as mpl
import numpy as np
import pymc3 as pm

from numpy.random import default_rng

plt = mpl.pyplot
%matplotlib inline

rng = default_rng(827)

# One dimensional column vectors of inputs
n1, n2 = (50, 30)
x1 = np.linspace(0, 5, n1)
x2 = np.linspace(0, 3, n2)

# make cartesian grid out of each dimension x1 and x2
X = pm.math.cartesian(x1[:, None], x2[:, None])

l1_true = 0.8
l2_true = 1.0
eta_true = 1.0

# Although we could, we don't exploit kronecker structure to draw the sample
cov = (
    eta_true ** 2
    * pm.gp.cov.Matern52(2, l1_true, active_dims=[0])
    * pm.gp.cov.Cosine(2, ls=l2_true, active_dims=[1])
)

K = cov(X).eval()
f_true = rng.multivariate_normal(np.zeros(X.shape[0]), K, 1).flatten()

sigma_true = 0.25
y = f_true + sigma_true * rng.standard_normal(X.shape[0])

The lengthscale along the x2 dimension is longer than the lengthscale along the x1 direction (l1_true < l2_true).

fig = plt.figure(figsize=(12, 6))
cmap = "terrain"
norm = mpl.colors.Normalize(vmin=-3, vmax=3)
plt.scatter(X[:, 0], X[:, 1], s=35, c=y, marker="o", norm=norm, cmap=cmap)
plt.colorbar()
plt.xlabel("x1"), plt.ylabel("x2")
plt.title("Simulated dataset");

There are 1500 data points in this data set. Without using the Kronecker factorization, finding the MAP estimate would be much slower.

Since the two covariances are a product, we only require one scale parameter eta to model the product covariance function.

# this implementation takes a list of inputs for each dimension as input
Xs = [x1[:, None], x2[:, None]]

with pm.Model() as model:
    # Set priors on the hyperparameters of the covariance
    ls1 = pm.Gamma("ls1", alpha=2, beta=2)
    ls2 = pm.Gamma("ls2", alpha=2, beta=2)
    eta = pm.HalfNormal("eta", sigma=2)

    # Specify the covariance functions for each Xi
    # Since the covariance is a product, only scale one of them by eta.
    # Scaling both overparameterizes the covariance function.
    cov_x1 = pm.gp.cov.Matern52(1, ls=ls1)  # cov_x1 must accept X1 without error
    cov_x2 = eta ** 2 * pm.gp.cov.Cosine(1, ls=ls2)  # cov_x2 must accept X2 without error

    # Specify the GP.  The default mean function is `Zero`.
    gp = pm.gp.MarginalKron(cov_funcs=[cov_x1, cov_x2])

    # Set the prior on the variance for the Gaussian noise
    sigma = pm.HalfNormal("sigma", sigma=2)

    # Place a GP prior over the function f.
    y_ = gp.marginal_likelihood("y", Xs=Xs, y=y, sigma=sigma)

with model:
    mp = pm.find_MAP(method="BFGS")

100.00% [90/90 00:02<00:00 logp = -124.24, ||grad|| = 3,375.9]

/Users/v7k/anaconda3/envs/gbi_env_py38/lib/python3.8/site-packages/theano/tensor/nlinalg.py:417: RuntimeWarning: divide by zero encountered in true_divide
  v[:, m] * V.T[n].dot(v[:, m]) / (w[n] - w[m])
/Users/v7k/anaconda3/envs/gbi_env_py38/lib/python3.8/site-packages/theano/tensor/nlinalg.py:417: RuntimeWarning: invalid value encountered in true_divide
  v[:, m] * V.T[n].dot(v[:, m]) / (w[n] - w[m])

mp

{'ls1_log__': array(0.3382134),
 'ls2_log__': array(0.00040654),
 'eta_log__': array(0.9907769),
 'sigma_log__': array(-1.39195968),
 'ls1': array(1.40243975),
 'ls2': array(1.00040662),
 'eta': array(2.69332612),
 'sigma': array(0.24858767)}

Next we use the map point mp to extrapolate in a region outside the original grid. We can also interpolate. There is no grid restriction on the new inputs where predictions are desired. It’s important to note that under the current implementation, having a grid structure in these points doesn’t produce any efficiency gains. The plot with the extrapolations is shown below. The original data is marked with circles as before, but the extrapolated posterior mean is marked with squares.

x1new = np.linspace(5.1, 7.1, 20)
x2new = np.linspace(-0.5, 3.5, 40)
Xnew = pm.math.cartesian(x1new[:, None], x2new[:, None])

mu, var = gp.predict(Xnew, point=mp, diag=True)

fig = plt.figure(figsize=(12, 6))
cmap = "terrain"
norm = mpl.colors.Normalize(vmin=-3, vmax=3)
m = plt.scatter(X[:, 0], X[:, 1], s=30, c=y, marker="o", norm=norm, cmap=cmap)
plt.colorbar(m)
plt.scatter(Xnew[:, 0], Xnew[:, 1], s=30, c=mu, marker="s", norm=norm, cmap=cmap)
plt.ylabel("x2"), plt.xlabel("x1")
plt.title("observed data 'y' (circles) with predicted mean (squares)");

LatentKron

Like the gp.Latent implementation, the LatentKron implementation specifies a Kronecker structured GP regardless of context. It can be used with any likelihood function, or can be used to model a variance or some other unobserved processes. The syntax follows that of gp.Latent exactly.

Example 1

To compare with MarginalLikelihood, we use same example as before where the noise is normal, but the GP itself is not marginalized out. Instead, it is sampled directly using NUTS. It is very important to note that LatentKron does not require a Gaussian likelihood like MarginalKron; rather, any likelihood is admissable.

with pm.Model() as model:
    # Set priors on the hyperparameters of the covariance
    ls1 = pm.Gamma("ls1", alpha=2, beta=2)
    ls2 = pm.Gamma("ls2", alpha=2, beta=2)
    eta = pm.HalfNormal("eta", sigma=2)

    # Specify the covariance functions for each Xi
    cov_x1 = pm.gp.cov.Matern52(1, ls=ls1)
    cov_x2 = eta ** 2 * pm.gp.cov.Cosine(1, ls=ls2)

    # Set the prior on the variance for the Gaussian noise
    sigma = pm.HalfNormal("sigma", sigma=2)

    # Specify the GP.  The default mean function is `Zero`.
    gp = pm.gp.LatentKron(cov_funcs=[cov_x1, cov_x2])

    # Place a GP prior over the function f.
    f = gp.prior("f", Xs=Xs)

    y_ = pm.Normal("y_", mu=f, sigma=sigma, observed=y)

with model:
    tr = pm.sample(500, chains=1, return_inferencedata=True, target_accept=0.90)

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Sequential sampling (1 chains in 1 job)
NUTS: [f_rotated_, sigma, eta, ls2, ls1]

100.00% [1500/1500 07:37<00:00 Sampling chain 0, 0 divergences]

/Users/v7k/anaconda3/envs/gbi_env_py38/lib/python3.8/site-packages/pymc3/math.py:252: RuntimeWarning: divide by zero encountered in log1p
  return np.where(x < 0.6931471805599453, np.log(-np.expm1(-x)), np.log1p(-np.exp(-x)))
Sampling 1 chain for 1_000 tune and 500 draw iterations (1_000 + 500 draws total) took 457 seconds.
Only one chain was sampled, this makes it impossible to run some convergence checks

The posterior distribution of the unknown lengthscale parameters, covariance scaling eta, and white noise sigma are shown below. The vertical lines are the true values that were used to generate the original data set.

az.plot_trace(
    tr,
    var_names=["ls1", "ls2", "eta", "sigma"],
    lines={"ls1": l1_true, "ls2": l2_true, "eta": eta_true, "sigma": sigma_true},
);

/Users/v7k/anaconda3/envs/gbi_env_py38/lib/python3.8/site-packages/arviz/plots/backends/matplotlib/traceplot.py:212: UserWarning: A valid var_name should be provided, found {'s', 'e', 'l'} expected from {'sigma', 'ls1', 'ls2', 'eta'}
  warnings.warn(

x1new = np.linspace(5.1, 7.1, 20)
x2new = np.linspace(-0.5, 3.5, 40)
Xnew = pm.math.cartesian(x1new[:, None], x2new[:, None])

with model:
    fnew = gp.conditional("fnew", Xnew)

with model:
    ppc = pm.sample_posterior_predictive(tr, 200, var_names=["fnew"])

/Users/v7k/anaconda3/envs/gbi_env_py38/lib/python3.8/site-packages/pymc3/sampling.py:1687: UserWarning: samples parameter is smaller than nchains times ndraws, some draws and/or chains may not be represented in the returned posterior predictive sample
  warnings.warn(

100.00% [200/200 02:59<00:00]

Below we show the original data set as colored circles, and the mean of the conditional samples as colored squares. The results closely follow those given by the MarginalKron implementation.

fig = plt.figure(figsize=(14, 7))
m = plt.scatter(X[:, 0], X[:, 1], s=30, c=y, marker="o", norm=norm, cmap=cmap)
plt.colorbar(m)
plt.scatter(
    Xnew[:, 0], Xnew[:, 1], s=30, c=np.mean(ppc["fnew"], axis=0), marker="s", norm=norm, cmap=cmap
)
plt.ylabel("x2"), plt.xlabel("x1")
plt.title("observed data 'y' (circles) with mean of conditional, or predicted, samples (squares)");

Next we plot the original data set indicated with circles markers, along with four samples from the conditional distribution over fnew indicated with square markers. As we can see, the level of variation in the predictive distribution leads to distinctly different patterns in the values of fnew. However, these samples display the correct correlation structure - we see distinct sinusoidal patterns in the y-axis and proximal correlation structure in the x-axis. The patterns displayed in the observed data seamlessly blend into the conditional distribution.

fig, axs = plt.subplots(2, 2, figsize=(24, 16))
axs = axs.ravel()

for i, ax in enumerate(axs):
    ax.axis("off")
    ax.scatter(X[:, 0], X[:, 1], s=20, c=y, marker="o", norm=norm, cmap=cmap)
    ax.scatter(Xnew[:, 0], Xnew[:, 1], s=20, c=ppc["fnew"][i], marker="s", norm=norm, cmap=cmap)
    ax.set_title(f"Sample {i+1}", fontsize=24)

%load_ext watermark
%watermark -n -u -v -iv -w

Last updated: Mon Apr 19 2021

Python implementation: CPython
Python version       : 3.8.8
IPython version      : 7.20.0

pymc3     : 3.11.0
numpy     : 1.19.2
matplotlib: 3.3.4
arviz     : 0.11.0

Watermark: 2.2.0