Manipulating kernels

GPflow comes with a range of kernels. In this notebook, we examine some of them, show how you can combine them to make new kernels, and discuss the active_dims feature.

import gpflow
import numpy as np
import matplotlib.pyplot as plt"ggplot")
import tensorflow as tf

%matplotlib inline

Standard kernels in GPflow

GPflow comes with lots of standard kernels. Some very simple kernels produce constant functions, linear functions, and white noise functions:

  • gpflow.kernels.Constant

  • gpflow.kernels.Linear

  • gpflow.kernels.White

Some stationary functions produce samples with varying degrees of smoothness:

  • gpflow.kernels.Exponential

  • gpflow.kernels.Matern12

  • gpflow.kernels.Matern32

  • gpflow.kernels.Matern52

  • gpflow.kernels.SquaredExponential (also known as gpflow.kernels.RBF)

  • gpflow.kernels.RationalQuadratic

Two kernels produce periodic samples:

  • gpflow.kernels.Cosine

  • gpflow.kernels.Periodic

Other kernels that are implemented in core GPflow include:

  • gpflow.kernels.Polynomial

  • gpflow.kernels.ArcCosine (“neural network kernel”)

  • gpflow.kernels.Coregion

Let’s define some plotting utils functions and have a look at samples from the prior for some of them:

def plotkernelsample(k, ax, xmin=-3, xmax=3):
    xx = np.linspace(xmin, xmax, 100)[:, None]
    K = k(xx)
    ax.plot(xx, np.random.multivariate_normal(np.zeros(100), K, 3).T)

f, axes = plt.subplots(2, 4, figsize=(12, 6), sharex=True, sharey=True)
plotkernelsample(gpflow.kernels.Matern12(), axes[0, 0])
plotkernelsample(gpflow.kernels.Matern32(), axes[0, 1])
plotkernelsample(gpflow.kernels.Matern52(), axes[0, 2])
plotkernelsample(gpflow.kernels.RBF(), axes[0, 3])
plotkernelsample(gpflow.kernels.Constant(), axes[1, 0])
plotkernelsample(gpflow.kernels.Linear(), axes[1, 1])
plotkernelsample(gpflow.kernels.Cosine(), axes[1, 2])
plotkernelsample(gpflow.kernels.Periodic(gpflow.kernels.SquaredExponential()), axes[1, 3])
_ = axes[0, 0].set_ylim(-3, 3)

First example: create a Matern 3/2 covariance kernel

Many kernels have hyperparameters, for example variance and lengthscales. You can change the value of these parameters from their default value of 1.0.

k = gpflow.kernels.Matern32(variance=10.0, lengthscales=2)

NOTE: The values specified for the variance and lengthscales parameters are floats.

To get information about the kernel, use print_summary(k) (plain text) or, in a notebook, pass the option fmt="notebook" to obtain a nicer rendering:

from gpflow.utilities import print_summary

print_summary(k, fmt="notebook")
# You can change the default format as follows:
│ name                  │ class     │ transform   │ prior   │ trainable   │ shape   │ dtype   │   value │
│ Matern32.variance     │ Parameter │ Softplus    │         │ True        │ ()      │ float64 │      10 │
│ Matern32.lengthscales │ Parameter │ Softplus    │         │ True        │ ()      │ float64 │       2 │
name class transform prior trainable shape dtype value
Matern32.variance ParameterSoftplus True () float64 10
Matern32.lengthscalesParameterSoftplus True () float64 2
name class transform prior trainable shape dtype value
Matern32.variance ParameterSoftplus True () float64 10
Matern32.lengthscalesParameterSoftplus True () float64 2

You can access the parameter values and assign new values with the same syntax as for models:

<Parameter: dtype=float64, shape=[], fn="softplus", numpy=2.0>
<Parameter: dtype=float64, shape=[], fn="softplus", numpy=0.5>

Finally, you can call the kernel object to compute covariance matrices:

X1 = np.array([[0.0]])
X2 = np.linspace(-2, 2, 101).reshape(-1, 1)

K21 = k(X2, X1)  # cov(f(X2), f(X1)): matrix with shape [101, 1]
K22 = k(X2)  # equivalent to k(X2, X2) (but more efficient): matrix with shape [101, 101]

# plotting
_ = plt.plot(X2, K21)

Combine kernels

Sums and products of kernels are also valid kernels. You can add or multiply instances of kernels to create a new composite kernel with the parameters of the old ones:

k1 = gpflow.kernels.Matern12()
k2 = gpflow.kernels.Linear()

k3 = k1 + k2
k4 = k1 * k2


def plotkernelfunction(k, ax, xmin=-3, xmax=3, other=0):
    xx = np.linspace(xmin, xmax, 200)[:, None]
    ax.plot(xx, k(xx, np.zeros((1, 1)) + other))
    ax.set_title(k.__class__.__name__ + " k(x, %f)" % other)

f, axes = plt.subplots(2, 2, figsize=(12, 6), sharex=True)
plotkernelfunction(k3, axes[0, 0], other=1.0)
plotkernelfunction(k4, axes[0, 1], other=1.0)
plotkernelsample(k3, axes[1, 0])
plotkernelsample(k4, axes[1, 1])
name class transform prior trainable shape dtype value
Sum.kernels[0].variance ParameterSoftplus True () float64 1
Sum.kernels[0].lengthscalesParameterSoftplus True () float64 1
Sum.kernels[1].variance ParameterSoftplus True () float64 1
name class transform prior trainable shape dtype value
Product.kernels[0].variance ParameterSoftplus True () float64 1
Product.kernels[0].lengthscalesParameterSoftplus True () float64 1
Product.kernels[1].variance ParameterSoftplus True () float64 1

Kernels for higher-dimensional input spaces

Kernels generalize to multiple dimensions straightforwardly. Stationary kernels support “Automatic Relevance Determination” (ARD), that is, having a different lengthscale parameter for each input dimension. Simply pass in an array of the same length as the number of input dimensions. NOTE: This means that the kernel object is then able to process only inputs of that dimension!

You can also initialize the lengthscales when the object is created:

k = gpflow.kernels.Matern52(lengthscales=[0.1, 0.2, 5.0])
name class transform prior trainable shape dtype value
Matern52.variance ParameterSoftplus True () float641.0
Matern52.lengthscalesParameterSoftplus True (3,) float64[0.1 0.2 5. ]

Specify active dimensions

When combining kernels, it’s often helpful to have bits of the kernel working on different dimensions. For example, to model a function that is linear in the first dimension and smooth in the second, we could use a combination of Linear and Matern52 kernels, one for each dimension.

To tell GPflow which dimension a kernel applies to, specify a list of integers as the value of the active_dims parameter.

k1 = gpflow.kernels.Linear(active_dims=[0])
k2 = gpflow.kernels.Matern52(active_dims=[1])
k = k1 + k2

active_dims makes it easy to create additive models. Here we build an additive Matern 5/2 kernel:

k = gpflow.kernels.Matern52(active_dims=[0], lengthscales=2) + gpflow.kernels.Matern52(
    active_dims=[1], lengthscales=2

Let’s plot this kernel and sample from it:

n_grid = 30
x = np.linspace(-10, 10, n_grid)
X, Y = np.meshgrid(x, x)
X = np.vstack((X.flatten(), Y.flatten())).T

x0 = np.array([[2.0, 2.0]])
# plot the kernel
KxX = k(X, x0).numpy().reshape(n_grid, n_grid)

fig, axes = plt.subplots(1, 3, figsize=(15, 5))
axes[0].imshow(KxX, extent=[-10, 10, -10, 10])
axes[0].set_title(f"$k((7, 5), (x1, x2))$")

# plot a GP sample
K = k(X).numpy()
Z = np.random.multivariate_normal(np.zeros(n_grid ** 2), K, 2)
axes[1].imshow(Z[0, :].reshape(n_grid, n_grid), extent=[-10, 10, -10, 10])
axes[1].set_title("GP sample 1")
axes[2].imshow(Z[1, :].reshape(n_grid, n_grid), extent=[-10, 10, -10, 10])
_ = axes[2].set_title("GP sample 2")

Define new covariance functions

GPflow makes it easy to define new covariance functions. See Kernel design for more information.