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.
[1]:
import gpflow
import numpy as np
import matplotlib.pyplot as plt
plt.style.use("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 asgpflow.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:
[2]:
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)
ax.set_title(k.__class__.__name__)
np.random.seed(27)
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)
2022-03-18 10:02:22.147823: I tensorflow/stream_executor/cuda/cuda_gpu_executor.cc:936] successful NUMA node read from SysFS had negative value (-1), but there must be at least one NUMA node, so returning NUMA node zero
2022-03-18 10:02:22.151138: W tensorflow/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcusolver.so.11'; dlerror: libcusolver.so.11: cannot open shared object file: No such file or directory
2022-03-18 10:02:22.151673: W tensorflow/core/common_runtime/gpu/gpu_device.cc:1850] Cannot dlopen some GPU libraries. Please make sure the missing libraries mentioned above are installed properly if you would like to use GPU. Follow the guide at https://www.tensorflow.org/install/gpu for how to download and setup the required libraries for your platform.
Skipping registering GPU devices...
2022-03-18 10:02:22.151891: I tensorflow/core/platform/cpu_feature_guard.cc:151] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations: AVX2 FMA
To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.
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
.
[3]:
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:
[4]:
from gpflow.utilities import print_summary
print_summary(k)
print_summary(k, fmt="notebook")
# You can change the default format as follows:
gpflow.config.set_default_summary_fmt("notebook")
print_summary(k)
╒═══════════════════════╤═══════════╤═════════════╤═════════╤═════════════╤═════════╤═════════╤═════════╕
│ 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 | Parameter | Softplus | True | () | float64 | 10 | |
Matern32.lengthscales | Parameter | Softplus | True | () | float64 | 2 |
name | class | transform | prior | trainable | shape | dtype | value |
---|---|---|---|---|---|---|---|
Matern32.variance | Parameter | Softplus | True | () | float64 | 10 | |
Matern32.lengthscales | Parameter | Softplus | True | () | float64 | 2 |
You can access the parameter values and assign new values with the same syntax as for models:
[5]:
print(k.lengthscales)
k.lengthscales.assign(0.5)
print(k.lengthscales)
<Parameter: name=softplus, dtype=float64, shape=[], fn="softplus", numpy=2.0>
<Parameter: name=softplus, dtype=float64, shape=[], fn="softplus", numpy=0.5>
Finally, you can call the kernel object to compute covariance matrices:
[6]:
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.figure()
_ = 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:
[7]:
k1 = gpflow.kernels.Matern12()
k2 = gpflow.kernels.Linear()
k3 = k1 + k2
k4 = k1 * k2
print_summary(k3)
print_summary(k4)
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 | Parameter | Softplus | True | () | float64 | 1 | |
Sum.kernels[0].lengthscales | Parameter | Softplus | True | () | float64 | 1 | |
Sum.kernels[1].variance | Parameter | Softplus | True | () | float64 | 1 |
name | class | transform | prior | trainable | shape | dtype | value |
---|---|---|---|---|---|---|---|
Product.kernels[0].variance | Parameter | Softplus | True | () | float64 | 1 | |
Product.kernels[0].lengthscales | Parameter | Softplus | True | () | float64 | 1 | |
Product.kernels[1].variance | Parameter | Softplus | 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:
[8]:
k = gpflow.kernels.Matern52(lengthscales=[0.1, 0.2, 5.0])
print_summary(k)
name | class | transform | prior | trainable | shape | dtype | value |
---|---|---|---|---|---|---|---|
Matern52.variance | Parameter | Softplus | True | () | float64 | 1.0 | |
Matern52.lengthscales | Parameter | Softplus | 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.
[9]:
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:
[10]:
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:
[11]:
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.