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#configure plotting
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
import matplotlib;matplotlib.rcParams['figure.figsize'] = (8,5)
from matplotlib import pyplot as plt
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import GPy
import numpy as np
For most kernels, the input dimension (domain) is the only mandatory parameter to define a kernel object. However, it is also possible to specify the values of the parameters. For example, the three following commands are valid for defining a squared exponential kernel (ie rbf or Gaussian)
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ker1 = GPy.kern.RBF(1) # Equivalent to ker1 = GPy.kern.rbf(input_dim=1, variance=1., lengthscale=1.)
ker2 = GPy.kern.RBF(input_dim=1, variance = .75, lengthscale=2.)
ker3 = GPy.kern.RBF(1, .5, .5)
A print and a plot function are implemented to represent kernel objects.
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print ker2
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_ = ker1.plot(ax=plt.gca())
_ = ker2.plot(ax=plt.gca())
_ = ker3.plot(ax=plt.gca())
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figure, axes = plt.subplots(3,3, figsize=(10,10), tight_layout=True)
kerns = [GPy.kern.RBF(1), GPy.kern.Exponential(1), GPy.kern.Matern32(1), GPy.kern.Matern52(1), GPy.kern.Brownian(1), GPy.kern.Bias(1), GPy.kern.Linear(1), GPy.kern.PeriodicExponential(1), GPy.kern.White(1)]
for k,a in zip(kerns, axes.flatten()):
k.plot(ax=a, x=1)
a.set_title(k.name.replace('_', ' '))
In GPy, kernel objects can be added or multiplied to create a mutlitude of kernel objects. Parameters and their gradients are handled automatically, and so appear in the combined objects. When kernels are used inside GP objects all the necessary graidents are automagically computed using the chain-rule.
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# Product of kernels
k1 = GPy.kern.RBF(1,1.,2.)
k2 = GPy.kern.Matern32(1, 0.5, 0.2)
k_prod = k1 *k2
print k_prod
k_prod.plot()
# Sum of kernels
k1 = GPy.kern.RBF(1,1.,2.)
k2 = GPy.kern.Matern32(1, 0.5, 0.2)
k_add = k1 + k2
print k_add
k_add.plot()
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Note that the kernels that have been added are pythonic in that the objects remain linked: changing parameters of an add kernel changes those of the constituent parts, and vice versa
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print k1, '\n'
k_add.rbf.variance = 12.
print k1
When multiplying and adding kernels, there are two general possibilites, one can assume that the kernels to add/multiply are defined on the same space or on different spaces:
To keep things as general as possible, in GPy kernels are assigned active_dims which tell the kernel what to work on. To create a kernel which is a product of krnels on different spaces, we can do
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k1 = GPy.kern.Linear(input_dim=1, active_dims=[0]) # works on the first column of X, index=0
k2 = GPy.kern.ExpQuad(input_dim=1, lengthscale=3, active_dims=[1]) # works on the second column of X, index=1
k = k1 * k2
k.plot(x=np.ones((1,2)))
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def plot_sample(k):
xx, yy = np.mgrid[-3:3:30j, -3:3:30j]
X = np.vstack((xx.flatten(), yy.flatten())).T
K = k.K(X)
s = np.random.multivariate_normal(np.zeros(X.shape[0]), K)
#plt.contourf(xx, yy, s.reshape(*xx.shape), cmap=plt.cm.hot)
plt.imshow(s.reshape(*xx.shape), interpolation='nearest')
plt.colorbar()
plot_sample(k)
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k1 = GPy.kern.PeriodicExponential(input_dim=1, active_dims=[0], period=6, lower=-10, upper=10)# + GPy.kern.Bias(1, variance=0, active_dims=[0])
k2 = GPy.kern.PeriodicExponential(input_dim=1, active_dims=[1], period=8, lower=-10, upper=10)# + GPy.kern.Bias(1, variance=0, active_dims=[0])
#k2 = GPy.kern.ExpQuad(1, active_dims=[1])
k = k1 * k2
plot_sample(k)
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