# Gaussian Process Regression with Shogun¶

Import all necessary modules from Shogun

In [1]:
from modshogun import RealFeatures, RegressionLabels, GaussianKernel, Math
from modshogun import GaussianLikelihood, ZeroMean, ExactInferenceMethod, GaussianProcessRegression


Generate some data, a 1d noisy sine wave, evaluated at random points

In [2]:
n=15
x_range=4*pi
y_noise_variance=0.1
y_amplitude=1
y_frequency=1

X=random.rand(1,n)*x_range
Y=sin(X*y_frequency)*y_amplitude+randn(n)*sqrt(y_noise_variance)
X_test=linspace(0,x_range, 200)
Y_true=sin(X_test)

plot(X_test,Y_true, 'b-')
plot(X,Y, 'ro')
_=legend(['data generating model', 'noisy observations'])


Convert data into Shogun representation, print dimensions to be sure data was passed in correct

In [3]:
labels=RegressionLabels(Y.ravel())
feats_train=RealFeatures(X)
feats_test=RealFeatures(reshape(X_test, (1, len(X_test))))

print feats_train.get_num_vectors()
print feats_train.get_num_features()
print feats_test.get_num_vectors()
print feats_test.get_num_features()

15
1
200
1



Specify a Shogun GP (exact GP-regression) with fixed hyper-parameters and pass it the data

In [4]:
kernel_sigma=1
gp_obs_noise=0.5

kernel=GaussianKernel(10, kernel_sigma)
mean=ZeroMean()
lik=GaussianLikelihood(gp_obs_noise)
inf=ExactInferenceMethod(kernel, feats_train, mean, labels, lik)
gp = GaussianProcessRegression(inf)


Train GP and plot its predictions

In [5]:
_=gp.train()


Perform inference and plot predictions on full range

In [6]:
predictions=gp.apply(feats_test)
Y_test=predictions.get_labels()

plot(X_test,Y_true, 'b')
plot(X_test, Y_test, 'r-')
plot(X,Y, 'ro')
_=legend(['data generating model', 'mean predictions', 'noisy observations'])


So far so good. The nice thing is: we have a distribution over the predictions

In [7]:
mean = gp.get_mean_vector(feats_test)
variance = gp.get_variance_vector(feats_test)

# print 95% confidence region
plot(X_test,Y_true, 'b')
plot(X_test, Y_test, 'r-')
plot(X,Y, 'ro')
error=1.96*sqrt(variance)
fill_between(X_test,mean-error,mean+error,color='grey')

ylim([-y_amplitude,y_amplitude+1])
_=legend(['data generating model', 'mean predictions', 'noisy observations'])


We can even visualise it more fancy

In [8]:
from scipy.stats import norm

means = gp.get_mean_vector(feats_test)
variances = gp.get_variance_vector(feats_test)

y_values=linspace(-y_amplitude-2*y_noise_variance, y_amplitude+2*y_noise_variance)
D=zeros((len(y_values), len(X_test)))

# evaluate normal distribution at every prediction point (column)
for i in range(shape(D)[1]):
norm.pdf(y_values, means[i], variances[i])
D[:,i]=norm.pdf(y_values, means[i], variances[i])

pcolor(X_test,y_values,D)
plot(X_test,Y_true, 'b')
plot(X_test, Y_test, 'r-')
_=plot(X,Y, 'ro')



Now lets learn the best model parameters with Maximum Likelihood II

In [9]:
from shogun.ModelSelection import GradientModelSelection, ModelSelectionParameters, R_LINEAR, R_EXP

kernel=GaussianKernel(10, kernel_sigma)
mean=ZeroMean()
lik=GaussianLikelihood(gp_obs_noise)
inf=ExactInferenceMethod(kernel, feats_train, mean, labels, lik)
gp = GaussianProcessRegression(inf)

# construct model selection parameter tree
root=ModelSelectionParameters();
c1=ModelSelectionParameters("inference_method", inf);
root.append_child(c1);

c2=ModelSelectionParameters("likelihood_model", lik);
c1.append_child(c2);

c3=ModelSelectionParameters("sigma");
c2.append_child(c3);
c3.build_values(-1.0, 1.0, R_EXP);

c4=ModelSelectionParameters("scale");
c1.append_child(c4);
c4.build_values(-1.0, 1.0, R_EXP);

c5=ModelSelectionParameters("kernel", kernel);
c1.append_child(c5);

c6=ModelSelectionParameters("width");
c5.append_child(c6);
c6.build_values(-1.0, 1.0, R_EXP);

# Evaluate our inference method for its derivatives

gp.print_modsel_params()
root.print_tree()

# gradient descent on marginal likelihood

# Set autolocking to false to get rid of warnings

# Search for best parameters

# apply them to gp
best_combination.apply_to_machine(gp)

# training and inference with learned parameters
gp.train()
predictions=gp.apply(feats_test)
Y_test=predictions.get_labels()

# visualise
means = gp.get_mean_vector(feats_test)
variances = gp.get_variance_vector(feats_test)

y_values=linspace(-y_amplitude-2*y_noise_variance, y_amplitude+2*y_noise_variance)
D=zeros((len(y_values), len(X_test)))

# evaluate normal distribution at every prediction point (column)
for i in range(shape(D)[1]):
norm.pdf(y_values, means[i], variances[i])
D[:,i]=norm.pdf(y_values, means[i], variances[i])

pcolor(X_test,y_values,D)
plot(X_test,Y_true, 'b')
plot(X_test, Y_test, 'r-')
_=plot(X,Y, 'ro')

# print best parameters
print "kernel width", kernel.get_width()
print "kernel scalling", inf.get_scale()
print "noise level", lik.get_sigma()

kernel width 3.6679925725
kernel scalling 0.785651526209
noise level 0.5


In [9]: