In [37]:
#configure plotting
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
import matplotlib;matplotlib.rcParams['figure.figsize'] = (8,5)
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import numpy as np
from matplotlib import pyplot as plt
import GPy
In [39]:
X = np.random.uniform(-3.,3.,(20,1))
Y = np.sin(X) + np.random.randn(20,1)*0.05
Note that the observations Y include some noise.
The first step is to define the covariance kernel we want to use for the model. We choose here a kernel based on Gaussian kernel (i.e. rbf or square exponential):
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kernel = GPy.kern.RBF(input_dim=1, variance=1., lengthscale=1.)
The parameter input_dim stands for the dimension of the input space. The parameters variance and lengthscale are optional, and default to 1. Many other kernels are implemented, type GPy.kern.<tab> to see a list
In [41]:
GPy.kern.
The inputs required for building the model are the observations and the kernel:
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m = GPy.models.GPRegression(X,Y,kernel)
By default, some observation noise is added to the model. The functions print and plot give an insight of the model we have just built:
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print m
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m.plot()
Out[44]:
The above cell shows our GP regression model before optimization of the parameters. The shaded region corresponds to ~95% confidence intervals (ie +/- 2 standard deviation).
The default values of the kernel parameters may not be optimal for the current data (for example, the confidence intervals seems too wide on the previous figure). A common approach is to find the values of the parameters that maximize the likelihood of the data. It as easy as calling m.optimize in GPy:
In [45]:
m.optimize()
If we want to perform some restarts to try to improve the result of the optimization, we can use the optimize_restarts function. This selects random (drawn from $N(0,1)$) initializations for the parameter values, optimizes each, and sets the model to the best solution found.
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m.optimize_restarts(num_restarts = 10)
In this simple example, the objective function (usually!) has only one local minima, and each of the found solutions are the same.
Once again, we can use print(m) and m.plot() to look at the resulting model resulting model. This time, the paraemters values have been optimized agains the log likelihood (aka the log marginal likelihood): the fit shoul dbe much better.
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print m
_ = m.plot()
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# sample inputs and outputs
X = np.random.uniform(-3.,3.,(50,2))
Y = np.sin(X[:,0:1]) * np.sin(X[:,1:2])+np.random.randn(50,1)*0.05
# define kernel
ker = GPy.kern.Matern52(2,ARD=True) + GPy.kern.White(2)
# create simple GP model
m = GPy.models.GPRegression(X,Y,ker)
# optimize and plot
m.optimize(max_f_eval = 1000)
m.plot()
print(m)
The flag ARD=True in the definition of the Matern kernel specifies that we want one lengthscale parameter per dimension (ie the GP is not isotropic). Note that for 2-d plotting, only the mean is shown.
To see the uncertaintly associated with the above predictions, we can plot slices through the surface. this is done by passing the optional fixed_inputs argument to the plot function. fixed_inputs is a list of tuples containing which of the inputs to fix, and to which value.
To get horixontal slices of the above GP, we'll fix second (index 1) input to -1, 0, and 1.5:
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figure, axes = plt.subplots(3,1)
for ax, y in zip(axes, [-1, 0, 1.5]):
m.plot(fixed_inputs=[(1,y)], ax=ax)
ax.set_ylabel('y=%d'%y)
plt.suptitle('horizontal slices through the function')
Out[34]:
A few things to note:
ax argument, to mnake the GP plot on a particular subplotTo get vertical slices, we simply fixed the other input. We'll turn the display of data off also:
In [35]:
figure, axes = plt.subplots(3,1)
for ax, y in zip(axes, [-1, 0, 1.5]):
m.plot(fixed_inputs=[(1,y)], ax=ax, which_data_rows=[])
ax.set_ylabel('y=%d'%y)
plt.suptitle('vertical slices through the function')
Out[35]:
You can find a host of other plotting options in the m.plot docstring. Type m.plot?<enter> to see.