Training hyperparameters for multidimensional Gaussian process regression

Question

Here is a simple working implementation of a code where I use Gaussian process regression (GPR) in Python's scikit-learn with 2-dimensional inputs (i.e grid over x1 and x2) and 1-dimensional outputs (y).

import numpy as np
from matplotlib import pyplot as plt 
from sklearn.gaussian_process import GaussianProcessRegressor 
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C
from mpl_toolkits.mplot3d import Axes3D

#  Example independent variable (observations)
X = np.array([[0.,0.], [1.,0.], [2.,0.], [3.,0.], [4.,0.], 
                [5.,0.], [6.,0.], [7.,0.], [8.,0.], [9.,0.], [10.,0.], 
                [11.,0.], [12.,0.], [13.,0.], [14.,0.],
                [0.,1.], [1.,1.], [2.,1.], [3.,1.], [4.,1.], 
                [5.,1.], [6.,1.], [7.,1.], [8.,1.], [9.,1.], [10.,1.], 
                [11.,1.], [12.,1.], [13.,1.], [14.,1.],
                [0.,2.], [1.,2.], [2.,2.], [3.,2.], [4.,2.], 
                [5.,2.], [6.,2.], [7.,2.], [8.,2.], [9.,2.], [10.,2.], 
                [11.,2.], [12.,2.], [13.,2.], [14.,2.]])#.T

# Example dependent variable (observations) - noiseless case 
y = np.array([4.0, 3.98, 4.01, 3.95, 3.9, 3.84,3.8,
              3.73, 2.7, 1.64, 0.62, 0.59, 0.3, 
              0.1, 0.1,
            4.4, 3.9, 4.05, 3.9, 3.5, 3.4,3.3,
              3.23, 2.6, 1.6, 0.6, 0.5, 0.32, 
              0.05, 0.02,
            4.0, 3.86, 3.88, 3.76, 3.6, 3.4,3.2,
              3.13, 2.5, 1.6, 0.55, 0.51, 0.23, 
              0.11, 0.01]) 

x1 = np.linspace(0, 14, 20)
x2 = np.linspace(0, 5, 100) 

i = 0 
inputs_x = []
while i < len(x1):
    j = 0
    while j < len(x2):
        inputs_x.append([x1[i],x2[j]])
        j = j + 1
    i = i + 1
inputs_x_array = np.array(inputs_x) 

# Instantiate a Gaussian Process model
kernel = C(1.0, (1e-3, 1e3)) * RBF((1e-2, 1e2), (1e-2, 1e2))
gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=20)

gp.fit(X, y.reshape(-1,1)) #removing reshape results in a different error

y_pred, sigma = gp.predict(inputs_x_array, return_std=True)

It works, but when defining the kernel, how can I ensure I set different hyperparameters (e.g. different scale lengths) for my different inputs (i.e. x1 and x2)? In the example above, the standard kernel used is a radial basis function (RBF) which appears to have a single length scale despite two input dimensions. But how could this kernel (or a custom kernel, e.g. hyperbolic tangent) be trained to account for different hyperparameters for the different input dimensions?

Answer 1

You'll need anisotropic kernels, which are only supported by a few kernels in sklearn for the moment. RBF is such an example where you can give a list as input for the length_scale parameter. For example, RBF(length_scale = [1, 10], length_scale_bounds=(1e-5, 1e5)) is perfectly valid, where 1 holds for x1 and 10 holds for x2.

Most kernels in sklearn however are isotropic, where the anisotropic case is -currently- not supported. If you want more freedom, I suggest you take a look at other packages (like GPy) or you can always try to implement your own anisotropic kernel.