How to correctly use scikit-learn's Gaussian Process for a 2D-inputs, 1D-output regression?

Question

Prior to posting I did a lot of searches and found this question which might be exactly my problem. However, I tried what is proposed in the answer but unfortunately this did not fix it, and I couldn't add a comment to request further explanation, as I am a new member here.

Anyway, I want to use the Gaussian Processes with scikit-learn in Python on a simple but real case to start (using the examples provided in scikit-learn's documentation). I have a 2D input set (8 couples of 2 parameters) called X. I have 8 corresponding outputs, gathered in the 1D-array y.

#  Inputs: 8 points 
X = np.array([[p1, q1],[p2, q2],[p3, q3],[p4, q4],[p5, q5],[p6, q6],[p7, q7],[p8, q8]])

# Observations: 8 couples
y = np.array([r1,r2,r3,r4,r5,r6,r7,r8])

I defined an input test space x:

# Input space
x1 = np.linspace(x1min, x1max) #p
x2 = np.linspace(x2min, x2max) #q
x = (np.array([x1, x2])).T

Then I instantiate the GP model, fit it to my training data (X,y), and make the 1D prediction y_pred on my input space x:

from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C

kernel = C(1.0, (1e-3, 1e3)) * RBF([5,5], (1e-2, 1e2))
gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=15)
gp.fit(X, y)
y_pred, MSE = gp.predict(x, return_std=True)

And then I make a 3D plot:

fig = pl.figure()
ax = fig.add_subplot(111, projection='3d')
Xp, Yp = np.meshgrid(x1, x2)
Zp = np.reshape(y_pred,50)

surf = ax.plot_surface(Xp, Yp, Zp, rstride=1, cstride=1, cmap=cm.jet,
linewidth=0, antialiased=False)
pl.show()

This is what I obtain:

When I modify the kernel parameters I get something like this, similar to what the poster I mentioned above got:

These plots don't even match the observation from the original training points (the lower response is obtained for [65.1,37] and the highest for [92.3,54]).

I am fairly new to GPs in 2D (also started Python not long ago) so I think I'm missing something here... Any answer would be helpful and greatly appreciated, thanks!

Answer 1

You're using two features to predict a third. Rather than a 3D plot like plot_surface, it's usually clearer if you use a 2D plot that's able to show information about a third dimension, like hist2d or pcolormesh. Here's a complete example using data/code similar to that in the question:

from itertools import product
import numpy as np
from matplotlib import pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RBF, ConstantKernel as C

X = np.array([[0,0],[2,0],[4,0],[6,0],[8,0],[10,0],[12,0],[14,0],[16,0],[0,2],
                    [2,2],[4,2],[6,2],[8,2],[10,2],[12,2],[14,2],[16,2]])

y = np.array([-54,-60,-62,-64,-66,-68,-70,-72,-74,-60,-62,-64,-66,
                    -68,-70,-72,-74,-76])

# Input space
x1 = np.linspace(X[:,0].min(), X[:,0].max()) #p
x2 = np.linspace(X[:,1].min(), X[:,1].max()) #q
x = (np.array([x1, x2])).T

kernel = C(1.0, (1e-3, 1e3)) * RBF([5,5], (1e-2, 1e2))
gp = GaussianProcessRegressor(kernel=kernel, n_restarts_optimizer=15)

gp.fit(X, y)

x1x2 = np.array(list(product(x1, x2)))
y_pred, MSE = gp.predict(x1x2, return_std=True)

X0p, X1p = x1x2[:,0].reshape(50,50), x1x2[:,1].reshape(50,50)
Zp = np.reshape(y_pred,(50,50))

# alternative way to generate equivalent X0p, X1p, Zp
# X0p, X1p = np.meshgrid(x1, x2)
# Zp = [gp.predict([(X0p[i, j], X1p[i, j]) for i in range(X0p.shape[0])]) for j in range(X0p.shape[1])]
# Zp = np.array(Zp).T

fig = plt.figure(figsize=(10,8))
ax = fig.add_subplot(111)
ax.pcolormesh(X0p, X1p, Zp)

plt.show()

Output:

Kinda plain looking, but so was my example data. In general, you shouldn't expect to get particular interesting resulting with this few data points.

Also, if you do want the surface plot, you can just replace the pcolormesh line with what you originally had (more or less):

ax = fig.add_subplot(111, projection='3d')            
surf = ax.plot_surface(X0p, X1p, Zp, rstride=1, cstride=1, cmap='jet', linewidth=0, antialiased=False)

Output:

Answer 2

I'm also fairly new using scikit-learn gaussian process. But after some effort, I managed to implement a 3-d gaussian process regression successfully. There are a lot of examples of 1-d regression but nothing on higher input dimensions.

Perhaps you could show the values that you are using.

I found that sometimes the format in which you send the inputs can produce some issues. Try formatting input X as:

X = np.array([param1, param2]).T

and format the output as:

gp.fit(X, y.reshape(-1,1))

Also, as I understood, the implementation assumes a mean function m=0. If the output you are trying to regress presents an average value which differs significantly from 0 you should normalize it (that will probably solve your problem). Standardizing the parameter space will help as well.