3D- Gaussian Process Regression

Question

I am very new to Gaussian processes and python as well. I am trying to produce a very simple Gaussian regression for a 3d model.

I have a very simple Python code for a function:

import numpy as np

def exponential_cov(x, y, params):

    return params[0] * np.exp( -0.5 * params[1] * np.subtract.outer(x, y)**2)

def conditional(x_new, x, y, params):

    B = exponential_cov(x_new, x, params)
    C = exponential_cov(x, x, params)
    A = exponential_cov(x_new, x_new, params)
    mu = np.linalg.inv(C).dot(B.T).T.dot(y)
    sigma = A - B.dot(np.linalg.inv(C).dot(B.T))
    return(mu.squeeze(), sigma.squeeze())

import matplotlib.pylab as plt

# GP PRIOR
tu = [1, 10]
Si_tu = exponential_cov(0, 0, tu)
xpts = np.arange(-5, 5, step=0.01)

plt.errorbar(xpts, np.zeros(len(xpts)), yerr=Si_tu, capsize=0, color='#95daed', alpha=0.5, label='error')  #error
plt.plot(xpts, np.zeros(len(xpts)), linestyle='dashed', color='#3105b2', linewidth=2.5, label='mu'); #mu

# GP FOR 1ST POINT
x = [1.]
y = np.sin(x)+np.cos(np.sqrt(15)*x)

Si_1 = exponential_cov(x, x, tu)

def predict(x, data, kernel, params, sigma, t):

    k = [kernel(x, y, params) for y in data]
    Sinv = np.linalg.inv(sigma)
    y_pred = np.dot(k, Sinv).dot(t)
    sigma_new = kernel(x, x, params) - np.dot(k, Sinv).dot(k)
    return y_pred, sigma_new

x_pred = np.linspace(-5, 5, 1000) #change step here!!
print "x_pred="
print(x_pred)
predictions = [predict(i, x, exponential_cov, tu, Si_1, y) for i in x_pred]
y_pred, sigmas = np.transpose(predictions)
print "y_pred ="
print(y_pred )
print "sigmas ="
print(sigmas )


# GP FOR 2ND POINT
m, s = conditional([-1], x, y, tu)
y2 = np.sin(-1)+np.cos(np.sqrt(15)*(-1))

x.append(-1)
y=np.append(y,y2)
Si_2 = exponential_cov(x, x, tu)
predictions = [predict(i, x, exponential_cov, tu, Si_2, y) for i in x_pred]
y_pred, sigmas = np.transpose(predictions)
print "y_pred ="
print(y_pred )
print "sigmas ="
print(sigmas )

By using this code I get very nice fitting results for the function np.sin(x) + np.cos(np.sqrt(15) * x), but what I really want to do is to try the same Gaussian process for the function Z = np.sin(2*X) * np.cos(2*Y) / 2.

I know that the idea is basically the same, but I cannot adapt my python code to the [x,y] input to obtain z.

I will really appreciate your help, hints or links!

score 1 · Answer 1 · answered Aug 14 '18 at 11:53

In the previous, the input of your function is 1-D, and then the new function is 2-D. So you have to change the covariance function, for example, use ard-based kernel, please refer to cook book for kernel. Also, you can do the isotropic kernel for 2-D, just make sure the suitable distance function (e.g. L2-norm) and the single lengthscale you choose.

score 0 · Answer 2 · answered Aug 10 '23 at 21:03

Now, you can develop your Gaussian Process (GP) for an N-dimensional case with custom kernels using the scikit-learn library. Other great libraries for GPs are GPy and GPflow (which leverages TensorFlow).

I have recently reply to a question where I've built a 3D GP using scikit-learn, take a look at the code here. Here are a few sample from the GP:

3D- Gaussian Process Regression

2 Answers2