Gaussian Process (GP) is a probabilistic model used for regression and classification tasks. It is particularly useful when dealing with small datasets or when a measure of uncertainty is required for predictions.
The Matern kernel is a versatile covariance function used in GP that can handle different levels of smoothness in the data. This kernel is beneficial for modeling functions that are not smooth and can adapt to various degrees of smoothness based on the parameter nu. The key hyperparameters for the Matern kernel are the length_scale, nu, and noise level. The length_scale controls the smoothness of the function, while nu determines the smoothness parameter, with common values like 1.5, 2.5, etc. The noise level is often set to a small value for numerical stability.
The Matern kernel is appropriate for both regression and classification problems, especially when the data’s smoothness is uncertain or varies.
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import Matern
from sklearn.model_selection import train_test_split
from sklearn.metrics import mean_squared_error
import numpy as np
# Prepare a synthetic dataset
X = np.random.uniform(low=-5, high=5, size=(100, 3))
y = np.sin(X[:, 0]) + np.cos(X[:, 1]) + np.random.normal(loc=0, scale=0.1, size=(100,))
# Create an instance of GaussianProcessRegressor with Matern kernel
kernel = Matern(length_scale=1.0, nu=1.5)
gp = GaussianProcessRegressor(kernel=kernel, random_state=0)
# Split the dataset into train and test portions
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.2, random_state=0)
# Fit the model on the training data
gp.fit(X_train, y_train)
# Evaluate the model's performance using mean squared error
y_pred = gp.predict(X_test)
mse = mean_squared_error(y_test, y_pred)
print(f"Mean Squared Error: {mse:.2f}")
# Make a prediction using the fitted model on a test sample
test_sample = np.array([[1, -2, 3]])
pred = gp.predict(test_sample)
print(f"Predicted value for test sample: {pred[0]:.2f}")
Running the example gives an output like:
Mean Squared Error: 0.16
Predicted value for test sample: 0.33
The key steps in this code example are:
Dataset preparation: A synthetic dataset is generated where the target variable is influenced by a non-linear combination of the input features and random noise.
Model instantiation and configuration: An instance of
GaussianProcessRegressoris created with theMaternkernel, and relevant hyperparameters are set.Model training: The dataset is split into training and test portions, and the model is fitted on the training data.
Model evaluation: The model’s performance is evaluated using mean squared error on the test set.
Inference on test sample(s): A prediction is made using the fitted model on one test sample, demonstrating how the model can be used for inference on new data.