Gaussian Process (GP) is a powerful 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 RationalQuadratic kernel is a covariance function used in GP that allows for varying smoothness of the function being modeled. This kernel is flexible and can adapt to different types of problems, making it a versatile choice for many applications.
The key hyperparameters for the RationalQuadratic kernel are length_scale and alpha. Length_scale controls the smoothness of the function, with larger values indicating a smoother function. Alpha determines the scale mixture, allowing for different levels of smoothness in different regions. Common values for length_scale are between 1 and 10, and alpha is typically set to values between 0.1 and 2.
The RationalQuadratic kernel is appropriate for regression problems where the relationship between inputs and outputs may vary in smoothness.
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import RationalQuadratic
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]) + X[:, 2] ** 2 + np.random.normal(loc=0, scale=0.5, size=(100,))
# Create an instance of GaussianProcessRegressor with RationalQuadratic kernel
kernel = RationalQuadratic(length_scale=1.0, alpha=0.5)
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: 14.19
Predicted value for test sample: 9.39
The key steps in this code example are:
Dataset preparation: A synthetic dataset is generated where the target variable has a nonlinear relationship with the input features, plus some random noise.
Model instantiation and configuration: An instance of
GaussianProcessRegressoris created with theRationalQuadratickernel, and relevant hyperparameters are set.Model training: The dataset is split into train 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.