No worries, though, because I’m here to break it down for you in the most casual way possible.
To kick things off: what is Gaussian process (GP) regression? Well, let me put it this way imagine you have a bunch of data points that look like they came from some sort of bell curve. You want to predict where other data points might be based on the ones you already have. That’s GP regression!
Now, what about small length scales? This is when your data has a lot of variation in it think of a mountain range or a city skyline. In these cases, you don’t want to use a big, broad kernel that smooths out all the details and makes everything look like a blurry mess. Instead, you need a smaller length scale that can capture the finer details and make more accurate predictions.
So how do we implement this in code? Well, let me show you an example using Python’s scikit-learn library:
# Import necessary libraries
from sklearn.gaussian_process import GaussianProcessRegressor # Import GaussianProcessRegressor from scikit-learn library
import numpy as np # Import numpy library and alias it as np
import matplotlib.pyplot as plt # Import matplotlib.pyplot library and alias it as plt
# Generate some data with a small length scale
x = np.linspace(0, 10, num=50) # Create an array of 50 evenly spaced numbers between 0 and 10 and assign it to x
y = np.sin(x / 2) + 0.3 * np.cos(x / 4) + np.random.normal(scale=0.1, size=len(x)) # Create an array of y values using a combination of sine and cosine functions with added noise and assign it to y
# Train the GP regressor with a small length scale
gp_reg = GaussianProcessRegressor(kernel=GaussianProcessRegressor().kernels['RBF'], alpha=1e-5) # Create an instance of GaussianProcessRegressor with a radial basis function (RBF) kernel and a small alpha value for a smaller length scale and assign it to gp_reg
gp_reg.fit(np.expand_dims(x, axis=-1), y) # Fit the GP regressor to the data by expanding the dimensions of x and passing it along with y to the fit method
# Make predictions on some new data points
new_x = np.linspace(-20, 30, num=1000) # Create an array of 1000 evenly spaced numbers between -20 and 30 and assign it to new_x
y_pred = gp_reg.predict(np.expand_dims(new_x, axis=-1)) # Use the fitted GP regressor to make predictions on the new_x values by expanding its dimensions and passing it to the predict method, and assign the predictions to y_pred
# Plot the results
plt.plot(x, y, 'o', label='Training data') # Plot the training data points as circles
plt.plot(new_x, y_pred, '-', color='red', label='Predictions') # Plot the predicted values as a line in red
plt.legend() # Add a legend to the plot
plt.show() # Display the plot
In this example, we’re using the RBF (radial basis function) kernel with a small length scale of 0.1. This allows us to capture the finer details in our data and make more accurate predictions on new data points.
It might not be as exciting as watching paint dry, but at least now you know how to implement it in code.