Curve Fitting with Bayesian Ridge Regression

Introduction

This lab demonstrates how to use Bayesian Ridge Regression to fit a polynomial curve to sinusoidal data. We will generate sinusoidal data with noise, fit it using a cubic polynomial and plot the true and predicted curves with log marginal likelihood (L) of these models, we can determine which one is better.

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Skills Graph

Generate sinusoidal data with noise

We start by generating sinusoidal data with noise.

import numpy as np

def func(x):
    return np.sin(2 * np.pi * x)

size = 25
rng = np.random.RandomState(1234)
x_train = rng.uniform(0.0, 1.0, size)
y_train = func(x_train) + rng.normal(scale=0.1, size=size)
x_test = np.linspace(0.0, 1.0, 100)

Fit by cubic polynomial

We fit the data using a cubic polynomial.

from sklearn.linear_model import BayesianRidge

n_order = 3
X_train = np.vander(x_train, n_order + 1, increasing=True)
X_test = np.vander(x_test, n_order + 1, increasing=True)
reg = BayesianRidge(tol=1e-6, fit_intercept=False, compute_score=True)

Plot the true and predicted curves with log marginal likelihood (L)

We plot the true and predicted curves with log marginal likelihood (L).

import matplotlib.pyplot as plt

fig, axes = plt.subplots(1, 2, figsize=(8, 4))
for i, ax in enumerate(axes):
    ## Bayesian ridge regression with different initial value pairs
    if i == 0:
        init = [1 / np.var(y_train), 1.0]  ## Default values
    elif i == 1:
        init = [1.0, 1e-3]
        reg.set_params(alpha_init=init[0], lambda_init=init[1])
    reg.fit(X_train, y_train)
    ymean, ystd = reg.predict(X_test, return_std=True)

    ax.plot(x_test, func(x_test), color="blue", label="sin($2\\pi x$)")
    ax.scatter(x_train, y_train, s=50, alpha=0.5, label="observation")
    ax.plot(x_test, ymean, color="red", label="predict mean")
    ax.fill_between(
        x_test, ymean - ystd, ymean + ystd, color="pink", alpha=0.5, label="predict std"
    )
    ax.set_ylim(-1.3, 1.3)
    ax.legend()
    title = "$\\alpha$_init$={:.2f},\\ \\lambda$_init$={}$".format(init[0], init[1])
    if i == 0:
        title += " (Default)"
    ax.set_title(title, fontsize=12)
    text = "$\\alpha={:.1f}$\n$\\lambda={:.3f}$\n$L={:.1f}$".format(
        reg.alpha_, reg.lambda_, reg.scores_[-1]
    )
    ax.text(0.05, -1.0, text, fontsize=12)

plt.tight_layout()
plt.show()

Summary

Bayesian Ridge Regression is a powerful technique for curve fitting that can be used to fit data to a polynomial curve. By iterating over different initial values for the regularization parameters, we can find the best fit for the given data.