Polynomial interpolation

This example demonstrates how to approximate a function with a polynomial of degree n_degree by using ridge regression. Concretely, from n_samples 1d points, it suffices to build the Vandermonde matrix, which is n_samples x n_degree+1 and has the following form:

[[1, x_1, x_1 ** 2, x_1 ** 3, ...],: [1, x_2, x_2 ** 2, x_2 ** 3, ...], ...]

Intuitively, this matrix can be interpreted as a matrix of pseudo features (the points raised to some power). The matrix is akin to (but different from) the matrix induced by a polynomial kernel.

This example shows that you can do non-linear regression with a linear model, using a pipeline to add non-linear features. Kernel methods extend this idea and can induce very high (even infinite) dimensional feature spaces.

../../_images/sphx_glr_plot_polynomial_interpolation_001.png

print(__doc__)
 
# Author: Mathieu Blondel
#         Jake Vanderplas
# License: BSD 3 clause
 
import numpy as np
import matplotlib.pyplot as plt
 
from sklearn.linear_model import Ridge
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline
 
 
def f(x):
    """ function to approximate by polynomial interpolation"""
    return x * np.sin(x)
 
 
# generate points used to plot
x_plot = np.linspace(0, 10, 100)
 
# generate points and keep a subset of them
x = np.linspace(0, 10, 100)
rng = np.random.RandomState(0)
rng.shuffle(x)
x = np.sort(x[:20])
y = f(x)
 
# create matrix versions of these arrays
X = x[:, np.newaxis]
X_plot = x_plot[:, np.newaxis]
 
colors = ['teal', 'yellowgreen', 'gold']
lw = 2
plt.plot(x_plot, f(x_plot), color='cornflowerblue', linewidth=lw,
         label="ground truth")
plt.scatter(x, y, color='navy', s=30, marker='o', label="training points")
 
for count, degree in enumerate([3, 4, 5]):
    model = make_pipeline(PolynomialFeatures(degree), Ridge())
    model.fit(X, y)
    y_plot = model.predict(X_plot)
    plt.plot(x_plot, y_plot, color=colors[count], linewidth=lw,
             label="degree %d" % degree)
 
plt.legend(loc='lower left')
 
plt.show()

Total running time of the script: (0 minutes 0.069 seconds)

Download Python source code: plot_polynomial_interpolation.py

Download IPython notebook: plot_polynomial_interpolation.ipynb

Links:

http://scikit-learn.org/stable/auto_examples/linear_model/plot_polynomial_interpolation.html

doc_scikit_learn

2025-01-10 15:47:30

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