3.6.10.10. Plot fitting a 9th order polynomialΒΆ

Fits data generated from a 9th order polynomial with model of 4th order and 9th order polynomials, to demonstrate that often simpler models are to be prefered

import numpy as np
from matplotlib import pyplot as plt
from matplotlib.colors import ListedColormap
from sklearn import linear_model
# Create color maps for 3-class classification problem, as with iris
cmap_light = ListedColormap(['#FFAAAA', '#AAFFAA', '#AAAAFF'])
cmap_bold = ListedColormap(['#FF0000', '#00FF00', '#0000FF'])
rng = np.random.RandomState(0)
x = 2*rng.rand(100) - 1
f = lambda t: 1.2 * t**2 + .1 * t**3 - .4 * t **5 - .5 * t ** 9
y = f(x) + .4 * rng.normal(size=100)
x_test = np.linspace(-1, 1, 100)

The data

plt.figure(figsize=(6, 4))
plt.scatter(x, y, s=4)
../../../_images/sphx_glr_plot_polynomial_regression_001.png

Fitting 4th and 9th order polynomials

For this we need to engineer features: the n_th powers of x:

plt.figure(figsize=(6, 4))
plt.scatter(x, y, s=4)
X = np.array([x**i for i in range(5)]).T
X_test = np.array([x_test**i for i in range(5)]).T
regr = linear_model.LinearRegression()
regr.fit(X, y)
plt.plot(x_test, regr.predict(X_test), label='4th order')
X = np.array([x**i for i in range(10)]).T
X_test = np.array([x_test**i for i in range(10)]).T
regr = linear_model.LinearRegression()
regr.fit(X, y)
plt.plot(x_test, regr.predict(X_test), label='9th order')
plt.legend(loc='best')
plt.axis('tight')
plt.title('Fitting a 4th and a 9th order polynomial')
../../../_images/sphx_glr_plot_polynomial_regression_002.png

Ground truth

plt.figure(figsize=(6, 4))
plt.scatter(x, y, s=4)
plt.plot(x_test, f(x_test), label="truth")
plt.axis('tight')
plt.title('Ground truth (9th order polynomial)')
plt.show()
../../../_images/sphx_glr_plot_polynomial_regression_003.png

Total running time of the script: ( 0 minutes 0.106 seconds)

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