Polynomial Regression

Toy Data

import numpy as np
import matplotlib.pyplot as plt
np.random.seed(0)

N = 100
X = np.linspace(-5,5,N)
noise = np.random.randn(N)
y = X**2+noise

plt.scatter(X, y, label='actual')
plt.legend();

Linear Regression

from sklearn.linear_model import LinearRegression
lin_reg = LinearRegression()
lin_reg.fit(X.reshape(-1,1), y)
y_l = lin_reg.predict(X.reshape(-1,1))

plt.scatter(X, y, label='actual')
plt.plot(X, y_l, label='linear model', c='orange')
plt.legend();

Polynomial Features

from sklearn.preprocessing import PolynomialFeatures

poly = PolynomialFeatures(degree=2, include_bias=False)

poly.fit(X.reshape(-1,1))

poly.get_feature_names_out()

array(['x0', 'x0^2'], dtype=object)

X_poly = poly.transform(X.reshape(-1,1))

X_poly.shape

(100, 2)

Polynomial Regression

lin_reg = LinearRegression()
lin_reg.fit(X_poly, y)
y_poly = lin_reg.predict(X_poly)

plt.scatter(X, y, label='actual')
plt.plot(X, y_l, label='linear model', c='orange')
plt.plot(X, y_poly, label='polynomial model', c='r')

plt.legend();

California Housing Data

from sklearn.datasets import fetch_california_housing
X, y = fetch_california_housing(return_X_y=True)

from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=0)

poly = PolynomialFeatures(degree=2, include_bias=False)

poly.fit(X_train)

poly.get_feature_names_out()

array(['x0', 'x1', 'x2', 'x3', 'x4', 'x5', 'x6', 'x7', 'x0^2', 'x0 x1',
       'x0 x2', 'x0 x3', 'x0 x4', 'x0 x5', 'x0 x6', 'x0 x7', 'x1^2',
       'x1 x2', 'x1 x3', 'x1 x4', 'x1 x5', 'x1 x6', 'x1 x7', 'x2^2',
       'x2 x3', 'x2 x4', 'x2 x5', 'x2 x6', 'x2 x7', 'x3^2', 'x3 x4',
       'x3 x5', 'x3 x6', 'x3 x7', 'x4^2', 'x4 x5', 'x4 x6', 'x4 x7',
       'x5^2', 'x5 x6', 'x5 x7', 'x6^2', 'x6 x7', 'x7^2'], dtype=object)

X_train_poly = poly.transform(X_train)
X_test_poly = poly.transform(X_test)

X_train_poly.shape, X_test_poly.shape

((15480, 44), (5160, 44))

lin_reg = LinearRegression()
lin_reg.fit(X_train_poly, y_train)

LinearRegression()

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lin_reg.score(X_train_poly, y_train)

0.685934216623096

lin_reg.score(X_test_poly, y_test)

-0.7296138933574114