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# Underfitting vs. Overfitting

This example demonstrates the problems of underfitting and overfitting and how we can use linear regression with polynomial features to approximate nonlinear functions. The plot shows the function that we want to approximate, which is a part of the cosine function. In addition, the samples from the real function and the approximations of different models are displayed. The models have polynomial features of different degrees. We can see that a linear function (polynomial with degree 1) is not sufficient to fit the training samples. This is called underfitting. A polynomial of degree 4 approximates the true function almost perfectly. However, for higher degrees the model will overfit the training data, i.e. it learns the noise of the training data. We evaluate quantitatively overfitting / underfitting by using cross-validation. We calculate the mean squared error (MSE) on the validation set, the higher, the less likely the model generalizes correctly from the training data.

    import matplotlib.pyplot as plt
    import numpy as np

    from sklearn.linear_model import LinearRegression
    from sklearn.model_selection import cross_val_score
    from sklearn.pipeline import Pipeline
    from sklearn.preprocessing import PolynomialFeatures

    def true_fun(X):
        return np.cos(1.5 * np.pi * X)

    np.random.seed(0)

    n_samples = 30
    degrees = [1, 4, 15]

    X = np.sort(np.random.rand(n_samples))
    y = true_fun(X) + np.random.randn(n_samples) * 0.1

    plt.figure(figsize=(14, 5))
    for i in range(len(degrees)):
        ax = plt.subplot(1, len(degrees), i + 1)
        plt.setp(ax, xticks=(), yticks=())

        polynomial_features = PolynomialFeatures(degree=degrees[i], include_bias=False)
        linear_regression = LinearRegression()
        pipeline = Pipeline(
            [
                ("polynomial_features", polynomial_features),
                ("linear_regression", linear_regression),
            ]
        )
        pipeline.fit(X[:, np.newaxis], y)

        # Evaluate the models using crossvalidation
        scores = cross_val_score(
            pipeline, X[:, np.newaxis], y, scoring="neg_mean_squared_error", cv=10
        )

        X_test = np.linspace(0, 1, 100)
        plt.plot(X_test, pipeline.predict(X_test[:, np.newaxis]), label="Model")
        plt.plot(X_test, true_fun(X_test), label="True function")
        plt.scatter(X, y, edgecolor="b", s=20, label="Samples")
        plt.xlabel("x")
        plt.ylabel("y")
        plt.xlim((0, 1))
        plt.ylim((-2, 2))
        plt.legend(loc="best")
        plt.title(
            "Degree {}\nMSE = {:.2e}(+/- {:.2e})".format(
                degrees[i], -scores.mean(), scores.std()
            )
        )
    plt.show()

Total running time of the script: (0 minutes 0.131 seconds)

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