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# Comparison of Calibration of Classifiers

Well calibrated classifiers are probabilistic classifiers for which the output of predict_proba can be directly interpreted as a confidence level. For instance, a well calibrated (binary) classifier should classify the samples such that for the samples to which it gave a predict_proba value close to 0.8, approximately 80% actually belong to the positive class.

In this example we will compare the calibration of four different models: Logistic regression, Gaussian Naive Bayes, Random Forest Classifier and Linear SVM.

Author: Jan Hendrik Metzen <jhm@informatik.uni-bremen.de> License: BSD 3 clause.

## Dataset

We will use a synthetic binary classification dataset with 100,000 samples and 20 features. Of the 20 features, only 2 are informative, 2 are redundant (random combinations of the informative features) and the remaining 16 are uninformative (random numbers). Of the 100,000 samples, 100 will be used for model fitting and the remaining for testing.

    from sklearn.datasets import make_classification
    from sklearn.model_selection import train_test_split

    X, y = make_classification(
        n_samples=100_000, n_features=20, n_informative=2, n_redundant=2, random_state=42
    )

    train_samples = 100  # Samples used for training the models
    X_train, X_test, y_train, y_test = train_test_split(
        X,
        y,
        shuffle=False,
        test_size=100_000 - train_samples,
    )

## Calibration curves

Below, we train each of the four models with the small training dataset, then plot calibration curves (also known as reliability diagrams) using predicted probabilities of the test dataset. Calibration curves are created by binning predicted probabilities, then plotting the mean predicted probability in each bin against the observed frequency (‘fraction of positives’). Below the calibration curve, we plot a histogram showing the distribution of the predicted probabilities or more specifically, the number of samples in each predicted probability bin.

    import numpy as np

    from sklearn.svm import LinearSVC

    class NaivelyCalibratedLinearSVC(LinearSVC):
        """LinearSVC with `predict_proba` method that naively scales
        `decision_function` output."""

        def fit(self, X, y):
            super().fit(X, y)
            df = self.decision_function(X)
            self.df_min_ = df.min()
            self.df_max_ = df.max()

        def predict_proba(self, X):
            """Min-max scale output of `decision_function` to [0,1]."""
            df = self.decision_function(X)
            calibrated_df = (df - self.df_min_) / (self.df_max_ - self.df_min_)
            proba_pos_class = np.clip(calibrated_df, 0, 1)
            proba_neg_class = 1 - proba_pos_class
            proba = np.c_[proba_neg_class, proba_pos_class]
            return proba

    from sklearn.calibration import CalibrationDisplay
    from sklearn.ensemble import RandomForestClassifier
    from sklearn.linear_model import LogisticRegression
    from sklearn.naive_bayes import GaussianNB

    # Create classifiers
    lr = LogisticRegression()
    gnb = GaussianNB()
    svc = NaivelyCalibratedLinearSVC(C=1.0, dual="auto")
    rfc = RandomForestClassifier()

    clf_list = [
        (lr, "Logistic"),
        (gnb, "Naive Bayes"),
        (svc, "SVC"),
        (rfc, "Random forest"),
    ]

    import matplotlib.pyplot as plt
    from matplotlib.gridspec import GridSpec

    fig = plt.figure(figsize=(10, 10))
    gs = GridSpec(4, 2)
    colors = plt.get_cmap("Dark2")

    ax_calibration_curve = fig.add_subplot(gs[:2, :2])
    calibration_displays = {}
    markers = ["^", "v", "s", "o"]
    for i, (clf, name) in enumerate(clf_list):
        clf.fit(X_train, y_train)
        display = CalibrationDisplay.from_estimator(
            clf,
            X_test,
            y_test,
            n_bins=10,
            name=name,
            ax=ax_calibration_curve,
            color=colors(i),
            marker=markers[i],
        )
        calibration_displays[name] = display

    ax_calibration_curve.grid()
    ax_calibration_curve.set_title("Calibration plots")

    # Add histogram
    grid_positions = [(2, 0), (2, 1), (3, 0), (3, 1)]
    for i, (_, name) in enumerate(clf_list):
        row, col = grid_positions[i]
        ax = fig.add_subplot(gs[row, col])

        ax.hist(
            calibration_displays[name].y_prob,
            range=(0, 1),
            bins=10,
            label=name,
            color=colors(i),
        )
        ax.set(title=name, xlabel="Mean predicted probability", ylabel="Count")

    plt.tight_layout()
    plt.show()

`LogisticRegression` returns well calibrated predictions as it directly optimizes log-loss. In contrast, the other methods return biased probabilities, with different biases for each method:

- `GaussianNB` tends to push probabilities to 0 or 1 (see histogram). This is mainly because the naive Bayes equation only provides correct estimate of probabilities when the assumption that features are conditionally independent holds [2]. However, features tend to be positively correlated and is the case with this dataset, which contains 2 features generated as random linear combinations of the informative features. These correlated features are effectively being ‘counted twice’, resulting in pushing the predicted probabilities towards 0 and 1 [3].

- `RandomForestClassifier` shows the opposite behavior: the histograms show peaks at approx. 0.2 and 0.9 probability, while probabilities close to 0 or 1 are very rare. An explanation for this is given by Niculescu-Mizil and Caruana [1]: “Methods such as bagging and random forests that average predictions from a base set of models can have difficulty making predictions near 0 and 1 because variance in the underlying base models will bias predictions that should be near zero or one away from these values. Because predictions are restricted to the interval [0,1], errors caused by variance tend to be one- sided near zero and one. For example, if a model should predict p = 0 for a case, the only way bagging can achieve this is if all bagged trees predict zero. If we add noise to the trees that bagging is averaging over, this noise will cause some trees to predict values larger than 0 for this case, thus moving the average prediction of the bagged ensemble away from 0. We observe this effect most strongly with random forests because the base-level trees trained with random forests have relatively high variance due to feature subsetting.” As a result, the calibration curve shows a characteristic sigmoid shape, indicating that the classifier is under-confident and could return probabilities closer to 0 or 1.

- To show the performance of `LinearSVC`, we naively scale the output of the decision_function into [0, 1] by applying min-max scaling, since SVC does not output probabilities by default. `LinearSVC` shows an even more sigmoid curve than the `RandomForestClassifier`, which is typical for maximum-margin methods [1] as they focus on difficult to classify samples that are close to the decision boundary (the support vectors).

## References

[1] (1,2)

Predicting Good Probabilities with Supervised Learning, A. Niculescu-Mizil & R. Caruana, ICML 2005

[2]

Beyond independence: Conditions for the optimality of the simple bayesian classifier Domingos, P., & Pazzani, M., Proc. 13th Intl. Conf. Machine Learning. 1996.

[3]

Obtaining calibrated probability estimates from decision trees and naive Bayesian classifiers Zadrozny, Bianca, and Charles Elkan. Icml. Vol. 1. 2001.

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