"""
==============================
Probability Calibration curves
==============================
When performing classification one often wants to predict not only the class
label, but also the associated probability. This probability gives some
kind of confidence on the prediction. This example demonstrates how to
visualize how well calibrated the predicted probabilities are using calibration
curves, also known as reliability diagrams. Calibration of an uncalibrated
classifier will also be demonstrated.
"""
# Author: Alexandre Gramfort
# Jan Hendrik Metzen
# 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, 10 are
# redundant (random combinations of the informative features) and the
# remaining 8 are uninformative (random numbers). Of the 100,000 samples, 1,000
# will be used for model fitting and the rest 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=10, random_state=42
)
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.99, random_state=42
)
# %%
# Calibration curves
# ------------------
#
# Gaussian Naive Bayes
# ^^^^^^^^^^^^^^^^^^^^
#
# First, we will compare:
#
# * :class:`~sklearn.linear_model.LogisticRegression` (used as baseline
# since very often, properly regularized logistic regression is well
# calibrated by default thanks to the use of the log-loss)
# * Uncalibrated :class:`~sklearn.naive_bayes.GaussianNB`
# * :class:`~sklearn.naive_bayes.GaussianNB` with isotonic and sigmoid
# calibration (see :ref:`User Guide `)
#
# Calibration curves for all 4 conditions are plotted below, with the average
# predicted probability for each bin on the x-axis and the fraction of positive
# classes in each bin on the y-axis.
import matplotlib.pyplot as plt
from matplotlib.gridspec import GridSpec
from sklearn.calibration import CalibratedClassifierCV, CalibrationDisplay
from sklearn.linear_model import LogisticRegression
from sklearn.naive_bayes import GaussianNB
lr = LogisticRegression(C=1.0)
gnb = GaussianNB()
gnb_isotonic = CalibratedClassifierCV(gnb, cv=2, method="isotonic")
gnb_sigmoid = CalibratedClassifierCV(gnb, cv=2, method="sigmoid")
clf_list = [
(lr, "Logistic"),
(gnb, "Naive Bayes"),
(gnb_isotonic, "Naive Bayes + Isotonic"),
(gnb_sigmoid, "Naive Bayes + Sigmoid"),
]
# %%
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 = {}
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),
)
calibration_displays[name] = display
ax_calibration_curve.grid()
ax_calibration_curve.set_title("Calibration plots (Naive Bayes)")
# 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()
# %%
# Uncalibrated :class:`~sklearn.naive_bayes.GaussianNB` is poorly calibrated
# because of
# the redundant features which violate the assumption of feature-independence
# and result in an overly confident classifier, which is indicated by the
# typical transposed-sigmoid curve. Calibration of the probabilities of
# :class:`~sklearn.naive_bayes.GaussianNB` with :ref:`isotonic` can fix
# this issue as can be seen from the nearly diagonal calibration curve.
# :ref:`Sigmoid regression ` also improves calibration
# slightly,
# albeit not as strongly as the non-parametric isotonic regression. This can be
# attributed to the fact that we have plenty of calibration data such that the
# greater flexibility of the non-parametric model can be exploited.
#
# Below we will make a quantitative analysis considering several classification
# metrics: :ref:`brier_score_loss`, :ref:`log_loss`,
# :ref:`precision, recall, F1 score ` and
# :ref:`ROC AUC `.
from collections import defaultdict
import pandas as pd
from sklearn.metrics import (
brier_score_loss,
f1_score,
log_loss,
precision_score,
recall_score,
roc_auc_score,
)
scores = defaultdict(list)
for i, (clf, name) in enumerate(clf_list):
clf.fit(X_train, y_train)
y_prob = clf.predict_proba(X_test)
y_pred = clf.predict(X_test)
scores["Classifier"].append(name)
for metric in [brier_score_loss, log_loss, roc_auc_score]:
score_name = metric.__name__.replace("_", " ").replace("score", "").capitalize()
scores[score_name].append(metric(y_test, y_prob[:, 1]))
for metric in [precision_score, recall_score, f1_score]:
score_name = metric.__name__.replace("_", " ").replace("score", "").capitalize()
scores[score_name].append(metric(y_test, y_pred))
score_df = pd.DataFrame(scores).set_index("Classifier")
score_df.round(decimals=3)
score_df
# %%
# Notice that although calibration improves the :ref:`brier_score_loss` (a
# metric composed
# of calibration term and refinement term) and :ref:`log_loss`, it does not
# significantly alter the prediction accuracy measures (precision, recall and
# F1 score).
# This is because calibration should not significantly change prediction
# probabilities at the location of the decision threshold (at x = 0.5 on the
# graph). Calibration should however, make the predicted probabilities more
# accurate and thus more useful for making allocation decisions under
# uncertainty.
# Further, ROC AUC, should not change at all because calibration is a
# monotonic transformation. Indeed, no rank metrics are affected by
# calibration.
#
# Linear support vector classifier
# ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
# Next, we will compare:
#
# * :class:`~sklearn.linear_model.LogisticRegression` (baseline)
# * Uncalibrated :class:`~sklearn.svm.LinearSVC`. Since SVC does not output
# probabilities by default, we naively scale the output of the
# :term:`decision_function` into [0, 1] by applying min-max scaling.
# * :class:`~sklearn.svm.LinearSVC` with isotonic and sigmoid
# calibration (see :ref:`User Guide `)
import numpy as np
from sklearn.svm import LinearSVC
class NaivelyCalibratedLinearSVC(LinearSVC):
"""LinearSVC with `predict_proba` method that naively scales
`decision_function` output for binary classification."""
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
# %%
lr = LogisticRegression(C=1.0)
svc = NaivelyCalibratedLinearSVC(max_iter=10_000)
svc_isotonic = CalibratedClassifierCV(svc, cv=2, method="isotonic")
svc_sigmoid = CalibratedClassifierCV(svc, cv=2, method="sigmoid")
clf_list = [
(lr, "Logistic"),
(svc, "SVC"),
(svc_isotonic, "SVC + Isotonic"),
(svc_sigmoid, "SVC + Sigmoid"),
]
# %%
fig = plt.figure(figsize=(10, 10))
gs = GridSpec(4, 2)
ax_calibration_curve = fig.add_subplot(gs[:2, :2])
calibration_displays = {}
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),
)
calibration_displays[name] = display
ax_calibration_curve.grid()
ax_calibration_curve.set_title("Calibration plots (SVC)")
# 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()
# %%
# :class:`~sklearn.svm.LinearSVC` shows the opposite
# behavior to :class:`~sklearn.naive_bayes.GaussianNB`; the calibration
# curve has a sigmoid shape, which is typical for an under-confident
# classifier. In the case of :class:`~sklearn.svm.LinearSVC`, this is caused
# by the margin property of the hinge loss, which focuses on samples that are
# close to the decision boundary (support vectors). Samples that are far
# away from the decision boundary do not impact the hinge loss. It thus makes
# sense that :class:`~sklearn.svm.LinearSVC` does not try to separate samples
# in the high confidence region regions. This leads to flatter calibration
# curves near 0 and 1 and is empirically shown with a variety of datasets
# in Niculescu-Mizil & Caruana [1]_.
#
# Both kinds of calibration (sigmoid and isotonic) can fix this issue and
# yield similar results.
#
# As before, we show the :ref:`brier_score_loss`, :ref:`log_loss`,
# :ref:`precision, recall, F1 score ` and
# :ref:`ROC AUC `.
scores = defaultdict(list)
for i, (clf, name) in enumerate(clf_list):
clf.fit(X_train, y_train)
y_prob = clf.predict_proba(X_test)
y_pred = clf.predict(X_test)
scores["Classifier"].append(name)
for metric in [brier_score_loss, log_loss, roc_auc_score]:
score_name = metric.__name__.replace("_", " ").replace("score", "").capitalize()
scores[score_name].append(metric(y_test, y_prob[:, 1]))
for metric in [precision_score, recall_score, f1_score]:
score_name = metric.__name__.replace("_", " ").replace("score", "").capitalize()
scores[score_name].append(metric(y_test, y_pred))
score_df = pd.DataFrame(scores).set_index("Classifier")
score_df.round(decimals=3)
score_df
# %%
# As with :class:`~sklearn.naive_bayes.GaussianNB` above, calibration improves
# both :ref:`brier_score_loss` and :ref:`log_loss` but does not alter the
# prediction accuracy measures (precision, recall and F1 score) much.
#
# Summary
# -------
#
# Parametric sigmoid calibration can deal with situations where the calibration
# curve of the base classifier is sigmoid (e.g., for
# :class:`~sklearn.svm.LinearSVC`) but not where it is transposed-sigmoid
# (e.g., :class:`~sklearn.naive_bayes.GaussianNB`). Non-parametric
# isotonic calibration can deal with both situations but may require more
# data to produce good results.
#
# References
# ----------
#
# .. [1] `Predicting Good Probabilities with Supervised Learning
# `_,
# A. Niculescu-Mizil & R. Caruana, ICML 2005