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"""
=================================================================
Test with permutations the significance of a classification score
=================================================================
This example demonstrates the use of
:func:`~sklearn.model_selection.permutation_test_score` to evaluate the
significance of a cross-validated score using permutations.
"""
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
# %%
# Dataset
# -------
#
# We will use the :ref:`iris_dataset`, which consists of measurements taken
# from 3 Iris species. Our model will use the measurements to predict
# the iris species.
from sklearn.datasets import load_iris
iris = load_iris()
X = iris.data
y = iris.target
# %%
# For comparison, we also generate some random feature data (i.e., 20 features),
# uncorrelated with the class labels in the iris dataset.
import numpy as np
n_uncorrelated_features = 20
rng = np.random.RandomState(seed=0)
# Use same number of samples as in iris and 20 features
X_rand = rng.normal(size=(X.shape[0], n_uncorrelated_features))
# %%
# Permutation test score
# ----------------------
#
# Next, we calculate the
# :func:`~sklearn.model_selection.permutation_test_score` for both, the original
# iris dataset (where there's a strong relationship between features and labels) and
# the randomly generated features with iris labels (where no dependency between features
# and labels is expected). We use the
# :class:`~sklearn.svm.SVC` classifier and :ref:`accuracy_score` to evaluate
# the model at each round.
#
# :func:`~sklearn.model_selection.permutation_test_score` generates a null
# distribution by calculating the accuracy of the classifier
# on 1000 different permutations of the dataset, where features
# remain the same but labels undergo different random permutations. This is the
# distribution for the null hypothesis which states there is no dependency
# between the features and labels. An empirical p-value is then calculated as
# the proportion of permutations, for which the score obtained by the model trained on
# the permutation, is greater than or equal to the score obtained using the original
# data.
from sklearn.model_selection import StratifiedKFold, permutation_test_score
from sklearn.svm import SVC
clf = SVC(kernel="linear", random_state=7)
cv = StratifiedKFold(n_splits=2, shuffle=True, random_state=0)
score_iris, perm_scores_iris, pvalue_iris = permutation_test_score(
clf, X, y, scoring="accuracy", cv=cv, n_permutations=1000
)
score_rand, perm_scores_rand, pvalue_rand = permutation_test_score(
clf, X_rand, y, scoring="accuracy", cv=cv, n_permutations=1000
)
# %%
# Original data
# ^^^^^^^^^^^^^
#
# Below we plot a histogram of the permutation scores (the null
# distribution). The red line indicates the score obtained by the classifier
# on the original data (without permuted labels). The score is much better than those
# obtained by using permuted data and the p-value is thus very low. This indicates that
# there is a low likelihood that this good score would be obtained by chance
# alone. It provides evidence that the iris dataset contains real dependency
# between features and labels and the classifier was able to utilize this
# to obtain good results. The low p-value can lead us to reject the null hypothesis.
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
ax.hist(perm_scores_iris, bins=20, density=True)
ax.axvline(score_iris, ls="--", color="r")
score_label = (
f"Score on original\niris data: {score_iris:.2f}\n(p-value: {pvalue_iris:.3f})"
)
ax.text(0.7, 10, score_label, fontsize=12)
ax.set_xlabel("Accuracy score")
_ = ax.set_ylabel("Probability density")
# %%
# Random data
# ^^^^^^^^^^^
#
# Below we plot the null distribution for the randomized data. The permutation
# scores are similar to those obtained using the original iris dataset
# because the permutation always destroys any feature-label dependency present.
# The score obtained on the randomized data in this case
# though, is very poor. This results in a large p-value, confirming that there was no
# feature-label dependency in the randomized data.
fig, ax = plt.subplots()
ax.hist(perm_scores_rand, bins=20, density=True)
ax.set_xlim(0.13)
ax.axvline(score_rand, ls="--", color="r")
score_label = (
f"Score on original\nrandom data: {score_rand:.2f}\n(p-value: {pvalue_rand:.3f})"
)
ax.text(0.14, 7.5, score_label, fontsize=12)
ax.set_xlabel("Accuracy score")
ax.set_ylabel("Probability density")
plt.show()
# %%
# Another possible reason for obtaining a high p-value could be that the classifier
# was not able to use the structure in the data. In this case, the p-value
# would only be low for classifiers that are able to utilize the dependency
# present. In our case above, where the data is random, all classifiers would
# have a high p-value as there is no structure present in the data. We might or might
# not fail to reject the null hypothesis depending on whether the p-value is high on a
# more appropriate estimator as well.
#
# Finally, note that this test has been shown to produce low p-values even
# if there is only weak structure in the data [1]_.
#
# .. rubric:: References
#
# .. [1] Ojala and Garriga. `Permutation Tests for Studying Classifier
# Performance
# <http://www.jmlr.org/papers/volume11/ojala10a/ojala10a.pdf>`_. The
# Journal of Machine Learning Research (2010) vol. 11
#
|
