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        "\n# Test with permutations the significance of a classification score\n\nThis example demonstrates the use of\n:func:`~sklearn.model_selection.permutation_test_score` to evaluate the\nsignificance of a cross-validated score using permutations.\n"
      ]
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      "execution_count": null,
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      "source": [
        "# Authors: The scikit-learn developers\n# SPDX-License-Identifier: BSD-3-Clause"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Dataset\n\nWe will use the `iris_dataset`, which consists of measurements taken\nfrom 3 types of irises.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "from sklearn.datasets import load_iris\n\niris = load_iris()\nX = iris.data\ny = iris.target"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "We will also generate some random feature data (i.e., 20 features),\nuncorrelated with the class labels in the iris dataset.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import numpy as np\n\nn_uncorrelated_features = 20\nrng = np.random.RandomState(seed=0)\n# Use same number of samples as in iris and 20 features\nX_rand = rng.normal(size=(X.shape[0], n_uncorrelated_features))"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## Permutation test score\n\nNext, we calculate the\n:func:`~sklearn.model_selection.permutation_test_score` using the original\niris dataset, which strongly predict the labels and\nthe randomly generated features and iris labels, which should have\nno dependency between features and labels. We use the\n:class:`~sklearn.svm.SVC` classifier and `accuracy_score` to evaluate\nthe model at each round.\n\n:func:`~sklearn.model_selection.permutation_test_score` generates a null\ndistribution by calculating the accuracy of the classifier\non 1000 different permutations of the dataset, where features\nremain the same but labels undergo different permutations. This is the\ndistribution for the null hypothesis which states there is no dependency\nbetween the features and labels. An empirical p-value is then calculated as\nthe percentage of permutations for which the score obtained is greater\nthat the score obtained using the original data.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
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      "outputs": [],
      "source": [
        "from sklearn.model_selection import StratifiedKFold, permutation_test_score\nfrom sklearn.svm import SVC\n\nclf = SVC(kernel=\"linear\", random_state=7)\ncv = StratifiedKFold(2, shuffle=True, random_state=0)\n\nscore_iris, perm_scores_iris, pvalue_iris = permutation_test_score(\n    clf, X, y, scoring=\"accuracy\", cv=cv, n_permutations=1000\n)\n\nscore_rand, perm_scores_rand, pvalue_rand = permutation_test_score(\n    clf, X_rand, y, scoring=\"accuracy\", cv=cv, n_permutations=1000\n)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Original data\n\nBelow we plot a histogram of the permutation scores (the null\ndistribution). The red line indicates the score obtained by the classifier\non the original data. The score is much better than those obtained by\nusing permuted data and the p-value is thus very low. This indicates that\nthere is a low likelihood that this good score would be obtained by chance\nalone. It provides evidence that the iris dataset contains real dependency\nbetween features and labels and the classifier was able to utilize this\nto obtain good results.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "import matplotlib.pyplot as plt\n\nfig, ax = plt.subplots()\n\nax.hist(perm_scores_iris, bins=20, density=True)\nax.axvline(score_iris, ls=\"--\", color=\"r\")\nscore_label = f\"Score on original\\ndata: {score_iris:.2f}\\n(p-value: {pvalue_iris:.3f})\"\nax.text(0.7, 10, score_label, fontsize=12)\nax.set_xlabel(\"Accuracy score\")\n_ = ax.set_ylabel(\"Probability density\")"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "### Random data\n\nBelow we plot the null distribution for the randomized data. The permutation\nscores are similar to those obtained using the original iris dataset\nbecause the permutation always destroys any feature label dependency present.\nThe score obtained on the original randomized data in this case though, is\nvery poor. This results in a large p-value, confirming that there was no\nfeature label dependency in the original data.\n\n"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "collapsed": false
      },
      "outputs": [],
      "source": [
        "fig, ax = plt.subplots()\n\nax.hist(perm_scores_rand, bins=20, density=True)\nax.set_xlim(0.13)\nax.axvline(score_rand, ls=\"--\", color=\"r\")\nscore_label = f\"Score on original\\ndata: {score_rand:.2f}\\n(p-value: {pvalue_rand:.3f})\"\nax.text(0.14, 7.5, score_label, fontsize=12)\nax.set_xlabel(\"Accuracy score\")\nax.set_ylabel(\"Probability density\")\nplt.show()"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Another possible reason for obtaining a high p-value is that the classifier\nwas not able to use the structure in the data. In this case, the p-value\nwould only be low for classifiers that are able to utilize the dependency\npresent. In our case above, where the data is random, all classifiers would\nhave a high p-value as there is no structure present in the data.\n\nFinally, note that this test has been shown to produce low p-values even\nif there is only weak structure in the data [1]_.\n\n.. rubric:: References\n\n.. [1] Ojala and Garriga. [Permutation Tests for Studying Classifier\n       Performance](http://www.jmlr.org/papers/volume11/ojala10a/ojala10a.pdf). The\n       Journal of Machine Learning Research (2010) vol. 11\n\n\n"
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