.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/model_selection/plot_roc.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code or to run this example in your browser via JupyterLite or Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_model_selection_plot_roc.py: ================================================== Multiclass Receiver Operating Characteristic (ROC) ================================================== This example describes the use of the Receiver Operating Characteristic (ROC) metric to evaluate the quality of multiclass classifiers. ROC curves typically feature true positive rate (TPR) on the Y axis, and false positive rate (FPR) on the X axis. This means that the top left corner of the plot is the "ideal" point - a FPR of zero, and a TPR of one. This is not very realistic, but it does mean that a larger area under the curve (AUC) is usually better. The "steepness" of ROC curves is also important, since it is ideal to maximize the TPR while minimizing the FPR. ROC curves are typically used in binary classification, where the TPR and FPR can be defined unambiguously. In the case of multiclass classification, a notion of TPR or FPR is obtained only after binarizing the output. This can be done in 2 different ways: - the One-vs-Rest scheme compares each class against all the others (assumed as one); - the One-vs-One scheme compares every unique pairwise combination of classes. In this example we explore both schemes and demo the concepts of micro and macro averaging as different ways of summarizing the information of the multiclass ROC curves. .. note:: See :ref:`sphx_glr_auto_examples_model_selection_plot_roc_crossval.py` for an extension of the present example estimating the variance of the ROC curves and their respective AUC. .. GENERATED FROM PYTHON SOURCE LINES 37-45 Load and prepare data ===================== We import the :ref:`iris_dataset` which contains 3 classes, each one corresponding to a type of iris plant. One class is linearly separable from the other 2; the latter are **not** linearly separable from each other. Here we binarize the output and add noisy features to make the problem harder. .. GENERATED FROM PYTHON SOURCE LINES 45-67 .. code-block:: Python import numpy as np from sklearn.datasets import load_iris from sklearn.model_selection import train_test_split iris = load_iris() target_names = iris.target_names X, y = iris.data, iris.target y = iris.target_names[y] random_state = np.random.RandomState(0) n_samples, n_features = X.shape n_classes = len(np.unique(y)) X = np.concatenate([X, random_state.randn(n_samples, 200 * n_features)], axis=1) ( X_train, X_test, y_train, y_test, ) = train_test_split(X, y, test_size=0.5, stratify=y, random_state=0) .. GENERATED FROM PYTHON SOURCE LINES 68-71 We train a :class:`~sklearn.linear_model.LogisticRegression` model which can naturally handle multiclass problems, thanks to the use of the multinomial formulation. .. GENERATED FROM PYTHON SOURCE LINES 71-77 .. code-block:: Python from sklearn.linear_model import LogisticRegression classifier = LogisticRegression() y_score = classifier.fit(X_train, y_train).predict_proba(X_test) .. GENERATED FROM PYTHON SOURCE LINES 78-97 One-vs-Rest multiclass ROC ========================== The One-vs-the-Rest (OvR) multiclass strategy, also known as one-vs-all, consists in computing a ROC curve per each of the `n_classes`. In each step, a given class is regarded as the positive class and the remaining classes are regarded as the negative class as a bulk. .. note:: One should not confuse the OvR strategy used for the **evaluation** of multiclass classifiers with the OvR strategy used to **train** a multiclass classifier by fitting a set of binary classifiers (for instance via the :class:`~sklearn.multiclass.OneVsRestClassifier` meta-estimator). The OvR ROC evaluation can be used to scrutinize any kind of classification models irrespectively of how they were trained (see :ref:`multiclass`). In this section we use a :class:`~sklearn.preprocessing.LabelBinarizer` to binarize the target by one-hot-encoding in a OvR fashion. This means that the target of shape (`n_samples`,) is mapped to a target of shape (`n_samples`, `n_classes`). .. GENERATED FROM PYTHON SOURCE LINES 97-104 .. code-block:: Python from sklearn.preprocessing import LabelBinarizer label_binarizer = LabelBinarizer().fit(y_train) y_onehot_test = label_binarizer.transform(y_test) y_onehot_test.shape # (n_samples, n_classes) .. rst-class:: sphx-glr-script-out .. code-block:: none (75, 3) .. GENERATED FROM PYTHON SOURCE LINES 105-106 We can as well easily check the encoding of a specific class: .. GENERATED FROM PYTHON SOURCE LINES 106-109 .. code-block:: Python label_binarizer.transform(["virginica"]) .. rst-class:: sphx-glr-script-out .. code-block:: none array([[0, 0, 1]]) .. GENERATED FROM PYTHON SOURCE LINES 110-115 ROC curve showing a specific class ---------------------------------- In the following plot we show the resulting ROC curve when regarding the iris flowers as either "virginica" (`class_id=2`) or "non-virginica" (the rest). .. GENERATED FROM PYTHON SOURCE LINES 115-120 .. code-block:: Python class_of_interest = "virginica" class_id = np.flatnonzero(label_binarizer.classes_ == class_of_interest)[0] class_id .. rst-class:: sphx-glr-script-out .. code-block:: none 2 .. GENERATED FROM PYTHON SOURCE LINES 121-138 .. code-block:: Python import matplotlib.pyplot as plt from sklearn.metrics import RocCurveDisplay display = RocCurveDisplay.from_predictions( y_onehot_test[:, class_id], y_score[:, class_id], name=f"{class_of_interest} vs the rest", color="darkorange", plot_chance_level=True, ) _ = display.ax_.set( xlabel="False Positive Rate", ylabel="True Positive Rate", title="One-vs-Rest ROC curves:\nVirginica vs (Setosa & Versicolor)", ) .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_roc_001.png :alt: One-vs-Rest ROC curves: Virginica vs (Setosa & Versicolor) :srcset: /auto_examples/model_selection/images/sphx_glr_plot_roc_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 139-150 ROC curve using micro-averaged OvR ---------------------------------- Micro-averaging aggregates the contributions from all the classes (using :func:`numpy.ravel`) to compute the average metrics as follows: :math:`TPR=\frac{\sum_{c}TP_c}{\sum_{c}(TP_c + FN_c)}` ; :math:`FPR=\frac{\sum_{c}FP_c}{\sum_{c}(FP_c + TN_c)}` . We can briefly demo the effect of :func:`numpy.ravel`: .. GENERATED FROM PYTHON SOURCE LINES 150-155 .. code-block:: Python print(f"y_score:\n{y_score[0:2,:]}") print() print(f"y_score.ravel():\n{y_score[0:2,:].ravel()}") .. rst-class:: sphx-glr-script-out .. code-block:: none y_score: [[0.38 0.05 0.57] [0.07 0.28 0.65]] y_score.ravel(): [0.38 0.05 0.57 0.07 0.28 0.65] .. GENERATED FROM PYTHON SOURCE LINES 156-159 In a multi-class classification setup with highly imbalanced classes, micro-averaging is preferable over macro-averaging. In such cases, one can alternatively use a weighted macro-averaging, not demoed here. .. GENERATED FROM PYTHON SOURCE LINES 159-173 .. code-block:: Python display = RocCurveDisplay.from_predictions( y_onehot_test.ravel(), y_score.ravel(), name="micro-average OvR", color="darkorange", plot_chance_level=True, ) _ = display.ax_.set( xlabel="False Positive Rate", ylabel="True Positive Rate", title="Micro-averaged One-vs-Rest\nReceiver Operating Characteristic", ) .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_roc_002.png :alt: Micro-averaged One-vs-Rest Receiver Operating Characteristic :srcset: /auto_examples/model_selection/images/sphx_glr_plot_roc_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 174-177 In the case where the main interest is not the plot but the ROC-AUC score itself, we can reproduce the value shown in the plot using :class:`~sklearn.metrics.roc_auc_score`. .. GENERATED FROM PYTHON SOURCE LINES 177-189 .. code-block:: Python from sklearn.metrics import roc_auc_score micro_roc_auc_ovr = roc_auc_score( y_test, y_score, multi_class="ovr", average="micro", ) print(f"Micro-averaged One-vs-Rest ROC AUC score:\n{micro_roc_auc_ovr:.2f}") .. rst-class:: sphx-glr-script-out .. code-block:: none Micro-averaged One-vs-Rest ROC AUC score: 0.77 .. GENERATED FROM PYTHON SOURCE LINES 190-193 This is equivalent to computing the ROC curve with :class:`~sklearn.metrics.roc_curve` and then the area under the curve with :class:`~sklearn.metrics.auc` for the raveled true and predicted classes. .. GENERATED FROM PYTHON SOURCE LINES 193-204 .. code-block:: Python from sklearn.metrics import auc, roc_curve # store the fpr, tpr, and roc_auc for all averaging strategies fpr, tpr, roc_auc = dict(), dict(), dict() # Compute micro-average ROC curve and ROC area fpr["micro"], tpr["micro"], _ = roc_curve(y_onehot_test.ravel(), y_score.ravel()) roc_auc["micro"] = auc(fpr["micro"], tpr["micro"]) print(f"Micro-averaged One-vs-Rest ROC AUC score:\n{roc_auc['micro']:.2f}") .. rst-class:: sphx-glr-script-out .. code-block:: none Micro-averaged One-vs-Rest ROC AUC score: 0.77 .. GENERATED FROM PYTHON SOURCE LINES 205-216 .. note:: By default, the computation of the ROC curve adds a single point at the maximal false positive rate by using linear interpolation and the McClish correction [:doi:`Analyzing a portion of the ROC curve Med Decis Making. 1989 Jul-Sep; 9(3):190-5.<10.1177/0272989x8900900307>`]. ROC curve using the OvR macro-average ------------------------------------- Obtaining the macro-average requires computing the metric independently for each class and then taking the average over them, hence treating all classes equally a priori. We first aggregate the true/false positive rates per class: .. GENERATED FROM PYTHON SOURCE LINES 216-238 .. code-block:: Python for i in range(n_classes): fpr[i], tpr[i], _ = roc_curve(y_onehot_test[:, i], y_score[:, i]) roc_auc[i] = auc(fpr[i], tpr[i]) fpr_grid = np.linspace(0.0, 1.0, 1000) # Interpolate all ROC curves at these points mean_tpr = np.zeros_like(fpr_grid) for i in range(n_classes): mean_tpr += np.interp(fpr_grid, fpr[i], tpr[i]) # linear interpolation # Average it and compute AUC mean_tpr /= n_classes fpr["macro"] = fpr_grid tpr["macro"] = mean_tpr roc_auc["macro"] = auc(fpr["macro"], tpr["macro"]) print(f"Macro-averaged One-vs-Rest ROC AUC score:\n{roc_auc['macro']:.2f}") .. rst-class:: sphx-glr-script-out .. code-block:: none Macro-averaged One-vs-Rest ROC AUC score: 0.78 .. GENERATED FROM PYTHON SOURCE LINES 239-240 This computation is equivalent to simply calling .. GENERATED FROM PYTHON SOURCE LINES 240-250 .. code-block:: Python macro_roc_auc_ovr = roc_auc_score( y_test, y_score, multi_class="ovr", average="macro", ) print(f"Macro-averaged One-vs-Rest ROC AUC score:\n{macro_roc_auc_ovr:.2f}") .. rst-class:: sphx-glr-script-out .. code-block:: none Macro-averaged One-vs-Rest ROC AUC score: 0.78 .. GENERATED FROM PYTHON SOURCE LINES 251-253 Plot all OvR ROC curves together -------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 253-293 .. code-block:: Python from itertools import cycle fig, ax = plt.subplots(figsize=(6, 6)) plt.plot( fpr["micro"], tpr["micro"], label=f"micro-average ROC curve (AUC = {roc_auc['micro']:.2f})", color="deeppink", linestyle=":", linewidth=4, ) plt.plot( fpr["macro"], tpr["macro"], label=f"macro-average ROC curve (AUC = {roc_auc['macro']:.2f})", color="navy", linestyle=":", linewidth=4, ) colors = cycle(["aqua", "darkorange", "cornflowerblue"]) for class_id, color in zip(range(n_classes), colors): RocCurveDisplay.from_predictions( y_onehot_test[:, class_id], y_score[:, class_id], name=f"ROC curve for {target_names[class_id]}", color=color, ax=ax, plot_chance_level=(class_id == 2), ) _ = ax.set( xlabel="False Positive Rate", ylabel="True Positive Rate", title="Extension of Receiver Operating Characteristic\nto One-vs-Rest multiclass", ) .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_roc_003.png :alt: Extension of Receiver Operating Characteristic to One-vs-Rest multiclass :srcset: /auto_examples/model_selection/images/sphx_glr_plot_roc_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 294-315 One-vs-One multiclass ROC ========================= The One-vs-One (OvO) multiclass strategy consists in fitting one classifier per class pair. Since it requires to train `n_classes` * (`n_classes` - 1) / 2 classifiers, this method is usually slower than One-vs-Rest due to its O(`n_classes` ^2) complexity. In this section, we demonstrate the macro-averaged AUC using the OvO scheme for the 3 possible combinations in the :ref:`iris_dataset`: "setosa" vs "versicolor", "versicolor" vs "virginica" and "virginica" vs "setosa". Notice that micro-averaging is not defined for the OvO scheme. ROC curve using the OvO macro-average ------------------------------------- In the OvO scheme, the first step is to identify all possible unique combinations of pairs. The computation of scores is done by treating one of the elements in a given pair as the positive class and the other element as the negative class, then re-computing the score by inversing the roles and taking the mean of both scores. .. GENERATED FROM PYTHON SOURCE LINES 315-321 .. code-block:: Python from itertools import combinations pair_list = list(combinations(np.unique(y), 2)) print(pair_list) .. rst-class:: sphx-glr-script-out .. code-block:: none [('setosa', 'versicolor'), ('setosa', 'virginica'), ('versicolor', 'virginica')] .. GENERATED FROM PYTHON SOURCE LINES 322-375 .. code-block:: Python pair_scores = [] mean_tpr = dict() for ix, (label_a, label_b) in enumerate(pair_list): a_mask = y_test == label_a b_mask = y_test == label_b ab_mask = np.logical_or(a_mask, b_mask) a_true = a_mask[ab_mask] b_true = b_mask[ab_mask] idx_a = np.flatnonzero(label_binarizer.classes_ == label_a)[0] idx_b = np.flatnonzero(label_binarizer.classes_ == label_b)[0] fpr_a, tpr_a, _ = roc_curve(a_true, y_score[ab_mask, idx_a]) fpr_b, tpr_b, _ = roc_curve(b_true, y_score[ab_mask, idx_b]) mean_tpr[ix] = np.zeros_like(fpr_grid) mean_tpr[ix] += np.interp(fpr_grid, fpr_a, tpr_a) mean_tpr[ix] += np.interp(fpr_grid, fpr_b, tpr_b) mean_tpr[ix] /= 2 mean_score = auc(fpr_grid, mean_tpr[ix]) pair_scores.append(mean_score) fig, ax = plt.subplots(figsize=(6, 6)) plt.plot( fpr_grid, mean_tpr[ix], label=f"Mean {label_a} vs {label_b} (AUC = {mean_score :.2f})", linestyle=":", linewidth=4, ) RocCurveDisplay.from_predictions( a_true, y_score[ab_mask, idx_a], ax=ax, name=f"{label_a} as positive class", ) RocCurveDisplay.from_predictions( b_true, y_score[ab_mask, idx_b], ax=ax, name=f"{label_b} as positive class", plot_chance_level=True, ) ax.set( xlabel="False Positive Rate", ylabel="True Positive Rate", title=f"{target_names[idx_a]} vs {label_b} ROC curves", ) print(f"Macro-averaged One-vs-One ROC AUC score:\n{np.average(pair_scores):.2f}") .. rst-class:: sphx-glr-horizontal * .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_roc_004.png :alt: setosa vs versicolor ROC curves :srcset: /auto_examples/model_selection/images/sphx_glr_plot_roc_004.png :class: sphx-glr-multi-img * .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_roc_005.png :alt: setosa vs virginica ROC curves :srcset: /auto_examples/model_selection/images/sphx_glr_plot_roc_005.png :class: sphx-glr-multi-img * .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_roc_006.png :alt: versicolor vs virginica ROC curves :srcset: /auto_examples/model_selection/images/sphx_glr_plot_roc_006.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-script-out .. code-block:: none Macro-averaged One-vs-One ROC AUC score: 0.78 .. GENERATED FROM PYTHON SOURCE LINES 376-379 One can also assert that the macro-average we computed "by hand" is equivalent to the implemented `average="macro"` option of the :class:`~sklearn.metrics.roc_auc_score` function. .. GENERATED FROM PYTHON SOURCE LINES 379-389 .. code-block:: Python macro_roc_auc_ovo = roc_auc_score( y_test, y_score, multi_class="ovo", average="macro", ) print(f"Macro-averaged One-vs-One ROC AUC score:\n{macro_roc_auc_ovo:.2f}") .. rst-class:: sphx-glr-script-out .. code-block:: none Macro-averaged One-vs-One ROC AUC score: 0.78 .. GENERATED FROM PYTHON SOURCE LINES 390-392 Plot all OvO ROC curves together -------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 392-423 .. code-block:: Python ovo_tpr = np.zeros_like(fpr_grid) fig, ax = plt.subplots(figsize=(6, 6)) for ix, (label_a, label_b) in enumerate(pair_list): ovo_tpr += mean_tpr[ix] ax.plot( fpr_grid, mean_tpr[ix], label=f"Mean {label_a} vs {label_b} (AUC = {pair_scores[ix]:.2f})", ) ovo_tpr /= sum(1 for pair in enumerate(pair_list)) ax.plot( fpr_grid, ovo_tpr, label=f"One-vs-One macro-average (AUC = {macro_roc_auc_ovo:.2f})", linestyle=":", linewidth=4, ) ax.plot([0, 1], [0, 1], "k--", label="Chance level (AUC = 0.5)") _ = ax.set( xlabel="False Positive Rate", ylabel="True Positive Rate", title="Extension of Receiver Operating Characteristic\nto One-vs-One multiclass", aspect="equal", xlim=(-0.01, 1.01), ylim=(-0.01, 1.01), ) .. image-sg:: /auto_examples/model_selection/images/sphx_glr_plot_roc_007.png :alt: Extension of Receiver Operating Characteristic to One-vs-One multiclass :srcset: /auto_examples/model_selection/images/sphx_glr_plot_roc_007.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 424-441 We confirm that the classes "versicolor" and "virginica" are not well identified by a linear classifier. Notice that the "virginica"-vs-the-rest ROC-AUC score (0.77) is between the OvO ROC-AUC scores for "versicolor" vs "virginica" (0.64) and "setosa" vs "virginica" (0.90). Indeed, the OvO strategy gives additional information on the confusion between a pair of classes, at the expense of computational cost when the number of classes is large. The OvO strategy is recommended if the user is mainly interested in correctly identifying a particular class or subset of classes, whereas evaluating the global performance of a classifier can still be summarized via a given averaging strategy. Micro-averaged OvR ROC is dominated by the more frequent class, since the counts are pooled. The macro-averaged alternative better reflects the statistics of the less frequent classes, and then is more appropriate when performance on all the classes is deemed equally important. .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.677 seconds) .. _sphx_glr_download_auto_examples_model_selection_plot_roc.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/model_selection/plot_roc.ipynb :alt: Launch binder :width: 150 px .. container:: lite-badge .. image:: images/jupyterlite_badge_logo.svg :target: ../../lite/lab/?path=auto_examples/model_selection/plot_roc.ipynb :alt: Launch JupyterLite :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_roc.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_roc.py ` .. include:: plot_roc.recommendations .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_