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.. 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.
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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.
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.. code-block:: default
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)
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We train a :class:`~sklearn.linear_model.LogisticRegression` model which can
naturally handle multiclass problems, thanks to the use of the multinomial
formulation.
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.. code-block:: default
from sklearn.linear_model import LogisticRegression
classifier = LogisticRegression()
y_score = classifier.fit(X_train, y_train).predict_proba(X_test)
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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`).
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.. code-block:: default
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)
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We can as well easily check the encoding of a specific class:
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.. code-block:: default
label_binarizer.transform(["virginica"])
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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).
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.. code-block:: default
class_of_interest = "virginica"
class_id = np.flatnonzero(label_binarizer.classes_ == class_of_interest)[0]
class_id
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.. code-block:: default
import matplotlib.pyplot as plt
from sklearn.metrics import RocCurveDisplay
RocCurveDisplay.from_predictions(
y_onehot_test[:, class_id],
y_score[:, class_id],
name=f"{class_of_interest} vs the rest",
color="darkorange",
)
plt.plot([0, 1], [0, 1], "k--", label="chance level (AUC = 0.5)")
plt.axis("square")
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.title("One-vs-Rest ROC curves:\nVirginica vs (Setosa & Versicolor)")
plt.legend()
plt.show()
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ROC curve using micro-averaged OvR
----------------------------------
Micro-averaging aggregates the contributions from all the classes (using
:func:`np.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:`np.ravel`:
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.. code-block:: default
print(f"y_score:\n{y_score[0:2,:]}")
print()
print(f"y_score.ravel():\n{y_score[0:2,:].ravel()}")
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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.
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.. code-block:: default
RocCurveDisplay.from_predictions(
y_onehot_test.ravel(),
y_score.ravel(),
name="micro-average OvR",
color="darkorange",
)
plt.plot([0, 1], [0, 1], "k--", label="chance level (AUC = 0.5)")
plt.axis("square")
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.title("Micro-averaged One-vs-Rest\nReceiver Operating Characteristic")
plt.legend()
plt.show()
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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`.
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.. code-block:: default
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}")
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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.
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.. code-block:: default
from sklearn.metrics import roc_curve, auc
# 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}")
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.. 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:
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.. code-block:: default
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}")
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This computation is equivalent to simply calling
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.. code-block:: default
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}")
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Plot all OvR ROC curves together
--------------------------------
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.. code-block:: default
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,
)
plt.plot([0, 1], [0, 1], "k--", label="ROC curve for chance level (AUC = 0.5)")
plt.axis("square")
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.title("Extension of Receiver Operating Characteristic\nto One-vs-Rest multiclass")
plt.legend()
plt.show()
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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.
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.. code-block:: default
from itertools import combinations
pair_list = list(combinations(np.unique(y), 2))
print(pair_list)
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.. code-block:: default
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",
)
plt.plot([0, 1], [0, 1], "k--", label="chance level (AUC = 0.5)")
plt.axis("square")
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.title(f"{target_names[idx_a]} vs {label_b} ROC curves")
plt.legend()
plt.show()
print(f"Macro-averaged One-vs-One ROC AUC score:\n{np.average(pair_scores):.2f}")
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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.
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.. code-block:: default
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}")
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Plot all OvO ROC curves together
--------------------------------
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.. code-block:: default
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]
plt.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))
plt.plot(
fpr_grid,
ovo_tpr,
label=f"One-vs-One macro-average (AUC = {macro_roc_auc_ovo:.2f})",
linestyle=":",
linewidth=4,
)
plt.plot([0, 1], [0, 1], "k--", label="chance level (AUC = 0.5)")
plt.axis("square")
plt.xlabel("False Positive Rate")
plt.ylabel("True Positive Rate")
plt.title("Extension of Receiver Operating Characteristic\nto One-vs-One multiclass")
plt.legend()
plt.show()
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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.
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