# 1.12. Multiclass and multilabel algorithms¶

Warning

All classifiers in scikit-learn do multiclass classification
out-of-the-box. You don’t need to use the `sklearn.multiclass`

module
unless you want to experiment with different multiclass strategies.

The `sklearn.multiclass`

module implements *meta-estimators* to solve
`multiclass`

and `multilabel`

classification problems
by decomposing such problems into binary classification problems. Multitarget
regression is also supported.

**Multiclass classification**means a classification task with more than two classes; e.g., classify a set of images of fruits which may be oranges, apples, or pears. Multiclass classification makes the assumption that each sample is assigned to one and only one label: a fruit can be either an apple or a pear but not both at the same time.**Multilabel classification**assigns to each sample a set of target labels. This can be thought as predicting properties of a data-point that are not mutually exclusive, such as topics that are relevant for a document. A text might be about any of religion, politics, finance or education at the same time or none of these.**Multioutput regression**assigns each sample a set of target values. This can be thought of as predicting several properties for each data-point, such as wind direction and magnitude at a certain location.**Multioutput-multiclass classification**and**multi-task classification**means that a single estimator has to handle several joint classification tasks. This is both a generalization of the multi-label classification task, which only considers binary classification, as well as a generalization of the multi-class classification task.*The output format is a 2d numpy array or sparse matrix.*The set of labels can be different for each output variable. For instance, a sample could be assigned “pear” for an output variable that takes possible values in a finite set of species such as “pear”, “apple”; and “blue” or “green” for a second output variable that takes possible values in a finite set of colors such as “green”, “red”, “blue”, “yellow”…

This means that any classifiers handling multi-output multiclass or multi-task classification tasks, support the multi-label classification task as a special case. Multi-task classification is similar to the multi-output classification task with different model formulations. For more information, see the relevant estimator documentation.

All scikit-learn classifiers are capable of multiclass classification,
but the meta-estimators offered by `sklearn.multiclass`

permit changing the way they handle more than two classes
because this may have an effect on classifier performance
(either in terms of generalization error or required computational resources).

Below is a summary of the classifiers supported by scikit-learn grouped by strategy; you don’t need the meta-estimators in this class if you’re using one of these, unless you want custom multiclass behavior:

**Inherently multiclass:**`sklearn.svm.LinearSVC`

(setting multi_class=”crammer_singer”)`sklearn.linear_model.LogisticRegression`

(setting multi_class=”multinomial”)`sklearn.linear_model.LogisticRegressionCV`

(setting multi_class=”multinomial”)

**Multiclass as One-Vs-One:**`sklearn.gaussian_process.GaussianProcessClassifier`

(setting multi_class = “one_vs_one”)

**Multiclass as One-Vs-All:**`sklearn.gaussian_process.GaussianProcessClassifier`

(setting multi_class = “one_vs_rest”)`sklearn.svm.LinearSVC`

(setting multi_class=”ovr”)`sklearn.linear_model.LogisticRegression`

(setting multi_class=”ovr”)`sklearn.linear_model.LogisticRegressionCV`

(setting multi_class=”ovr”)

**Support multilabel:****Support multiclass-multioutput:**

Warning

At present, no metric in `sklearn.metrics`

supports the multioutput-multiclass classification task.

## 1.12.1. Multilabel classification format¶

In multilabel learning, the joint set of binary classification tasks is
expressed with label binary indicator array: each sample is one row of a 2d
array of shape (n_samples, n_classes) with binary values: the one, i.e. the non
zero elements, corresponds to the subset of labels. An array such as
`np.array([[1, 0, 0], [0, 1, 1], [0, 0, 0]])`

represents label 0 in the first
sample, labels 1 and 2 in the second sample, and no labels in the third sample.

Producing multilabel data as a list of sets of labels may be more intuitive.
The `MultiLabelBinarizer`

transformer can be used to convert between a collection of collections of
labels and the indicator format.

```
>>> from sklearn.preprocessing import MultiLabelBinarizer
>>> y = [[2, 3, 4], [2], [0, 1, 3], [0, 1, 2, 3, 4], [0, 1, 2]]
>>> MultiLabelBinarizer().fit_transform(y)
array([[0, 0, 1, 1, 1],
[0, 0, 1, 0, 0],
[1, 1, 0, 1, 0],
[1, 1, 1, 1, 1],
[1, 1, 1, 0, 0]])
```

## 1.12.2. One-Vs-The-Rest¶

This strategy, also known as **one-vs-all**, is implemented in
`OneVsRestClassifier`

. The strategy consists in fitting one classifier
per class. For each classifier, the class is fitted against all the other
classes. In addition to its computational efficiency (only `n_classes`

classifiers are needed), one advantage of this approach is its
interpretability. Since each class is represented by one and only one classifier,
it is possible to gain knowledge about the class by inspecting its
corresponding classifier. This is the most commonly used strategy and is a fair
default choice.

### 1.12.2.1. Multiclass learning¶

Below is an example of multiclass learning using OvR:

```
>>> from sklearn import datasets
>>> from sklearn.multiclass import OneVsRestClassifier
>>> from sklearn.svm import LinearSVC
>>> X, y = datasets.load_iris(return_X_y=True)
>>> OneVsRestClassifier(LinearSVC(random_state=0)).fit(X, y).predict(X)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2])
```

### 1.12.2.2. Multilabel learning¶

`OneVsRestClassifier`

also supports multilabel classification.
To use this feature, feed the classifier an indicator matrix, in which cell
[i, j] indicates the presence of label j in sample i.

Examples:

## 1.12.3. One-Vs-One¶

`OneVsOneClassifier`

constructs one classifier per pair of classes.
At prediction time, the class which received the most votes is selected.
In the event of a tie (among two classes with an equal number of votes), it
selects the class with the highest aggregate classification confidence by
summing over the pair-wise classification confidence levels computed by the
underlying binary classifiers.

Since it requires to fit `n_classes * (n_classes - 1) / 2`

classifiers,
this method is usually slower than one-vs-the-rest, due to its
O(n_classes^2) complexity. However, this method may be advantageous for
algorithms such as kernel algorithms which don’t scale well with
`n_samples`

. This is because each individual learning problem only involves
a small subset of the data whereas, with one-vs-the-rest, the complete
dataset is used `n_classes`

times. The decision function is the result
of a monotonic transformation of the one-versus-one classification.

### 1.12.3.1. Multiclass learning¶

Below is an example of multiclass learning using OvO:

```
>>> from sklearn import datasets
>>> from sklearn.multiclass import OneVsOneClassifier
>>> from sklearn.svm import LinearSVC
>>> X, y = datasets.load_iris(return_X_y=True)
>>> OneVsOneClassifier(LinearSVC(random_state=0)).fit(X, y).predict(X)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2])
```

References:

“Pattern Recognition and Machine Learning. Springer”, Christopher M. Bishop, page 183, (First Edition)

## 1.12.4. Error-Correcting Output-Codes¶

Output-code based strategies are fairly different from one-vs-the-rest and one-vs-one. With these strategies, each class is represented in a Euclidean space, where each dimension can only be 0 or 1. Another way to put it is that each class is represented by a binary code (an array of 0 and 1). The matrix which keeps track of the location/code of each class is called the code book. The code size is the dimensionality of the aforementioned space. Intuitively, each class should be represented by a code as unique as possible and a good code book should be designed to optimize classification accuracy. In this implementation, we simply use a randomly-generated code book as advocated in 3 although more elaborate methods may be added in the future.

At fitting time, one binary classifier per bit in the code book is fitted. At prediction time, the classifiers are used to project new points in the class space and the class closest to the points is chosen.

In `OutputCodeClassifier`

, the `code_size`

attribute allows the user to
control the number of classifiers which will be used. It is a percentage of the
total number of classes.

A number between 0 and 1 will require fewer classifiers than
one-vs-the-rest. In theory, `log2(n_classes) / n_classes`

is sufficient to
represent each class unambiguously. However, in practice, it may not lead to
good accuracy since `log2(n_classes)`

is much smaller than n_classes.

A number greater than 1 will require more classifiers than one-vs-the-rest. In this case, some classifiers will in theory correct for the mistakes made by other classifiers, hence the name “error-correcting”. In practice, however, this may not happen as classifier mistakes will typically be correlated. The error-correcting output codes have a similar effect to bagging.

### 1.12.4.1. Multiclass learning¶

Below is an example of multiclass learning using Output-Codes:

```
>>> from sklearn import datasets
>>> from sklearn.multiclass import OutputCodeClassifier
>>> from sklearn.svm import LinearSVC
>>> X, y = datasets.load_iris(return_X_y=True)
>>> clf = OutputCodeClassifier(LinearSVC(random_state=0),
... code_size=2, random_state=0)
>>> clf.fit(X, y).predict(X)
array([0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1,
1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2])
```

References:

“Solving multiclass learning problems via error-correcting output codes”, Dietterich T., Bakiri G., Journal of Artificial Intelligence Research 2, 1995.

- 3
“The error coding method and PICTs”, James G., Hastie T., Journal of Computational and Graphical statistics 7, 1998.

“The Elements of Statistical Learning”, Hastie T., Tibshirani R., Friedman J., page 606 (second-edition) 2008.

## 1.12.5. Multioutput regression¶

Multioutput regression support can be added to any regressor with
`MultiOutputRegressor`

. This strategy consists of fitting one
regressor per target. Since each target is represented by exactly one
regressor it is possible to gain knowledge about the target by
inspecting its corresponding regressor. As
`MultiOutputRegressor`

fits one regressor per target it can not
take advantage of correlations between targets.

Below is an example of multioutput regression:

```
>>> from sklearn.datasets import make_regression
>>> from sklearn.multioutput import MultiOutputRegressor
>>> from sklearn.ensemble import GradientBoostingRegressor
>>> X, y = make_regression(n_samples=10, n_targets=3, random_state=1)
>>> MultiOutputRegressor(GradientBoostingRegressor(random_state=0)).fit(X, y).predict(X)
array([[-154.75474165, -147.03498585, -50.03812219],
[ 7.12165031, 5.12914884, -81.46081961],
[-187.8948621 , -100.44373091, 13.88978285],
[-141.62745778, 95.02891072, -191.48204257],
[ 97.03260883, 165.34867495, 139.52003279],
[ 123.92529176, 21.25719016, -7.84253 ],
[-122.25193977, -85.16443186, -107.12274212],
[ -30.170388 , -94.80956739, 12.16979946],
[ 140.72667194, 176.50941682, -17.50447799],
[ 149.37967282, -81.15699552, -5.72850319]])
```

## 1.12.6. Multioutput classification¶

Multioutput classification support can be added to any classifier with
`MultiOutputClassifier`

. This strategy consists of fitting one
classifier per target. This allows multiple target variable
classifications. The purpose of this class is to extend estimators
to be able to estimate a series of target functions (f1,f2,f3…,fn)
that are trained on a single X predictor matrix to predict a series
of responses (y1,y2,y3…,yn).

Below is an example of multioutput classification:

```
>>> from sklearn.datasets import make_classification
>>> from sklearn.multioutput import MultiOutputClassifier
>>> from sklearn.ensemble import RandomForestClassifier
>>> from sklearn.utils import shuffle
>>> import numpy as np
>>> X, y1 = make_classification(n_samples=10, n_features=100, n_informative=30, n_classes=3, random_state=1)
>>> y2 = shuffle(y1, random_state=1)
>>> y3 = shuffle(y1, random_state=2)
>>> Y = np.vstack((y1, y2, y3)).T
>>> n_samples, n_features = X.shape # 10,100
>>> n_outputs = Y.shape[1] # 3
>>> n_classes = 3
>>> forest = RandomForestClassifier(random_state=1)
>>> multi_target_forest = MultiOutputClassifier(forest, n_jobs=-1)
>>> multi_target_forest.fit(X, Y).predict(X)
array([[2, 2, 0],
[1, 2, 1],
[2, 1, 0],
[0, 0, 2],
[0, 2, 1],
[0, 0, 2],
[1, 1, 0],
[1, 1, 1],
[0, 0, 2],
[2, 0, 0]])
```

## 1.12.7. Classifier Chain¶

Classifier chains (see `ClassifierChain`

) are a way of combining a
number of binary classifiers into a single multi-label model that is capable
of exploiting correlations among targets.

For a multi-label classification problem with N classes, N binary classifiers are assigned an integer between 0 and N-1. These integers define the order of models in the chain. Each classifier is then fit on the available training data plus the true labels of the classes whose models were assigned a lower number.

When predicting, the true labels will not be available. Instead the predictions of each model are passed on to the subsequent models in the chain to be used as features.

Clearly the order of the chain is important. The first model in the chain has no information about the other labels while the last model in the chain has features indicating the presence of all of the other labels. In general one does not know the optimal ordering of the models in the chain so typically many randomly ordered chains are fit and their predictions are averaged together.

References:

- Jesse Read, Bernhard Pfahringer, Geoff Holmes, Eibe Frank,
“Classifier Chains for Multi-label Classification”, 2009.

## 1.12.8. Regressor Chain¶

Regressor chains (see `RegressorChain`

) is analogous to
ClassifierChain as a way of combining a number of regressions
into a single multi-target model that is capable of exploiting
correlations among targets.