# 1.13. Feature selection¶

The classes in the sklearn.feature_selection module can be used for feature selection/dimensionality reduction on sample sets, either to improve estimators’ accuracy scores or to boost their performance on very high-dimensional datasets.

## 1.13.1. Removing features with low variance¶

VarianceThreshold is a simple baseline approach to feature selection. It removes all features whose variance doesn’t meet some threshold. By default, it removes all zero-variance features, i.e. features that have the same value in all samples.

As an example, suppose that we have a dataset with boolean features, and we want to remove all features that are either one or zero (on or off) in more than 80% of the samples. Boolean features are Bernoulli random variables, and the variance of such variables is given by

$\mathrm{Var}[X] = p(1 - p)$

so we can select using the threshold .8 * (1 - .8):

>>> from sklearn.feature_selection import VarianceThreshold
>>> X = [[0, 0, 1], [0, 1, 0], [1, 0, 0], [0, 1, 1], [0, 1, 0], [0, 1, 1]]
>>> sel = VarianceThreshold(threshold=(.8 * (1 - .8)))
>>> sel.fit_transform(X)
array([[0, 1],
[1, 0],
[0, 0],
[1, 1],
[1, 0],
[1, 1]])


As expected, VarianceThreshold has removed the first column, which has a probability $$p = 5/6 > .8$$ of containing a zero.

## 1.13.2. Univariate feature selection¶

Univariate feature selection works by selecting the best features based on univariate statistical tests. It can be seen as a preprocessing step to an estimator. Scikit-learn exposes feature selection routines as objects that implement the transform method:

For instance, we can perform a $$\chi^2$$ test to the samples to retrieve only the two best features as follows:

>>> from sklearn.datasets import load_iris
>>> from sklearn.feature_selection import SelectKBest
>>> from sklearn.feature_selection import chi2
>>> X.shape
(150, 4)
>>> X_new = SelectKBest(chi2, k=2).fit_transform(X, y)
>>> X_new.shape
(150, 2)


These objects take as input a scoring function that returns univariate scores and p-values (or only scores for SelectKBest and SelectPercentile):

The methods based on F-test estimate the degree of linear dependency between two random variables. On the other hand, mutual information methods can capture any kind of statistical dependency, but being nonparametric, they require more samples for accurate estimation.

Feature selection with sparse data

If you use sparse data (i.e. data represented as sparse matrices), chi2, mutual_info_regression, mutual_info_classif will deal with the data without making it dense.

Warning

Beware not to use a regression scoring function with a classification problem, you will get useless results.

## 1.13.3. Recursive feature elimination¶

Given an external estimator that assigns weights to features (e.g., the coefficients of a linear model), recursive feature elimination (RFE) is to select features by recursively considering smaller and smaller sets of features. First, the estimator is trained on the initial set of features and the importance of each feature is obtained either through any specific attribute (such as coef_, feature_importances_) or callable. Then, the least important features are pruned from current set of features.That procedure is recursively repeated on the pruned set until the desired number of features to select is eventually reached.

RFECV performs RFE in a cross-validation loop to find the optimal number of features.

Examples:

## 1.13.4. Feature selection using SelectFromModel¶

SelectFromModel is a meta-transformer that can be used along with any estimator that importance of each feature through a specific attribute (such as coef_, feature_importances_) or callable after fitting. The features are considered unimportant and removed, if the corresponding importance of the feature values are below the provided threshold parameter. Apart from specifying the threshold numerically, there are built-in heuristics for finding a threshold using a string argument. Available heuristics are “mean”, “median” and float multiples of these like “0.1*mean”. In combination with the threshold criteria, one can use the max_features parameter to set a limit on the number of features to select.

For examples on how it is to be used refer to the sections below.

Examples

### 1.13.4.1. L1-based feature selection¶

Linear models penalized with the L1 norm have sparse solutions: many of their estimated coefficients are zero. When the goal is to reduce the dimensionality of the data to use with another classifier, they can be used along with feature_selection.SelectFromModel to select the non-zero coefficients. In particular, sparse estimators useful for this purpose are the linear_model.Lasso for regression, and of linear_model.LogisticRegression and svm.LinearSVC for classification:

>>> from sklearn.svm import LinearSVC
>>> from sklearn.feature_selection import SelectFromModel
>>> X.shape
(150, 4)
>>> lsvc = LinearSVC(C=0.01, penalty="l1", dual=False).fit(X, y)
>>> model = SelectFromModel(lsvc, prefit=True)
>>> X_new = model.transform(X)
>>> X_new.shape
(150, 3)


With SVMs and logistic-regression, the parameter C controls the sparsity: the smaller C the fewer features selected. With Lasso, the higher the alpha parameter, the fewer features selected.

Examples:

L1-recovery and compressive sensing

For a good choice of alpha, the Lasso can fully recover the exact set of non-zero variables using only few observations, provided certain specific conditions are met. In particular, the number of samples should be “sufficiently large”, or L1 models will perform at random, where “sufficiently large” depends on the number of non-zero coefficients, the logarithm of the number of features, the amount of noise, the smallest absolute value of non-zero coefficients, and the structure of the design matrix X. In addition, the design matrix must display certain specific properties, such as not being too correlated.

There is no general rule to select an alpha parameter for recovery of non-zero coefficients. It can by set by cross-validation (LassoCV or LassoLarsCV), though this may lead to under-penalized models: including a small number of non-relevant variables is not detrimental to prediction score. BIC (LassoLarsIC) tends, on the opposite, to set high values of alpha.

Reference Richard G. Baraniuk “Compressive Sensing”, IEEE Signal Processing Magazine  July 2007 http://users.isr.ist.utl.pt/~aguiar/CS_notes.pdf

### 1.13.4.2. Tree-based feature selection¶

Tree-based estimators (see the sklearn.tree module and forest of trees in the sklearn.ensemble module) can be used to compute impurity-based feature importances, which in turn can be used to discard irrelevant features (when coupled with the sklearn.feature_selection.SelectFromModel meta-transformer):

>>> from sklearn.ensemble import ExtraTreesClassifier
>>> from sklearn.feature_selection import SelectFromModel
>>> X.shape
(150, 4)
>>> clf = ExtraTreesClassifier(n_estimators=50)
>>> clf = clf.fit(X, y)
>>> clf.feature_importances_
array([ 0.04...,  0.05...,  0.4...,  0.4...])
>>> model = SelectFromModel(clf, prefit=True)
>>> X_new = model.transform(X)
>>> X_new.shape
(150, 2)


Examples:

## 1.13.5. Feature selection as part of a pipeline¶

Feature selection is usually used as a pre-processing step before doing the actual learning. The recommended way to do this in scikit-learn is to use a sklearn.pipeline.Pipeline:

clf = Pipeline([
('feature_selection', SelectFromModel(LinearSVC(penalty="l1"))),
('classification', RandomForestClassifier())
])
clf.fit(X, y)


In this snippet we make use of a sklearn.svm.LinearSVC coupled with sklearn.feature_selection.SelectFromModel to evaluate feature importances and select the most relevant features. Then, a sklearn.ensemble.RandomForestClassifier is trained on the transformed output, i.e. using only relevant features. You can perform similar operations with the other feature selection methods and also classifiers that provide a way to evaluate feature importances of course. See the sklearn.pipeline.Pipeline examples for more details.