Feature selection

In machine learning, feature selection is the process of selecting a subset of relevant features for use in model construction. Feature selection techniques are used for several reasons:

simplification of models to make them easier to interpret,
shorter training times,
to avoid the curse of dimensionality,
improve the compatibility of the data with a certain learning model class,
to encode inherent symmetries present in the input space.

The central premise when using feature selection is that data sometimes contains features that are redundant or irrelevant, and can thus be removed without incurring much loss of information. Redundancy and irrelevance are two distinct notions, since one relevant feature may be redundant in the presence of another relevant feature with which it is strongly correlated.
Feature extraction creates new features from functions of the original features, whereas feature selection finds a subset of the features. Feature selection techniques are often used in domains where there are many features and comparatively few samples.

Introduction

A feature selection algorithm can be seen as the combination of a search technique for proposing new feature subsets, along with an evaluation measure which scores the different feature subsets. The simplest algorithm is to test each possible subset of features finding the one which minimizes the error rate. This is an exhaustive search of the space, and is computationally intractable for all but the smallest of feature sets. The choice of evaluation metric heavily influences the algorithm, and it is these evaluation metrics which distinguish between the three main categories of feature selection algorithms: wrappers, filters and embedded methods.

Wrapper methods use a predictive model to score feature subsets. Each new subset is used to train a model, which is tested on a hold-out set. Counting the number of mistakes made on that hold-out set gives the score for that subset. As wrapper methods train a new model for each subset, they are very computationally intensive, but usually provide the best performing feature set for that particular type of model or typical problem.
Filter methods use a proxy measure instead of the error rate to score a feature subset. This measure is chosen to be fast to compute, while still capturing the usefulness of the feature set. Common measures include the mutual information, the pointwise mutual information, Pearson product-moment correlation coefficient, Relief-based algorithms, and inter/intra class distance or the scores of significance tests for each class/feature combinations. Filters are usually less computationally intensive than wrappers, but they produce a feature set which is not tuned to a specific type of predictive model. This lack of tuning means a feature set from a filter is more general than the set from a wrapper, usually giving lower prediction performance than a wrapper. However the feature set doesn't contain the assumptions of a prediction model, and so is more useful for exposing the relationships between the features. Many filters provide a feature ranking rather than an explicit best feature subset, and the cut off point in the ranking is chosen via cross-validation. Filter methods have also been used as a preprocessing step for wrapper methods, allowing a wrapper to be used on larger problems. One other popular approach is the Recursive Feature Elimination algorithm, commonly used with Support Vector Machines to repeatedly construct a model and remove features with low weights.
Embedded methods are a catch-all group of techniques which perform feature selection as part of the model construction process. The exemplar of this approach is the LASSO method for constructing a linear model, which penalizes the regression coefficients with an L1 penalty, shrinking many of them to zero. Any features which have non-zero regression coefficients are 'selected' by the LASSO algorithm. Improvements to the LASSO include Bolasso which bootstraps samples; Elastic net regularization, which combines the L1 penalty of LASSO with the L2 penalty of ridge regression; and FeaLect which scores all the features based on combinatorial analysis of regression coefficients. AEFS further extends LASSO to nonlinear scenario with autoencoders. These approaches tend to be between filters and wrappers in terms of computational complexity.

In traditional regression analysis, the most popular form of feature selection is stepwise regression, which is a wrapper technique. It is a greedy algorithm that adds the best feature at each round. The main control issue is deciding when to stop the algorithm. In machine learning, this is typically done by cross-validation. In statistics, some criteria are optimized. This leads to the inherent problem of nesting. More robust methods have been explored, such as branch and bound and piecewise linear network.

Subset selection

Subset selection evaluates a subset of features as a group for suitability. Subset selection algorithms can be broken up into wrappers, filters, and embedded methods. Wrappers use a search algorithm to search through the space of possible features and evaluate each subset by running a model on the subset. Wrappers can be computationally expensive and have a risk of over fitting to the model. Filters are similar to wrappers in the search approach, but instead of evaluating against a model, a simpler filter is evaluated. Embedded techniques are embedded in, and specific to, a model.
Many popular search approaches use greedy hill climbing, which iteratively evaluates a candidate subset of features, then modifies the subset and evaluates if the new subset is an improvement over the old. Evaluation of the subsets requires a scoring metric that grades a subset of features. Exhaustive search is generally impractical, so at some implementor defined stopping point, the subset of features with the highest score discovered up to that point is selected as the satisfactory feature subset. The stopping criterion varies by algorithm; possible criteria include: a subset score exceeds a threshold, a program's maximum allowed run time has been surpassed, etc.
Alternative search-based techniques are based on targeted projection pursuit which finds low-dimensional projections of the data that score highly: the features that have the largest projections in the lower-dimensional space are then selected.
Search approaches include:

Two popular filter metrics for classification problems are correlation and mutual information, although neither are true metrics or 'distance measures' in the mathematical sense, since they fail to obey the triangle inequality and thus do not compute any actual 'distance' – they should rather be regarded as 'scores'. These scores are computed between a candidate feature and the desired output category. There are, however, true metrics that are a simple function of the mutual information; see here.
Other available filter metrics include:

Class separability
* Error probability
* Inter-class distance
* Probabilistic distance
* Entropy
Consistency-based feature selection
Correlation-based feature selection
Optimality criteria

The choice of optimality criteria is difficult as there are multiple objectives in a feature selection task. Many common criteria incorporate a measure of accuracy, penalised by the number of features selected. Examples include Akaike information criterion and Mallows's C_p, which have a penalty of 2 for each added feature. AIC is based on information theory, and is effectively derived via the maximum entropy principle.
Other criteria are Bayesian information criterion, which uses a penalty of for each added feature, minimum description length which asymptotically uses, Bonferroni / RIC which use, maximum dependency feature selection, and a variety of new criteria that are motivated by false discovery rate, which use something close to. A maximum entropy rate criterion may also be used to select the most relevant subset of features.

Structure learning

Filter feature selection is a specific case of a more general paradigm called structure learning. Feature selection finds the relevant feature set for a specific target variable whereas structure learning finds the relationships between all the variables, usually by expressing these relationships as a graph. The most common structure learning algorithms assume the data is generated by a Bayesian Network, and so the structure is a directed graphical model. The optimal solution to the filter feature selection problem is the Markov blanket of the target node, and in a Bayesian Network, there is a unique Markov Blanket for each node.

Information Theory Based Feature Selection Mechanisms

There are different Feature Selection mechanisms around that utilize mutual information for scoring the different features. They usually use all the same algorithm:

Calculate the mutual information as score for between all features and the target class
Select the feature with the largest score and add it to the set of selected features
Calculate the score which might be derived from the mutual information
Select the feature with the largest score and add it to the set of select features
Repeat 3. and 4. until a certain number of features is selected

The simplest approach uses the mutual information as the "derived" score.
However, there are different approaches, that try to reduce the redundancy between features.

Minimum-redundancy-maximum-relevance (mRMR) feature selection

Peng et al. proposed a feature selection method that can use either mutual information, correlation, or distance/similarity scores to select features. The aim is to penalise a feature's relevancy by its redundancy in the presence of the other selected features. The relevance of a feature set for the class is defined by the average value of all mutual information values between the individual feature and the class as follows:
The redundancy of all features in the set is the average value of all mutual information values between the feature and the feature :
The mRMR criterion is a combination of two measures given above and is defined as follows:
Suppose that there are full-set features. Let be the set membership indicator function for feature, so that indicates presence and indicates absence of the feature in the globally optimal feature set. Let and. The above may then be written as an optimization problem:
The mRMR algorithm is an approximation of the theoretically optimal maximum-dependency feature selection algorithm that maximizes the mutual information between the joint distribution of the selected features and the classification variable. As mRMR approximates the combinatorial estimation problem with a series of much smaller problems, each of which only involves two variables, it thus uses pairwise joint probabilities which are more robust. In certain situations the algorithm may underestimate the usefulness of features as it has no way to measure interactions between features which can increase relevancy. This can lead to poor performance when the features are individually useless, but are useful when combined. Overall the algorithm is more efficient than the theoretically optimal max-dependency selection, yet produces a feature set with little pairwise redundancy.
mRMR is an instance of a large class of filter methods which trade off between relevancy and redundancy in different ways.