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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 AIC 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. A maximum entropy rate criterion may also be used to select the most relevant subset of features.
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.
There are different Feature Selection mechanisms around that utilize mutual information for scoring the different features. They usually use all the same algorithm:.
The simplest approach uses the mutual information as the "derived" score. Peng et al. 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 S for the class c is defined by the average value of all mutual information values between the individual feature f i and the class c as follows:.
The redundancy of all features in the set S is the average value of all mutual information values between the feature f i and the feature f j :.
Suppose that there are n full-set features. 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 a pathological case is found when the class is a parity function of the features.
Overall the algorithm is more efficient in terms of the amount of data required than the theoretically optimal max-dependency selection, yet produces a feature set with little pairwise redundancy.
While mRMR could be optimized using floating search to reduce some features, it might also be reformulated as a global quadratic programming optimization problem as follows: .
QPFS is solved via quadratic programming. Another score derived for the mutual information is based on the conditional relevancy: .
In a study of different scores Brown et al. The score tries to find the feature, that adds the most new information to the already selected features, in order to avoid redundancy.
The score is formulated as follows:. For high-dimensional and small sample data e. HSIC always takes a non-negative value, and is zero if and only if two random variables are statistically independent when a universal reproducing kernel such as the Gaussian kernel is used.
The optimization problem is a Lasso problem, and thus it can be efficiently solved with a state-of-the-art Lasso solver such as the dual augmented Lagrangian method.
The correlation feature selection CFS measure evaluates subsets of features on the basis of the following hypothesis: "Good feature subsets contain features highly correlated with the classification, yet uncorrelated to each other".
The CFS criterion is defined as follows:. Hall's dissertation uses neither of these, but uses three different measures of relatedness, minimum description length MDL , symmetrical uncertainty , and relief.
Let x i be the set membership indicator function for feature f i ; then the above can be rewritten as an optimization problem:.
The combinatorial problems above are, in fact, mixed 0—1 linear programming problems that can be solved by using branch-and-bound algorithms.
The features from a decision tree or a tree ensemble are shown to be redundant. A recent method called regularized tree  can be used for feature subset selection.
Regularized trees penalize using a variable similar to the variables selected at previous tree nodes for splitting the current node.
Regularized trees only need build one tree model or one tree ensemble model and thus are computationally efficient. Regularized trees naturally handle numerical and categorical features, interactions and nonlinearities.
They are invariant to attribute scales units and insensitive to outliers , and thus, require little data preprocessing such as normalization.
Regularized random forest RRF  is one type of regularized trees. A metaheuristic is a general description of an algorithm dedicated to solve difficult typically NP-hard problem optimization problems for which there is no classical solving methods.
Generally, a metaheuristic is a stochastic algorithm tending to reach a global optimum. There are many metaheuristics, from a simple local search to a complex global search algorithm.
The feature selection methods are typically presented in three classes based on how they combine the selection algorithm and the model building.
Filter type methods select variables regardless of the model. They are based only on general features like the correlation with the variable to predict.
Filter methods suppress the least interesting variables. The other variables will be part of a classification or a regression model used to classify or to predict data.
These methods are particularly effective in computation time and robust to overfitting. Filter methods tend to select redundant variables when they do not consider the relationships between variables.
However, more elaborate features try to minimize this problem by removing variables highly correlated to each other, such as the FCBF algorithm.
Wrapper methods evaluate subsets of variables which allows, unlike filter approaches, to detect the possible interactions between variables.
Embedded methods have been recently proposed that try to combine the advantages of both previous methods. A learning algorithm takes advantage of its own variable selection process and performs feature selection and classification simultaneously, such as the FRMT algorithm.
This is a survey of the application of feature selection metaheuristics lately used in the literature. This survey was realized by J.
Hammon in her thesis. Some learning algorithms perform feature selection as part of their overall operation.
These include:. From Wikipedia, the free encyclopedia. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations.
Please help to improve this article by introducing more precise citations. July Learn how and when to remove this template message. Dimensionality reduction.
Structured prediction. Graphical models Bayes net Conditional random field Hidden Markov. Anomaly detection. Artificial neural network.
Reinforcement learning. Machine-learning venues. Glossary of artificial intelligence. Related articles.
List of datasets for machine-learning research Outline of machine learning. An Introduction to Statistical Learning.
Bibcode : NatSR A comparative study on feature selection in text categorization PDF. Journal of Biomedical Informatics.
Journal of Machine Learning Research. Machine Learning. Bolasso: model consistent lasso estimation through the bootstrap.
Proceedings of the 25th International Conference on Machine Learning. BMC Genomics. Autoencoder inspired unsupervised feature selection.
Bibcode : PLoSO.. Computers in Industry. Expert Systems with Applications. Knowledge-Based Systems. Garcia-Lopez, M. Garcia-Torres, B. Melian, J.
Moreno-Perez, J. Garcia-Torres, F. Gomez-Vela, B. High-dimensional feature selection via feature grouping: A Variable Neighborhood Search approach , Information Sciences , vol.
Bibcode : q. Advances in Neural Information Processing Systems : — Neural Computation. University of Waikato. Vancouver, Canada. The concept of adjustment factors for the rock mass before and after excavation is very attractive.
In addition, this allows for quantification of the influence of excavation and excavation induced stresses, excavation methods, and the influence of past and future weathering of the rock mass.
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In Stewart, D. Design and operation of caving and sub-level storing mines. In Hustrulid, W. Categories : Rock mass classification.
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