# User:Atry/特征选择

• 简化模型，使之更易于被研究人员或用户理解，[1]
• 缩短训练时间，
• 改善通用性、降低过拟合[2]（即降低方差[1]

## 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. 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 (or operator) 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 (or set of features) and the desired output category. There are, however, true metrics that are a simple function of the mutual information;[15] 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 ones incorporate a measure of accuracy, penalised by the number of features selected (e.g. the Bayesian information criterion). The oldest are Mallows's Cp statistic and Akaike information criterion (AIC). These add variables if the t -statistic is bigger than ${\displaystyle {\sqrt {2}}}$.

Other criteria are Bayesian information criterion (BIC) which uses ${\displaystyle {\sqrt {\log {n}}}}$, minimum description length (MDL) which asymptotically uses ${\displaystyle {\sqrt {\log {n}}}}$, Bonferroni / RIC which use ${\displaystyle {\sqrt {2\log {p}}}}$, maximum dependency feature selection, and a variety of new criteria that are motivated by false discovery rate (FDR) which use something close to ${\displaystyle {\sqrt {2\log {\frac {p}{q}}}}}$.

## 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.[16]

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

Peng et al. [17] 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 S for the class c is defined by the average value of all mutual information values between the individual feature fi and the class c as follows:

${\displaystyle D(S,c)={\frac {1}{|S|}}\sum _{f_{i}\in S}I(f_{i};c)}$.

The redundancy of all features in the set S is the average value of all mutual information values between the feature fi and the feature fj:

${\displaystyle R(S)={\frac {1}{|S|^{2}}}\sum _{f_{i},f_{j}\in S}I(f_{i};f_{j})}$

The mRMR criterion is a combination of two measures given above and is defined as follows:

${\displaystyle \mathrm {mRMR} =\max _{S}\left[{\frac {1}{|S|}}\sum _{f_{i}\in S}I(f_{i};c)-{\frac {1}{|S|^{2}}}\sum _{f_{i},f_{j}\in S}I(f_{i};f_{j})\right].}$

Suppose that there are n full-set features. Let xi be the set membership indicator function for feature fi, so that xi=1 indicates presence and xi=0 indicates absence of the feature fi in the globally optimal feature set. Let ${\displaystyle c_{i}=I(f_{i};c)}$ and ${\displaystyle a_{ij}=I(f_{i};f_{j})}$. The above may then be written as an optimization problem:

${\displaystyle \mathrm {mRMR} =\max _{x\in \{0,1\}^{n}}\left[{\frac {\sum _{i=1}^{n}c_{i}x_{i}}{\sum _{i=1}^{n}x_{i}}}-{\frac {\sum _{i,j=1}^{n}a_{ij}x_{i}x_{j}}{(\sum _{i=1}^{n}x_{i})^{2}}}\right].}$

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[18] 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.

mRMR is an instance of a large class of filter methods which trade off between relevancy and redundancy in different ways.[18][19]

### Global optimization formulations

mRMR is a typical example of an incremental greedy strategy for feature selection: once a feature has been selected, it cannot be deselected at a later stage. 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:[20]

${\displaystyle \mathrm {QPFS} :\min _{\mathbf {x} }\left\{\alpha \mathbf {x} ^{T}H\mathbf {x} -\mathbf {x} ^{T}F\right\}\quad {\mbox{s.t.}}\ \sum _{i=1}^{n}x_{i}=1,x_{i}\geq 0}$

where ${\displaystyle F_{n\times 1}=[I(f_{1};c),\ldots ,I(f_{n};c)]^{T}}$ is the vector of feature relevancy assuming there are n features in total, ${\displaystyle H_{n\times n}=[I(f_{i};f_{j})]_{i,j=1\ldots n}}$ is the matrix of feature pairwise redundancy, and ${\displaystyle \mathbf {x} _{n\times 1}}$ represents relative feature weights. QPFS is solved via quadratic programming. It is recently shown that QFPS is biased towards features with smaller entropy,[21] due to its placement of the feature self redundancy term ${\displaystyle I(f_{i};f_{i})}$ on the diagonal of H.

Another global formulation for the mutual information based feature selection problem is based on the conditional relevancy:[21]

${\displaystyle \mathrm {SPEC_{CMI}} :\max _{\mathbf {x} }\left\{\mathbf {x} ^{T}Q\mathbf {x} \right\}\quad {\mbox{s.t.}}\ \|\mathbf {x} \|=1,x_{i}\geq 0}$

where ${\displaystyle Q_{ii}=I(f_{i};c)}$ and ${\displaystyle Q_{ij}=I(f_{i};c|f_{j}),i\neq j}$.

An advantage of SPECCMI is that it can be solved simply via finding the dominant eigenvector of Q, thus is very scalable. SPECCMI also handles second-order feature interaction.

For high-dimensional and small sample data (e.g., dimensionality > 105 and the number of samples < 103), the Hilbert-Schmidt Independence Criterion Lasso (HSIC Lasso) is useful.[22] HSIC Lasso optimization problem is given as

${\displaystyle \mathrm {HSIC_{Lasso}} :\min _{\mathbf {x} }{\frac {1}{2}}\sum _{k,l=1}^{n}x_{k}x_{l}{\mbox{HSIC}}(f_{k},f_{l})-\sum _{k=1}^{n}x_{k}{\mbox{HSIC}}(f_{k},c)+\lambda \|\mathbf {x} \|_{1},\quad {\mbox{s.t.}}\ x_{1},\ldots ,x_{n}\geq 0,}$

where ${\displaystyle {\mbox{HSIC}}(f_{k},c)={\mbox{tr}}({\bar {\mathbf {K} }}^{(k)}{\bar {\mathbf {L} }})}$ is a kernel-based independence measure called the (empirical) Hilbert-Schmidt independence criterion (HSIC), ${\displaystyle {\mbox{tr}}(\cdot )}$ denotes the trace, ${\displaystyle \lambda }$ is the regularization parameter, ${\displaystyle {\bar {\mathbf {K} }}^{(k)}=\mathbf {\Gamma } \mathbf {K} ^{(k)}\mathbf {\Gamma } }$ and ${\displaystyle {\bar {\mathbf {L} }}=\mathbf {\Gamma } \mathbf {L} \mathbf {\Gamma } }$ are input and output centered Gram matrices, ${\displaystyle K_{i,j}^{(k)}=K(u_{k,i},u_{k,j})}$ and ${\displaystyle L_{i,j}=L(c_{i},c_{j})}$ are Gram matrices, ${\displaystyle K(u,u')}$ and ${\displaystyle L(c,c')}$ are kernel functions, ${\displaystyle \mathbf {\Gamma } =\mathbf {I} _{m}-{\frac {1}{m}}\mathbf {1} _{m}\mathbf {1} _{m}^{T}}$ is the centering matrix, ${\displaystyle \mathbf {I} _{m}}$ is the m-dimensional identity matrix (m: the number of samples), ${\displaystyle \mathbf {1} _{m}}$ is the m-dimensional vector with all ones, and ${\displaystyle \|\cdot \|_{1}}$ is the ${\displaystyle \ell _{1}}$-norm. 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 HSIC Lasso can be written as

${\displaystyle \mathrm {HSIC_{Lasso}} :\min _{\mathbf {x} }{\frac {1}{2}}\left\|{\bar {\mathbf {L} }}-\sum _{k=1}^{n}x_{k}{\bar {\mathbf {K} }}^{(k)}\right\|_{F}^{2}+\lambda \|\mathbf {x} \|_{1},\quad {\mbox{s.t.}}\ x_{1},\ldots ,x_{n}\geq 0,}$

where ${\displaystyle \|\cdot \|_{F}}$ is the Frobenius norm. 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.

## Correlation feature selection

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".[23][24] The following equation gives the merit of a feature subset S consisting of k features:

${\displaystyle Merit_{S_{k}}={\frac {k{\overline {r_{cf}}}}{\sqrt {k+k(k-1){\overline {r_{ff}}}}}}.}$

Here, ${\displaystyle {\overline {r_{cf}}}}$ is the average value of all feature-classification correlations, and ${\displaystyle {\overline {r_{ff}}}}$ is the average value of all feature-feature correlations. The CFS criterion is defined as follows:

${\displaystyle \mathrm {CFS} =\max _{S_{k}}\left[{\frac {r_{cf_{1}}+r_{cf_{2}}+\cdots +r_{cf_{k}}}{\sqrt {k+2(r_{f_{1}f_{2}}+\cdots +r_{f_{i}f_{j}}+\cdots +r_{f_{k}f_{1}})}}}\right].}$

The ${\displaystyle r_{cf_{i}}}$ and ${\displaystyle r_{f_{i}f_{j}}}$ variables are referred to as correlations, but are not necessarily Pearson's correlation coefficient or Spearman's ρ. Dr. Mark Hall's dissertation uses neither of these, but uses three different measures of relatedness, minimum description length (MDL), symmetrical uncertainty, and relief.

Let xi be the set membership indicator function for feature fi ; then the above can be rewritten as an optimization problem:

${\displaystyle \mathrm {CFS} =\max _{x\in \{0,1\}^{n}}\left[{\frac {(\sum _{i=1}^{n}a_{i}x_{i})^{2}}{\sum _{i=1}^{n}x_{i}+\sum _{i\neq j}2b_{ij}x_{i}x_{j}}}\right].}$

The combinatorial problems above are, in fact, mixed 0–1 linear programming problems that can be solved by using branch-and-bound algorithms.[25]

## Regularized trees

The features from a decision tree or a tree ensemble are shown to be redundant. A recent method called regularized tree[26] 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)[27] is one type of regularized trees. The guided RRF is an enhanced RRF which is guided by the importance scores from an ordinary random forest.

## Overview on metaheuristics methods

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 stochastics algorithm tending to reach a global optima. There are many metaheuristics, from a simple local search to a complex global search algorithm.

### Main principles

The feature selection methods are typically presented in three classes based on how they combine the selection algorithm and the model building.

#### Filter Method

Filter Method for feature selection

Filter-based feature selection has become crucial in many classification settings, especially object recognition, recently faced with feature learning strategies that originate thousands of cues.[28] Filter methods analyze intrinsic properties of data, ignoring the classifier. Most of these methods can perform two operations, ranking and subset selection: in the former, the importance of each individual feature is evaluated, usually by neglecting potential interactions among the elements of the joint set; in the latter, the final subset of features to be selected is provided. In some cases, these two operations are performed sequentially (first the ranking, then the selection); in other cases, only the selection is carried out.[28] Filter methods suppress the least interesting variables. These methods are particularly effective in computation time and robust to overfitting.[29]

However, filter methods tend to select redundant variables because they do not consider the relationships between variables. Therefore, they are mainly used as a pre-process method.

#### Wrapper Method

Wrapper Method for Feature selection

Wrapper methods evaluate subsets of variables which allows, unlike filter approaches, to detect the possible interactions between variables.[30] The two main disadvantages of these methods are :

• The increasing overfitting risk when the number of observations is insufficient.
• The significant computation time when the number of variables is large.

#### Embedded Method

Embedded method for Feature selection

Recently, embedded methods have been proposed to reduce the classification of learning. They try to combine the advantages of both previous methods. The learning algorithm takes advantage of its own variable selection algorithm. So, it needs to know preliminary what a good selection is, which limits their exploitation.[31]

### Application of feature selection metaheuristics

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.[29]

SNPs Feature Selection using Feature Similarity 过滤 ŕ Phuong 2005 [30]
SNPs 这些理论包括： {0}wrapper{/0}{1}.{/1} Decision Tree Classification accuracy (10-fold) Shah 2004 [32]
SNPs HillClimbing Filter + Wrapper Naive Bayesian Predicted residual sum of squares
SNPs Simulated Annealing Naive bayesian Classification accuracy (5-fold) Ustunkar 2011 [33]
Segments parole Ants colony {0}wrapper{/0}{1}.{/1} Artificial Neural Network MSE Al-ani 2005 [來源請求]

Spectral Mass 这些理论包括： {0}wrapper{/0}{1}.{/1} Multiple Linear Regression, Partial Least Squares root-mean-square error of prediction Broadhurst 2007 [36]
Microarray Tabu Search + PSO {0}wrapper{/0}{1}.{/1} Support Vector Machine, K Nearest Neighbors Euclidean Distance Chuang 2009 [37]
Microarray PSO + Genetic Algorithm {0}wrapper{/0}{1}.{/1} Support Vector Machine Classification accuracy (10-fold) Alba 2007 [38]
Microarray Genetic Algorithm + Iterated Local Search EMBEDDED Support Vector Machine Classification accuracy (10-fold) Duval 2009 [31]
Microarray Iterated Local Search {0}wrapper{/0}{1}.{/1} 回归 Posterior Probability Hans 2007 [39]
Microarray 这些理论包括： {0}wrapper{/0}{1}.{/1} K Nearest Neighbors Classification accuracy (Leave-one-out cross-validation) Jirapech-Umpai 2005 [40]
Microarray Hybrid Genetic Algorithm {0}wrapper{/0}{1}.{/1} K Nearest Neighbors Classification accuracy (Leave-one-out cross-validation)
Microarray 这些理论包括： {0}wrapper{/0}{1}.{/1} Support Vector Machine Sensitivity and specificity Xuan 2011 [41]
Microarray 这些理论包括： {0}wrapper{/0}{1}.{/1} All paired Support Vector Machine Classification accuracy (Leave-one-out cross-validation) Peng 2003 [42]
Microarray 这些理论包括： EMBEDDED Support Vector Machine Classification accuracy (10-fold) Hernandez 2007 [43]
Microarray 这些理论包括： 混合式 Support Vector Machine Classification accuracy (Leave-one-out cross-validation) Huerta 2006 [44]
Microarray 这些理论包括： Support Vector Machine Classification accuracy (10-fold) Muni 2006 [45]
Microarray 这些理论包括： {0}wrapper{/0}{1}.{/1} Support Vector Machine EH-DIALL, CLUMP Jourdan 2004 [46]
Object Recognition Infinite Feature Selection 过滤 Support Vector Machine Mean Average Precision (mAP) Roffo 2015 [28]

## Feature selection embedded in learning algorithms

Some learning algorithms perform feature selection as part of their overall operation. 这些心理素质包括:

• ${\displaystyle l_{1}}$-regularization techniques, such as sparse regression, LASSO, and ${\displaystyle l_{1}}$-SVM
• Regularized trees,[26] e.g. regularized random forest implemented in the RRF package[27]
• Decision tree[來源請求]
• Memetic algorithm
• Random multinomial logit (RMNL)
• Auto-encoding networks with a bottleneck-layer
• Submodular feature selection[47][48][49]

## 参考文献

1. Gareth James; Daniela Witten; Trevor Hastie; Robert Tibshirani. An Introduction to Statistical Learning. Springer. 2013: 204.
2. Bermingham, Mairead L.; Pong-Wong, Ricardo; Spiliopoulou, Athina; Hayward, Caroline; Rudan, Igor; Campbell, Harry; Wright, Alan F.; Wilson, James F.; Agakov, Felix; Navarro, Pau; Haley, Chris S. Application of high-dimensional feature selection: evaluation for genomic prediction in man. Sci. Rep. 2015, 5.
3. Guyon, Isabelle; Elisseeff, André. An Introduction to Variable and Feature Selection. JMLR. 2003, 3.
4. Yang, Yiming; Pedersen, Jan O. A comparative study on feature selection in text categorization. ICML. 1997.
5. ^ Forman, George. An extensive empirical study of feature selection metrics for text classification. Journal of Machine Learning Research. 2003, 3: 1289–1305.
6. ^ Bach, Francis R. Bolasso: model consistent lasso estimation through the bootstrap. Proceedings of the 25th international conference on Machine learning. 2008: 33–40. doi:10.1145/1390156.1390161.
7. ^ Zare, Habil. Scoring relevancy of features based on combinatorial analysis of Lasso with application to lymphoma diagnosis. BMC Genomics. 2013, 14: S14. doi:10.1186/1471-2164-14-S1-S14.
8. ^ Figueroa, Alejandro. Exploring effective features for recognizing the user intent behind web queries. Computers in Industry. 2015, 68: 162–169. doi:10.1016/j.compind.2015.01.005.
9. ^ Figueroa, Alejandro; Guenter Neumann. Learning to Rank Effective Paraphrases from Query Logs for Community Question Answering. AAAI. 2013.
10. ^ Figueroa, Alejandro; Guenter Neumann. Category-specific models for ranking effective paraphrases in community Question Answering. Expert Systems with Applications. 2014, 41: 4730–4742. doi:10.1016/j.eswa.2014.02.004.
11. ^ Zhang, Y.; Wang, S.; Phillips, P. Binary PSO with Mutation Operator for Feature Selection using Decision Tree applied to Spam Detection. Knowledge-Based Systems. 2014, 64: 22–31. doi:10.1016/j.knosys.2014.03.015.
12. ^ F.C. Garcia-Lopez, M. Garcia-Torres, B. Melian, J.A. Moreno-Perez, J.M. Moreno-Vega. Solving feature subset selection problem by a Parallel Scatter Search, European Journal of Operational Research , vol. 169, no. 2, pp. (2006年).
13. ^ F.C. Garcia-Lopez, M. Garcia-Torres, B. Melian, J.A. Moreno-Perez, J.M. Moreno-Vega. Solving Feature Subset Selection Problem by a Hybrid Metaheuristic. In First International Workshop on Hybrid Metaheuristics , pp. 59–68, 2004.
14. ^ M. Garcia-Torres, F. Gomez-Vela, B. Melian, J.M. Moreno-Vega. High-dimensional feature selection via feature grouping: A Variable Neighborhood Search approach, Information Sciences , vol. 326, pp. {0}8.{/0} {1}{/1}
15. ^ Alexander Kraskov, Harald Stögbauer, Ralph G. Andrzejak, and Peter Grassberger, "Hierarchical Clustering Based on Mutual Information", (2003) ArXiv q-bio/0311039
16. ^ Aliferis, Constantin. Local causal and markov blanket induction for causal discovery and feature selection for classification part I: Algorithms and empirical evaluation (PDF). Journal of Machine Learning Research. 2010, 11: 171–234.
17. ^ Peng, H. C.; Long, F.; Ding, C. Feature selection based on mutual information: criteria of max-dependency, max-relevance, and min-redundancy. IEEE Transactions on Pattern Analysis and Machine Intelligence. 2005, 27 (8): 1226–1238. PMID 16119262. doi:10.1109/TPAMI.2005.159.教学大纲
18. Brown, G., Pocock, A., Zhao, M.-J., Lujan, M. (2012). "Conditional Likelihood Maximisation: A Unifying Framework for Information Theoretic Feature Selection", In the Journal of Machine Learning Research (JMLR). [1]
19. ^ Nguyen, H., Franke, K., Petrovic, S. (2010). "Towards a Generic Feature-Selection Measure for Intrusion Detection", In Proc. International Conference on Pattern Recognition (ICPR), Istanbul, Turkey. [2]
20. ^ Rodriguez-Lujan, I.; Huerta, R.; Elkan, C.; Santa Cruz, C. Quadratic programming feature selection (PDF). JMLR. 2010, 11: 1491–1516.
21. Nguyen X. Vinh, Jeffrey Chan, Simone Romano and James Bailey, "Effective Global Approaches for Mutual Information based Feature Selection". Proceeedings of the 20th ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD'14), August 24–27, New York City, 2014. "[3]"
22. ^ M. Yamada, W. Jitkrittum, L. Sigal, E. P. Xing, M. Sugiyama, High-Dimensional Feature Selection by Feature-Wise Non-Linear Lasso. Neural Computation, vol.26, no.1, pp.185-207, 2014.
23. ^
24. ^ Senliol, Baris, et al. "Fast Correlation Based Filter (FCBF) with a different search strategy." Computer and Information Sciences, 2008. ISCIS'08. 23rd International Symposium on. 978-1-4244-2108-4/08 /©2008电气与电子工程师协会[4]
25. ^ Hai Nguyen, Katrin Franke, and Slobodan Petrovic, Optimizing a class of feature selection measures, Proceedings of the NIPS 2009 Workshop on Discrete Optimization in Machine Learning: Submodularity, Sparsity & Polyhedra (DISCML), Vancouver, Canada, December 2009. [5]
26. H. Deng, G. Runger, "Feature Selection via Regularized Trees", Proceedings of the 2012 International Joint Conference on Neural Networks (IJCNN), IEEE, 2012
27. RRF: Regularized Random Forest, R package on CRAN
28. Roffo, Giorgio; Melzi, Simone; Cristani, Marco. Infinite Feature Selection. www.cv-foundation.org. International Conference on Computer Vision. 2015 [2016-01-25].
29. T. M. Phuong, Z. Lin et R. B. Altman. Choosing SNPs using feature selection. Proceedings / IEEE Computational Systems Bioinformatics Conference, CSB. IEEE Computational Systems Bioinformatics Conference, pages 301-309, 2005. PMID:
30. B. Duval, J.-K. Hao et J. C. Hernandez Hernandez. A memetic algorithm for gene selection and molecular classification of an cancer. In Proceedings of the 11th Annual conference on Genetic and evolutionary computation, GECCO '09, pages 201-208, New York, NY, USA, 2009. （u A/cm2）
31. ^ S. C. Shah et A. Kusiak. Data mining and genetic algorithm based gene/SNP selection. Artificial intelligence in medicine, vol. 31, no. 3, pages 183-196, July 2004. PMID:
32. ^ G. Ustunkar, S. Ozogur-Akyuz, G. W. Weber, C. M. Friedrich et Yesim Aydin Son. Selection of representative SNP sets for genome-wide association studies : a metaheuristic approach. Optimization Letters, November 2011.
33. ^ R. Meiri et J. Zahavi. Using simulated annealing to optimize the feature selection problem in marketing applications. European Journal of Operational Research, vol. 171, no. 3, pages 842-858, Juin 2006
34. ^ G. Kapetanios. Variable Selection using Non-Standard Optimisation of Information Criteria. Working Paper 533, Queen Mary, University of London, School of Economics and Finance, 2005.
35. ^ D. Broadhurst, R. Goodacre, A. Jones, J. J. Rowland et D. B. Kell. Genetic algorithms as a method for variable selection in multiple linear regression and partial least squares regression, with applications to pyrolysis mass spectrometry. Analytica Chimica Acta, vol. 348, no. 1-3, pages 71-86, August 1997.
36. ^ Chuang, L.-Y.; Yang, C.-H. Tabu search and binary particle swarm optimization for feature selection using microarray data. Journal of computational biology. 2009, 16 (12): 1689–1703. PMID 20047491. doi:10.1089/cmb.2007.0211.
37. ^ E. Alba, J. Garia-Nieto, L. Jourdan et E.-G. Talbi. Gene Selection in Cancer Classification using PSO-SVM and GA-SVM Hybrid Algorithms. Congress on Evolutionary Computation, Singapor : Singapore (2007), 2007
38. ^ C. Hans, A. Dobra et M. West. Shotgun stochastic search for 'large p' regression. Journal of the American Statistical Association, 2007.
39. ^ T. Jirapech-Umpai et S. Aitken. Feature selection and classification for microarray data analysis : Evolutionary methods for identifying predictive genes. BMC bioinformatics, vol. 6, no. 1, page 148, 2005.
40. ^ Xuan, P.; Guo, M. Z.; Wang, J.; Liu, X. Y.; Liu, Y. Genetic algorithm-based efficient feature selection for classification of pre-miRNAs. Genetics and Molecular Research. 2011, 10 (2): 588–603. PMID 21491369. doi:10.4238/vol10-2gmr969.
41. ^ S. Peng. Molecular classification of cancer types from microarray data using the combination of genetic algorithms and support vector machines. FEBS Letters, vol. 555, no. 2, pages 358-362, December 2003.
42. ^ J. C. H. Hernandez, B. Duval et J.-K. Hao. A genetic embedded approach for gene selection and classification of microarray data. In Proceedings of the 5th European conference on Evolutionary computation, machine learning and data mining in bioinformatics, EvoBIO'07, pages 90-101, Berlin, Heidelberg, 2007. SpringerVerlag.
43. ^ E. B. Huerta, B. Duval et J.-K. Hao. A hybrid GA/SVM approach for gene selection and classification of microarray data. evoworkshops 2006, LNCS, vol. 3907, pages 34-44, 2006.
44. ^ D. P. Muni, N. R. Pal et J. Das. Genetic programming for simultaneous feature selection and classifier design. IEEE Transactions on Systems, Man, and Cybernetics, Part B : Cybernetics, vol. 36, no. 1, pages 106-117, February 2006.
45. ^ L. Jourdan, C. Dhaenens et E.-G. Talbi. Linkage disequilibrium study with a parallel adaptive GA. International Journal of Foundations of Computer Science, 2004.
46. ^
47. ^
48. ^

## 外部链接

[[Category:{0}3{/0}{1}     {/1}{0}模型选择{/0}]]