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**Variable selection and updating in model-based discriminant analysis for high dimensional data with food authenticity applications.**
*(English)*
Zbl 1189.62105

Summary: Food authenticity studies are concerned with determining if food samples have been correctly labeled or not. Discriminant analysis methods are an integral part of the methodology for food authentication. Motivated by food authenticity applications, a model-based discriminant analysis method that includes variable selection is presented. The discriminant analysis model is fitted in a semi-supervised manner using both labeled and unlabeled data. The method is shown to give excellent classification performance on several high-dimensional multiclass food authenticity data sets with more variables than observations. The variables selected by the proposed method provide information about which variables are meaningful for classification purposes. A headlong search strategy for variable selection is shown to be efficient in terms of computation and achieves excellent classification performance. In applications to several food authenticity data sets, our proposed method outperformed default implementations of Random Forests, AdaBoost, transductive SVMs and Bayesian Multinomial Regression by substantial margins.

### MSC:

62H30 | Classification and discrimination; cluster analysis (statistical aspects) |

62P99 | Applications of statistics |

65C60 | Computational problems in statistics (MSC2010) |

### Keywords:

food authenticity studies; headlong search; model-based discriminant analysis; normal mixture models; semi-supervised learning; updating classification rules; variable selection
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\textit{T. B. Murphy} et al., Ann. Appl. Stat. 4, No. 1, 396--421 (2010; Zbl 1189.62105)

### References:

[1] | Arnalds, T., McElhinney, J., Fearn, T. and Downey, G. (2004). A hierarchical discriminant analysis for species identification in raw meat by visible and near infrared spectroscopy. Journal of Near Infrared Spectroscopy 12 183-188. |

[2] | Badsberg, J. H. (1992). Model search in contingency tables by CoCo. In Computational Statistics (Y. Dodge and J. Whittaker, eds.) 1 251-256. Physica, Heidelberg. |

[3] | Banfield, J. D. and Raftery, A. E. (1993). Model-based Gaussian and non-Gaussian clustering. Biometrics 49 803-821. · Zbl 0794.62034 |

[4] | Bensmail, H. and Celeux, G. (1996). Regularized Gaussian discriminant analysis through eigenvalue decomposition. J. Amer. Statist. Assoc. 91 1743-1748. · Zbl 0885.62068 |

[5] | Breiman, L. (2001). Random Forests. Mach. Learn. 45 5-32. · Zbl 1007.68152 |

[6] | Chang, W.-C. (1983). On using principal components before separating a mixture of two multivariate normal distributions. J. Roy. Statist. Soc. Ser. C. 32 267-275. · Zbl 0538.62050 |

[7] | Chapelle, O., Schölkopf, B. and Zien, A. (2006). Semi-Supervised Learning . MIT Press, Cambridge. Available at . |

[8] | Chiang, L. H. and Pell, R. J. (2004). Genetic algorithms combined with discriminant analysis for key variable identification. J. Process Control 14 143-155. |

[9] | Collobert, R., Sinz, F., Weston, J. and Bottou, L. (2006). Large scale transductive SVMs. J. Mach. Learn. Res. 7 1687-1712. · Zbl 1222.68173 |

[10] | Connolly, C. (2006). Spectroscopic and Analytical Developments Ltd fingerprints brand spirits with ultraviolet spectrophotometry. Sensor Review 26 94-97. |

[11] | Cortés, E. A., Martínez, M. G. and Rubio, N. G. (2007). adabag: Applies adaboost.M1 and bagging. R package version 1.1. |

[12] | Dash, D. and Cooper, G. F. (2004). Model averaging for prediction with discrete Bayesian networks. J. Mach. Learn. Res. 5 1177-1203. · Zbl 1222.68178 |

[13] | Dean, N., Murphy, T. B. and Downey, G. (2006). Using unlabelled data to update classification rules with applications in food authenticity studies. J. Roy. Statist. Soc. Ser. C 55 1-14. · Zbl 05188723 |

[14] | Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1-38. · Zbl 0364.62022 |

[15] | Downey, G. (1996). Authentication of food and food ingredients by near infrared spectroscopy. Journal of Near Infrared Spectroscopy 4 47-61. |

[16] | Downey, G., McIntyre, P. and Davies, A. N. (2003). Geographical classification of extra virgin olive oils from the eastern Mediterranean by chemometric analysis of visible and near infrared spectroscopic data. Applied Spectroscopy 57 158-163. |

[17] | Fraley, C. and Raftery, A. E. (1998). How many clusters? Which clustering method? Answers via model-based cluster analysis. Computer Journal 41 578-588. · Zbl 0920.68038 |

[18] | Fraley, C. and Raftery, A. E. (1999). MCLUST: Software for model-based clustering. J. Classification 16 297-306. · Zbl 0951.91500 |

[19] | Fraley, C. and Raftery, A. E. (2002). Model-based clustering, discriminant analysis, and density estimation. J. Amer. Statist. Assoc. 97 611-631. · Zbl 1073.62545 |

[20] | Fraley, C. and Raftery, A. E. (2003). Enhanced model-based clustering, density estimation and discriminant analysis software: MCLUST. J. Classification 20 263-296. · Zbl 1055.62071 |

[21] | Fraley, C. and Raftery, A. E. (2007). mclust: Model-based clustering/normal mixture modeling. R package version 3.1-1. · Zbl 1159.62302 |

[22] | Freund, Y. and Schapire, R. E. (1997). A decision-theoretic generalization of on-line learning and an application to boosting. J. Comp. System Sci. 55 119-139. · Zbl 0880.68103 |

[23] | Ganesalingam, S. and McLachlan, G. J. (1978). The efficiency of a linear discriminant function based on unclassified initial samples. Biometrika 65 658-662. · Zbl 0389.62045 |

[24] | Genkin, A., Lewis, D. D. and Madigan, D. (2005). BMR: Bayesian multinomial regression software. Available at . |

[25] | Greenshtein, E. (2006). Best subset selection, persistence in high-dimensional statistical learning and optimization under l 1 constraint. Ann. Statist. 34 2367-2386. · Zbl 1106.62022 |

[26] | Guyon, I. and Elisseeff, A. (2003). An introduction to variable and feature selection. J. Mach. Learn. Res. 3 1157-1182. · Zbl 1102.68556 |

[27] | Hastie, T. and Tibshirani, R. (1998). Classification by pairwise coupling. Ann. Statist. 26 451-471. · Zbl 0932.62071 |

[28] | Hastie, T., Tibshirani, R. and Friedman, J. H. (2001). The Elements of Statistical Learning . Springer, New York. · Zbl 0973.62007 |

[29] | Hoos, H. H. and Stützle, T. (2005). Stochastic Local Search: Foundations and Applications . Morgan Kaufmann, San Francisco. · Zbl 1126.68032 |

[30] | Indahl, U. and Naes, T. (2004). A variable selection strategy for supervised classification with continuous spectroscopic data. Journal of Chemometrics 18 53-61. |

[31] | Joachims, T. (1999). Transductive inference for text classification using support vector machines. In ICML’99: Proceedings of the Sixteenth International Conference on Machine Learning 200-209. Morgan Kaufmann, San Francisco. |

[32] | Kohavi, R. and John, G. (1997). Wrappers for feature selection. Artificial Intelligence 91 273-324. · Zbl 0904.68143 |

[33] | Liang, F., Mukherjee, S. and West, M. (2007). The use of unlabeled data in predictive modeling. Statist. Sci. 22 189-205. · Zbl 1246.62157 |

[34] | Liaw, A. and Wiener, M. (2002). Classification and regression by randomForest. R News 2 18-22. |

[35] | Liu, Y. and Chen, Y. R. (2000). Two-dimensional correlation spectroscopy study of visible and near-infrared spectral variations of chicken meats in cold storage. Applied Spectroscopy 54 1458-1470. |

[36] | Liu, Y., Chen, Y. R. and Ozaki, Y. (2000). Two-dimensional visible/near infrared correlation spectroscopy study of thermal treatment of chicken meat. Journal of Agricultural and Food Chemistry 48 901-908. |

[37] | Louw, N. and Steep, S. J. (2006). Variable selection in kernel Fisher discriminant analysis by means of recursive feature elimination. Comput. Statist. Data Anal. 51 2043-2055. · Zbl 1157.62440 |

[38] | Madigan, D. and Raftery, A. E. (1994). Model selection and accounting for model uncertainty in graphical models using Occam’s window. J. Amer. Statist. Assoc. 89 1535-1546. · Zbl 0814.62030 |

[39] | Madigan, D., Genkin, A., Lewis, D. D. and Fradkin, D. (2005). Bayesian multinomial logistic regression for author identification. In Bayesian Inference and Maximum Entropy Methods in Science and Engineering (K. H. Knuth, A. E. Abbas, R. D. Morris and J. P. Castle, eds.). AIP Conf. Proc. 803 509-516. Institute of Physics, London. |

[40] | Mary-Huard, T., Robin, S. and Daudin, J.-J. (2007). A penalized criterion for variable selection in classification. J. Multivariate Anal. 98 695-705. · Zbl 1118.62066 |

[41] | McElhinney, J., Downey, G. and Fearn, T. (1999). Chemometric processing of visible and near infrared reflectance spectra for species identification in selected raw homogenised meats. Journal of Near Infrared Spectroscopy 7 145-154. |

[42] | McLachlan, G. J. (1992). Discriminant Analysis and Statistical Pattern Recognition . Wiley, New York. · Zbl 1108.62317 |

[43] | McLachlan, G. J. and Peel, D. (2000). Finite Mixture Models . Wiley, New York. · Zbl 0963.62061 |

[44] | Munita, C. S., Barroso, L. P. and Oliveira, P. M. S. (2006). Stopping rule for variable selection using stepwise discriminant analysis. Journal of Radioanalytical and Nuclear Chemistry 269 335-338. |

[45] | Murphy, T. B., Dean, N. and Raftery, A. E. (2009). Supplement to “Variable selection and updating in model-based discriminant analysis for high dimensional data with food authenticity applications.” DOI: . · Zbl 1189.62105 |

[46] | O’Neill, T. J. (1978). Normal discrimination with unclassified observations. J. Amer. Statist. Assoc. 73 821-826. · Zbl 0409.62047 |

[47] | Osborne, B. G., Fearn, T. and Hindle, P. H. (1993). Practical NIR Spectroscopy With Applications in Food and Beverage Analysis . Longman Scientific & Technical, Harlow, UK. |

[48] | Osborne, B. G., Fearn, T., Miller, A. R. and Douglas, S. (1984). Application of near infrared reflectance spectroscopy to the compositional analysis of biscuits and biscuit doughs. Journal of the Science of Food and Agriculture 35 99-105. |

[49] | Pacheco, J., Casado, S., Núñez, L. and Gómez, O. (2006). Analysis of new variable selection methods for discriminant analysis. Comput. Statist. Data Anal. 51 1463-1478. · Zbl 1157.62442 |

[50] | R Development Core Team (2007). R: A Language and Environment for Statistical Computing . R Foundation for Statistical Computing. Vienna, Austria. |

[51] | Raftery, A. E. and Dean, N. (2006). Variable selection for model-based clustering. J. Amer. Statist. Assoc. 101 168-178. · Zbl 1118.62339 |

[52] | Reid, L. M., O’Donnell, C. P. and Downey, G. (2006). Recent technological advances in the determination of food authenticity. Trends in Food Science and Technology 17 344-353. |

[53] | Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461-464. · Zbl 0379.62005 |

[54] | Sinz, F. and Roffilli, M. (2007). UniverSVM software. Version 1.1. Available at . |

[55] | Szepannek, G. and Weihs, C. (2006). Variable selection for discrimination of more than two classes where data are sparse. In From Data and Information Analysis to Knowledge Engineering (M. Spiliopoulou, R. Kruse, C. Borgelt, A. Nurnberger and W. Gaul, eds.) 700-707. Springer, Berlin. |

[56] | Toher, D., Downey, G. and Murphy, T. B. (2007). A comparison of model-based and regression classification techniques applied to near infrared spectroscopic data in food authentication studies. Chemometrics and Intelligent Laboratory Systems 89 102-115. |

[57] | Trendafilov, N. T. and Jolliffe, I. T. (2007). DALASS: Variable selection in discriminant analysis via the LASSO. Comput. Statist. Data Anal. 51 3718-3736. · Zbl 1161.62379 |

[58] | Vapnik, V. (1995). The Nature of Statistical Learning Theory , 2nd ed. Springer, New York. · Zbl 0833.62008 |

[59] | Wang, L. and Xiatong, S. (2007). On L 1 -norm multiclass support vector machines: Methodology and theory. J. Amer. Statist. Assoc. 102 583-594. · Zbl 1172.62317 |

[60] | West, M. (2003). Bayesian factor regression models in the “large p , small n ” paradigm. In Bayesian Statistics 7 723-732. Oxford Univ. Press, Oxford. |

[61] | Yeung, K. Y., Bumgarner, R. and Raftery, A. E. (2005). Bayesian model averaging: Development of an improved multi-class, gene selection and classification tool for microarray data. Bioinformatics 21 2394-2402. |

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