Statistical challenges with high dimensionality: feature selection in knowledge discovery.

*(English)* Zbl 1117.62137
Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume III: Invited lectures. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 595-622 (2006).

Summary: Technological innovations have revolutionized the process of scientific research and knowledge discovery. The availability of massive data and challenges from frontiers of research and development have reshaped statistical thinking, data analysis and theoretical studies. The challenges of high-dimensionality arise in diverse fields of sciences and the humanities, ranging from computational biology and health studies to financial engineering and risk management. In all of these fields, variable selection and feature extraction are crucial for knowledge discovery. We first give a comprehensive overview of statistical challenges with high dimensionality in these diverse disciplines. We then approach the problem of variable selection and feature extraction using a unified framework: penalized likelihood methods. Issues relevant to the choice of penalty functions are addressed. We demonstrate that for a host of statistical problems, as long as the dimensionality is not excessively large, we can estimate the model parameters as well as if the best model is known in advance. The persistence property in risk minimization is also addressed. The applicability of such a theory and method to diverse statistical problems is demonstrated. Other related problems with high-dimensionality are also discussed. For the entire collection see [

Zbl 1095.00006].

##### MSC:

62P99 | Applications of statistics |

62A01 | Foundations and philosophical topics in statistics |

62J99 | Linear statistical inference |

62F12 | Asymptotic properties of parametric estimators |

##### Keywords:

AIC, BIC, LASSO, bioinformatics, financial econometrics, model selection, oracle property, penalized likelihood, persistent, SCAD, statistical learning