# The Large P small n Project

A scientific project for designing sound statistical methods to make inference from high-dimensional multivariate data.

# Overview

The Large $$P$$ small $$n$$ project – or $$Pn$$ project for short – aims at providing inferential tools for the statistical analysis of data sets characterized by a large number of features observed on a small number of sample units, often referred to in the literature as high-dimensional multivariate data. During the last decades, this kind of data have become very common in many active fields of research, such as medicine, engineering, climatology, and economics. The first application of the newly developed inferential procedures will be the statistical analysis of the brain vascular system with the final goal of detecting statistical associations between geometry, topology, hemodynamic space-time patterns and the onset and rupture of cerebral aneurysms.

# Consortium

The scientific team behind the Large $$P$$ small $$n$$ project is composed of $$4$$ researchers from three different institutions:

Name Role Position Institution
Simone Vantini Principal Investigator Associate Professor in Statistics MOX, Department of Mathematics F. Brioschi, Politecnico di Milano, Italy
Tiziano Passerini Associate Investigator Post-Doctoral Fellow Emory University, USA
Aymeric Stamm Associate Investigator Ph.D. Student IRISA, Univ. of Rennes I, France
Alessia Pini Ph.D. Student Ph.D. Student MOX, Department of Mathematics F. Brioschi, Politecnico di Milano, Italy

# Scopes

During the last decades, the phenomena investigated in many active fields of research (e.g., medicine, engineering, climatology, economics) have become increasingly complex, providing large amount of information to be statistically analyzed. Consequently, the typical data set - provided by scientific experiments or observations - has drifted from being characterized by a number $$n$$ of sample units much larger than the number $$p$$ of variables (small $$p$$ large $$n$$ data), to its large $$p$$ small $$n$$ counterpart (i.e. data sets characterized by a number $$p$$ of variables much larger than the number $$n$$ of sample units).

Inferential statistical analysis of traditional small $$p$$ large $$n$$ data often relies on the well known central limit theorem: this theorem, under very weak assumptions (i.e. independence and identical distribution of the observations) guarantees the reliability of many commonly used inferential statistical procedures (e.g. hypothesis testing, confidence intervals, …) when the number $$n$$ of sample units is far larger than the number $$p$$ of variables.

The Large $$p$$ Small $$n$$ project aims at providing a new $$p$$-asymptotic central limit theorem (i.e. $$p$$ goes to infinity with $$n$$ finite) enabling inferential analysis of modern large $$p$$ small $$n$$ data: both in the $$p$$-discrete case (random vectors with a very large number of components) and in the $$p$$-continuous case (random functions). No reliable inferential results are nowadays available to this purpose.

The achievement of such results and the development of the consequent inferential tools (that are going to be jointly carried out by the Department of Mathematics of Politecnico di Milano and by the IRISA Institut de Recherche en Informatique et Systèmes Aléatoires, Rennes, France) will involve expertise in advanced fields of mathematics (theory of stochastic processes, multivariate statistics and data analysis, functional real analysis, and operator theory). In spite of this, the present project aims at providing tools that will be reasonably simple and readily usable by any member of the scientific community (an automated R library and a Microsoft Excel macro are expected to be provided at the end of the project).

The first application of the newly developed inferential procedures will be the statistical analysis of the brain vascular system of different groups of patients hospitalized at the Neuroradiology Department of Niguarda Ca’ Granda Hospital – Milan. The final goal of the analysis is the detection of statistical associations between geometry, topology, hemodynamic space-time patterns and the onset and rupture of cerebral aneurysms. These associations, if found, would help neurosurgeons in the choice of the patient’s best treatment, possibly decreasing the mortality rate due to both this pathology and to its medical treatments.

Raw data collected at the Neuroradiology Department of Niguarda Ca’ Granda Hospital – Milan are going to be preprocessed by the Department of Mathematics and Computer Science of Emory University – Atlanta, GA, USA.

# State of the project

The Pn project was a $$2$$-year project that started in november 2011 until december 2013 and is now over. The project partners were Politecnico di Milano, Unversity of Rennes I and Emory University. The project funder was the 5x1000 donation program of the Politecnico di Milano. As for the team members, here is what they became since the end of the project:

• Simone is now Full Professor in Statistics at the Politecnico di Milano, Italy.
• Tiziano is now a research scientist at Siemens Corporation, Princeton, USA.
• I am now a research engineer, expert in statistical information, at the National Center for Scientific Research (CNRS), France.
• Alessia is now Associate Professor in Statistics at the Università del Sacro Cuore, Milano, Italy.
Pini, Alessia, Aymeric Stamm, and Simone Vantini. 2018. “Hotelling’s T2 in Separable Hilbert Spaces.” Journal of Multivariate Analysis 167: 284–305. https://doi.org/10.1016/j.jmva.2018.05.007.
Secchi, Piercesare, Aymeric Stamm, and Simone Vantini. 2013. “Inference for the Mean of Large p Small n Data: A Finite-Sample High-Dimensional Generalization of Hotelling’s Theorem.” Electronic Journal of Statistics 7: 2005–31. https://doi.org/10.1214/13-EJS833.