Typical Yet Unlikely and Normally Abnormal: The Intuition Behind High-Dimensional Statistics

TL;DR

This paper explores the peculiar behaviors of high-dimensional data, introduces the concept of typicality for better normality characterization, and demonstrates its application in outlier detection contrasting with traditional methods.

Contribution

It provides intuition on high-dimensional data peculiarities and introduces typicality as a novel, information-theoretic approach for normality assessment and outlier detection.

Findings

High-dimensional data exhibit severe peculiarities even at low dimensions.

Typicality offers a better characterization of normality than traditional measures.

Typicality can effectively identify outliers, outperforming Mahalanobis distance.

Abstract

Normality, in the colloquial sense, has historically been considered an aspirational trait, synonymous with ideality. The arithmetic average and, by extension, statistics including linear regression coefficients, have often been used to characterize normality, and are often used as a way to summarize samples and identify outliers. We provide intuition behind the behavior of such statistics in high dimensions, and demonstrate that even for datasets with a relatively low number of dimensions, data start to exhibit a number of peculiarities which become severe as the number of dimensions increases. Whilst our main goal is to familiarize researchers with these peculiarities, we also show that normality can be better characterized with `typicality', an information theoretic concept relating to entropy. An application of typicality to both synthetic and real-world data concerning political values reveals that in multi-dimensional space, to be `normal' is actually to be atypical. We briefly explore the ramifications for outlier detection, demonstrating how typicality, in contrast with the popular Mahalanobis distance, represents a viable method for outlier detection.