Tilted Euler characteristic densities for Central Limit random fields, with application to "bubbles"

TL;DR

This paper develops a saddlepoint approximation for Euler characteristic densities of nearly Gaussian random fields, improving thresholding accuracy in statistical tests, especially for non-Gaussian data like 'bubbles' with applications in brain imaging.

Contribution

It introduces a saddlepoint-based expansion for EC densities applicable to non-Gaussian fields approximated by CLT, enhancing thresholding precision in statistical field analysis.

Findings

Derived a saddlepoint expansion for EC densities of non-Gaussian fields.

Showed improved asymptotic accuracy over Gaussian EC densities when thresholds grow with sample size.

Applied the method to 'bubbles' data demonstrating practical utility.

Abstract

Local increases in the mean of a random field are detected (conservatively) by thresholding a field of test statistics at a level $u$ chosen to control the tail probability or $p$-value of its maximum. This $p$-value is approximated by the expected Euler characteristic (EC) of the excursion set of the test statistic field above $u$, denoted $\mathbb{E}\varphi(A_u)$. Under isotropy, one can use the expansion $\mathbb{E}\varphi(A_u)=\sum_k\mathcal{V}_k\rho_k(u)$, where $\mathcal{V}_k$ is an intrinsic volume of the parameter space and $\rho_k$ is an EC density of the field. EC densities are available for a number of processes, mainly those constructed from (multivariate) Gaussian fields via smooth functions. Using saddlepoint methods, we derive an expansion for $\rho_k(u)$ for fields which are only approximately Gaussian, but for which higher-order cumulants are available. We focus on linear combinations of $n$ independent non-Gaussian fields, whence a Central Limit theorem is in force. The threshold $u$ is allowed to grow with the sample size $n$, in which case our expression has a smaller relative asymptotic error than the Gaussian EC density. Several illustrative examples including an application to "bubbles" data accompany the theory.