Estimating thresholding levels for random fields via Euler characteristics

Understand

We introduce Lipschitz-Killing curvature (LKC) regression, a new method to produce $(1-\alpha)$ thresholds for signal detection in random fields that does not require knowledge of the spatial correlation structure.

• The idea is to fit observed empirical Euler characteristics to the Gaussian kinematic formula via generalized least squares, which quickly and easily provides statistical estimates of the LKCs --- complex topological quantities that can be extremely challenging to compute, both theoretically and numerically.
• With these estimates, we can then make use of a powerful parametric approximation via Euler characteristics for Gaussian random fields to generate accurate $(1-\alpha)$ thresholds and $p$-values.
• The main features of our proposed LKC regression method are easy implementation, conceptual simplicity, and facilitated diagnostics, which we demonstrate in a variety of simulations and applications.