[Paper Review] Two-dimensional Kac-Rice formula. Application to shot noise processes excursions

Raphaël Lachièze-Rey|arXiv (Cornell University)|Jul 19, 2016

Simulation Techniques and Applications4 citations

TL;DR

This paper establishes a two-dimensional Kac-Rice formula that expresses the Euler characteristic of excursion sets of a deterministic $ C^2 $ function $ f: \mathbb{R}^2 \to \mathbb{R} $ as an integral over $ \mathbb{R}^2 $, involving $ f $, its gradient $ \nabla f $, and its Hessian. The formula enables the computation of the expected Euler characteristic for random fields, including shot noise processes, without requiring density assumptions on the field or its derivatives, and ensures continuity and moment finiteness under mild regularity conditions.

ABSTRACT

Given a deterministic function f:R^2->R atisfying suitable assump- tions, we show that for h smooth with compact support, the integral of the Euler characteristic of the excursion set of f above some level u against a test function h corresponds to the Lebesgue integral on R^2 of a bounded quantity depending on grad(f)(x),h(f(x)),h'(f(x)) and \\partial_{ii}f(x),i = 1,2. This formula can be seen as a 2-dimensional analogue of Kac-Rice formula. It yields in particular that the left hand member is continuous in the argument f, for an appropriate norm on the space of C2 functions. If f is a random field, the expectation can be passed under integrals in this identity under minimal requirements, not involving any density assumptions on the marginals of f or his derivatives. We apply these results to give a weak expression of the mean Euler characteristic of a shot noise process, and the finiteness of its moments.

Motivation & Objective

• To derive a deterministic integral formula for the Euler characteristic of excursion sets in two dimensions, extending the classical Kac-Rice approach.
• To establish continuity of the Euler characteristic functional in the $ C^2 $ topology, enabling robust analysis of perturbations in random fields.
• To provide a weak formulation of the mean Euler characteristic for random $ C^2 $ fields, particularly shot noise processes, under minimal probabilistic assumptions.
• To avoid reliance on density conditions for the field or its derivatives, thus broadening applicability to non-Gaussian and non-regular fields.
• To demonstrate the finiteness of moments of the Euler characteristic functional for shot noise processes via the derived formula.

Proposed method

• Derives a two-dimensional Kac-Rice formula by applying co-area and variographic techniques to express the Euler characteristic of excursion sets as an integral over $ \mathbb{R}^2 $.
• Introduces a functional $ \chi_f(h) = \int_\mathbb{R} h(u) \chi(\{f \geq u\}) du $, the Euler primitive, and shows it equals a Lebesgue integral involving $ h(f(x)) $, $ h'(f(x)) $, $ \partial_i f(x) $, and $ \partial_{ii}^2 f(x) $.
• Defines quarter-plane regions $ Q_1, Q_2 \subset \mathbb{R}^2 $ based on the sign of the gradient components to localize contributions to the Euler characteristic.
• Uses $ \mathcal{C}^{1,1} $ regularity (almost everywhere twice differentiability) rather than Morse conditions, broadening applicability beyond critical point counting.
• Applies the formula to shot noise processes by leveraging stationarity and isotropy to compute characteristic functions and derive moment finiteness.
• Uses improper integral representations of sign functions via $ \int_0^\infty \frac{\sin(ux)}{u} du = \frac{\pi}{2} \text{sign}(u) $ to handle expectation expressions involving products of sign functions of derivatives.

Experimental results

Research questions

• RQ1 Can the Euler characteristic of two-dimensional excursion sets be expressed as a smooth integral over the domain without relying on critical point counting?
• RQ2 Does the Euler characteristic functional remain continuous under $ C^2 $-topology convergence, even without Morse or density assumptions?
• RQ3 Can the mean Euler characteristic of a random field be computed without requiring the density of its marginals or derivatives?
• RQ4 What conditions ensure the finiteness of moments of the Euler characteristic for shot noise processes?
• RQ5 How can the Kac-Rice formula be adapted to non-Gaussian, non-Morse fields using weak formulations?

Key findings

• The Euler characteristic of the excursion set $ \{f \geq u\} $ is expressed as a Lebesgue integral over $ \mathbb{R}^2 $, involving $ h'(f(x)) \partial_i f(x)^2 $ and $ h(f(x)) \partial_{ii}^2 f(x) $, weighted by indicator functions on gradient regions $ Q_i $.
• The formula holds under $ \mathcal{C}^{1,1} $ regularity, allowing it to apply to non-Morse functions, and ensures continuity of the Euler primitive in the $ C^2 $ norm.
• For random $ C^2 $ fields, the expectation of the Euler primitive can be computed without assuming density of $ f(x) $ or its derivatives, enabling analysis of non-Gaussian fields.
• The mean Euler characteristic of shot noise processes is shown to be finite, and its moments are proven to be finite under mild moment conditions on the field and its derivatives.
• For isotropic, stationary shot noise fields, the characteristic function of the Euler primitive is expressed via derivatives of the characteristic function of $ (f(0), \nabla f(0), \partial_{ii}^2 f(0)) $, enabling explicit computation in specific cases.
• The formula allows the derivation of the asymptotic behavior of the Euler characteristic in large domains via weak convergence and $ o(|W_n|) $ error control.