Expected number of zeros of a random Gaussian process — predicted by spectral moments
Kac formula E[zeros]—
Observed (avg)—
λ₀ (variance)—
λ₂ (spectral 2nd moment)—
The Kac-Rice formula gives the expected number of zeros of a smooth random function:
E[N₀] = (L/π)·√(λ₂/λ₀), where λ_k = ∫ω^k S(ω) dω are spectral moments of the power spectrum S(ω).
For a stationary Gaussian process with covariance C(τ) = exp(−τ²/2ℓ²), we get λ₂/λ₀ = 1/ℓ².
Shorter correlation length → more oscillations → more zeros. The formula is exact for Gaussian processes.