Exogenous cascades versus self-excitation: distinguishing Markov-Switching-Multifractal Cox processes from Hawkes processes, with an application to musical onsets

Abstract

Bursty, hierarchically clustered event streams are almost universally modeled as self-exciting Hawkes processes. We study an alternative mechanism — exogenous hierarchical regime switching — realized as a doubly-stochastic (Cox) process whose intensity is a discrete-time Markov-Switching Multifractal (MSM): N_t ~ Poisson(λ0 · G_t^κ) with G_t the product of k mirror-balanced two-state multipliers on a geometric rate ladder. We derive the closed-form stationary moments, overdispersion, and autocovariance of the MSM-Cox counts, and show the autocovariance is an exact sum of geometrically-spaced exponentials — the 1/f mechanism. We then prove a mimicry result: because a Hawkes count autocovariance is itself an exponential sum, a Hawkes kernel can be fit to match the MSM-Cox second-order structure to R² > 0.999. The processes nonetheless separate at higher order: MSM-Cox is genuinely multifractal (MFDFA width h(-2)-h(2) > 0), has heavier inter-burst quiet-time tails, and — decisively — has an intensity that is autonomous of realized events, which the time-rescaling residual test detects. A likelihood cross-fitting study yields a strongly diagonal model-selection matrix at 10^4+ events, including for near-critical Hawkes (branching ratio 0.9), the regime where second-order mimicry is strongest. Applied to onset streams from the Fenophone generative instrument (an MSM-Cox by construction), the tools select MSM-Cox and attribute the decision to the discriminating statistics. Applied to real music (first corpus run 2026-07-12; seeded 120-piece samples per corpus on a 12-ticks/beat grid), min-BIC selection chooses MSM-Cox for 84% of MAESTRO piano performances — real piano onset streams prefer exogenous regime-switching over self-excitation — while Groove drum-kit streams on the beat grid select Poisson/Hawkes (58%/37%), consistent with the companion finding that the drum idiom's cascade lives in velocity, not count density. The earthquake positive control (run 2026-07-13, SCEDC M≥2.5 2010–2020): handed a Hawkes-true catalog, min-BIC selects Hawkes — decisively with a two-exponential (Omori-approximating) kernel — with the instructive twist that MSM-Cox out-fits a single-exponential Hawkes on those same aftershocks, quantifying on real data that likelihood selection between mechanism classes is fair only when each class is given comparable timescale flexibility (mimicry cuts both ways; §7a).