Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2411.14297

Limitations of the Generalized Pareto Distribution-based estimators for the local dimension

Ignacio del Amo·George Datseris·Mark Holland
View original source ↗Read source document ↗Export metadata ↓

Abstract

Two dynamical indicators, the local dimension and the extremal index, used to quantify persistence in phase space have been developed and applied to different data across various disciplines. These are computed using the asymptotic limit of exceedances over a threshold, which turns to be a Generalized Pareto Distribution in many cases. However the derivation of the asymptotic distribution requires mathematical properties which are not present even in highly idealized dynamical systems, and unlikely to be present in real data. Here we examine in detail issues that arise when estimating these quantities for some known dynamical systems with a particular focus on how the geometry of an invariant set can affect the regularly varying properties of the invariant measure. We demonstrate that singular measures supported on sets of non-integer dimension are typically not regularly varying and that the absence of regular variation makes the estimates resolution dependent. We show as well that the most common extremal index estimation method is ambiguous for continuous time processes sampled at fixed time steps, which is an underlying assumption in its application to data.

Explore related subjects

math.DSnlin.CD

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ignacio del Amo, George Datseris, Mark Holland. 2024-11-25. Limitations of the Generalized Pareto Distribution-based estimators for the local dimension. https://arxiv.org/abs/2411.14297

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Asymmetry of a class of Mellin transforms

We introduce the quantity $μ_η$, defined for every complex $s$ in the critical strip, as a transformation of the Mellin transform associated to the functions $η$. We establish a sufficient condition on $η$ under which $μ_η(s)$ and $μ_η(1-s)$ cannot both vanish outside the critical line. An application is given to the case in which $η$ is the fractional part function, and the zeros of $μ_η$ coincide with the zeros of the Riemann zeta function.

math.DS↗
arXiv2026

Infinite Set of Resonances in the Linear Damped Oscillator Subject to Harmonic Forcing with Non-standard Frequency Modulation

It is shown that harmonic signals incorporating a type of weak non-standard frequency modulation (wNSFM) have interesting spectral properties, namely, time-dependent bandwidths that become increasingly broader with increasing time. As such, they represent a class of signals with frequency-time coupling in their spectra. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are persistent to changes in the parameters of the wNSFM. Our results reveal interesting infinite sets of resonances in linear SDOF resonators under frequency-modulated excitations.

math.DS↗