Record statistics for random walks and Lévy flights with resetting

We compute exactly the mean number of records $\langle R_N \rangle$ for a time-series of size $N$ whose entries represent the positions of a discrete time random walker on the line with resetting. At each time step, the walker jumps by a length $\eta$ drawn independently from a symmetric and continuous distribution $f(\eta)$ with probability $1-r$ (with $0\leq r < 1$) and with the complementary probability $r$ it resets to its starting point $x=0$. This is an exactly solvable example of a weakly correlated time-series that interpolates between a strongly correlated random walk series (for $r=0$) and an uncorrelated time-series (for $(1-r) \ll 1$). Remarkably, we found that for every fixed $r \in [0,1[$ and any $N$, the mean number of records $\langle R_N \rangle$ is completely universal, i.e., independent of the jump distribution $f(\eta)$. In particular, for large $N$, we show that $\langle R_N \rangle$ grows very slowly with increasing $N$ as $\langle R_N \rangle \approx (1/\sqrt{r})\, \ln N$ for $0<r <1$. We also computed the exact universal crossover scaling functions for $\langle R_N \rangle$ in the two limits $r \to 0$ and $r \to 1$. Our analytical predictions are in excellent agreement with numerical simulations.

The Moment Properties of Order, Reversed Order and Upper Record Statistics for the Power Ailamujia Distribution

The power Ailamujia distribution has been successfully developed in statistics, both theoretically and practically, performing well in the fitting of various types of data. This paper investigates the moment properties of the associated order, reversed order and upper record statistics, which are indeed unexplored aspects of this distribution. In particular, the exact expressions for the single moments of the order and reversed order statistics are provided. Some recurrence relationships for both single and product moments for the order and upper record statistics are proved. For additional goals, certain joint distributions are also given.

Objective Bayesian Prediction of Future Record Statistics Based on the Exponentiated Gumbel Distribution: Comparison with Time-Series Prediction

The interest in the study of record statistics has been increasing in recent years in the context of predicting stock markets and addressing global warming and climate change problems such as cyclones and floods. However, because record values are mostly rare observed, its probability distribution may be skewed or asymmetric. In this case, the Bayesian approach with a reasonable choice of the prior distribution can be a good alternative. This paper presents an objective Bayesian method for predicting future record values when observed record values have a two-parameter exponentiated Gumbel distribution with the scale and shape parameters. For objective Bayesian analysis, objective priors such as the Jeffreys and reference priors are first derived from the Fisher information matrix for the scale and shape parameters, and an analysis of the resulting posterior distribution is then performed to examine its properness and validity. In addition, under the derived objective prior distributions, a simple algorithm using a pivotal quantity is proposed to predict future record values. To validate the proposed approach, it was applied to a real dataset. For a closer examination and demonstration of the superiority of the proposed predictive method, it was compared to time-series models such as the autoregressive integrated moving average and dynamic linear model in an analysis of real data that can be observed from an infinite time series comprising independent sample values.

0442 Experimenting Automatic Sleep Analysis Application in a Clinical Context

Introduction Multiplication of publications describing groundbreaking automatic sleep analysis processes and algorithms push for real-life experimentation in clinical context, outside of controlled research environments. Methods Various automatic sleep analysis processes from the literature were implemented and orchestrated in a streamlined workflow. Artificial Intelligence algorithms using regular statistical learning or deep learning were re-trained on our own data after repeating the ad-hoc pre-processing steps described in the corresponding articles. For this, we used polysomnographic records previously taped in our clinic, subject to adequate legal authorizations and agreements: 500 nights from single patients with various pathologies. Those trained models were then applied to newly recorded polysomnographies through a platform developed and hosted on premise. For each polysomnography, a standardized and automatized report were generated and transmitted to the clinician in charge of the analysis. This report contains algorithms outputs, including automatic staging and related statistics such as hypnodensity, quantitative electroencephalography (EEG) analysis, spindles detection and automatic diagnosis. Aggregated record statistics are displayed next to our database statistics for benchmarking purposes. Results For sleep staging, we not only reproduced the results of the selected literature but obtained better metrics: a 0.76 Kappa agreement vs 0.69 in the literature. This may be due to our larger training database or the quality of physiologic signals in our data. Clinicians showed interest in the automatic staging part of the analysis. They noticed algorithm errors are mostly focused on ambiguous epochs, just like visual scoring. However, they found help into automated output and explanatory variables (hypnodensity) to score those ambiguous epochs. Conclusion Automatic sleep analysis algorithms used as decision helping tools shows real potential and should be generalized, as long as underlying processes are published and understood by users and clinicians. Support Banque Publique d’Investissement.

Record statistics of bursts signals the onset of acceleration towards failure

Forecasting the imminent catastrophic failure has a high importance for a large variety of systems from the collapse of engineering constructions, through the emergence of landslides and earthquakes, to volcanic eruptions. Failure forecast methods predict the lifetime of the system based on the time-to-failure power law of observables describing the final acceleration towards failure. We show that the statistics of records of the event series of breaking bursts, accompanying the failure process, provides a powerful tool to detect the onset of acceleration, as an early warning of the impending catastrophe. We focus on the fracture of heterogeneous materials using a fiber bundle model, which exhibits transitions between perfectly brittle, quasi-brittle, and ductile behaviors as the amount of disorder is increased. Analyzing the lifetime of record size bursts, we demonstrate that the acceleration starts at a characteristic record rank, below which record breaking slows down due to the dominance of disorder in fracturing, while above it stress redistribution gives rise to an enhanced triggering of bursts and acceleration of the dynamics. The emergence of this signal depends on the degree of disorder making both highly brittle fracture of low disorder materials, and ductile fracture of strongly disordered ones, unpredictable.