Reproducibility of a deep-learning fully convolutional neural network is evaluated by training several times the same network on identical conditions (database, hyperparameters, and hardware) with nondeterministic graphics processing unit operations. The network is trained to model three typical time–space-evolving physical systems in two dimensions: heat, Burgers’, and wave equations. The behavior of the networks is evaluated on both recursive and nonrecursive tasks. Significant changes in models’ properties (weights and feature fields) are observed. When tested on various benchmarks, these models systematically return estimations with a high level of deviation, especially for the recurrent analysis which strongly amplifies variability due to the nondeterminism. Trainings performed with double floating-point precision provide slightly better estimations and a significant reduction of the variability of both the network parameters and its testing error range.

We study the asymptotic behaviour of the expected exit time from an interval for the ARMA process, when the noise level approaches 0.

Artificial Neural Network based Nonlinear Autoregressive Model is designed to reconstruct and predict Forbush Decrease (FD) Data obtained from Izmiran, Russia. Result indicates that the model seems adequate for short term prediction of the FD data.

This paper discusses an autoregressive model for the analysis of irregularly observed time series. The properties of this model are studied and a maximum likelihood estimation procedure is proposed. The finite sample performance of this estimator is assessed by Monte Carlo simulations, showing accurate estimators. We implement this model to the residuals after fitting an harmonic model to light-curves from periodic variable stars from the Optical Gravitational Lensing Experiment (OGLE) and Hipparcos surveys, showing that the model can identify time dependency structure that remains in the residuals when, for example, the period of the light-curves was not properly estimated.

In this paper, we consider a Hilbert-space-valued autoregressive stochastic sequence (Xn) with several regimes. We suppose that the underlying process (In) which drives the evolution of (Xn) is stationary. Under some dependence assumptions on (In), we prove the existence of a unique stationary solution, and with a symmetric compact autocorrelation operator, we can state a law of large numbers with rates and the consistency of the covariance estimator. An overall hypothesis states that the regimes where the autocorrelation operator's norm is greater than 1 should be rarely visited.

A stationary sequence of random variables with exponential marginal distributions and the correlation structure of an ARMA (1, 1) process is defined. The process is formed as a random linear combination of i.i.d. exponential random variables and is very simple to generate on a computer. Moments and joint distributions for the sequence are obtained, as well as limiting properties of sums of the random variables and of the point process whose intervals have the EARMA (1, 1) structure.

Expressions are obtained for the determinant and inverse of the covariance matrix of a set of n consecutive observations on a mixed autoregressive moving average process. Explicit formulae for the inverse of this matrix are given for the general autoregressive process of order p (n ≧ p), and for the first order mixed autoregressive moving average process.