scholarly journals Multiple drawing multi-colour urns by stochastic approximation

2018 ◽  
Vol 55 (1) ◽  
pp. 254-281 ◽  
Author(s):  
Nabil Lasmar ◽  
Cécile Mailler ◽  
Olfa Selmi
Keyword(s):  
Markov Process ◽  
Stochastic Approximation ◽  
Discrete Time ◽  
Main Idea ◽  
Approximation Methods ◽  
Asymptotic Results ◽  
Time Step ◽  
Urn Scheme ◽  
First Time ◽  
Replacement Rule

Abstract A classical Pólya urn scheme is a Markov process where the evolution is encoded by a replacement matrix (Ri, j)1 ≤ i, j ≤ d. At every discrete time-step, we draw a ball uniformly at random, denote its colour c, and replace it in the urn together with Rc, j balls of colour j (for all 1 ≤ j ≤ d). We study multiple drawing Pólya urns, where the replacement rule depends on the random drawing of a set of m balls from the urn (with or without replacement). Many particular examples of this situation have been studied in the literature, but the only general results are due to Kuba and Mahmoud (2017). These authors proved second-order asymptotic results in the two-colour case, under the so-called balance and affinity assumptions, the latter being somewhat artificial. The main idea of this work is to apply stochastic approximation methods to this problem, which enables us to prove analogous results to Kuba and Mahmoud, but without the artificial affinity hypothesis, and, for the first time in the literature, in the d-colour case (d ≥ 3). We also provide some partial results in the two-colour nonbalanced case, the novelty here being that the only results for this case currently in the literature are for particular examples.

Unbalanced multi-drawing urn with random addition matrix

2019 ◽  
Vol 26 (1/2) ◽  
pp. 57-74
Author(s):  
Aguech Rafik ◽  
Selmi Olfa
Keyword(s):  
Asymptotic Behavior ◽  
Approximation Algorithm ◽  
Stochastic Approximation ◽  
Discrete Time ◽  
Random Number ◽  
Random Variables ◽  
Urn Model ◽  
Time Step ◽  
Stochastic Approximation Algorithm ◽  
Replacement Rule

In this paper, we consider a two color multi-drawing urn model. At each discrete time step, we draw uniformly at random a sample of m balls (m≥1) and note their color, they will be returned to the urn together with a random number of balls depending on the sample’s composition. The replacement rule is a 2 × 2 matrix depending on bounded discrete positive random variables. Using a stochastic approximation algorithm and martingales methods, we investigate the asymptotic behavior of the urn after many draws.

scholarly journals An Advance-Tracing Boundary Element Method in the Time Domain for Solving the Radiating Problem of an Arbitrary Obstacle Accelerating Randomly

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Jui-Hsiang Kao
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Time Domain ◽  
Boundary Integral ◽  
Normal Velocity ◽  
Time Step ◽  
Random Acceleration ◽  
The Time Domain ◽  
First Time ◽  
Element Method

This research develops an Advance-Tracing Boundary Element Method in the time domain to calculate the waves that radiate from an immersed obstacle moving with random acceleration. The moving velocity of the immersed obstacle is multifrequency and is projected along the normal direction of every element on the obstacle. The projected normal velocity of every element is presented by the Fourier series and includes the advance-tracing time, which is equal to a quarter period of the moving velocity. The moving velocity is treated as a known boundary condition. The computing scheme is based on the boundary integral equation in the time domain, and the approach process is carried forward in a loop from the first time step to the last. At each time step, the radiated pressure on each element is updated until obtaining a convergent result. The Advance-Tracing Boundary Element Method is suitable for calculating the radiating problem from an arbitrary obstacle moving with random acceleration in the time domain and can be widely applied to the shape design of an immersed obstacle in order to attain security and confidentiality.

scholarly journals Asymptotic results for the first and second moments and numerical computations in discrete-time bulk-renewal process

2019 ◽  
Vol 29 (1) ◽  
pp. 135-144
Author(s):  
James Kim ◽  
Mohan Chaudhry ◽  
Abdalla Mansur
Keyword(s):  
Discrete Time ◽  
Renewal Process ◽  
Generating Functions ◽  
Continuous Time ◽  
Constant Term ◽  
Numerical Results ◽  
Asymptotic Results ◽  
Numerical Computations ◽  
Second Moments ◽  
Simplified Solution

This paper introduces a simplified solution to determine the asymptotic results for the renewal density. It also offers the asymptotic results for the first and second moments of the number of renewals for the discrete-time bulk-renewal process. The methodology adopted makes this study distinguishable compared to those previously published where the constant term in the second moment is generated. In similar studies published in the literature, the constant term is either missing or not clear how it was obtained. The problem was partially solved in the study by Chaudhry and Fisher where they provided a asymptotic results for the non-bulk renewal density and for both the first and second moments using the generating functions. The objective of this work is to extend their results to the bulk-renewal process in discrete-time, including some numerical results, give an elegant derivation of the asymptotic results and derive continuous-time results as a limit of the discrete-time results.

Assimilation of inundation extent observations into a flood forecasting system:  a tempered particle filter for combatting degeneracy and sample impoverishment.

2021 ◽  
Author(s):  
Concetta Di Mauro ◽  
Renaud Hostache ◽  
Patrick Matgen ◽  
Peter Jan van Leeuwen ◽  
Nancy Nichols ◽  
...  
Keyword(s):  
Particle Filter ◽  
Main Idea ◽  
Water Levels ◽  
Smooth Transition ◽  
Observation Data ◽  
Flood Inundation ◽  
State Variables ◽  
Time Step ◽  
Radar Images ◽  
Forecasting System

<p>Data assimilation uses observation for updating model variables and improving model output accuracy. In this study, flood extent information derived from Earth Observation data (namely Synthetic Aperture Radar images) are assimilated into a loosely coupled flood inundation forecasting system via a Particle Filter (PF). A previous study based on a synthetic experiment has shown the validity and efficiency of a recently developed PF-based assimilation framework allowing to effectively integrate remote sensing-derived probabilistic flood inundation maps into a coupled hydrologic-hydraulic model. One of the main limitations of this recent framework based on sequential importance sampling is the sample degeneracy and impoverishment, as particles loose diversity and only few of them keep a substantial importance weight in the posterior distribution. In order to circumvent this limitation, a new methodology is adopted and evaluated: a tempered particle filter. The main idea is to update a set of state variables, namely through a smooth transition (iterative and adaptative process). To do so, the likelihood is factorized using small tempering factors. Each iteration includes subsequent resampling and mutation steps using a Monte Carlo Metropolis Hasting algorithm. The mutation step is required to regain diversity between the particles after the resampling. The new methodology is tested using synthetic twin experiments and the results are compared to the one obtained with the previous approach. The new proposed method enables to substantially improve the predictions of streamflow and water levels within the hydraulic domain at the assimilation time step. Moreover, the preliminary results show that these improvements are longer lasting. The proposed tempered particle filter also helps in keeping more diversity within the ensemble.</p>

A method of hidden messages embedding in graphlike structures of a social network

2018 ◽  
Author(s):  
И.В. Нечта
Keyword(s):  
Social Networks ◽  
Social Network ◽  
Main Idea ◽  
The Other ◽  
Secret Message ◽  
Automatic Registration ◽  
Connected Component ◽  
Single Node ◽  
Empty Container ◽  
First Time

Предложен новый метод передачи скрытых сообщений в социальных сетях на примере сети “Вконтакте”, позволяющий через структуру графа друзей пользователя внедрять секретные сообщения. Получены количественные оценки объема внедряемого сообщения в графы различного размера. Показана необходимость добавления избыточности во внедряемое сообщение. Представленный метод позволяет использовать другие графоподобные структуры социальной сети для внедрения скрытых сообщений. Purpose. This article addresses the construction of a new method for transmission of hidden messages in social networks. Methodology. The research employs methods of information theory, probability theory and mathematical statistics The Shannon entropy is used as the statistics for the analysis of an embedded message. Findings. The author proposed using the graphical structures of social networks as a container for the secret message transmission for the first time. As an example, the popular Vkontakte network is considered. The main idea of the method involves using the structure of the user’s friends graph to embed a secret message. Based on the available vertices (friends’ accounts), a complete graph is constructed, and its edges are enumerated. Each edge of the graph corresponds to one bit of the message being embedded: the bit is “1”, if the edge is present in the graph (one account in friends of the other), the bit is “0” if the edge is missing. To transfer the graph from one person to another, a key vertex is used. The specified vertex is connected by an edge with each connected component of the graph, which allows the graph to be transmitted using a single node and take into account all the vertices (including isolated ones). When retrieving a message, the key vertex and the edges connected to it are not considered. Conclusions. During the experimental research, it was shown that messages extracted from an empty container differ from the encrypted message by the probability distribution of bits. The necessity of adding redundancy to transmitted secret messages is shown with the purpose of “leveling” the statistical properties of an empty and filled container. The results of the experiment have showed that this method of steganography allows embedding a large amount of information into various social network structures represented in the form of a graph. It was noted in the paper that potentially “narrow” place of the algorithm is registration of new accounts. The restrictions imposed by the administration of some social networks for security purposes do not always allow automatic registration of new accounts, which makes the process of message embedding more difficult.

scholarly journals The Robust Consensus of a Noisy Deffuant-Weisbuch Model

2018 ◽  
Vol 2018 ◽  
pp. 1-10
Author(s):  
Jiangbo Zhang ◽  
Yiyi Zhao
Keyword(s):  
Discrete Time ◽  
Model Parameters ◽  
Positive Probability ◽  
Noise Strength ◽  
Opinion Formation ◽  
Time Step ◽  
Absolute Value ◽  
The Difference ◽  
Robust Consensus

We construct a new opinion formation of the Deffuant-Weisbuch model with the interference of the outer noise, where there are finite n agents and the evolution is discrete-time. The opinion interaction occurs by one randomly chosen pair at each time step. The difference to the original Deffuant-Weisbuch model is that communications of any selected pairs will be affected by noises. The aim of this paper is to study the robust consensus of this noisy Deffuant-Weisbuch model. We first define the noise strength as the maximum noise absolute value. We will then show that when the noise strength is less than a certain threshold, this noisy model will achieve T-robust consensus when t is sufficiently large; next we prove that the noisy model achieves robust consensus with a positive probability; finally, we demonstrate these results and provide numerical relations among the noise strength and some model parameters.

On a fixed point theorem and its stochastic equivalent

1975 ◽  
Vol 12 (03) ◽  
pp. 605-611 ◽  
Author(s):  
Joseph A. Yahav
Keyword(s):  
Fixed Point ◽  
Markov Process ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Limiting Distribution

A discrete-time Markov process on the interval [0, 1] is considered. Sufficient conditions for the existence of a unique stationary limiting distribution are given.

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