Asymptotic expansions for the moments of the Gaussian random walk with two barriers

2004 ◽  
Vol 69 (1) ◽  
pp. 91-103 ◽  
Author(s):  
Tahir Khaniyev ◽  
Zafer Kucuk
Keyword(s):  
Random Walk ◽  
Asymptotic Expansions ◽  
Gaussian Random Walk ◽  
Two Barriers

A point-based Bayesian hierarchical model to predict the outcome of tennis matches

2019 ◽  
Vol 15 (4) ◽  
pp. 313-325 ◽  
Author(s):  
Martin Ingram
Keyword(s):  
Random Walk ◽  
Hierarchical Model ◽  
Bayesian Hierarchical Model ◽  
Bayesian Hierarchical ◽  
Gaussian Random Walk ◽  
Over Time ◽  
Modelling Approach ◽  
Independent And Identically Distributed

Abstract A well-established assumption in tennis is that point outcomes on each player’s serve in a match are independent and identically distributed (iid). With this assumption, it is enough to specify the serve probabilities for both players to derive a wide variety of event distributions, such as the expected winner and number of sets, and number of games. However, models using this assumption, which we will refer to as “point-based”, have typically performed worse than other models in the literature at predicting the match winner. This paper presents a point-based Bayesian hierarchical model for predicting the outcome of tennis matches. The model predicts the probability of winning a point on serve given surface, tournament and match date. Each player is given a serve and return skill which is assumed to follow a Gaussian random walk over time. In addition, each player’s skill varies by surface, and tournaments are given tournament-specific intercepts. When evaluated on the ATP’s 2014 season, the model outperforms other point-based models, predicting match outcomes with greater accuracy (68.8% vs. 66.3%) and lower log loss (0.592 vs. 0.641). The results are competitive with approaches modelling the match outcome directly, demonstrating the forecasting potential of the point-based modelling approach.

scholarly journals Some approximate results in renewal and dam theories

1971 ◽  
Vol 12 (4) ◽  
pp. 425-432 ◽  
Author(s):  
R. M. Phatarfod
Keyword(s):  
Random Walk ◽  
Sequential Analysis ◽  
Random Variables ◽  
Renewal Theory ◽  
Slight Modification ◽  
Exact Results ◽  
Fundamental Identity ◽  
Gambler’S Ruin ◽  
Gambler's Ruin Problem ◽  
Two Barriers

It is well known that Wald's Fundamental Identity (F.I.) in sequential analysis can be used to derive approximate (and, sometimes exact) results in most situations wherein we have essentially a random walk phenomenon. Bartlett [2] used it for the gambler's ruin problem and also for a simple renewal problem. Phatarfod [18] used it for a problem in dam theory. It is the purpose of this paper to show how a generalization of the Fundamental Identity to Markovian variables, (Phatarfod [19]) can be used to derive approximate results in some problems in dam and renewal theories where the random variables involved have Markovian dependence. The reason for considering both the theories together is that the models usually proposed for both the theories — input distribution for dam theory, and lifedistribution for renewal theory — are similar, and only a slight modification (to account for the ‘release rules’ in dam theory, plus the fact that we have two barriers) is necessary to derive results in dam theory from those of renewal theory.

scholarly journals A Tent Marine Predators Algorithm with Estimation Distribution Algorithm and Gaussian Random Walk for Continuous Optimization Problems

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Chang-Jian Sun ◽  
Fang Gao
Keyword(s):  
Random Walk ◽  
Population Distribution ◽  
Optimization Problems ◽  
Continuous Optimization ◽  
Population Diversity ◽  
Initial Population ◽  
Local Optimum ◽  
Marine Predators ◽  
Gaussian Random Walk ◽  
Estimation Distribution Algorithm

The marine predators algorithm (MPA) is a novel population-based optimization method that has been widely used in real-world optimization applications. However, MPA can easily fall into a local optimum because of the lack of population diversity in the late stage of optimization. To overcome this shortcoming, this paper proposes an MPA variant with a hybrid estimation distribution algorithm (EDA) and a Gaussian random walk strategy, namely, HEGMPA. The initial population is constructed using cubic mapping to enhance the diversity of individuals in the population. Then, EDA is adapted into MPA to modify the evolutionary direction using the population distribution information, thus improving the convergence performance of the algorithm. In addition, a Gaussian random walk strategy with medium solution is used to help the algorithm get rid of stagnation. The proposed algorithm is verified by simulation using the CEC2014 test suite. Simulation results show that the performance of HEGMPA is more competitive than other comparative algorithms, with significant improvements in terms of convergence accuracy and convergence speed.

GAUSSIAN RANDOM POLYGONS ARE GLOBALLY KNOTTED

1994 ◽  
Vol 03 (04) ◽  
pp. 455-464 ◽  
Author(s):  
DOUGLAS JUNGREIS
Keyword(s):  
Random Walk ◽  
Small Perturbation ◽  
High Probability ◽  
Random Variables ◽  
Solid Torus ◽  
Line Segments ◽  
Random Polygons ◽  
Straight Line ◽  
Gaussian Random Walk ◽  
Satellite Knot

A Gaussian random walk is a random walk in which each step is a vector whose coordinates are Gaussian random variables. In 3-space, if a Gaussian random walk of n steps begins and ends at the origin, then we can join successive points by straight line segments to get a knot. It is known that if n is large, then the knot is non-trivial with high probability. We give a new proof of this fact. Our proof shows in addition that with high probability the knot is contained as an essential loop in a fat, knotted, solid torus. Therefore the knot is a satellite knot and cannot be unknotted by any small perturbation.

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