Star-shaped order for distributions characterized by several parameters and some applications

2021 ◽  
pp. 1-11
Author(s):  
Idir Arab ◽  
Milto Hadjikyriakou ◽  
Paulo Eduardo Oliveira ◽  
Beatriz Santos
Keyword(s):  
Complex Systems ◽  
Probability Distributions ◽  
Real Parameter ◽  
Applied Probability ◽  
Aging Properties ◽  
Multidimensional Parameter ◽  
Complex Models

Abstract The star-shaped ordering between probability distributions is a common way to express aging properties. A well-known criterion was proposed by Saunders and Moran [(1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability 15(2): 426–432], to order families of distributions depending on one real parameter. However, the lifetime of complex systems usually depends on several parameters, especially when considering heterogeneous components. We extend the Saunders and Moran criterion characterizing the star-shaped order when the multidimensional parameter moves along a given direction. A few applications to the lifetime of complex models, namely parallel and series models assuming different individual components behavior, are discussed.

A Computationally Efficient Evolutionary Algorithm for Real-Parameter Optimization

2002 ◽  
Vol 10 (4) ◽  
pp. 371-395 ◽  
Author(s):  
Kalyanmoy Deb ◽  
Ashish Anand ◽  
Dhiraj Joshi
Keyword(s):  
Parameter Optimization ◽  
Optimization Problems ◽  
Research Effort ◽  
Probability Distributions ◽  
Scale Up ◽  
Extensive Study ◽  
Evolution Strategy ◽  
Real Parameter ◽  
Test Problems ◽  
Main Research

Due to increasing interest in solving real-world optimization problems using evolutionary algorithms (EAs), researchers have recently developed a number of real-parameter genetic algorithms (GAs). In these studies, the main research effort is spent on developing an efficient recombination operator. Such recombination operators use probability distributions around the parent solutions to create an offspring. Some operators emphasize solutions at the center of mass of parents and some around the parents. In this paper, we propose a generic parent-centric recombination operator (PCX) and a steady-state, elite-preserving, scalable, and computationally fast population-alteration model (we call the G3 model). The performance of the G3 model with the PCX operator is investigated on three commonly used test problems and is compared with a number of evolutionary and classical optimization algorithms including other real-parameter GAs with the unimodal normal distribution crossover (UNDX) and the simplex crossover (SPX) operators, the correlated self-adaptive evolution strategy, the covariance matrix adaptation evolution strategy (CMA-ES), the differential evolution technique, and the quasi-Newton method. The proposed approach is found to consistently and reliably perform better than all other methods used in the study. A scale-up study with problem sizes up to 500 variables shows a polynomial computational complexity of the proposed approach. This extensive study clearly demonstrates the power of the proposed technique in tackling real-parameter optimization problems.

scholarly journals Stochastic Hydrology: Is Scientific Understanding Growing?

2020 ◽  
Vol 163 ◽  
pp. 06001
Author(s):  
Mikhail Bolgov
Keyword(s):  
Climate Changes ◽  
Probability Distributions ◽  
Random Processes ◽  
Stochastic Hydrology ◽  
One Dimensional ◽  
Probabilistic Forecasts ◽  
Markovian Property ◽  
New Models ◽  
Complex Models ◽  
With Memory

Among many problems of stochastic hydrology, several major problems may be singled out. (1) The methodology problem – may fluctuation of hydro-meteorological values be considered within the framework of probabilities and random processes? Was this topic discussed after 1953? (2) One-dimensional probability distributions – is there progress? Are there new models? (3) Random Processes: Is Markovian property sufficient or more complex models with memory are needed? (4) Lack of stability resulting from climate changes: Is there progress in understanding the approaches to probabilistic forecasts?

scholarly journals Bounds for some entropies and special functions

2018 ◽  
Vol 34 (1) ◽  
pp. 09-15
Author(s):  
ADINA BARAR ◽  
◽  
GABRIELA RALUCA MOCANU ◽  
IOAN RASA ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Convex Function ◽  
Negative Binomial ◽  
Special Functions ◽  
Probability Distributions ◽  
Real Parameter ◽  
Binomial Distributions

We consider a family of probability distributions depending on a real parameter and including the binomial, Poisson and negative binomial distributions. The corresponding index of coincidence satisfies a Heun differential equation and is a logarithmically convex function. Combining these facts we get bounds for the index of coincidence, and consequently for Renyi and Tsallis entropies of order 2.

Statistical Inference, Occam's Razor, and Statistical Mechanics on the Space of Probability Distributions

1997 ◽  
Vol 9 (2) ◽  
pp. 349-368 ◽  
Author(s):  
Vijay Balasubramanian
Keyword(s):  
Statistical Mechanics ◽  
Probability Distributions ◽  
Bayesian Posterior Probability ◽  
Model Parameters ◽  
Jeffreys Prior ◽  
Precise Understanding ◽  
Occam’S Razor ◽  
Complex Models ◽  
Model Family ◽  
Occam's Razor

The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of low-temperature expansions, I arrive at a systematic series for the Bayesian posterior probability of a model family that significantly extends known results in the literature. In particular, I arrive at a precise understanding of how Occam's razor, the principle that simpler models should be preferred until the data justify more complex models, is automatically embodied by probability theory. These results require a measure on the space of model parameters and I derive and discuss an interpretation of Jeffreys' prior distribution as a uniform prior over the distributions indexed by a family. Finally, I derive a theoretical index of the complexity of a parametric family relative to some true distribution that I call the razor of the model. The form of the razor immediately suggests several interesting questions in the theory of learning that can be studied using the techniques of statistical mechanics.

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