Mathematics

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17 pages, 381 KiB

Open AccessArticle

Analyzing Non-Markovian Systems by Using a Stochastic Process Calculus and a Probabilistic Model Checker

by Gabriel Ciobanu

Mathematics 2023, 11(2), 302; https://doi.org/10.3390/math11020302 - 6 Jan 2023

Cited by 1 | Viewed by 1416

The non-Markovian systems represent almost all stochastic processes, except of a small class having the Markov property; it is a real challenge to analyze these systems. In this article, we present a general method of analyzing non-Markovian systems. The novel viewpoint is given [...] Read more.

The non-Markovian systems represent almost all stochastic processes, except of a small class having the Markov property; it is a real challenge to analyze these systems. In this article, we present a general method of analyzing non-Markovian systems. The novel viewpoint is given by the use of a compact stochastic process calculus developed in the formal framework of computer science for describing concurrent systems. Since phase-type distributions can approximate non-Markovian systems with arbitrary precision, we approximate a non-Markovian system by describing it easily in our stochastic process calculus, which employs phase-type distributions. The obtained process (in our calculus) are then translated into the probabilistic model checker PRISM; by using this free software tool, we can analyze several quantitative properties of the Markovian approximation of the initial non-Markovian system. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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14 pages, 1412 KiB

Open AccessArticle

Two Gaussian Bridge Processes for Mapping Continuous Trait Evolution along Phylogenetic Trees

by Dwueng-Chwuan Jhwueng

Mathematics 2021, 9(16), 1998; https://doi.org/10.3390/math9161998 - 20 Aug 2021

Cited by 1 | Viewed by 2448

Abstract

Gaussian processes are powerful tools for modeling trait evolution along phylogenetic trees. As the value of a trait may change randomly throughout the evolution, two Gaussian bridge processes, the Brownian bridge (BB) and the Ornstein–Uhlenbeck bridge (OUB), are proposed for mapping continuous trait [...] Read more.

Gaussian processes are powerful tools for modeling trait evolution along phylogenetic trees. As the value of a trait may change randomly throughout the evolution, two Gaussian bridge processes, the Brownian bridge (BB) and the Ornstein–Uhlenbeck bridge (OUB), are proposed for mapping continuous trait evolution for a group of related species along a phylogenetic tree, respectively. The corresponding traitgrams to the two bridge processes are created to display the evolutionary trajectories. The novel models are applied to study the body mass evolution of a group of marsupial species. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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20 pages, 311 KiB

Open AccessArticle

Affine Differential Geometric Control Tools for Statistical Manifolds

by Iulia-Elena Hirica, Cristina-Liliana Pripoae, Gabriel-Teodor Pripoae and Vasile Preda

Mathematics 2021, 9(14), 1654; https://doi.org/10.3390/math9141654 - 14 Jul 2021

Cited by 4 | Viewed by 2080

Abstract

The paper generalizes and extends the notions of dual connections and of statistical manifold, with and without torsion. Links with the deformation algebras and with the Riemannian Rinehart algebras are established. The semi-Riemannian manifolds admitting flat dual connections with torsion are characterized, thus [...] Read more.

The paper generalizes and extends the notions of dual connections and of statistical manifold, with and without torsion. Links with the deformation algebras and with the Riemannian Rinehart algebras are established. The semi-Riemannian manifolds admitting flat dual connections with torsion are characterized, thus solving a problem suggested in 2000 by S. Amari and H. Nagaoka. New examples of statistical manifolds are constructed, within and beyond the classical setting. The invariant statistical structures on Lie groups are characterized and the dimension of their set is determined. Examples for the new defined geometrical objects are found in the theory of Information Geometry. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

22 pages, 755 KiB

Open AccessArticle

Tsallis Log-Scale-Location Models. Moments, Gini Index and Some Stochastic Orders

by Vasile Preda and Luigi-Ionut Catana

Mathematics 2021, 9(11), 1216; https://doi.org/10.3390/math9111216 - 27 May 2021

Cited by 3 | Viewed by 2046

Abstract

In this article we give theoretical results for different stochastic orders of a log-scale-location family which uses Tsallis statistics functions. These results describe the inequalities of moments or Gini index according to parameters. We also compute the mean in the case of q-Weibull [...] Read more.

In this article we give theoretical results for different stochastic orders of a log-scale-location family which uses Tsallis statistics functions. These results describe the inequalities of moments or Gini index according to parameters. We also compute the mean in the case of q-Weibull and q-Gaussian distributions. The paper is aimed at analyzing the order between survival functions, Lorenz curves and (as consequences) the moments together with the Gini index (respectively a generalized Gini index). A real data application is presented in the last section. This application uses only the survival function because the stochastic order implies the order of moments. Given some supplementary conditions, we prove that the stochastic order implies the Lorenz order in the log-scale-location model and this implies the order between Gini coefficients. The application uses the estimated parameters of a Pareto distribution computed from a real data set in a log-scale-location model, by specifying the Kolmogorov–Smirnov p-value. The examples presented in this application highlight the stochastic order between four models in several cases using survival functions. As direct consequences, we highlight the inequalities between the moments and the generalized Gini coefficients by using the stochastic order and the Lorenz order. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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24 pages, 11244 KiB

Open AccessArticle

Stochastic Process-Based Inversion of Electromagnetic Data for Hydrocarbon Resistivity Estimation in Seabed Logging

by Muhammad Naeim Mohd Aris, Hanita Daud, Khairul Arifin Mohd Noh and Sarat Chandra Dass

Mathematics 2021, 9(9), 935; https://doi.org/10.3390/math9090935 - 23 Apr 2021

Cited by 4 | Viewed by 1666

Abstract

This work proposes a stochastic process-based inversion to estimate hydrocarbon resistivity based on multifrequency electromagnetic (EM) data. Currently, mesh-based algorithms are used for processing the EM responses which cause high time-consuming and unable to quantify uncertainty. Gaussian process (GP) is utilized as the [...] Read more.

This work proposes a stochastic process-based inversion to estimate hydrocarbon resistivity based on multifrequency electromagnetic (EM) data. Currently, mesh-based algorithms are used for processing the EM responses which cause high time-consuming and unable to quantify uncertainty. Gaussian process (GP) is utilized as the alternative forward modeling approach to evaluate the EM profiles with uncertainty quantification. For the optimization, gradient descent is used to find the optimum by minimizing its loss function. The prior EM profiles are evaluated using finite element (FE) through computer simulation technology (CST) software. For validation purposes, mean squared deviation and its root between EM profiles evaluated by the GP and FE at the unobserved resistivities are computed. Time taken for the GP and CST to evaluate the EM profiles is compared, and absolute error between the estimate and its simulation input is also computed. All the resulting deviations were significantly small, and the GP took lesser time to evaluate the EM profiles compared to the software. The observational datasets also lied within the 95% confidence interval (CI) where the resistivity inputs were estimated by the proposed inversion. This indicates the stochastic process-based inversion can effectively estimate the hydrocarbon resistivity in the seabed logging. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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16 pages, 352 KiB

Open AccessArticle

On Similarity Measures for Stochastic and Statistical Modeling

by Konstantinos Makris, Ilia Vonta and Alex Karagrigoriou

Mathematics 2021, 9(8), 840; https://doi.org/10.3390/math9080840 - 12 Apr 2021

Viewed by 1689

Abstract

In this work, our goal is to present and discuss similarity techniques for ordered observations between time series and non-time dependent data. The purpose of the study was to measure whether ordered observations of data sets are displayed at or close to, the [...] Read more.

In this work, our goal is to present and discuss similarity techniques for ordered observations between time series and non-time dependent data. The purpose of the study was to measure whether ordered observations of data sets are displayed at or close to, the same time points for the case of time series and with the same or similar frequencies for the case of non-time dependent data sets. A simultaneous time pairing and comparison can be achieved effectively via indices, advanced indices and the associated index matrices based on statistical functions of ordered observations. Hence, in this work we review some previously defined standard indices and propose new advanced dimensionless indices and the associated index matrices which are both easily interpreted and provide efficient comparison of the series involved. Furthermore, the proposed methodology allows the analysis of data with different units of measurement as the indices presented are dimensionless. The applicability of the proposed methodology is explored through an epidemiological data set on influenza-like-illness (ILI). We finally provide a thorough discussion on all parameters involved in the proposed indices for practical purposes along with examples. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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19 pages, 341 KiB

Open AccessArticle

A Robust Version of the Empirical Likelihood Estimator

by Amor Keziou and Aida Toma

Mathematics 2021, 9(8), 829; https://doi.org/10.3390/math9080829 - 10 Apr 2021

Cited by 3 | Viewed by 1865

Abstract

In this paper, we introduce a robust version of the empirical likelihood estimator for semiparametric moment condition models. This estimator is obtained by minimizing the modified Kullback–Leibler divergence, in its dual form, using truncated orthogonality functions. We prove the robustness and the consistency [...] Read more.

In this paper, we introduce a robust version of the empirical likelihood estimator for semiparametric moment condition models. This estimator is obtained by minimizing the modified Kullback–Leibler divergence, in its dual form, using truncated orthogonality functions. We prove the robustness and the consistency of the new estimator. The performance of the robust empirical likelihood estimator is illustrated through examples based on Monte Carlo simulations. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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23 pages, 1057 KiB

Open AccessArticle

On the Computation of Some Interval Reliability Indicators for Semi-Markov Systems

by Guglielmo D’Amico, Raimondo Manca, Filippo Petroni and Dharmaraja Selvamuthu

Mathematics 2021, 9(5), 575; https://doi.org/10.3390/math9050575 - 8 Mar 2021

Cited by 6 | Viewed by 1871

Abstract

In this paper, we computed general interval indicators of availability and reliability for systems modelled by time non-homogeneous semi-Markov chains. First, we considered duration-dependent extensions of the Interval Reliability and then, we determined an explicit formula for the availability with a given window [...] Read more.

In this paper, we computed general interval indicators of availability and reliability for systems modelled by time non-homogeneous semi-Markov chains. First, we considered duration-dependent extensions of the Interval Reliability and then, we determined an explicit formula for the availability with a given window and containing a given point. To make the computation of the window availability, an explicit formula was derived involving duration-dependent transition probabilities and the interval reliability function. Both interval reliability and availability functions were evaluated considering the local behavior of the system through the recurrence time processes. The results are illustrated through a numerical example. They show that the considered indicators can describe the duration effects and the age of the multi-state system and be useful in real-life problems. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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15 pages, 295 KiB

Open AccessArticle

Ordering Awad–Varma Entropy and Applications to Some Stochastic Models

by Răzvan-Cornel Sfetcu, Sorina-Cezarina Sfetcu and Vasile Preda

Mathematics 2021, 9(3), 280; https://doi.org/10.3390/math9030280 - 31 Jan 2021

Cited by 6 | Viewed by 2160

Abstract

We consider a generalization of Awad–Shannon entropy, namely Awad–Varma entropy, introduce a stochastic order on Awad–Varma residual entropy and study some properties of this order, like closure, reversed closure and preservation in some stochastic models (the proportional hazard rate model, the proportional reversed [...] Read more.

We consider a generalization of Awad–Shannon entropy, namely Awad–Varma entropy, introduce a stochastic order on Awad–Varma residual entropy and study some properties of this order, like closure, reversed closure and preservation in some stochastic models (the proportional hazard rate model, the proportional reversed hazard rate model, the proportional odds model and the record values model). Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

25 pages, 11968 KiB

Open AccessArticle

Prognostic and Classification of Dynamic Degradation in a Mechanical System Using Variance Gamma Process

by Marwa Belhaj Salem, Mitra Fouladirad and Estelle Deloux

Mathematics 2021, 9(3), 254; https://doi.org/10.3390/math9030254 - 27 Jan 2021

Cited by 5 | Viewed by 2177

Abstract

Recently, maintaining a complex mechanical system at the appropriate times is considered a significant task for reliability engineers and researchers. Moreover, the development of advanced mechanical systems and the dynamics of the operating environments raises the complexity of a system’s degradation behaviour. In [...] Read more.

Recently, maintaining a complex mechanical system at the appropriate times is considered a significant task for reliability engineers and researchers. Moreover, the development of advanced mechanical systems and the dynamics of the operating environments raises the complexity of a system’s degradation behaviour. In this aspect, an efficient maintenance policy is of great importance, and a better modelling of the operating system’s degradation is essential. In this study, the non-monotonic degradation of a centrifugal pump system operating in the dynamic environment is considered and modelled using variance gamma stochastic process. The covariates are introduced to present the dynamic environmental effects and are modelled using a finite state Markov chain. The degradation of the system in the presence of covariates is modelled and prognostic results are analysed. Two machine learning algorithms k-nearest-neighbour (KNN) and neural network (NN) are applied to identify the various characteristics of degradation and the environmental conditions. A predefined degradation threshold is assigned and used to propose a prognostic result for each classification state. It was observed that this methodology shows promising prognostic results. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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14 pages, 310 KiB

Open AccessArticle

A Dependent Lindeberg Central Limit Theorem for Cluster Functionals on Stationary Random Fields

by José G. Gómez-García and Christophe Chesneau

Mathematics 2021, 9(3), 212; https://doi.org/10.3390/math9030212 - 21 Jan 2021

Cited by 1 | Viewed by 1843

Abstract

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes

{(Z_{n} (f))}_{f \in F}

whose index set

F

is a family of cluster functionals valued on blocks of values of [...] Read more.

In this paper, we provide a central limit theorem for the finite-dimensional marginal distributions of empirical processes

{(Z_{n} (f))}_{f \in F}

whose index set

F

is a family of cluster functionals valued on blocks of values of a stationary random field. The practicality and applicability of the result depend mainly on the usual Lindeberg condition and on a sequence

T_{n}

which summarizes the dependence between the blocks of the random field values. Finally, in application, we use the previous result in order to show the Gaussian asymptotic behavior of the proposed iso-extremogram estimator. Full article

(This article belongs to the Special Issue Stochastic Models and Methods with Applications)

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