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16 pages, 13345 KB

Open AccessArticle

Amortized Parameter Inference for the Arbitrary-Order Hidden Markov Model

by Sixiang Zhang and Liming Cai

Axioms 2026, 15(4), 289; https://doi.org/10.3390/axioms15040289 - 14 Apr 2026

Viewed by 408

Abstract

The arbitrary-order hidden Markov model (

α

-HMM) is a nontrivial generalization of the standard HMM, designed to model stochastic processes with higher-order dependences among arbitrarily distant random events. The

α

-HMM admits an efficient Viterbi-style optimal decoding algorithm, making it feasible to [...] Read more.

The arbitrary-order hidden Markov model (

α

-HMM) is a nontrivial generalization of the standard HMM, designed to model stochastic processes with higher-order dependences among arbitrarily distant random events. The

α

-HMM admits an efficient Viterbi-style optimal decoding algorithm, making it feasible to discover higher-order dependences among data objects in observed sequential data. Because the

α

-HMM exceeds the expressive power of standard HMMs, fixed

k th

-order HMMs, and stochastic context-free grammars, effective probabilistic parameter estimation approaches are required to translate this theoretical expressiveness of the

α

-HMM into practical utility. This paper introduces a principled methodology for effective estimation of probabilistic parameters of the

α

-HMM from observed data. In large-scale sequential datasets, higher-order dependencies can vary widely across instances, so a single global parameter set may be inadequate. Instead, an amortized parameter inference approach is proposed for the

α

-HMM, in which an input-conditioned parameter estimator is learned from data and used to infer instance-specific parameters for each input instance to the decoding algorithm. Specifically, the neural parameter estimator is trained using a composite learning objective that is partially enabled by the optimal decoding algorithm. The effectiveness of the proposed parameter estimation method is demonstrated through empirical results of the application of the

α

-HMM in biomolecular structure modeling and prediction. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

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5 pages, 201 KB

Open AccessArticle

Any Distribution on the Positive Real Line as a Limit of Random Sums

by Lev Klebanov and Michal Šumbera

Axioms 2026, 15(2), 84; https://doi.org/10.3390/axioms15020084 - 23 Jan 2026

Viewed by 543

Abstract

We prove that any positive random variable may be the limit of the random sums of independent identically distributed random variables. Thus, the limit distribution in the case of summing a random number of random variables may not be stable. The lack of [...] Read more.

We prove that any positive random variable may be the limit of the random sums of independent identically distributed random variables. Thus, the limit distribution in the case of summing a random number of random variables may not be stable. The lack of stability in the limit distribution significantly distinguishes the summation scheme for a random number of random variables from the classical summation scheme. Moreover, the analogs of stable distributions in the summation of a random number of terms also turn out to be limit distributions in a suitable summation scheme. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

23 pages, 350 KB

Open AccessArticle

Application of Stochastic Elements in the Universality of the Periodic Zeta-Function: The Case of Short Intervals

by Marius Grigaliūnas, Antanas Laurinčikas and Darius Šiaučiūnas

Axioms 2026, 15(1), 58; https://doi.org/10.3390/axioms15010058 - 14 Jan 2026

Viewed by 466

Abstract

Let

a = {a_{m} : m \in N}

be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip [...] Read more.

Let

a = {a_{m} : m \in N}

be a multiplicative periodic sequence of complex numbers. In this paper, we consider the approximation of analytic functions defined in the strip

{s = σ + i t : 1 / 2 < σ < 1}

by shifts

ζ (s + i τ; a)

of the zeta-function defined, for

σ > 1

, by

ζ (s; a) = \sum_{m = 1}^{\infty} a_{m} m^{- s}

and by analytic continuation elsewhere. Using stochastic techniques, we obtain that the set of the above shifts approximating a given analytic function has a positive lower density (or density with at most countably many exceptions) in the interval

[T, T + V]

with

T^{23 / 70} ⩽ V ⩽ T^{1 / 2}

as

T \to \infty

. The proofs are based on a limit theorem with an explicitly given limit probability measure in the space of analytic functions. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

32 pages, 1584 KB

Open AccessArticle

Adaptive Sparse Clustering of Mixed Data Using Azzalini-Encoded Ordinal Variables

by Ismail Arjdal, Mohamed Alahiane, Echarif Elharfaoui and Mustapha Rachdi

Axioms 2025, 14(12), 902; https://doi.org/10.3390/axioms14120902 - 7 Dec 2025

Viewed by 453

Abstract

In this paper, we propose a novel sparse clustering method designed for high-dimensional mixed-type data, integrating Azzalini’s score-based encoding for ordinal variables. Our approach aims to retain the inherent nature of each variable type—continuous, ordinal, and nominal—while enhancing clustering quality and interpretability. To [...] Read more.

In this paper, we propose a novel sparse clustering method designed for high-dimensional mixed-type data, integrating Azzalini’s score-based encoding for ordinal variables. Our approach aims to retain the inherent nature of each variable type—continuous, ordinal, and nominal—while enhancing clustering quality and interpretability. To this end, we extend classical distance metrics and adapt the Davies–Bouldin Index (DBI) to better reflect the structure of mixed data. We also introduce a weighted formulation that accounts for the distinct contributions of variable types in the clustering process. Empirical results on simulated and real-world datasets demonstrate that our method consistently achieves better separation and coherence of clusters compared to traditional techniques, while effectively identifying the most informative variables. This work opens promising directions for clustering in complex, high-dimensional settings such as marketing analytics and customer segmentation. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

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28 pages, 1067 KB

Open AccessArticle

Inference Based on Progressive-Stress Accelerated Life-Testing for Extended Distribution via the Marshall-Olkin Family Under Progressive Type-II Censoring with Optimality Techniques

by Ehab M. Almetwally, Osama M. Khaled and Haroon M. Barakat

Axioms 2025, 14(4), 244; https://doi.org/10.3390/axioms14040244 - 23 Mar 2025

Cited by 3 | Viewed by 1226

Abstract

This paper explores a progressive-stress accelerated life test under progressive type-II censoring with binomial random removal. It assumes a cumulative exposure model in which the lifetimes of test units follow a Marshall–Olkin length-biased exponential distribution. The study derives maximum likelihood and Bayes estimates [...] Read more.

This paper explores a progressive-stress accelerated life test under progressive type-II censoring with binomial random removal. It assumes a cumulative exposure model in which the lifetimes of test units follow a Marshall–Olkin length-biased exponential distribution. The study derives maximum likelihood and Bayes estimates of the model parameters and constructs Bayes estimates of the unknown parameters under various loss functions. In addition, this study provides approximate, credible, and bootstrapping confidence intervals for the estimators. Moreover, it evaluates three optimal test methods to determine the most effective censoring approach based on various optimality criteria. A real-life dataset is analyzed to demonstrate the proposed procedures and simulation studies used to compare two different designs of the progressive-stress test. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

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31 pages, 2537 KB

Open AccessArticle

A Novel Framework for Belief and Plausibility Measures in Intuitionistic Fuzzy Sets: Belief and Plausibility Distance, Similarity, and TOPSIS for Multicriteria Decision Making

by Shahid Hussain, Zahid Hussain, Rashid Hussain, Ahmad Bakhet, Hussain Arafat, Mohammed Zakarya, Amirah Ayidh I Al-Thaqfan and Maha Ali

Axioms 2024, 13(12), 858; https://doi.org/10.3390/axioms13120858 - 7 Dec 2024

Cited by 1 | Viewed by 2188

Abstract

Dempster–Shafer Theory (DST) relies significantly on belief and plausibility measures to handle ambiguity and uncertainty; however, DST has been extended to fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs) with only a few extensions focusing on belief and plausibility intuitionistic fuzzy distance (BP-distance) [...] Read more.

Dempster–Shafer Theory (DST) relies significantly on belief and plausibility measures to handle ambiguity and uncertainty; however, DST has been extended to fuzzy sets (FSs) and intuitionistic fuzzy sets (IFSs) with only a few extensions focusing on belief and plausibility intuitionistic fuzzy distance (BP-distance) and similarity (BP-similarity) until now. In this work, we propose a novel framework for the belief and plausibility of intuitionistic fuzzy sets (BP-IFSs) and their BP-distance and BP-similarity measures. We modified steps 4 and 5 of the classical TOPSIS method, utilizing both distance and similarity measures to rank the alternatives that satisfy all necessary axioms of distance and similarity. We present numerical examples involving pattern recognition, linguistic variables, and clustering to illustrate the efficiency of these measures, and we develop belief and plausibility TOPSIS (BP-TOPSIS) using the proposed criteria and apply it to complex multicriteria decision-making (MCDM) challenges. The results demonstrate the practicality and effectiveness of our approach. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

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15 pages, 310 KB

Open AccessArticle

Mathematical Optimization of Wind Turbine Maintenance Using Repair Rate Thresholds

by Nataša Kontrec, Stefan Panić, Jelena Vujaković, Dejan Stošović and Sergei Khotnenok

Axioms 2024, 13(11), 809; https://doi.org/10.3390/axioms13110809 - 20 Nov 2024

Cited by 2 | Viewed by 2086

Abstract

As reliance on wind energy intensifies globally, optimizing the efficiency and reliability of wind turbines is becoming vital. This paper explores sophisticated maintenance strategies, crucial for enhancing the operational sustainability of wind turbines. It introduces an innovative approach to maintenance scheduling that utilizes [...] Read more.

As reliance on wind energy intensifies globally, optimizing the efficiency and reliability of wind turbines is becoming vital. This paper explores sophisticated maintenance strategies, crucial for enhancing the operational sustainability of wind turbines. It introduces an innovative approach to maintenance scheduling that utilizes a mathematical model incorporating an alternating renewal process for accurately determining repair rate thresholds. These thresholds are important for identifying optimal maintenance timings, thereby averting failures and minimizing downtime. Central to this study are the obtained generalized analytical expressions that can be used to predict the total repair time for an observed entity. Four key lemmas are developed to establish formal proofs for the probability density function (PDF) and cumulative distribution function (CDF) of repair rates, both above and below critical repair rate thresholds. The core innovation of this study lies in the methodological application of PDFs and CDFs to set repair time thresholds that refine maintenance schedules. The model’s effectiveness is illustrated using simulated data based on typical wind turbine components such as gearboxes, generators, and converters, validating its potential for improving system availability and operational readiness. By establishing measurable repair rate thresholds, the model effectively prioritizes maintenance tasks, extending the life of crucial turbine components and ensuring consistent energy output. Beyond enhancing theoretical understanding, this research provides practical insights that could inform broader maintenance strategies across various renewable energy systems, marking a significant advancement in the field of maintenance engineering Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

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21 pages, 391 KB

Open AccessArticle

Randomly Stopped Sums, Minima and Maxima for Heavy-Tailed and Light-Tailed Distributions

by Remigijus Leipus, Jonas Šiaulys, Svetlana Danilenko and Jūratė Karasevičienė

Axioms 2024, 13(6), 355; https://doi.org/10.3390/axioms13060355 - 25 May 2024

Cited by 3 | Viewed by 1511

Abstract

This paper investigates the randomly stopped sums, minima and maxima of heavy- and light-tailed random variables. The conditions on the primary random variables, which are independent but generally not identically distributed, and counting random variable are given in order that the randomly stopped [...] Read more.

This paper investigates the randomly stopped sums, minima and maxima of heavy- and light-tailed random variables. The conditions on the primary random variables, which are independent but generally not identically distributed, and counting random variable are given in order that the randomly stopped sum, random minimum and maximum is heavy/light tailed. The results generalize some existing ones in the literature. The examples illustrating the results are provided. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

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