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20 pages, 3355 KB

Open AccessArticle

Forecasting Repeated-Measures Trajectories Using Nonlinear Mixed-Effects Models: A Comparison of Population-Averaged, Subject-Specific, and Autocorrelation-Based Predictions

by Suborna Ahmed, Valerie LeMay, Andrew Robinson, Peter Marshall and Gary Bull

Mathematics 2026, 14(16), 3010; https://doi.org/10.3390/math14163010 - 20 Aug 2026

Viewed by 199

Abstract

Nonlinear mixed-effects models (NLMMs) provide a flexible framework for modeling repeated-measures trajectories. However, how best to forecast future observations, especially at ages well beyond those represented in the data, remains relatively underexamined. In this study, we develop a Chapman–Richards NLMM with a spatial-power [...] Read more.

Nonlinear mixed-effects models (NLMMs) provide a flexible framework for modeling repeated-measures trajectories. However, how best to forecast future observations, especially at ages well beyond those represented in the data, remains relatively underexamined. In this study, we develop a Chapman–Richards NLMM with a spatial-power autocorrelation structure for irregularly spaced repeated measures and compare three forecasting strategies: (i) population-averaged forecasts based on the fixed-effects component only; (ii) subject-specific forecasts in which empirical best linear unbiased predictors (EBLUPs) of the random effects are obtained via a first-order Taylor series expansion with an iterative Newton–Raphson update, including the case of new progenies not used in model fitting; and (iii) forecasts that combine the population-averaged prediction with prior repeated measures through the fitted autocorrelation matrix. Forecast accuracy was assessed with progeny-level validation under fully held-out and partially observed scenarios, using root mean square prediction error (RMSPE) and mean absolute error (MAE), and was examined as a function of: (i) the number of available prior measures and (ii) the accuracy of the fixed-effects component of the NLMM. The methods were illustrated with repeated-measures data from hybrid spruce (Picea engelmannii Parry ex Engelmann × Picea glauca (Moench) Voss) progeny trials at three planting sites in British Columbia, Canada, with measurement ages from 2 to 42 years. Subject-specific forecasts had the lowest prediction errors when sufficient prior measures were available and were also the least affected by misspecification of the fixed-effects component. With only two prior measurements, autocorrelation-based forecasts had the lowest or tied-lowest observed errors, although differences among the three approaches were small. Using all measurements taken before age 42, subject-specific forecasts of height at age 42 achieved an RMSPE of 0.50 m. With only two prior measurements, the corresponding RMSPEs were approximately 1.31–1.33 m across the forecasting approaches. Although demonstrated with a single hybrid spruce dataset from three planting sites, the comparison is, in principle, applicable to other repeated-measures settings in which long-horizon predictions are required from short observation histories; broader applicability remains to be confirmed. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applied Statistics)

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23 pages, 3153 KB

Open AccessArticle

Exact Reliability Model for a Mixed Redundant System with Heterogeneous Components and Component Sequencing

by Heungseob Kim

Mathematics 2026, 14(16), 2925; https://doi.org/10.3390/math14162925 - 13 Aug 2026

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Abstract

A mixed redundancy, in which some components of a subsystem operate under active redundancy while the others wait in cold standby, has recently been shown to achieve a higher system reliability than the traditional active and standby strategies within equivalent resources. Existing reliability [...] Read more.

A mixed redundancy, in which some components of a subsystem operate under active redundancy while the others wait in cold standby, has recently been shown to achieve a higher system reliability than the traditional active and standby strategies within equivalent resources. Existing reliability models for the strategy, however, either provide only a lower bound of the subsystem reliability or restrict the components to be identical with exponential or Erlang lifetimes. This study proposes an exact reliability model for a mixed redundant system composed of heterogeneous components. The time to failure of every candidate component is described by a generalized phase-type distribution (PHD), and a structured continuous-time Markov chain (CTMC) integrates the active redundant module, the standby components—installed in a specified sequence recorded by an ordered component-index set—and an imperfect fault detector/switch. Because the model yields the infinitesimal generator of the subsystem lifetime, it provides not only the exact reliability but also the hazard function and the moments of the system lifetime. Building on a pre-computed database that enumerates every feasible subsystem structure, the redundancy allocation problem determining the component types, their number and sequence, and the number of active redundancies is formulated as a compact binary integer linear program and solved to optimality. Benchmark experiments show that the previous approximate function underestimates the achievable design by an average maximum possible improvement (MPI) of 15.6%, and that permitting heterogeneous components raises the optimal system reliability by a further 7.3% on average—up to 18.7% as the resource budget grows. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applied Statistics)

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29 pages, 823 KB

Open AccessArticle

On Optimality and Robustness in Linear Dynamic System Identification

by Marko Živković, Zoran Banjac, Miloš Pavlović, Tomislav Unkašević and Branko Kovačević

Mathematics 2026, 14(14), 2663; https://doi.org/10.3390/math14142663 - 22 Jul 2026

Viewed by 558

Abstract

Strong consistency and asymptotic error distribution for a new class of nonlinear recursive parameter estimation algorithms of an approximate Newton–Raphson type are established. The system model is given in the discrete-time domain by a linear difference equation with constant parameters. The parameter estimator [...] Read more.

Strong consistency and asymptotic error distribution for a new class of nonlinear recursive parameter estimation algorithms of an approximate Newton–Raphson type are established. The system model is given in the discrete-time domain by a linear difference equation with constant parameters. The parameter estimator design is based on martingale theory and the Cramér–Rao (CR) theorem, providing a maximum likelihood (ML)-type optimal recursive parameter identification algorithm, whose minimum asymptotic estimation error covariance matrix achieves the CR bound under the worst-case pdf within a specified class; this worst-case pdf simultaneously yields the maximum asymptotic error covariance matrix within the class. However, the worst-case pdf does not generally exist, making the min–max optimal design indeterminable. Therefore, an approximate ML (AML)-type optimal on a class design, based on a suboptimal worst-case pdf, minimizing the scalar Fisher information within the specified class, has also been developed. The proposed design minimizes the conditional estimation error covariance under the specified suboptimal worst-case pdf. Such an approach results in Huber’s M-robustified version of the AML-based optimal design on the class of contaminated Gaussian pdfs. The practical performance of the proposed approach is analyzed through statistical validation, based on the relative asymptotic estimation efficiency measure and Monte Carlo simulations. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applied Statistics)

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26 pages, 2000 KB

Open AccessArticle

Mathematical Modeling of Degradation Data Using a Proportional Hazard Gumbel Type-II Distribution Under Generalized Progressive Hybrid Censoring

by Mohamed Aboshady, Hanan Haj Ahmad and Ridab Adlan

Mathematics 2026, 14(14), 2496; https://doi.org/10.3390/math14142496 - 10 Jul 2026

Viewed by 291

Abstract

Mathematical modeling of degradation data is essential for quantifying the lifetime, reliability, and long-term stability of advanced materials when a direct experimental assessment is costly or limited. This paper develops an applied statistical framework based on the proportional hazard Gumbel Type-II (PHGT-II) distribution [...] Read more.

Mathematical modeling of degradation data is essential for quantifying the lifetime, reliability, and long-term stability of advanced materials when a direct experimental assessment is costly or limited. This paper develops an applied statistical framework based on the proportional hazard Gumbel Type-II (PHGT-II) distribution for modeling positive degradation times under a generalized progressive hybrid censoring scheme. The proposed model extends the baseline Gumbel Type-II distribution through a proportional hazard structure, providing additional flexibility for representing non-monotone hazard behavior, heavy-tailed lifetime patterns, and heterogeneous degradation mechanisms. The probability density, survival, hazard, and mean time to failure functions were derived, and the likelihood function was formulated under generalized progressive hybrid censoring. Parameter estimation was performed using maximum likelihood estimation and Bayesian inference with independent Gamma priors. Bayesian estimates were obtained under squared error and general entropy loss functions using a Metropolis–Hastings algorithm. The model was applied to thermal degradation data of the hydroxylated fullerene nanocomposite Sc₃N@C₈₀(OH)₁₈, where the degradation time was defined through a 2% weight-loss threshold obtained from a thermogravimetric analysis. The PHGT-II model was compared with other distributions using several goodness-of-fit measures. The results show that the PHGT-II distribution provides the best fit to the observed degradation data and yields consistent reliability estimates across maximum likelihood and Bayesian approaches. The proposed framework offers a flexible and interpretable tool for modeling censored degradation data and can be extended to other reliability and lifetime applications in engineering and material science. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applied Statistics)

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38 pages, 882 KB

Open AccessFeature PaperArticle

The Nature of Mathematical Models

by Andrea De Gaetano

Mathematics 2026, 14(11), 1882; https://doi.org/10.3390/math14111882 - 28 May 2026

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Abstract

Mathematical modeling has become pervasive in applications, not only in physics or economics, but also in biomedicine and other “soft” sciences. To the conceptual formulation of a model, there often follows its identification by statistical parameter estimation, given available observations. While the nature [...] Read more.

Mathematical modeling has become pervasive in applications, not only in physics or economics, but also in biomedicine and other “soft” sciences. To the conceptual formulation of a model, there often follows its identification by statistical parameter estimation, given available observations. While the nature of the modeling process as well as its relationship with the attending statistical computations could both appear obvious to the practitioner, it may be useful to formalize them in a precise way. Insight into the process of (linear and nonlinear) model parameter estimation can be obtained from the description of the geometry of estimation in case space. The objective then is to describe the geometry of modeling in the abstract, and to show how the correspondence between the conceptual context of the model as an operator in the Hilbert space of finite-variance random variables and the computational context in

R^{n}

can be formally represented. This work formalizes the geometric correspondence between model manifolds in the Hilbert space of random variables and the geometry of statistical estimation in case space, integrating classical tools (Hilbert spaces, manifolds, projections) into a unified framework for understanding modeling and estimation. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applied Statistics)

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39 pages, 1089 KB

Open AccessArticle

Generalized Kinematic Modeling of Wheeled Mobile Robots: A Unified Framework for Heterogeneous Architectures

by Jesús Said Pantoja-García, Alejandro Rodríguez-Molina, Miguel Gabriel Villarreal-Cervantes, Andrés Abraham Palma-Huerta, Mario Aldape-Pérez and Jacobo Sandoval-Gutiérrez

Mathematics 2026, 14(3), 415; https://doi.org/10.3390/math14030415 - 25 Jan 2026

Cited by 4 | Viewed by 2416

Abstract

The increasing heterogeneity of wheeled mobile robot (WMR) architectures, including differential-drive, Ackermann, omnidirectional, and reconfigurable platforms, poses a major challenge for defining a unified, scalable kinematic representation. Most existing formulations are tailored to specific mechanical layouts, limiting analytical coherence, cross-platform interoperability, and the [...] Read more.

The increasing heterogeneity of wheeled mobile robot (WMR) architectures, including differential-drive, Ackermann, omnidirectional, and reconfigurable platforms, poses a major challenge for defining a unified, scalable kinematic representation. Most existing formulations are tailored to specific mechanical layouts, limiting analytical coherence, cross-platform interoperability, and the systematic reuse of modeling, odometry, and motion-related algorithms. This work introduces a generalized kinematic modeling framework that provides a mathematically consistent formulation applicable to arbitrary WMR configurations. Wheel–ground velocity relationships and non-holonomic constraints are expressed through a concise vector formulation that maps wheel motions to chassis velocities, ensuring consistency with established models while remaining independent of the underlying mechanical structure. A parameterized wheel descriptor encodes all relevant geometric and kinematic properties, enabling the modular assembly of complete robot models by aggregating wheel-level relations. The framework is evaluated through numerical simulations on four structurally distinct platforms: differential-drive, Ackermann, three-wheel omnidirectional (3, 0), and 4WD. Results show that the proposed formulation accurately reproduces the expected kinematic behavior across these fundamentally different architectures and provides a coherent and consistent representation of their motion. The unified representation further provides a common kinematic backbone that is directly compatible with odometry, motion-control, and simulation pipelines, facilitating the systematic retargeting of algorithms across heterogeneous robot platforms without architecture-specific reformulation. Additional simulation studies under realistic physics-based conditions show that the proposed formulation preserves coherent kinematic behavior during complex trajectory execution and supports the explicit incorporation of geometric imperfections, such as wheel mounting misalignments, when such parameters are available. By consolidating traditionally separate derivations into a single coherent formulation, this work establishes a rigorous, scalable, and architecture-agnostic foundation for unified kinematic modeling of wheeled mobile robots, with particular relevance for modular, reconfigurable, and cross-architecture robotic systems. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applied Statistics)

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19 pages, 560 KB

Open AccessArticle

Modeling PM2.5 Pollution Using a Truncated Positive Student’s-t Distribution: A Case Study in Chile

by Héctor J. Gómez, Karol I. Santoro, Diego I. Gallardo, Paola E. Leal and Tiago M. Magalhães

Mathematics 2025, 13(23), 3838; https://doi.org/10.3390/math13233838 - 30 Nov 2025

Viewed by 698

Abstract

This study revisits a recently proposed member of the truncated positive family of distributions, referred to as the positively truncated Student’s-t distribution. The distribution retains the structure of the classical Student’s-t distribution while explicitly incorporating a kurtosis parameter, yielding a flexible three-parameter formulation [...] Read more.

This study revisits a recently proposed member of the truncated positive family of distributions, referred to as the positively truncated Student’s-t distribution. The distribution retains the structure of the classical Student’s-t distribution while explicitly incorporating a kurtosis parameter, yielding a flexible three-parameter formulation that governs location, scale, and tail behavior. A closed-form quantile function is derived, allowing a novel reparameterization based on the pth quantile and thereby facilitating integration into quantile regression models. The analytical tractability of the quantile function also enables efficient random number generation via the inverse transform method, which supports a comprehensive simulation study demonstrating the strong performance of the proposed estimators, particularly for the degrees-of-freedom parameter. The entire methodology is implemented in the tpn package for the R software. Finally, two real-data applications involving PM2.5 measurements—one without covariates and another with covariates—highlight the model’s robustness and its ability to capture heavy-tailed behavior. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applied Statistics)

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24 pages, 7349 KB

Open AccessArticle

Return Level Prediction with a New Mixture Extreme Value Model

by Emrah Altun, Hana N. Alqifari and Kadir Söyler

Mathematics 2025, 13(17), 2705; https://doi.org/10.3390/math13172705 - 22 Aug 2025

Cited by 1 | Viewed by 1481

Abstract

The generalized Pareto distribution is frequently used for modeling extreme values above an appropriate threshold level. Since the process of determining the appropriate threshold value is difficult, a mixture of extreme value models rises to prominence. In this study, mixture extreme value models [...] Read more.

The generalized Pareto distribution is frequently used for modeling extreme values above an appropriate threshold level. Since the process of determining the appropriate threshold value is difficult, a mixture of extreme value models rises to prominence. In this study, mixture extreme value models based on exponentiated Pareto distribution are proposed. The Weibull, gamma, and log-normal models are used as bulk densities. The parameter estimates of the proposed models are obtained using the maximum likelihood approach. Two different approaches based on maximization of the log-likelihood and Kolmogorov–Smirnov p-value are used to determine the appropriate threshold value. The effectiveness of these methods is compared using simulation studies. The proposed models are compared with other mixture models through an application study on earthquake data. The GammaEP web application is developed to ensure the reproducibility of the results and the usability of the proposed model. Full article

(This article belongs to the Special Issue Mathematical Modelling and Applied Statistics)

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