ABSTRACT Higher-order Markov chains are frequently used to model categorical time series. However, a major problem with fitting such models is the exponentially growing number of parameters in the model order. A popular approach to parsimonious modeling is to use a Variable Length Markov Chain (VLMC), which determines relevant contexts (recent pasts) of variable orders and forms a context tree. A more general parsimonious modeling approach is given by Sparse Markov Models (SMMs), where the possible histories of order m are partitioned such that transition probability vectors are identical for histories belonging to the same group. In this paper, we develop an elegant method of fitting SMMs based on convex clustering and regularization. The regularization parameter is selected using the BIC criterion. Theoretical results establish model selection consistency for large sample sizes. Extensive simulation results under different set-ups are presented to study finite sample performance of the method. Real data analysis on modeling and classifying disease sub-types demonstrates the applicability of our method.

Cellular respiration toxicity: A novel comprehensive water quality index based on universal biological processes.

This study presents a novel numerical framework for simulating glucose-insulin regulatory dynamics using the Caputo-Fabrizio (CF) fractal-fractional operator with both constant and variable fractional orders. The model incorporates an exponential decay kernel to capture memory and hereditary effects in metabolic regulation. A Newton interpolation-based numerical scheme is developed to approximate the CF-FF derivatives, ensuring computational stability and accuracy. For the variable-order formulation, the fractional order dynamically evolves with time, reflecting physiological variability typically observed during intravenous glucose tolerance tests (IVGTT). Numerical experiments reproduce physiologically realistic glucose-insulin oscillations and demonstrate how feedback control stabilizes chaotic metabolic behavior. The results are based entirely on simulation evidence calibrated within clinically reported parameter ranges, providing conceptual validation rather than direct patient-data comparison. The proposed approach bridges mathematical fractional calculus with biomedical applications, offering new insights for personalized diabetes management and adaptive glucose control strategies.•Fractal-fractional model formulation capturing glucose-insulin memory and adaptation•Stable numerical scheme using Newton interpolation for accurate fractional integration•Linear feedback control applied to regulate chaotic glucose-insulin dynamics•Numerical Methodology for glucose-insulin dynamics. Our investigation of the fractal-fractional glucose-insulin system employs the following analytical framework:•Model Development: We formulate a fractal-fractional-order extension of the minimal glucose insulin model, incorporating an exponential decay type kernel to capture the system's memory effects and anomalous diffusion characteristics inherent in metabolic processes. The model accounts for both insulin-dependent and independent glucose utilization dynamics.•Computational Implementation: We develop a novel numerical solver based on Newton's interpolation polynomials, implementing the Atangana-Seda fractal-fractional derivative formulation. This method provides an efficient computational framework for solving the coupled nonlinear fractional differential equations while maintaining numerical stability across different fractional orders.•The purpose of this section is to define a mathematical model to study the dynamic behavior of glucose-insulin physiology.•With the Adams-Bashforth-Moulton numerical scheme, we compute the Lyapunov exponent of the system, which is useful for studying dissipative.•In a generalized numerical method, we simulate the solutions of the system using the time-fractal fractional derivative of Atangana-Seda.

• Epistemic uncertainty based reliability assessment of reconfigurable manufacturing systems • Reliability performance models built by optimistic and pessimistic assessment approaches • Numerical analysis of the optimistic and pessimistic based reliability models The importance of assessing the reliability of a complex system paradigm such as a reconfigurable manufacturing system arises from its relationship with quality, efficiency, flexibility, complexity, safety and application. However, during practical reliability analysis of a reconfigurable manufacturing system, epistemic uncertainty can emerge since the degradation information of the complex system can be insufficient, making it critical to develop a response approach. Therefore, this study develops an approach which takes into account the two extreme possibilities - optimistic and pessimistic - to set the boundary for the reliability performance of the system. The approach is proposed specifically for a reconfigurable manufacturing system with variable configuration orders. The optimistic assessment assumes that the system does not have a usage history when reconfigured with a new tool. In contrast, the pessimistic assessment assumes that the usage history has full impact on the newly reconfigured system. An algorithm is developed for the optimistic and pessimistic assessment model based on Monte Carlo simulation. The simulation result demonstrates the ability of the approach to analyze different levels of optimism and pessimism, and to quantify the uncertainty of reliability performance of a reconfigurable manufacturing system.

Data-driven causal structure discovery to generate causal hypotheses and strengthen causal inference in exposome studies.

Existence results for a variable order Hadamard fractional problem under weak topology features

This paper introduces a novel memristor-based hyperchaotic system in which the integer-order derivatives are replaced by a variable-order fractal-fractional operator. The dynamical properties of the system, including equilibrium points, Lyapunov exponents, bifurcation diagrams with respect to the variable orders, and the Kaplan–Yorke dimension, are analyzed. A synchronization scheme based on active control is designed for the master–slave configuration, and global Mittag–Leffler stability of the error dynamics is established using a suitable variable-order Lyapunov function. The synchronized states are then applied to an image encryption algorithm. Numerical simulations, security analyses, and NIST randomness tests demonstrate the effectiveness and enhanced performance of the proposed framework compared to existing fixed-order and classical fractional-order methods.

Cluster lag synchronization and control for multiplex and directed network systems.

ABSTRACT The k ‐out‐of‐ n phased‐mission systems (PMSs) widely exist in the areas such as cloud computing, smart grids, industrial manufacturing, and so on. Accurate and efficient reliability analysis methods are required for such systems in critical applications. Multiple‐valued decision diagram (MDD) is a common model for reliability analysis of k ‐out‐of‐ n PMSs. To construct the system MDD, each variable corresponding to a system component is assigned a different order. Different input variable orders can lead to dramatically different sizes of final system MDD, which in turn affects the reliability analysis efficiency for such systems. This paper aims to reduce the nonsink nodes during system MDD generation by investigating optimal MDD variable ordering strategies for the systems. Unlike those variable ordering strategies based on fault tree traversal or PMS‐BDD variable ordering, the proposed strategies take full advantage of the structural characteristics of the k ‐out‐of‐ n PMSs and incorporate all interrelationships among the number of components shared by all phases, and the system structure parameters k and n . The proposed optimal strategies are validated through both theoretical analysis and experimental evaluations.

Background: The metamorphopsia questionnaire (MeMoQ) is an established patient-reported outcome measure (PROM) in the context of macular disease. However, its performance has not been proved in those being treated for various macular conditions with intravitreal anti-vascular endothelial growth factor (Anti-VEGF). The objective was to eliminate misfitting items, enhance measurement precision, and ensure optimal response categorisation. Methods: Rasch analysis was performed iteratively on 2286 responses from patients with macular diseases being treated with Anti-VEGF to optimise the MeMoQ. Fit statistics, reliability indices, person and item separation measures, and principal component analysis (PCA) of residuals were assessed to determine the optimal model. This study was conducted in an outpatient clinic specialising in retinal diseases in Hywel Dda University Health Board. Results: Misfitting items were removed in successive iterations, leading to optimised category probability curves and stable fit statistics for the MeMoQ. The resulting model for all responses included two final items, with person separation remaining inadequate reducing from 1.23 to 1.12 and reliability from 0.60 to 0.56. Category probability curves demonstrated good ordering of response variables with Andrich thresholds separated by >1.2 logits. In the subgroups of neovascular age-related macular degeneration and diabetic macular oedema person separation remained below two and reliability remained low. Conclusions: Rasch analysis demonstrated that the MeMoQ was not a valid or reliable PROM in this patient population. Therefore, the MeMoQ may not provide a reliable index of patient's perception and visual experience when undergoing Anti-VEGF treatment.

Abstract Many studies have investigated the failure mechanisms of centrifugal pumps, and several time-to-failure predictive models have been developed, but the relative impact of the variables affecting their expected life is rarely quantified. This paper addresses the issue by quantifying the influence of key variables through the application of four different methods. Three are based on statistical techniques: partial Likelihood Ratio χ2 analysis, Bayesian coefficient magnitudes, and variable inclusion order in Lasso regression applied to a Cox Proportional Hazards Model. The fourth method employs a machine learning technique, permutation importance applied to a Random Survival Forest model. To enhance the robustness and consistency of the findings, a bootstrapping procedure and a Copeland-Llull voting strategy were implemented on a real-world dataset from an oil refinery, consisting of 675 pumps with a set of 27 potential predictors.

This work presents a space-time variable-order (V-O) fractional framework for the analytical investigation of nonlinear longitudinal wave propagation in magneto-electro-elastic (MEE) materials. The model contains Caputo-type V-O derivatives, which account for spatial and temporal memory effects and nonlocal interactions inherent to coupled mechanical, electrical, and magnetic fields. On this basis, exact traveling-wave solutions of the resulting nonlinear fractional wave equation were derived using an exponential expansion methodology, in the form of periodic, kink-type, and solitary waves. The effect of the V-O parameters on the wave amplitude, dispersion characteristics, and stability behavior is systematically analyzed. Stability and bifurcation analyses show parameter-dependent transition of both stable and unstable regimes, whereas insertion of random perturbations illustrates the development of chaotic dynamics. The proposed V-O model offers increased modeling flexibility and parameter-dependent stability regimes in the reduced system compared to constant-order (C-O) fractional formulations. This enhancement provides increased analytical flexibility at the theoretical level for coupled field interactions in MEE media. These results suggest that the V-O fractional modeling represents a valid tool for the analysis of complex nonlinear electromechanical phenomena and could lead to a better understanding of the wave dynamics at a theoretical level in multifunctional material systems.

Variable order fractional model for describing the mechanical behaviour of amorphous polymers

In this study, we incorporate the Caputo Fractional Variable Order Derivative (CFVOD) to account for memory effects and nonlocal interactions, improving the mathematical description of the transmission of water-based diseases. Understanding fluid and contaminant movement requires a more complex description of the dynamic processes occurring with inporous media, which is made possible by the application of CFVOD. We prove Ulam–Hyers (UH) stability and investigate the existence and uniqueness of solutions for the model to guarantee robustness against small perturbations. The Euler approach is applied to numerical simulations in order to investigate the temporal spread of illnesses. The incorporation of CFVOD exhibits enhanced precision in capturing disease dynamics, underscoring its possible use in forecasting and controlling waterborne infections inporous settings.

A rheological model based on the new formulation of fractional calculus with variable order is developed to study the viscoelastic behavior of wood. The model, which uses only two rheological elements, exhibits an enhanced memory effect compared to constant-order fractional derivative models, demonstrated by a satisfactory fit to the experimental deformations observed in four-point bending tests on \textit{Pericopsis elata} samples. By determining the parameters of the linear function of the fractional order used, a physical significance emerges that explains the changes occurring within the material during the tests. This type of model, therefore, provides wood engineers with additional information about the behavior of the material under stress.

ABSTRACT Several methods in the literature address the discovery, from data, of a directed acyclic graph that is the structure of a Bayesian network. This is a challenging task, due to the combinatorial nature of the space of graphs, which grows exponentially with the number of variables. This paper proposes a new method to learn Bayesian network structures, based on Markov blankets. The proposed method, called DMBBN (Dynamic Markov Blanket Bayesian Network), builds a graph from a set of local structures that are induced based on the Markov blanket of each variable of interest. The local structures are combined to generate a single Bayesian network structure, without repetition of nodes and without cycles. The experiments carried out show that DMBBN is promising, especially in large datasets, as it does not depend on a prior ordering of the variables.

Correction: Analysis of COVID-19 Model Using Atangana-Baleanu Variable Order Operator

This paper considers the solution behavior and dynamical properties of the variable-order fractional Newton–Leipnik system defined via Liouville–Caputo derivatives of variable order. In contrast to integer-order models, the presence of variable-order fractional operators in the Newton–Leipnik structure enriches the model by providing memory-dependent effects that vary with time; hence, it is capable of a broader and more flexible range of nonlinear responses. Numerical simulations have been conducted to study how different order functions influence the trajectory and qualitative dynamics: clear transitions in oscillatory patterns have been identified by phase portraits, time-series profiles, and three-dimensional state evolution. The work goes further by considering the development of bifurcations and chaotic regimes and stability shifts and confirms the occurrence of several phenomena unattainable in fixed-order and/or integer-order formulations. Analysis of Lyapunov exponents confirms strong sensitivity to the initial conditions and further details how the memory effects either reinforce or prevent chaotic oscillations according to the type of order function. The results, in fact, show that the variable-order fractional Newton–Leipnik framework allows for more expressive and realistic modeling of complex nonlinear phenomena and points out the crucial role played by evolving memory in controlling how the system moves between periodic, quasi-periodic, and chaotic states.

ABSTRACT This paper investigates the inverse problem of determining spatially dependent source term in diffusion equations with variable‐order time fractional derivatives from the knowledge of integral‐type measurements. We first construct an integral equation that combines unknown sources and integral‐type observations. With the aid of Mittag‐Leffler functions, we will use the Fredholm alternative for compact operators to study the existence, uniqueness, and regularity estimates of the solutions to the forward problem. Finally, we developed an algorithm utilizing Tikhonov regularization technique and validated its superior performance in accuracy and efficiency through a series of numerical experiments.

This paper investigates the existence and uniqueness of solutions to nonlinear Volterra integral equations of variable fractional order in Fréchet spaces. The variable‐order fractional derivative is considered in the Riemann–Liouville sense, which extends classical approaches and is central to the paper’s novelty. By employing a nonlinear alternative of the Frigon–Granas fixed‐point theorem for contraction mappings, a rigorous mathematical framework is provided, suitable for problems on semi‐infinite intervals and for functions with variable fractional order, where classical Banach space approaches may fail. Illustrative examples demonstrate the applicability of the main results. The approach highlights the flexibility and generality of the method, paving the way for future extensions such as stability analysis and numerical schemes based on the established theory.