Ecologists and other statistical practitioners with access to high-resolution spatial data often lack guidance on best approaches for discerning meaningful spatial scales for environmental covariates which is necessary when spatial factors influence environmental processes. Recently developed methods have attempted to automate investigating spatial scales for covariates by evaluating models for which potential explanatory variables are derived from circular extents of increasing size centered at survey locations. However, these methods make a strong assumption on the inclusion of the covariate and do not help discern whether a covariate should be included in the model. We present an approach that utilizes researcher guidance to create informative priors on the model space that, along with parallelizable Reversible Jump Markov chain Monte Carlo techniques, enables efficient estimation of posterior model probabilities to assist with the choice of meaningful spatial scales for environmental covariates.

The asymptotic equipartition property (AEP) plays an important role in establishing lossless source coding theorems and asymptotic coding theorems through the concepts of typical sets and typical sequences in information theory. In this paper, we shall study the strong law of large numbers and the AEP for hidden Markov tree models. First, we give a strict definition of hidden Markov tree models, and study their key properties and equivalent characterizations. In fact, hidden Markov tree models are closely related to the tree-indexed Markov chains, therefore, we also introduce the definition of tree-indexed Markov chains and their equivalent properties. We also establish a strong law of large numbers for hidden Markov tree models indexed by a Cayley tree. As corollaries, we obtain some strong laws of large numbers for the parameters and the AEP for these models.

This study aims to investigate the effect of seasonal influenza vaccination on hospitalization rates among patients presenting to the emergency department with influenza-like illness. A retrospective, single-center observational study was conducted, involving adult patients with influenza (ICD-10 codes J10 and J11) diagnosed in the emergency department between May 2024 and April 2025. Clinical and demographic information was collected from electronic records, and vaccination status was confirmed through follow-up phone calls. To tackle the "zero event" problem-no hospitalizations among vaccinated individuals-advanced statistical modeling was employed, including standard logistic regression, and Bayesian logistic regression using Markov Chain Monte Carlo (MCMC) simulations. Odds ratios (OR) and 95% Highest Density Intervals (HDI) were calculated to assess the effectiveness of vaccination. A total of 878 patients were enrolled: 3.3% (n = 29) received vaccinations, while 2.7% (n = 24) required hospitalization. None of the vaccine recipients were hospitalized. Standard logistic regression indicated that age was a significant indicator of hospitalization. Furthermore, Bayesian logistic regression followed, which confirmed vaccination's statistically significant protective effect. The OR for vaccination was 0.526 (95% HDI: 0.336-0.739), indicating a 47% reduction in hospitalization risk among vaccinated individuals. Seasonal influenza vaccination was significantly associated with a lower risk of hospitalization in patients presenting with influenza-like illness to the emergency department. These findings support public health initiatives to enhance influenza vaccine coverage, particularly for the elderly.

We simulate the dynamics of pelagic Sargassum rafts as systems of finite-size floating particles, governed by a Maxey-Riley law with nonlinear elastic interactions. Using surface ocean currents and wind data from reanalysis systems for clump transport, we computed trajectories within a domain covering the tropical and subtropical north Atlantic. The subsequent motion is reduced using Ulam's discretization method into a time-inhomogeneous Markov chain that simulates a background Sargassum concentration. Bayesian inversion, combined with nonautonomous transition path theory, was used to infer the origin of the first significant recorded bloom in the tropical North Atlantic, which unfolded in April 2011. Both methodologies independently identified the bloom's origin as near the West African coast, up to 2 years before it was detectable via satellite imagery on the basin's western side. This finding supports anecdotal evidence of Sargassum strandings on the Ghanaian coast in 2009. Moreover, it correlates with unusual environmental conditions-such as increased nutrient loads from significant upwelling linked to a pronounced Dakar Niña and Saharan dust deposition-that promote bloom proliferation. Additionally, it aligns with the observation that the species of Sargassum in the 2011 bloom differ from those in the Sargasso Sea, which might otherwise be considered a natural origin.

Applications of Markov chains in climate change modelling: A comprehensive review of advances, challenges, and future directions

Lie algebraic duality for some Markov processes

ABSTRACT The properties of moiré excitons in twisted bilayers of transition metal dichalcogenides (TMDCs) vary significantly with the twist angle, ranging from quasi‐localized excitons with flat dispersions for small twist angles to delocalized excitons for larger ones. This twist‐angle dependence directly impacts the exciton–phonon coupling, which plays a significant role for the optical properties of these materials. In this work, we theoretically investigate the twist‐angle‐dependent influence of phonons on absorption spectra of intralayer moiré excitons in a twisted TMDC heterobilayer. For the lowest‐lying intralayer moiré exciton, we find that the exciton–phonon coupling interpolates between two physically distinct regimes when tuning the twist angle. At small twist angles non‐Markovian polarization dynamics and phonon sidebands dominate the properties of absorption spectra for localized excitons. For larger twist angles Markovian processes become more important, leading to additional line broadening. Furthermore, the absorption spectra here show a characteristic asymmetric peak similar to monolayer TMDCs. When taking into account multiple bright moiré exciton bands, we find that intraband scattering due to optical phonons has a significant impact on absorption spectra, effectively suppressing absorption peaks of higher lying bands when their bandwidth surpasses the optical phonon energy.

Traditional Markov chain models are commonly used to represent cancer progression but are limited by the memoryless assumption and reduced ability to capture nonlinear temporal dependencies. This study proposes a hybrid LSTM-assisted Markov chain framework that integrates sequential deep learning with probabilistic state-transition modeling. A clinically informed simulation framework generated 30,000 synthetic longitudinal patient trajectories across five disease states (Carcinoma in situ, Early/Localized, Locally Advanced, Regionally Advanced, and Metastatic). Transition probabilities were parameterized to reflect realistic progression dynamics, including intensified forward transitions, rare backward transitions (<1.5%), and an absorbing metastatic state. Each sequence contained 5–15 time steps with 17 clinical features. Due to irreversible progression, advanced stages were moderately overrepresented. Data were stratified into training (64%, n=19,200), validation (16%, n=4,800), and independent test (20%, n=6,000) sets while preserving class proportions. Hyperparameters were tuned on the validation set. Performance uncertainty was estimated using 1,000 bootstrap resamples to compute 95% confidence intervals (CIs). On the independent test set, the hybrid model achieved an accuracy of 0.919 (95% CI: 0.912–0.926), precision of 0.883 (95% CI: 0.874–0.892), recall of 0.919 (95% CI: 0.912–0.926), and F1-score of 0.896 (95% CI: 0.887–0.905). Compared with a traditional Markov baseline, the hybrid framework demonstrated improved predictive stability and better discrimination of advanced disease states.These findings demonstrate methodological robustness within a simulation-based environment encoding realistic cancer progression patterns. However, results are derived solely from synthetic data and require validation on real-world longitudinal clinical datasets before clinical applicability can be established.

In this study, we explore the late-time cosmic acceleration using the [Formula: see text] gravity framework, where the non-metricity Q is coupled to the matter Lagrangian [Formula: see text]. We investigate a nonlinear model defined by [Formula: see text]. To reconstruct the cosmological evolution, we employ a kinematic ansatz with a logarithmic parametrization of the deceleration parameter, [Formula: see text]. We constrain the model parameters using the latest observational data from Cosmic Chronometers (CC) and Baryon Acoustic Oscillations (BAO) via the Dark Energy Spectroscopic Instrument (DESI). The Markov Chain Monte Carlo (MCMC) analysis yields a current Hubble parameter of [Formula: see text] and indicates a transition from deceleration to acceleration at [Formula: see text]. Solving the modified Friedmann equations allows us to determine the evolution of the Hubble parameter and effective thermodynamic quantities. Further, we analyze the energy density ([Formula: see text]), pressure (p), and Equation of State (EoS) parameter ([Formula: see text]). Additionally, we distinguish the proposed [Formula: see text] model from the standard [Formula: see text]CDM paradigm using geometrical diagnostics, including the Statefinder hierarchy [Formula: see text] and [Formula: see text], as well as cosmographic parameters like Jerk, lerk and Snap. Finally, we test the model’s validity against standard Energy Conditions. The results demonstrate that while the model violates the Strong Energy Condition (SEC) to support cosmic acceleration, it satisfies the Null Energy Condition (NEC) within the observational range, ensuring the stability of the cosmic fluid.

The current maintenance practices at commercial nuclear power plants (NPPs) adhere to a fixed schedule designed to preclude any form of degradation or failure of structures, systems, and components (SSCs). This preventive maintenance paradigm, although effective in enhancing reliability, could result in considerable operational inefficiencies. This paper presents a methodology for addressing the current scheduled maintenance practices through a risk-informed approach that provides component health and condition assessment to inform condition-based maintenance practices. In this work component risk parameters are estimated and updated utilizing historical maintenance data at NPPs, and Markov chain–based models are developed to represent healthy, degraded, and failed states of SSCs. The transition rates of Markov models are estimated using Bayesian estimation that utilizes historical maintenance data of components at a commercial NPP. By integrating historical maintenance data of components, this approach offers a nuanced understanding of SSC condition, beyond the simplistic preventive maintenance schedules. The parameter estimation in this work facilitates the transition from the prevailing fixed-schedule preventive maintenance paradigms to a more efficient, component-specific, condition-based maintenance framework. Such an advanced maintenance paradigm would not only fill a significant gap in current practices but would also align with the industry’s objective to maintain high safety standards while improving operational cost-effectiveness and sustainability.

This tutorial introduces factored structural equation modeling (FSEM), an alternative to multivariate structural equation modeling, and demonstrates its implementation in the Blimp software and the rblimp R package. FSEM reconceptualizes the joint distribution of observed and latent variables as a set of univariate and multivariate submodels-each specified via simple regression equations-and treats latent variables as missing data to be imputed via Bayesian data augmentation. This approach seamlessly accommodates combinations of continuous (normal and nonnormal), binary, ordinal, nominal, count, and two-part variables; interactions; nonlinear effects; heteroscedasticity; and multilevel data structures without violating distributional assumptions. We first outline the theoretical foundations of factored regression specification and its connection to the probability chain rule. We then describe the Markov chain Monte Carlo estimation and missing-data imputation algorithms that underlie FSEM. Next, we present a series of illustrative models-ranging from basic confirmatory factor analysis to complex dynamic, multilevel, and hybrid generalized-linear-structural equation modeling applications-providing Blimp syntax excerpts and a real-data example. We conclude by discussing practical considerations, limitations, and directions for future methodological research. This tutorial aims to equip researchers with a flexible, user-friendly framework for modeling complex data in behavioral and social sciences. (PsycInfo Database Record (c) 2026 APA, all rights reserved).

Within clinical research and healthcare decision-making, stochastic modeling methods are becoming increasingly more necessary due to the complexity of predicting the results of clinical processes, disease progression, and analyzing the effectiveness of various treatments. Markov chain models in particular present a good mix of accuracy and simplicity for modeling healthcare outcomes. This study presents a detailed overview of the theoretical foundations of Markov chain models while also discussing their application in patient risk stratification, clinical decision-making, and cost-effectiveness analysis of treatments. Both the advantages and disadvantages of Markov chain models like the memoryless assumption, data requirements needed, and state complexity particularly in healthcare contexts, are examined. Possible future directions for Markov chain modeling, namely hybrid modeling approaches and Markov decision processes (MDPs), are assessed to compare their ability to improve predictive accuracy and influence healthcare policies with regular Markov chain models. Combined all the elements, this study offers clinical researchers and policymakers a comprehensive reference on the strengths and weaknesses of Markov chain modeling specifically in healthcare applications.

Multimorbidity is a growing public health concern globally, yet remains under-researched in fragile and conflict-affected settings, including Somalia, a country in the Horn of Africa with limited health infrastructure. This study investigates the prevalence, regional patterns, and individual- and contextual-level determinants of multimorbidity among women aged 15–49 years in Somalia. A cross-sectional analytical study design was applied using nationally representative data from the 2020 Somalia Demographic and Health Survey (DHS). The final sample included 8,172 women after excluding cases with missing or incomplete information. Multimorbidity was defined as the presence of ≥ 2 self-reported chronic conditions. Bayesian multilevel logistic regression was employed to account for hierarchical data structure— individuals (Level 1) nested within regions and residence types (Level 2). Random intercept and slope models were compared using Deviance Information Criterion (DIC), with posterior distributions estimated via Markov Chain Monte Carlo (MCMC) simulation. Associations were assessed using odds ratios (ORs) and 95% Bayesian credible intervals (CrIs). The prevalence of single morbidity was 7.2% (95%CI: 6.64%–7.76%), while multimorbidity was 2.1% (95%CI: 1.78%–2.40%), with higher burden among urban residents and certain regions (e.g., Banadir, Bay, Sool). Hypertension, kidney disease, and chronic headache were the most common conditions. Model III (random intercept and slopes) provided the best fit (DIC = 1525.051). Key determinants of higher multimorbidity risk included food insecurity (OR = 2.939, 95%CrI: 2.280–3.775), obesity (OR = 1.601, 95%CrI: 1.175–2.093), older maternal age (OR = 1.047, 95%CrI: 1.026–1.066), lack of internet access (OR = 1.254, 95%CrI: 1.084–1.471), and longer water-fetching time (Up to 30 min, OR = 1.798). Protective factors included rural (OR = 0.658, 95%CrI: 0.464–0.809) and nomadic (OR = 0.349, 95%CrI: 0.250–0.453) residence, non-smoking (OR = 0.641, 95%CrI: 0.434–0.988), lack of agricultural land ownership (OR = 0.563, 95%CrI: 0.484–0.653), and toilet non-sharing (OR = 0.664, 95%CrI: 0.570–0.813). Multimorbidity affects a notable proportion of Somali women, with considerable geographic and socioeconomic variation. The findings underscore the need for regionally tailored, multisectoral interventions targeting modifiable risk factors such as food insecurity, obesity, and access to healthcare infrastructure.

Communication through scent-marking plays a crucial role in animal fitness, particularly for solitary species such as the Andean bear, in which it fulfills the function of advertising individual attributes to conspecifics and managing social interactions. However, research on the scent-marking behavior of Andean bears and the impact of free-ranging dogs on their chemical communication remains limited. This paper aims to characterize Andean bears’ scent-marking behavior and to study the overlap of their daily activity patterns with free-ranging dogs in a key biological corridor linking the Andes to the Amazon region. We collected data using camera traps in front of 10 trees selected for rubbing by Andean bears. Cameras were deployed in the Guácharos-Puracé Biological Corridor Regional Natural Park (PNRGP), Colombia, from April 2023 to April 2024. We determined the transition probabilities of marking behaviors using finite Markov chains. We compared the time budget for the behaviors displayed by Andean bears at rub trees by age and sex. Finally, we described the temporal overlap of Andean bears and free-ranging dogs and analyzed whether it differed between rainfall seasons. Results revealed significant intersexual differences in the time budget of scent-marking, with adult males investing more time on marking than females and subadults. This behavior was prevalent during the dry season. The daily activity patterns of Andean bears and free-ranging dogs were predominantly diurnal. Both species showed high temporal overlap during the dry season (77%), decreasing during the general analysis (75%) and the wet season (69%). We also suggest an increased potential for interspecific interactions between Andean bears and free-ranging dogs that could negatively impact the former, through competition, altered communication patterns and the potential transmission of diseases. We emphasize the need for further studies on the impact of free-ranging dogs on wildlife, particularly on the intraspecific communication of Andean bears.

In ecologically fragile and geomorphologically complex mountainous regions, ensuring a smooth transition from poverty alleviation to multidimensional sustainable rural development remains a key issue in regional governance. Focusing on the Qinba Mountains, a typical former contiguous poverty-stricken region in China covering 18 prefecture-level cities in six provinces, this study uses 2009–2023 prefecture-level panel data to examine the spatiotemporal evolution and driving mechanisms of coordinated rural revitalization. An integrated framework of “multi-dimensional evaluation–spatiotemporal tracking–attribution diagnosis” is developed by combining the improved AHP–entropy-weight TOPSIS method, the Coupling Coordination Degree (CCD) model, spatial Markov chains, spatial autocorrelation, and the Geodetector. The results show pronounced subsystem asynchrony. Livelihood and Well-being Security (U5) improves steadily, while Level of Industrial Development (U1), Civic Virtues and Cultural Vibrancy (U3), and Rural Governance (U4) also rise but with clear spatial differentiation; by contrast, Quality of Human Settlements (U2) fluctuates in stages under ecological fragility. Overall, the coupling coordination level advances from the Verge of Imbalance to Intermediate Coordination, yet the regional pattern remains uneven, with eastern basin cities leading and western deep mountainous cities lagging. State transitions display both policy responsiveness and path dependence: the probability of retaining the original state ranges from 50.0% to 90.5%; low-level neighborhoods reduce the upward transition probability to 25%, whereas medium-to-high-level neighborhoods raise the upward transition probability of low-level cities from 36.36% to 53.33%. Spatial dependence is also evident, with Global Moran’s I increasing, with fluctuations, from 0.331 in 2009 to 0.536 in 2023; high-value clusters extend along the Guanzhong Plain–Han River Valley corridor, while low-value clusters remain relatively locked in mountainous border areas. Driving mechanisms show clear stage-wise succession. At the single-factor level, the explanatory power of Road Network Density (F6) declines from 0.639 to 0.287, whereas Terrain Relief Amplitude (F1) becomes the dominant background constraint in the later stage (q = 0.772). Multi-factor interactions are generally enhanced. In particular, the traditional infrastructure-led pathway weakens markedly, with F1 ∩ F6 = 0.055 in 2023, while the interaction between terrain and consumer market vitality becomes dominant, with F1 ∩ F7 = 0.987 in 2023. On this basis, three major pathways are identified: government fiscal intervention and transportation accessibility improvement, capital agglomeration and market demand stimulation, and human–earth system adaptation and ecological value realization. These findings provide quantitative evidence for breaking spatial lock-in and improving cross-regional resource allocation in ecologically constrained mountainous regions.

The standard survival models for time-to-event data assume that every subject in the study population is susceptible to the event of interest and will eventually experience the event. However, many diseases are curable and a portion of patients will never experience the event of interest. In addition, unobservable effects or correlation among the subjects within the same cluster may exist and careful attention is needed. In this paper, we developed the EM-based likelihood inference for semiparametric mixture cure frailty model with Generalized-Gamma frailty distribution. The proposed model contains most existing mixture cure frailty model as special cases. Since the expectations in the E-step lack closed-form solutions, we employ a Markov chain Monte Carlo (MCMC) method for numerical approximation. The bootstrap method is adopted for variance estimation. We conducted comprehensive simulation study to assess the performance of our proposed model and its associated inference methods. To demonstrate practical applicability, we subsequently applied this model on a real data examining blood purification status and estimated amount effects on acute diquat poisoning cases.

In this paper, we develop a unified theory of IPA estimators for steady-state performance characteristics. Our goal is to clarify how infinitesimal perturbation analysis can be systematically connected to a measure-valued differentiation framework for the underlying Markov chain dynamics. Traditionally, the asymptotic behavior of gradient estimators has been studied by analyzing the estimators themselves. Here, we take a different perspective. We show that by clearly separating the level at which arguments are made, either at the sample-path level, where IPA is usually formulated, or at the operator/measure level, where Markov chain differentiation is naturally expressed, one can combine the strengths of both approaches. Our framework helps bridge two established streams of research and contributes a more general understanding of IPA for steady-state problems.

Abstract In this paper, we revisit a fundamental technical issue within the theory of stochastic approximation (SA) in a Markovian framework, first proposed in the book by Derevitskii and Fradkov (Applied theory of discrete adaptive control systems, Nauka, 1981), and further developed in much detail in the book by Benveniste, Métivier, and Priouret (Adaptive algorithms and stochastic approximations, Springer, Berlin, 1990). This theory is instrumental in many application areas such as the statistical analysis of Hidden Markov Models arising in telecommunication, quantized linear stochastic systems, and reinforcement learning. The problem at hand is the verification of the existence, uniqueness, and Lipschitz continuity of the solution of a parameter-dependent Poisson equation, in an appropriate weighted sup-norm, associated with a collection of Markov chains on general state spaces. Verification of the above facts is vital in the analysis of SA processes presented in the cited book via the ODE (ordinary differential equations) method, requiring substantial technical effort. The motivation and focus of the paper is to address this technical issue, by presenting a simple set of conditions, under which the above properties of the Poisson equation at hand can be conveniently established. A distinctive feature of our work is that it is based on a remarkable result of Hairer and Mattingly (2011), proving that by tilting standard conditions of mainstream stability theory for Markov chains, the transition kernels prove to be contractions in the space of differences of probability measures in a suitable metric. To demonstrate the applicability of our results, the proposed conditions are verified for a class of queuing system with open-loop control.

Computerized adaptive tests (CATs) play a crucial role in educational assessment and diagnostic screening in behavioral health. Unlike traditional linear tests that administer a fixed set of pre-assembled items, CATs adaptively tailor the test to an examinee's latent trait level based on their previous responses. We introduce a novel CAT system that builds on recent advances in Bayesian multivariate IRT. Our approach leverages direct sampling from the latent factor posterior distributions, significantly accelerating existing information-theoretic item-selection methods by eliminating the need for computationally intensive Markov chain Monte Carlo simulations. To address the potential suboptimality of one-step-ahead item-selection rules, we also develop a double deep Q-learning algorithm that efficiently learns an optimal item-selection policy offline using a calibrated item bank. Through simulation and real-data studies, we demonstrate that our approach not only accelerates existing item-selection methods but also highlights the potential of reinforcement learning (RL) in CATs. Notably, our Q-learning-based strategy consistently achieves the fastest posterior variance reduction, leading to earlier test termination. These results demonstrate the promise of combining exact posterior sampling with RL to deliver scalable, high-precision CATs.

We present an analysis of parton distribution functions (PDFs) of the proton using Markov chain Monte Carlo (MCMC) methods. The MCMC approach naturally implements Bayes’ theorem and, thus, provides a means to directly sample the underlying probability distribution—in this case, the probability distribution of the PDF parameters. This allows for a straightforward propagation of the resulting uncertainties into any PDF-dependent observable, preserving their simple probabilistic interpretation. In our analysis we include a broad set of deep inelastic scattering data from HERA, BCDMS and NMC experiments along with the Drell-Yan, W and Z boson data from LHC and Tevatron experiments, which combined with theoretical calculations at next-to-next-to-leading order in QCD allow for realistic determination of PDFs. The main focus of this analysis is to explore alternative methods for PDF uncertainty estimation that are more firmly grounded in statistical principles. We show that the flexibility of the Bayes framework, allowing one, e.g., to account for non-Gaussianity or inconsistencies of datasets, is crucial to extract realistic uncertainties when such assumptions are not fulfilled. We also demonstrate that MCMC allows one to determine the Δ χ 2 value corresponding to a given confidence level in the sample, which can, in turn, be used as a statistically well-founded tolerance criterion used in the Hessian method, thus addressing one of its main long-standing drawbacks.