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Revealing the mechanisms underlying community organization has long been a central interest for ecologists and biogeographers. Recent findings have suggested that different dimensions of biodiversity may be shaped by contrasting ecological processes, offering complementary insights about community assembly. However, studies integrating multiple diversity facets across the Qinghai‐Tibetan Plateau (QTP) remain insufficient. We combined multiple analytical frameworks to unravel the patterns (taxonomic, functional and phylogenetic) and correlates (local environment, land use and dispersal pathways) of aquatic arthropod diversity in lotic (stream) and lentic (wetland) ecosystems of the QTP. We observed strong phylogenetic signals in most functional traits, pointing to the influence of evolutionary processes on these faunas. Null models indicated that community structure in most streams and wetlands was mostly the result of random draws from the regional functional and phylogenetic species pool. The prevalence of stochasticity was most likely associated with the interplay of the paleogeographical history, the extremely harsh environmental conditions at high elevations and the subsequent impoverishment of the regional species pool. However, some streams and wetlands also exhibited phylogenetic overdispersion and functional clustering, emphasizing the potential importance of competitive exclusion and habitat filtering, respectively. Variation partitioning further revealed that both environmental filtering and dispersal limitation accounted for the spatial variation of diversity measures, with local environment overriding the effects of space and catchment land use. Overall, this study improves our understanding of community organization and diversity patterns in environmentally extreme alpine catchments, with broad implications for the conservation and management of one of the world’s most important high‐altitude treasures. [ABSTRACT FROM AUTHOR]

This article presents a robust security‐based model predictive control (MPC) framework for a class of discrete‐time nonlinear cyber‐physical systems (CPSs) suffering random deception attacks and persistent actuator saturation. In CPSs, the attacks can significantly degrade the control performance. Naturally, a larger degree of control is required, prompting the consideration of actuator saturation in CPS's structure. In our work, the cyberattacks on the controller‐actuator (C‐A) channel are described as a stochastic process with specific probability and bounded energy. The security‐related constraint is introduced to improve the resistance and robustness against attacks. Mean‐square quadratic boundedness is derived to estimate the robustly invariant set. By placing the saturated input into a convex hull, a robust security‐based MPC optimization problem is addressed less conservatively with linear matrix inequality technique. Furthermore, the feasibility of the proposed method is proved recursively, and the stability and security of the closed‐loop system are established in mean‐square sense. In the end, the laboratory tank and reactor‐separator process are applied to validate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The Strauss point process is a very popular model for describing the random cellular networks, yet several key statistical properties such as intensity, empty space function, and probability generating functional have remained elusive. This article addresses these issues by first leveraging the Poisson saddle point method to approximate the distance‐conditioned intensity for Strauss point processes. Subsequently, the author derives an analytically tractable expression for the distribution of empty space distance based on a conditional thinning mechanism. Additionally, the author establishes an upper bound for the probability generating functional in Strauss point processes, which is crucial for evaluating the Laplace transform of cumulative interference in relevant cellular networks. These findings facilitate the systematic derivation of spatially averaged probability of coverage, and the accuracy of analytic results is validated through simulations. [ABSTRACT FROM AUTHOR]

We derive consistency and asymptotic normality results for quasi‐maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous‐time setting. The special feature of our analysis is that the stochastic integral part is unobserved and nonparametric. Additionally, the drift may depend on the (unknown and unobserved) stochastic integrand. Our results hold for ergodic semi‐parametric diffusions and backward SDEs. Simulation studies confirm that the methods proposed yield good convergence results. [ABSTRACT FROM AUTHOR]

This study proposes a novel velocity divergence‐generalized adaptive probability density evolution method (VD‐GAPDEM) for calculating the probability density function of the stochastic response process of stochastic structures under stochastic dynamic loads. First, based on the principle of probability conservation, the velocity divergence‐generalized adaptive probability density evolution equation (VD‐GAPDEE) is derived for a stochastic system that can effectively consider the shape and location changes of the joint transitional probability density of representative points (RPs) in the stochastic response process. Second, a novel VD‐GAPDEM is proposed to solve the VD‐GAPDEE directly using the point selection technique based on the generalized F discrepancy and the second‐order Runge–Kutta method with a smoothing kernel method (Runge–Kutta‐SKFAM). Furthermore, the differences and connections between VD‐GAPDEM and the existing probability density evolution method are analyzed. Additionally, the high computational efficiency and accuracy of the proposed VD‐GAPDEM are demonstrated through three typical examples of stochastic response analysis, involving stochastic systems subjected to stochastic dynamic loads. [ABSTRACT FROM AUTHOR]

In [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let r⩾3$$ r\geqslant 3 $$ and let n$$ n $$ be divisible by r$$ r $$. Then, in the random r$$ r $$‐uniform hypergraph process on n$$ n $$ vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we prove the analogue of this result for clique factors in the random graph process: at the time that the last vertex joins a copy of the complete graph Kr$$ {K}_r $$, the random graph process contains a Kr$$ {K}_r $$‐factor. Our proof draws on a novel sequence of couplings which embeds the random hypergraph process into the cliques of the random graph process. An analogous result is proved for clique factors in the s$$ s $$‐uniform hypergraph process (s⩾3$$ s\geqslant 3 $$). [ABSTRACT FROM AUTHOR]

This research explores the reliability of deep learning, specifically Long Short‐Term Memory (LSTM) networks, for estimating the Hurst parameter in fractional stochastic processes. The study focuses on three types of processes: fractional Brownian motion (fBm), fractional Ornstein–Uhlenbeck (fOU) process, and linear fractional stable motions (lfsm). The work involves a fast generation of extensive datasets for fBm and fOU to train the LSTM network on a large volume of data in a feasible time. The study analyses the accuracy of the LSTM network's Hurst parameter estimation regarding various performance measures like root mean squared error (RMSE), mean absolute error (MAE), mean relative error (MRE), and quantiles of the absolute and relative errors. It finds that LSTM outperforms the traditional statistical methods in the case of fBm and fOU processes; however, it has limited accuracy on lfsm processes. The research also delves into the implications of training length and valuation sequence length on the LSTM's performance. The methodology is applied to estimating the Hurst parameter in li‐ion battery degradation data and obtaining confidence bounds for the estimation. The study concludes that while deep learning methods show promise in parameter estimation of fractional processes, their effectiveness is contingent on the process type and the quality of training data. [ABSTRACT FROM AUTHOR]

Time‐varying group formation tracking control problems for multi‐agent systems are investigated based on distributed multi‐sensor multi‐target filtering with intermittent observations. First, in order to estimate the states of multiple targets under the phenomenon of intermittent observations accurately, a distributed multi‐sensor multi‐target filtering algorithm is proposed based on cubature Kalman filtering. Second, a time‐varying group formation tracking protocol is designed for multi‐agent systems by using the state estimations obtained from the filtering algorithm and the neighboring interaction. The protocol enables multi‐agent systems to form time‐varying subformations and track multiple targets in the same subgroups, respectively. Third, the boundedness of the error covariance matrices is proved under the condition that the observation probability is higher than the minimum threshold. Then the estimation errors of the filtering algorithm are proved to be stochastically bounded by introducing a stochastic process. Furthermore, the boundedness of the group formation tracking errors is proved. Finally, a numerical example is used to verify the performance of the proposed algorithm and protocol. [ABSTRACT FROM AUTHOR]

Subsampling is a practical strategy for analyzing vast survival data, which are progressively encountered across diverse research domains. While the optimal subsampling method has been applied to inferences for Cox models and parametric accelerated failure time (AFT) models, its application to semi‐parametric AFT models with rank‐based estimation have received limited attention. The challenges arise from the non‐smooth estimating function for regression coefficients and the seemingly zero contribution from censored observations in estimating functions in the commonly seen form. To address these challenges, we develop optimal subsampling probabilities for both event and censored observations by expressing the estimating functions through a well‐defined stochastic process. Meanwhile, we apply an induced smoothing procedure to the non‐smooth estimating functions. As the optimal subsampling probabilities depend on the unknown regression coefficients, we employ a two‐step procedure to obtain a feasible estimation method. An additional benefit of the method is its ability to resolve the issue of underestimation of the variance when the subsample size approaches the full sample size. We validate the performance of our estimators through a simulation study and apply the methods to analyze the survival time of lymphoma patients in the surveillance, epidemiology, and end results program. [ABSTRACT FROM AUTHOR]

Continuous time Bayesian networks (CTBNs) represent a class of stochastic processes, which can be used to model complex phenomena, for instance, they can describe interactions occurring in living processes, social science models or medicine. The literature on this topic is usually focused on a case when a dependence structure of a system is known and we are to determine conditional transition intensities (parameters of a network). In the paper, we study a structure learning problem, which is a more challenging task and the existing research on this topic is limited. The approach, which we propose, is based on a penalized likelihood method. We prove that our algorithm, under mild regularity conditions, recognizes a dependence structure of a graph with high probability. We also investigate properties of the procedure in numerical studies. [ABSTRACT FROM AUTHOR]

In the analysis of clustered failure time data, Cox frailty models have been extensively studied by incorporating frailty with a prespecified distribution to address potential correlation of data within clusters. In this paper, we propose a frailty proportional mean residual life regression model to analyze such data. A novel conditional quasi‐likelihood inference procedure is developed, utilizing a stochastic process and the inverse probability of censoring weighting (IPCW) to form estimating equations for regression parameters. Our proposal employs conditional inference based on a penalized quasi‐likelihood to address within‐cluster correlation without need to specify the frailty distribution, bringing the method closer to what suffices for real‐world applications. By adopting the Buckley–James estimator in the IPCW, the method further allows for dependent censoring. We establish asymptotic properties of the proposed estimator and evaluate its finite sample performance via simulation studies. An application to the data from a multi‐institutional breast cancer study is presented for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

An integro‐differential equation for the probability density of the generalized stochastic Ornstein–Uhlenbeck process with jump diffusion is considered for a special case of the Laplacian distribution of jumps. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly, as a finite sum. Alternatively, the fundamental solution can be represented as converging power series. The properties of this solution are investigated. The fundamental solution makes it possible to obtain explicit formulas for the density at each instant of time, which is important, for example, for testing numerical methods. [ABSTRACT FROM AUTHOR]

This paper deals with the probabilistic analysis of a class of compartmental models formulated via a system of linear differential equations with time‐dependent non‐homogeneous terms. For the sake of generality, we assume that initial conditions and rates between compartments are random variables with arbitrary distributions while the source terms defining the flows entering the compartments are stochastic processes. We then take extensive advantage of the so‐called random variable transformation (RVT) technique to determine the first probability distribution of the solution of such a randomized model under very general hypotheses. In the simplest but relevant case of time‐independent source terms, we also obtain the probability distribution of the equilibrium, which is a random vector. Furthermore, we first particularize the aforementioned results for different types of integrable source terms that permit obtaining an explicit solution of the compartmental model: random constants, a train of Dirac delta impulses with random intensities, and a Wiener process. Secondly, we show an alternative approach based on the Liouville–Gibbs equation, which is useful when dealing with source terms that do not admit a closed‐form primitive. All the previous theoretical results are first illustrated through several numerical examples and simulations where a wide range of different probability distributions are assumed for the model parameters. The paper concludes by applying a compartmental model that describes the dynamics of oral drug administration through multiple chronologically spaced doses using synthetic data generated according to pharmacological references. [ABSTRACT FROM AUTHOR]

Punctuated equilibrium theory (PET) posits that policy change is incremental for long periods of time with occasional shifts in problem definition and governing authority that produce large changes. Incrementalism results from shared norms and standard operating procedures within policy subsystems. The U.S. Congress experienced a series of institutional changes in the 1990s that both introduced more partisanship into subsystem interactions and made it easier for majority party leaders to enact their agenda. These changes have disrupted many of the processes that produce incrementalism. Using data from 1949 to 2022, I find that U.S. policy outputs, measured through public laws, have experienced fewer incremental changes from one term to the next and more moderate and very large changes since 1995. The shift in outputs has been driven primarily by increased partisan conflict within congressional committees. This study highlights the utility of using the stochastic process approach to compare distributions of outputs across time and not only across countries or institutions. [ABSTRACT FROM AUTHOR]

In the recent past, the Gravity Recovery and Climate Experiment (GRACE) satellite mission and its successor GRACE Follow‐On (GRACE‐FO), have become invaluable tools for characterizing drought through measurements of Total Water Storage Anomaly (TWSA). However, the existing approaches have often overlooked the uncertainties in TWSA that stem from GRACE orbit configuration, background models, and intrinsic data errors. Here we introduce a fresh view on this problem which incorporates the uncertainties in the data: the Probabilistic Storage‐based Drought Index (PSDI). Our method leverages Monte Carlo simulations to yield realistic realizations for the stochastic process of the TWSA time series. These realizations depict a range of plausible drought scenarios that later on are used to characterize drought. This approach provides probability for each drought category instead of selecting a single final category at each epoch. We have compared PSDI with the deterministic approach (Storage‐based Drought Index, SDI) over major global basins. Our results show that the deterministic approach often leans toward an overestimation of storage‐based drought severity. Furthermore, we scrutinize the performance of PSDI across diverse hydrologic events, spanning continents from the United States to Europe, the Middle East, Southern Africa, South America, and Australia. In each case, PSDI emerges as a reliable indicator for characterizing drought conditions, providing a more comprehensive perspective than conventional deterministic indices. In contrast to the common deterministic view, our probabilistic approach provides a more realistic characterization of the TWS drought, making it more suited for adaptive strategies and realistic risk management. Plain Language Summary: Total Water Storage (TWS) is defined as the sum of water stored as surface water (e.g., lakes and rivers), groundwater, soil moisture, snow, ice, and vegetation biomass. Since its launch in 2002, the Gravity Recovery and Climate Experiment (GRACE) satellite mission has provided unique TWS change measurements with many applications in hydrology, including characterizing drought events. Scientists have been using satellites like GRACE and its successor, GRACE‐FO, to understand drought by measuring the Total Water Storage Anomaly (TWSA). However, previous methods didn't consider uncertainties from satellite orbits, models, and data errors. This study offers a novel probabilistic approach for characterizing drought, Probabilistic Storage‐based Drought Index (PSDI), which acknowledges the uncertainties in the GRACE TWS change. We use simulations to create different drought scenarios, offering probabilities for each category instead of one fixed category. Comparing PSDI to traditional methods, we found that traditional methods tend to overestimate drought severity. We tested PSDI across different regions, and it consistently proved to be a reliable way to understand drought conditions, offering a more comprehensive perspective. Our probabilistic approach offers a more realistic view of TWS drought, making it suitable for adaptive strategies and risk management. Key Points: A novel probabilistic framework is introduced to characterize drought using Gravity Recovery and Climate Experiment (GRACE) and GRACE Follow‐On observations and propagating their stochasticsOur study suggests a tendency of deterministic approaches to overestimate storage‐based drought severityThe probabilistic approach captures global hydrological droughts while delivering more realistic results suited for risk management [ABSTRACT FROM AUTHOR]

The article will focus on vector autoregressive‐moving average (VARMA) models with time‐dependent coefficients (td) to represent general nonstationary time series, not necessarily Gaussian. The coefficients depend on time, possibly on the length of the series n, hence the name tdVARMA (n) for the models, but not necessarily on the rescaled time t/n. As a consequence of the dependency on n of the model, we need to consider array processes instead of stochastic processes. Under appropriate assumptions, it is shown that a Gaussian quasi‐maximum likelihood estimator is consistent in probability and asymptotically normal. The theoretical results are illustrated using three examples of bivariate processes, the first two with marginal heteroscedasticity. The first example is a tdVAR (n)(1) process while the second example is a tdVMA (n)(1) process. In these two cases, the finite‐sample behavior is checked via a Monte Carlo simulation study. The results are compatible with the asymptotic properties even for small n. A third example shows the application of the tdVARMA (n) models for a real time series. [ABSTRACT FROM AUTHOR]

One of the frequently used drought metrics in scientific research is the consecutive dry days (CDDs) because it effectively indicates short‐term droughts important to ecosystems and agriculture. CDDs are expected to increase in many parts of the world in the future. In Serbia, both the frequency and severity of droughts have increased in recent decades, with most droughts being caused by a lack of precipitation during the warmer months of the year and an increase in evapotranspiration due to higher temperatures. In this study, the frequency and duration of extreme CDDs in the growing season in Serbia were analysed for the past (1950–2019) and the future (2020–2100) period. The Threshold Level Method over precipitation data series was used to analyse CDD events, where extreme CDDs are defined as at least 15 consecutive days without precipitation. In contrast to the original definition of the CDD as the maximum number of consecutive days with precipitation less than 1 mm, here we defined the threshold that is more suitable for agriculture because field crops can experience water stress after 15 days of no rainfall or irrigation. An approach for modelling the stochastic process of extreme CDDs based on the Zelenhasić–Todorović (ZT) method was applied in this research. The ZT method was modified by selecting a different distribution function for modelling the durations of the longest CDD events, enabling a more reliable calculation of probabilities of occurrences. According to the results, future droughts in Serbia are likely to be more frequent and severe than those in the past. The duration of the longest CDDs in a growing season will be extended in the future, lasting up to 62 days with a 10‐year return period and up to 94 days with a 100‐year return period. Results indicate a worsening of drought conditions, especially in the eastern and northern parts of Serbia. The results can help decision‐makers adapt agricultural strategies to climate change by providing information on the expected durations of extreme rainless periods in future growing seasons. Although the analysis was performed in Serbia, it can be applied to any other region. [ABSTRACT FROM AUTHOR]

We prove that the spatial Wiener–Hopf factorisation of a Lévy process or random walk without killing is unique. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of assessing the homogeneity of the disease transmission process in stochastic epidemic models in populations that are partitioned into social groups. We develop a classical hypothesis test for completed epidemics which assesses whether or not there is significant within‐group transmission during an outbreak. The test is based on time‐ordered group labels of individuals. The null hypothesis is that of homogeneity of disease transmission among individuals, a hypothesis under which the discrete random vector of groups labels has a known sampling distribution that is independent of any model parameters. The test exhibits excellent performance when applied to various scenarios of simulated data and is also illustrated using two real‐life epidemic data sets. We develop some asymptotic theory including a central limit theorem. The test is practically very appealing, being computationally cheap and straightforward to implement, as well as being applicable to a wide range of real‐life outbreak settings and to related problems in other fields. [ABSTRACT FROM AUTHOR]

Stationarity, a cornerstone in hydraulic design, is now under scrutiny due to anthropogenic activities and climate change. Numerous studies have sought to identify non‐stationarity (NS); however, a comprehensive assessment of time invariance in all statistical properties of a time series is less explored. This study presents a non‐overlapping block‐stratified random sampling (NBRS) framework leveraging the strengths of several nonparametric tests to assess NS. The NBRS approach exclusively detects NS and distinguishes between various forms of stationarity, including weak and strict. A variant of NBRS is proposed in this study to identify the underlying stochastic process(es) influencing NS in hydroclimatic extremes. Furthermore, a nonparametric clustering approach is used to unveil spatial clusters showcasing NS due to shifts in mean, variance, distribution of time series or a combination of these factors. A comparative assessment of the modified NBRS approach with traditional trend and change point methods is also performed. The proposed methodology is applied to assess the presence of NS in 28 hydroclimatic indices derived for the west‐central river basins of India, exhibiting diverse physio‐climatic settings, for the study period 1973–2021. The modified NBRS approach rigorously explores NS within extreme hydroclimatic indices, conclusively pinpointing its root causes and profound implications for hydrologic design. The applicability of the modified NBRS approach to gridded and point datasets is also demonstrated. The findings highlight the limitations of conventional trend and change point tests in capturing time‐invariant characteristics in heteroscedastic variables (such as streamflow and rainfall extremes) compared to the NBRS approach. The research reveals that NS in rainfall and streamflow extremes primarily results from distributional shifts, whilst temperature extremes are influenced by changes in mean and distribution properties. This research deepens our understanding of the evolving patterns in hydroclimatic extremes in a changing climate. [ABSTRACT FROM AUTHOR]

Environmental change exerts a profound effect on soil microbial domains—including bacteria, fungi, and protists—that each perform vital ecological processes. While these microbial domains are ubiquitous and extremely diverse, little is known about how they respond to environmental changes in urban soil ecosystems and what ecological processes shape them. Here we investigated the community assembly processes governing bacteria, fungi, and protists through the lens of four distinct subcommunities: abundant, conditionally rare, conditionally abundant, and rare taxa. We show that transient taxa, including the conditionally rare and conditionally rare or abundant taxa, were the predominant subcommunities. Deterministic processes (e.g., environmental filtering) had major roles in structuring all subcommunities of fungi, as well as conditionally rare and abundant protists. Stochastic processes had strong effects in structuring all subcommunities of bacteria (except rare taxa) and conditionally rare protists. Overall, our study underscores the importance of complementing the traditional taxonomy of microbial domains with the subcommunity approach when investigating microbial communities in urban soil ecosystems. [ABSTRACT FROM AUTHOR]

Degradation modeling serves as a valuable tool for assessing the lifetime information of highly reliable products. One frequently employed approach for describing the degradation phenomenon involves the use of a degradation model that relies on stochastic processes. In a stochastic‐process‐based degradation model, it is assumed that the increments follow a distribution with the additivity property. This property makes the further inferences mathematically and statistically tractable. However, it limits the choices of the distributions. This paper aims to use those distributions without the additivity property to model the increments and explores distributions from the log‐location‐scale family. Under the frame of Bayesian analysis, Markov Chain Monte Carlo algorithms are developed for executing the necessary computations. Given that the proposed degradation models do not adhere to the additivity property, this paper tackles the challenges involved in predicting the lifetime of both on‐line and off‐line products. Two illustrative examples are subsequently analyzed to demonstrate the procedural steps outlined. The suitability of the proposed model is finally validated through a simulation study. [ABSTRACT FROM AUTHOR]

We aimed to explore, in a sample of systematic reviews (SRs) with meta‐analyses of the association between food/diet and health‐related outcomes, whether systematic reviewers selectively included study effect estimates in meta‐analyses when multiple effect estimates were available. We randomly selected SRs of food/diet and health‐related outcomes published between January 2018 and June 2019. We selected the first presented meta‐analysis in each review (index meta‐analysis), and extracted from study reports all study effect estimates that were eligible for inclusion in the meta‐analysis. We calculated the Potential Bias Index (PBI) to quantify and test for evidence of selective inclusion. The PBI ranges from 0 to 1; values above or below 0.5 suggest selective inclusion of effect estimates more or less favourable to the intervention, respectively. We also compared the index meta‐analytic estimate to the median of a randomly constructed distribution of meta‐analytic estimates (i.e., the estimate expected when there is no selective inclusion). Thirty‐nine SRs with 312 studies were included. The estimated PBI was 0.49 (95% CI 0.42–0.55), suggesting that the selection of study effect estimates from those reported was consistent with a process of random selection. In addition, the index meta‐analytic effect estimates were similar, on average, to what we would expect to see in meta‐analyses generated when there was no selective inclusion. Despite this, we recommend that systematic reviewers report the methods used to select effect estimates to include in meta‐analyses, which can help readers understand the risk of selective inclusion bias in the SRs. [ABSTRACT FROM AUTHOR]

We derive embeddability conditions for the integrated CARMA (2, 1) stochastic process with single and double zero roots in the case of stock variables. [ABSTRACT FROM AUTHOR]

Functional data analysis is an evolving field focused on analyzing data that reveals insights into curves, surfaces, or entities within a continuous domain. This type of data is typically distinguished by the inherent dependence and smoothness observed within each data curve. Traditional functional data analysis approaches have predominantly relied on linear models, which, while foundational, often fall short in capturing the intricate, nonlinear relationships within the data. This paper seeks to bridge this gap by reviewing the integration of deep neural networks into functional data analysis. Deep neural networks present a transformative approach to navigating these complexities, excelling particularly in high‐dimensional spaces and demonstrating unparalleled flexibility in managing diverse data constructs. This review aims to advance functional data regression, classification, and representation by integrating deep neural networks with functional data analysis, fostering a harmonious and synergistic union between these two fields. The remarkable ability of deep neural networks to adeptly navigate the intricate functional data highlights a wealth of opportunities for ongoing exploration and research across various interdisciplinary areas. This article is categorized under:Data: Types and Structure > Time Series, Stochastic Processes, and Functional DataStatistical Learning and Exploratory Methods of the Data Sciences > Deep LearningStatistical Learning and Exploratory Methods of the Data Sciences > Neural Networks [ABSTRACT FROM AUTHOR]

Dynamic fluctuation is a common phenomenon in degradation processes. Hence, how to model it properly has a great impact on the degradation modeling as well as the remaining useful lifetime prediction. To capture the dynamic features and to avoid the risk of the model mis‐specification, a nonparametric degradation model based on functional variance process is proposed in this article. The model is composed of a unit‐specific mean trend and a degradation fluctuation which follows a stochastic process. The mean trend is estimated by the local smoother method, while the stochastic fluctuation is estimated by the functional principal component analysis method. The asymptotic properties of the estimators are proved. Also, the prediction for the remaining useful lifetime is discussed and the estimator is proved to converge in distribution. Moreover, a Bayesian scheme is developed to forecast the remaining useful lifetime for units with incomplete degradation observations. Simulation results show the superiority of the proposed method by comparing it with some existing methods. Finally, two real data sets are analyzed and used to illustrate the application of the method. [ABSTRACT FROM AUTHOR]

Spectral density analysis plays an important role in studying a stationary random process on a real line. In this paper, we extend this discussion for the random process with stationary increments. We investigate the properties of the method of moments structure function estimation, and propose a nonparametric spectral density function estimator. Our numerical results show that the proposed spectral density estimator performs comparable with the parametric counterpart when the underlying process is assumed to be band‐limited. Additionally, this method is applied to analyze US Housing Starts Data, where the hidden periodicities are detected, providing consistent conclusions with previous economic studies. [ABSTRACT FROM AUTHOR]

Real‐world traffic flow is influenced by various random factors. Gaussian white noise is employed to describe the inherent randomness of traffic flow. A stochastic optimization velocity model is established for dynamic analysis. To enhance the traffic flow stability, the velocity difference of two successive vehicles ahead is considered. The velocity difference feedback control strategy is proposed, utilizing the velocity differences between the vehicle and the two targeted vehicles as control signals. Stability conditions for both the stochastic optimization velocity model and the feedback control model are derived by using the moment stability theory. The study demonstrates that the random factors can destabilize traffic flow, while the proposed feedback control strategy effectively stabilizes traffic flow. Additionally, the influences of random factors on the stability of traffic flow are discussed extensively, aiding in understanding actual traffic congestion mechanisms and offering insights for traffic control strategies. [ABSTRACT FROM AUTHOR]

The middle and lower reaches of the Yangtze River serve as principal rice production bases in China, yet the biodiversity and ecological processes of bacterial communities in paddy soils are not well understood. This study explores the diversity, composition, ecological function, and assembly processes of abundant and rare bacterial communities in paddy soils. A total of 129 paddy soil samples from 43 sites along the middle and lower reaches of the Yangtze River were collected and analyzed using NovaSeq sequencing. The results showed that the dominant phylum for both abundant and rare taxa was Proteobacteria, with a greater relative abundance of the abundant taxa. The diversity of the abundant community was lower than that of the rare community. Soil properties and geographic variables explained more of the variation in the abundant community than in the rare community. The rare community exhibited a significant distance‐decay relationship. The assembly of the abundant community was more influenced by stochastic processes, although both the abundant and rare communities were governed by stochastic processes. It is concluded that both abundant and rare bacterial communities exhibit differing biogeographic patterns, yet they undergo similar ecological processes in the paddy soils along the middle and lower reaches of the Yangtze River. These observations offer a theoretical framework for a deeper comprehension of the function of both abundant and rare bacteria, as well as the development and preservation of soil bacterial diversity within agricultural ecosystems. [ABSTRACT FROM AUTHOR]

Warming temperatures are altering communities and trophic networks across Earth's ecosystems. While the overall influence of warming on food webs is often context‐dependent, increasing temperatures are predicted to change communities in two fundamental ways: (1) by reducing average body size and (2) by increasing individual metabolic rates. These warming‐induced changes have the potential to influence the distribution of food web fluxes, food web stability, and the relative importance of deterministic and stochastic ecological processes shaping community assembly. Here, we quantified patterns and the relative distribution of organic matter fluxes through stream food webs spanning a broad natural temperature gradient (5–27°C). We then related these patterns to species and community trait distributions of mean body size and population biomass turnover (P:B) within and across streams. We predicted that (1) communities in warmer streams would exhibit smaller body size and higher P:B and (2) organic matter fluxes within warmer communities would increasingly skew toward smaller, higher P:B populations. Across the temperature gradient, warmer communities were characterized by smaller body size (~9% per °C) and higher P:B (~7% faster turnover per °C) populations on average. Additionally, organic matter fluxes within warmer streams were increasingly skewed toward higher P:B populations, demonstrating that warming can restructure organic matter fluxes in both an absolute and relative sense. With warming, the relative distribution of organic matter fluxes was decreasingly likely to arise through the random sorting of species, suggesting stronger selection for traits driving high turnover with increasing temperature. Our study suggests that a warming world will favor energy fluxes through "smaller and faster" populations, and that these changes may be more predictable than previously thought. [ABSTRACT FROM AUTHOR]

The core principle shared by most theories and models of succession is that, following a major disturbance, plant–environment feedback dynamics drive a directional change in the plant community. The most commonly studied feedback loops are those in which the regrowth of the plant community causes changes to the abiotic (e.g. soil nutrients) or biotic (e.g. dispersers) environment, which differentially affect species availability or performance. This, in turn, leads to shifts in the species composition of the plant community. However, there are many other PE feedback loops that potentially drive succession, each of which can be considered a model of succession. While plant–environment feedback loops in principle generate predictable successional trajectories, succession is generally observed to be highly variable. Factors contributing to this variability are the stochastic processes involved in feedback dynamics, such as individual mortality and seed dispersal, and extrinsic causes of succession, which are not affected by changes in the plant community but do affect species performance or availability. Both can lead to variation in the identity of dominant species within communities. This, in turn, leads to further contingencies if these species differ in their effect on their environment (priority effects). Predictability and variability are thus intrinsically linked features of ecological succession. We present a new conceptual framework of ecological succession that integrates the propositions discussed above. This framework defines seven general causes: landscape context, disturbance and land‐use, biotic factors, abiotic factors, species availability, species performance, and the plant community. When involved in a feedback loop, these general causes drive succession and when not, they are extrinsic causes that create variability in successional trajectories and dynamics. The proposed framework provides a guide for linking these general causes into causal pathways that represent specific models of succession. Our framework represents a systematic approach to identifying the main feedback processes and causes of variation at different successional stages. It can be used for systematic comparisons among study sites and along environmental gradients, to conceptualise studies, and to guide the formulation of research questions and design of field studies. Mapping an extensive field study onto our conceptual framework revealed that the pathways representing the study's empirical outcomes and conceptual model had important differences, underlining the need to move beyond the conceptual models that currently dominate in specific fields and to find ways to examine the importance of and interactions among alternative causal pathways of succession. To further this aim, we argue for integrating long‐term studies across environmental and anthropogenic gradients, combined with controlled experiments and dynamic modelling. [ABSTRACT FROM AUTHOR]

The modeling and identification of time series data with a long memory are important in various fields. The streamflow discharge is one such example that can be reasonably described as an aggregated stochastic process of randomized affine processes where the probability measure, we call it reversion measure, for the randomization is not directly observable. Accurate identification of the reversion measure is critical because of its omnipresence in the aggregated stochastic process. However, the modeling accuracy is commonly limited by the available real‐world data. We resolve this issue by proposing the novel upper and lower bounds of a statistic of interest subject to ambiguity of the reversion measure. Here, we use the Tsallis value‐at‐risk (TsVaR) as a convex risk functional to generalize the widely used entropic value‐at‐risk (EVaR) as a sharp statistical indicator. We demonstrate that the EVaR cannot be used for evaluating key statistics, such as mean and variance, of the streamflow discharge due to the blowup of some exponential integrand. We theoretically show that the TsVaR can avoid this issue because it requires only the existence of some polynomial moment, not exponential moment. As a demonstration, we apply the semi‐implicit gradient descent method to calculate the TsVaR and corresponding Radon–Nikodym derivative for time series data of actual streamflow discharges in mountainous river environments. [ABSTRACT FROM AUTHOR]

Summary: This paper replicates the results of the stochastic volatility–Dirichlet process mixture (SV‐DPM) models proposed in Jensen and Maheu (2010) in both a narrow and a wide sense. By using a normal‐Wishart prior and the collapsed Gibbs sampling method, our algorithm can be applied for more general settings, and it is more efficient for sampling the Dirichlet process mixture. For the stochastic volatility component, we adopt the method in Chan (2017) to further increase the overall efficiency of our algorithm. Using the same dataset, we obtain mixed results. Some of the results have significant differences. If we use recent time period dataset, which includes the COVID‐19 pandemic period, the log market portfolio volatility seems to increase in terms of the number of clusters and size of magnitude. [ABSTRACT FROM AUTHOR]

Polyploidy, resulting from whole‐genome duplication (WGD), is ubiquitous in nature and reportedly associated with extreme environments and biological invasions. However, WGD usually comes with great costs, raising questions about the establishment chance of newly formed polyploids. The surprisingly high number of polyploid and mixed‐ploidy species observed in nature may be a consequence of their continuous emergence or may reflect stable polyploid persistence and even coexistence with the ancestral ploidy under certain circumstances. However, empirical studies on contemporary polyploid establishment often neglect the cost–benefit balances of polyploid characteristics, tradeoffs between phenotypic characteristics, intercytotype interactions, recurrent polyploid formation, and stochastic processes. Here, we advocate for considering population‐level success, combining the aforementioned factors that affect polyploid establishment and long‐term coexistence with their ancestors. We approach the paradox of polyploid establishment despite high costs from a modern coexistence theory perspective and give an overview of the diversity of mechanisms and their timing that may potentially enable stable rather than transient persistence. [ABSTRACT FROM AUTHOR]

A relativistic diffusion model with cylindrical symmetry, which propagates an initial state based on quantum chromodynamics in time toward a thermal equilibrium limit, is derived from nonequilibrium–statistical considerations: adapting an existing framework for Markovian stochastic processes representing relativistic phase‐space trajectories, a Fokker–Planck equation is obtained for the time evolution of particle‐number distribution functions with respect to transverse and longitudinal rapidity. The resulting partially‐evolved distribution functions are transformed to transverse‐momentum and pseudorapidity space, and compared with charged‐hadron data from the CERN Large Hadron Collider. [ABSTRACT FROM AUTHOR]

This paper presents a differential equation including both random matrices and a Liu process. Then we demonstrate that the solution to this equation exists and is unique. Under the framework of chance theory, problems of optimal control and two‐person zero‐sum game subject to differential equations are considered. An equation of optimality is provided for solving a problem of optimal control. Then equilibrium equations are proposed to identify the saddle‐point of a two‐person zero‐sum game problem. As an extension, we generalize the obtained results to the problems subject to differential equations including both random matrices and multiple Liu processes. Finally, we utilize the acquired theoretical results to analyze a portfolio selection game problem. [ABSTRACT FROM AUTHOR]

Flip processes, introduced in [Garbe, Hladký, Šileikis, Skerman: From flip processes to dynamical systems on graphons], are a class of random graph processes defined using a rule which is just a function ℛ:ℋk→ℋk$$ \mathcal{R}:{\mathscr{H}}_k\to {\mathscr{H}}_k $$ from all labelled graphs of a fixed order k$$ k $$ into itself. The process starts with an arbitrary given n$$ n $$‐vertex graph G0$$ {G}_0 $$. In each step, the graph Gi$$ {G}_i $$ is obtained by sampling k$$ k $$ random vertices v1,...,vk$$ {v}_1,\dots, {v}_k $$ of Gi−1$$ {G}_{i-1} $$ and replacing the induced graph Gi−1[v1,...,vk]$$ {G}_{i-1}\left[{v}_1,\dots, {v}_k\right] $$ by ℛ(Gi−1[v1,...,vk])$$ \mathcal{R}\left({G}_{i-1}\left[{v}_1,\dots, {v}_k\right]\right) $$. Using the formalism of dynamical systems on graphons associated to each such flip process from ibid. we study several specific flip processes, including the triangle removal flip process and its generalizations, 'extremist flip processes' (in which ℛ(H)$$ \mathcal{R}(H) $$ is either a clique or an independent set, depending on whether e(H)$$ e(H) $$ has less or more than half of all potential edges), and 'ignorant flip processes' in which the output ℛ(H)$$ \mathcal{R}(H) $$ does not depend on H$$ H $$. [ABSTRACT FROM AUTHOR]

The 풟‐process is a single player game in which the player is initially presented the empty graph on n$$ n $$ vertices. In each step, a subset of edges X$$ X $$ is independently sampled according to a distribution 풟. The player then selects one edge e$$ e $$ from X$$ X $$, and adds e$$ e $$ to its current graph. For a fixed monotone increasing graph property 풫, the objective of the player is to force the graph to satisfy 풫 in as few steps as possible. The 풟‐process generalizes both the Achlioptas process and the semi‐random graph process. We prove a sufficient condition for the existence of a sharp threshold for 풫 in the 풟‐process. Using this condition, in the semi‐random process we prove the existence of a sharp threshold when 풫 corresponds to being Hamiltonian or to containing a perfect matching. This resolves two of the open questions proposed by Ben‐Eliezer et al. (RSA, 2020). [ABSTRACT FROM AUTHOR]

We propose a non‐stationary spatial model based on a normal‐inverse‐Wishart framework, conditioning on a set of nearest‐neighbors. The model, called nearest‐neighbor Gaussian process with random covariance matrices is developed for both univariate and multivariate spatial settings and allows for fully flexible covariance structures that impose no stationarity or isotropic restrictions. In addition, the model can handle duplicate observations and missing data. We consider an approach based on integrating out the spatial random effects that allows fast inference for the model parameters. We also consider a full hierarchical approach that leverages the sparse structures induced by the model to perform fast Monte Carlo computations. Strong computational efficiency is achieved by leveraging the adaptive localized structure of the model that allows for a high level of parallelization. We illustrate the performance of the model with univariate and bivariate simulations, as well as with observations from two stationary satellites consisting of albedo measurements. [ABSTRACT FROM AUTHOR]

In degradation modeling, stochastic processes often do not meet the classical properties necessary for traditional goodness‐of‐fit tests. This paper presents an initial investigation into employing the ACH depth function and its potential in degradation model selection. We commence by presenting various stochastic processes as degradation models and their selection criteria. Subsequently, we delve into the ACH depth function, highlighting its potential in this context. Through simulated data, we assess the application of this functional depth measure for model selection. The methodology's validity is further reinforced by its application to real‐world data, underscoring its effectiveness. [ABSTRACT FROM AUTHOR]

This paper proposes a new time scaled Wiener process with random effects that has been specially designed to allow the description of non‐monotonic degradation phenomena with bathtub shaped degradation rate, here intended as derivative of the mean function. Two different parameterizations of the proposed Wiener process are suggested, and the main features of the process are illustrated and discussed. The maximum likelihood estimation of its parameters is addressed. A failure threshold model is adopted to formulate and estimate the probability distribution function of the remaining useful life, which constitutes the core prognostic tool in condition‐based maintenance. As a motivating example, the proposed model is applied to a set of real degradation data of metal‐oxide‐semiconductor field‐effect transistors, where, as a result of preliminary analyses, some of the process parameters are assumed to be random in order to describe the considerable heterogeneity observed between the degradation paths of different units. Obtained results demonstrate the affordability and utility of the proposed model, especially for the estimation of unit‐specific features performed on the basis of degradation data collected in the early phase of the life of a unit belonging to the considered heterogeneous population. [ABSTRACT FROM AUTHOR]

The basic idea in the estimation of distribution algorithms is the replacement of heuristic operators with machine learning models such as regression models, clustering models, or classification models. So, recently, the model‐based evolutionary algorithms (MBEAs) have been suggested in three groups: The estimation of distribution algorithms (EDAs), surrogate assisted evolutionary algorithms, and the inversed models to map from the objective space to the decision space. In this article, a new approach, based on an inversed model of Gaussian process and random forest framework, is proposed. The main idea is applying the process of random forest variable importance with a random grouping that determines some of the best assignment of decision variables to objective functions in order to form a Gaussian process in inverse models that maps to decision space the rich solutions which are discovered from objective space. Then these inverse models through sampling the objective space generate offspring. The proposed algorithm has been tested on the benchmark test suite for evolutionary algorithms (modified Deb K, Thiele L, Laumanns M, Zitzler E (DTLZ), and Walking Fish Group (WFG)) and indicates that the proposed method is a competitive and promising approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of octonion‐valued stochastic shunting inhibitory cellular neural networks with delays. First, we give an estimate of the distance between two different moments of finite‐dimensional distributions of a stochastic process. Then, based on this and by using fixed point theorems and inequality techniques, we establish the existence and global exponential stability of Besicovitch almost periodic (B$$ \mathcal{B} $$‐almost periodic for short) solutions in finite‐dimensional distributions for this kind of networks. Our results are new even if the networks we consider in the paper are real‐valued ones. At the same time, the method proposed in this paper can be applied to study the existence of Besicovitch almost periodic solutions in finite‐dimensional distributions for other types of stochastic neural networks. Finally, an example is given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

The dynamics of an overdamped Brownian particle in a thermal bath that contains a dilute solution of active particles is studied. The particle moves in a harmonic potential and experiences Poisson shot‐noise kicks with specified amplitude distribution due to moving active particles in the bath. From the Fokker–Planck equation for the particle dynamics, the stationary solution for the displacement distribution is derived along with the moments characterizing mean, variance, skewness, and kurtosis, as well as finite‐time first and second moments. An effective temperature is also computed through the fluctuation–dissipation theorem and show that equipartition theorem holds for all zero‐mean kick distributions, including those leading to non‐Gaussian stationary statistics. For the case of Gaussian‐distributed active kicks, a re‐entrant behavior from non‐Gaussian to Gaussian stationary states and a heavy‐tailed leptokurtic distribution across a wide range of parameters are found as seen in recent experimental studies. Further analysis reveals statistical signatures of the irreversible dynamics of the particle displacement in terms of the time asymmetry of cross‐correlation functions. Fruits of the work is the development of an compact inference scheme that may allow experimentalists to extract the rate and moments of underlying shot‐noise solely from the statistics the particle position. [ABSTRACT FROM AUTHOR]

In classical stochastic theory, the joint probability distributions of a stochastic process obey by definition the Kolmogorov consistency conditions. Interpreting such a process as a sequence of physical measurements with probabilistic outcomes, these conditions reflect that the measurements do not alter the state of the underlying physical system. Prominently, this assumption has to be abandoned in the context of quantum mechanics, yet there are also classical processes in which measurements influence the measured system. Here, conditions that characterize uniquely classical processes that are probed by a reasonable class of such invasive measurements are derived. We then analyze under what circumstances such classical processes can simulate the statistics arising from quantum processes associated with informationally‐complete measurements. It is expected that this investigation will help build a bridge between two fundamental traits of non‐classicality, namely, coherence and contextuality. [ABSTRACT FROM AUTHOR]

Revealing the mechanisms underlying soil microbial community assembly is a fundamental objective in molecular ecology. However, despite increasing body of research on overall microbial community assembly mechanisms, our understanding of subcommunity assembly mechanisms for different prokaryotic and fungal taxa remains limited. Here, soils were collected from more than 100 sites across southwestern China. Based on amplicon high‐throughput sequencing and iCAMP analysis, we determined the subcommunity assembly mechanisms for various microbial taxa. The results showed that dispersal limitation and homogenous selection were the primary drivers of soil microbial community assembly in this region. However, the subcommunity assembly mechanisms of different soil microbial taxa were highly variable. For instance, the contribution of homogenous selection to Crenarchaeota subcommunity assembly was 70%, but it was only around 10% for the subcommunity assembly of Actinomycetes, Gemmatimonadetes and Planctomycetes. The assembly of subcommunities including microbial taxa with higher occurrence frequencies, average relative abundance and network degrees, as well as wider niches tended to be more influenced by homogenizing dispersal and drift, but less affected by heterogeneous selection and dispersal limitation. The subcommunity assembly mechanisms also varied substantially among different functional guilds. Notably, the subcommunity assembly of diazotrophs, nitrifiers, saprotrophs and some pathogens were predominantly controlled by homogenous selection, while that of denitrifiers and fungal pathogens were mainly affected by stochastic processes such as drift. These findings provide novel insights into understanding soil microbial diversity maintenance mechanisms, and the analysis pipeline holds significant value for future research. [ABSTRACT FROM AUTHOR]

Zn0.995Cu0.005TiNb2O8@TiO2 mixed ceramics and Zn0.995Cu0.005TiNb2O8-TiO2-Zn0.995Cu0.005TiNb2O8 layered ceramics with different TiO2 contents were designed and manufactured to adjust the microwave dielectric performances of Zn0.995Cu0.005TiNb2O8/TiO2 composite ceramics. The effects of random distribution and laminated co-firing processes on the microwave dielectric properties of the ZnTiNb2O8 systems were fully investigated and compared. During the random distribution process, the compound transforms from the original ZnTiNb2O8 phase to the Zn0.17Nb0.33Ti0.5O2 phase with the increase of TiO2 content. Its τƒ value can be adjusted to near zero, but its Qxƒ value decreases severely. For the laminated co-firing process, the distinctive tri-layer architecture allows the reaction between Zn0.995Cu0.005TiNb2O8 and TiO2 to be limited to a narrow region of about 20 µm width and can play a role similar to that of "glue" to connect the layers well. The process significantly minimizes the probability of Qxƒ values deteriorating. Compared with the random distribution Zn0.995Cu0.005TiNb2O8@TiO2, the Qxƒ value of the tri-layer architecture ceramics is improved by about 85% with no significant decrease in dielectric permittivity while ensuring high-temperature stability. When stacked with 0.04 wt% TiO2, the Zn0.995Cu0.005TiNb2O8-TiO2-Zn0.995Cu0.005TiNb2O8 tri-layer architecture ceramic exhibits superior dielectric properties: εr = 39.96, Qxƒ = 51,987 GHz, and τƒ = -1.31 ppm/°C. [ABSTRACT FROM AUTHOR]

Background: Following the first locally transmitted case in Sukhbaatar soum, Selenge Province, we aimed to investigate the ultimate scale of the epidemic in the scenario of uninterrupted transmission. Methods: This was a prospective case study following the locally modified WHO FFX cases generic protocol. A rapid response team collected data from November 14 to 29, 2020. We created a stochastic process to draw many transmission chains from this greater distribution to better understand and make inferences regarding the outbreak under investigation. Results: The majority of the cases involved household transmissions (35, 52.2%), work transmissions (20, 29.9%), index (5, 7.5%), same apartment transmissions (2, 3.0%), school transmissions (2, 3.0%), and random contacts between individuals transmissions (1, 1.5%). The posterior means of the basic reproduction number of both the asymptomatic cases R0Asy$$ {R}_0^{Asy} $$ and the presymptomatic cases R0Pre$$ {R}_0^{Pre} $$ (1.35 [95% CrI 0.88–1.86] and 1.29 [95% CrI 0.67–2.10], respectively) were lower than that of the symptomatic cases (2.00 [95% Crl 1.38–2.76]). Conclusion: Our study highlights the heterogeneity of COVID‐19 transmission across different symptom statuses and underscores the importance of early identification and isolation of symptomatic cases in disease control. Our approach, which combines detailed contact tracing data with advanced statistical methods, can be applied to other infectious diseases, facilitating a more nuanced understanding of disease transmission dynamics. [ABSTRACT FROM AUTHOR]

Human papillomavirus (HPV) infection poses a significant risk to women's health by causing cervical cancer. In addition to HPV, cervical cancer incidence rates can be influenced by various factors, including human immunodeficiency virus and herpes, as well as screening policy. In this study, a mathematical model with stochastic processes was developed to analyze HPV transmission between genders and its subsequent impact on cervical cancer incidence. The model simulations suggest that both‐gender vaccination is far more effective than female‐only vaccination in preventing an increase in cervical cancer incidence. With increasing stochasticity, the difference between the number of patients in the vaccinated group and the number in the nonvaccinated group diminishes. To distinguish the patient population distribution of the vaccinated from the nonvaccinated, we calculated effect size (Cohen's distance) in addition to Student's t‐test. The model analysis suggests a threshold vaccination rate for both genders for a clear reduction of cancer incidence when significant stochastic factors are present. [ABSTRACT FROM AUTHOR]