The Markov method is a common reliability assessment method. It is often used to describe the dynamic characteristics of a system, such as its repairability, fault sequence and multiple degradation states. However, the \"curse of dimensionality\", which refers to the exponential growth of the system state space with the increase in system complexity, presents a challenge to reliability assessments for complex systems based on the Markov method. In response to this challenge, a novel reliability assessment method for complex systems based on non-homogeneous Markov processes is proposed. This method entails the decomposition of a complex system into multilevel subsystems, each with a relatively small state space, in accordance with the system function. The homogeneous Markov model or the non-homogeneous Markov model is established for each subsystem/system from bottom to top. In order to utilize the outcomes of the lower-level subsystem models as inputs to the upper-level subsystem model, an algorithm is proposed for converting the unavailability curve of a subsystem into its corresponding 2×2 dynamic state transition probability matrix (STPM). The STPM is then employed as an input to the upper-level system\'s non-homogeneous Markov model. A case study is presented using the reliability assessment of the Reactor Protection System (RPS) based on the proposed method, which is then compared with the models based on the other two contrast methods. This comparison verifies the effectiveness and accuracy of the proposed method.

The curse-of-dimensionality taxes computational resources heavily with exponentially increasing computational cost as the dimension increases. This poses great challenges in solving high-dimensional partial differential equations (PDEs), as Richard E. Bellman first pointed out over 60 years ago. While there has been some recent success in solving numerical PDEs in high dimensions, such computations are prohibitively expensive, and true scaling of general nonlinear PDEs to high dimensions has never been achieved. We develop a new method of scaling up physics-informed neural networks (PINNs) to solve arbitrary high-dimensional PDEs. The new method, called Stochastic Dimension Gradient Descent (SDGD), decomposes a gradient of PDEs\' and PINNs\' residual into pieces corresponding to different dimensions and randomly samples a subset of these dimensional pieces in each iteration of training PINNs. We prove theoretically the convergence and other desired properties of the proposed method. We demonstrate in various diverse tests that the proposed method can solve many notoriously hard high-dimensional PDEs, including the Hamilton-Jacobi-Bellman (HJB) and the Schrödinger equations in tens of thousands of dimensions very fast on a single GPU using the PINNs mesh-free approach. Notably, we solve nonlinear PDEs with nontrivial, anisotropic, and inseparable solutions in less than one hour for 1000 dimensions and in 12 h for 100,000 dimensions on a single GPU using SDGD with PINNs. Since SDGD is a general training methodology of PINNs, it can be applied to any current and future variants of PINNs to scale them up for arbitrary high-dimensional PDEs.

The causes of many complex human diseases are still largely unknown. Genetics plays an important role in uncovering the molecular mechanisms of complex human diseases. A key step to characterize the genetics of a complex human disease is to unbiasedly identify disease-associated gene transcripts on a whole-genome scale. Confounding factors could cause false positives. Paired design, such as measuring gene expression before and after treatment for the same subject, can reduce the effect of known confounding factors. However, not all known confounding factors can be controlled in a paired/match design. Model-based clustering, such as mixtures of hierarchical models, has been proposed to detect gene transcripts differentially expressed between paired samples. To the best of our knowledge, no model-based gene clustering methods have the capacity to adjust for the effects of covariates yet. In this article, we proposed a novel mixture of hierarchical models with covariate adjustment in identifying differentially expressed transcripts using high-throughput whole-genome data from paired design. Both simulation study and real data analysis show the good performance of the proposed method.

The marginality principle guides analysts to avoid omitting lower-order terms from models in which higher-order terms are included as covariates. Lower-order terms are viewed as \"marginal\" to higher-order terms. We consider how this principle applies to three cases: regression models that may include the ratio of two measured variables; polynomial transformations of a measured variable; and factorial arrangements of defined interventions. For each case, we show that which terms or transformations are considered to be lower-order, and therefore marginal, depends on the scale of measurement, which is frequently arbitrary. Understanding the implications of this point leads to an intuitive understanding of the curse of dimensionality. We conclude that the marginality principle may be useful to analysts in some specific cases but caution against invoking it as a context-free recipe.

We reformulate and reframe a series of increasingly complex parametric statistical topics into a framework of response-vs.-covariate (Re-Co) dynamics that is described without any explicit functional structures. Then we resolve these topics\' data analysis tasks by discovering major factors underlying such Re-Co dynamics by only making use of data\'s categorical nature. The major factor selection protocol at the heart of Categorical Exploratory Data Analysis (CEDA) paradigm is illustrated and carried out by employing Shannon\'s conditional entropy (CE) and mutual information (I[Re;Co]) as the two key Information Theoretical measurements. Through the process of evaluating these two entropy-based measurements and resolving statistical tasks, we acquire several computational guidelines for carrying out the major factor selection protocol in a do-and-learn fashion. Specifically, practical guidelines are established for evaluating CE and I[Re;Co] in accordance with the criterion called [C1:confirmable]. Following the [C1:confirmable] criterion, we make no attempts on acquiring consistent estimations of these theoretical information measurements. All evaluations are carried out on a contingency table platform, upon which the practical guidelines also provide ways of lessening the effects of the curse of dimensionality. We explicitly carry out six examples of Re-Co dynamics, within each of which, several widely extended scenarios are also explored and discussed.

The goal of this paper is to provide a way for Bayesian statisticians to incorporate subsampling directly into the Bayesian hierarchical model of their choosing without imposing additional restrictive model assumptions. We are motivated by the fact that the rise of \"big data\" has created difficulties for statisticians to directly apply their methods to big datasets. We introduce a \"data subset model\" to the popular \"data model, process model, and parameter model\" framework used to summarize Bayesian hierarchical models. The hyperparameters of the data subset model are specified constructively in that they are chosen such that the implied size of the subset satisfies pre-defined computational constraints. Thus, these hyperparameters effectively calibrate the statistical model to the computer itself to obtain predictions/estimations in a pre-specified amount of time. Several properties of the data subset model are provided including: propriety, partial sufficiency, and semi-parametric properties. Simulated datasets will be used to assess the consequences of subsampling, and results will be presented across different computers to show the effect of the computer on the statistical analysis. Additionally, we provide a joint analysis of a high-dimensional dataset (roughly 10 gigabytes) consisting of 2018 5-year period estimates from the US Census Bureau\'s Public Use Micro-Sample (PUMS).

The foundations of geometric morphometrics were worked out about 30 years ago and have continually been refined and extended. What has remained as a central thrust and source of debate in the morphometrics community is the shared goal of meaningful biological inference through a tight connection between biological theory, measurement, multivariate biostatistics, and geometry. Here we review the building blocks of modern geometric morphometrics: the representation of organismal geometry by landmarks and semilandmarks, the computation of shape or form variables via superimposition, the visualization of statistical results as actual shapes or forms, the decomposition of shape variation into symmetric and asymmetric components and into different spatial scales, the interpretation of various geometries in shape or form space, and models of the association between shape or form and other variables, such as environmental, genetic, or behavioral data. We focus on recent developments and current methodological challenges, especially those arising from the increasing number of landmarks and semilandmarks, and emphasize the importance of thorough exploratory multivariate analyses rather than single scalar summary statistics. We outline promising directions for further research and for the evaluation of new developments, such as \"landmark-free\" approaches. To illustrate these methods, we analyze three-dimensional human face shape based on data from the Avon Longitudinal Study of Parents and Children (ALSPAC).

We propose a method for inference on moderately high-dimensional, nonlinear, non-Gaussian, partially observed Markov process models for which the transition density is not analytically tractable. Markov processes with intractable transition densities arise in models defined implicitly by simulation algorithms. Widely used particle filter methods are applicable to nonlinear, non-Gaussian models but suffer from the curse of dimensionality. Improved scalability is provided by ensemble Kalman filter methods, but these are inappropriate for highly nonlinear and non-Gaussian models. We propose a particle filter method having improved practical and theoretical scalability with respect to the model dimension. This method is applicable to implicitly defined models having analytically intractable transition densities. Our method is developed based on the assumption that the latent process is defined in continuous time and that a simulator of this latent process is available. In this method, particles are propagated at intermediate time intervals between observations and are resampled based on a forecast likelihood of future observations. We combine this particle filter with parameter estimation methodology to enable likelihood-based inference for highly nonlinear spatiotemporal systems. We demonstrate our methodology on a stochastic Lorenz 96 model and a model for the population dynamics of infectious diseases in a network of linked regions.

We present a new heuristic feature-selection (FS) algorithm that integrates in a principled algorithmic framework the three key FS components: relevance, redundancy, and complementarity. Thus, we call it relevance, redundancy, and complementarity trade-off (RRCT). The association strength between each feature and the response and between feature pairs is quantified via an information theoretic transformation of rank correlation coefficients, and the feature complementarity is quantified using partial correlation coefficients. We empirically benchmark the performance of RRCT against 19 FS algorithms across four synthetic and eight real-world datasets in indicative challenging settings evaluating the following: (1) matching the true feature set and (2) out-of-sample performance in binary and multi-class classification problems when presenting selected features into a random forest. RRCT is very competitive in both tasks, and we tentatively make suggestions on the generalizability and application of the best-performing FS algorithms across settings where they may operate effectively.

Despite the numerous band selection (BS) algorithms reported in the field, most if not all have exhibited maximal accuracy when more spectral bands are utilized for classification. This apparently disagrees with the theoretical model of the \'curse of dimensionality\' phenomenon, without apparent explanations. If it were true, then BS would be deemed as an academic piece of research without real benefits to practical applications. This paper presents a spatial spectral mutual information (SSMI) BS scheme that utilizes a spatial feature extraction technique as a preprocessing step, followed by the clustering of the mutual information (MI) of spectral bands for enhancing the efficiency of the BS. Through the SSMI BS scheme, a sharp \'bell\'-shaped accuracy-dimensionality characteristic that peaks at about 20 bands has been observed for the very first time. The performance of the proposed SSMI BS scheme has been validated through 6 hyperspectral imaging (HSI) datasets (Indian Pines, Botswana, Barrax, Pavia University, Salinas, and Kennedy Space Center (KSC)), and its classification accuracy is shown to be approximately 10% better than seven state-of-the-art BS schemes (Saliency, HyperBS, SLN, OCF, FDPC, ISSC, and Convolution Neural Network (CNN)). The present result confirms that the high efficiency of the BS scheme is essentially important to observe and validate the Hughes\' phenomenon in the analysis of HSI data. Experiments also show that the classification accuracy can be affected by as much as approximately 10% when a single \'crucial\' band is included or missed out for classification.