Random Variables Research Articles

Accurate estimation of the normalized mutual information of multidimensional data.

While the linear Pearson correlation coefficient represents a well-established normalized measure to quantify the inter-relation of two stochastic variables X and Y, it fails for multidimensional variables, such as Cartesian coordinates. Avoiding any assumption about the underlying data, the mutual information I(X, Y) does account for multidimensional correlations. However, unlike the normalized Pearson correlation, it has no upper bound (I ∈ [0, ∞)), i.e., it is not clear if say, I = 0.4 corresponds to a low or a high correlation. Moreover, the mutual information (MI) involves the estimation of high-dimensional probability densities (e.g., six-dimensional for Cartesian coordinates), which requires a k nearest-neighbor algorithm, such as the estimator by Kraskov et al. [Phys. Rev. E 69, 066138 (2004)]. As existing methods to normalize the MI cannot be used in connection with this estimator, a new approach is presented, which uses an entropy estimation method that is invariant under variable transformations. The algorithm is numerically efficient and does not require more effort than the calculation of the (un-normalized) MI. After validating the method by applying it to various toy models, the normalized MI between the Cα-coordinates of T4 lysozyme is considered and compared to a correlation analysis of inter-residue contacts.

Using joint probability density to create most informative unidimensional indices: a new method using pain and psychiatric severity as examples

BackgroundDimension reduction methods do not always reduce their underlying indicators to a single composite score. Furthermore, such methods are usually based on optimality criteria that require discarding some information. We suggest, under some conditions, to use the joint probability density function (joint pdf or JPD) of p-dimensional random variable (the p indicators), as an index or a composite score. It is proved that this index is more informative than any alternative composite score. In two examples, we compare the JPD index with some alternatives constructed from traditional methods.MethodsWe develop a probabilistic unsupervised dimension reduction method based on the probability density of multivariate data. We show that the conditional distribution of the variables given JPD is uniform, implying that the JPD is the most informative scalar summary under the most common notions of information. B. We show under some widely plausible conditions, JPD can be used as an index. To use JPD as an index, in addition to having a plausible interpretation, all the random variables should have approximately the same direction(unidirectionality) as the density values (codirectionality). We applied these ideas to two data sets: first, on the 7 Brief Pain Inventory Interference scale (BPI-I) items obtained from 8,889 US Veterans with chronic pain and, second, on a novel measure based on administrative data for 912 US Veterans. To estimate the JPD in both examples, among the available JPD estimation methods, we used its conditional specifications, identified a well-fitted parametric model for each factored conditional (regression) specification, and, by maximizing the corresponding likelihoods, estimated their parameters. Due to the non-uniqueness of conditional specification, the average of all estimated conditional specifications was used as the final estimate. Since a prevalent common use of indices is ranking, we used measures of monotone dependence [e.g., Spearman’s rank correlation (rho)] to assess the strength of unidirectionality and co-directionality. Finally, we cross-validate the JPD score against variance–covariance-based scores (factor scores in unidimensional models), and the “person’s parameter” estimates of (Generalized) Partial Credit and Graded Response IRT models. We used Pearson Divergence as a measure of information and Shannon’s entropy to compare uncertainties (informativeness) in these alternative scores.ResultsAn unsupervised dimension reduction was developed based on the joint probability density (JPD) of the multi-dimensional data. The JPD, under regularity conditions, may be used as an index. For the well-established Brief Pain Interference Inventory (BPI-I (the short form with 7 Items) and for a new mental health severity index (MoPSI) with 6 indicators, we estimated the JPD scoring. We compared, assuming unidimensionality, factor scores, Person’s scores of the Partial Credit model, the Generalized Partial Credit model, and the Graded Response model with JPD scoring. As expected, all scores’ rankings in both examples were monotonically dependent with various strengths. Shannon entropy was the smallest for JPD scores. Pearson Divergence of the estimated densities of different indices against uniform distribution was maximum for JPD scoring.ConclusionsAn unsupervised probabilistic dimension reduction is possible. When appropriate, the joint probability density function can be used as the most informative index. Model specification and estimation and steps to implement the scoring were demonstrated. As expected, when the required assumption in factor analysis and IRT models are satisfied, JPD scoring agrees with these established scores. However, when these assumptions are violated, JPD scores preserve all the information in the indicators with minimal assumption.

Stress–Strength Reliability Estimation for Gamma Distributions in a Noisy System

When assessing the reliability of a system or the viability of a material, it is necessary to consider the stress conditions of the operating system. Let X be the strength of the system and Y be the stress of the system caused by the environment. In a stress–strength model, the system’s reliability is defined as [Formula: see text], and due to ignoring environmental shocks or human failure, it is usually overestimated. To underestimate reliability, we assume that when X goes to the environmental conditions, the system’s strength reduces to [Formula: see text], where Z is a random variable on [Formula: see text]. We propose a modified reliability model with [Formula: see text] for the gamma distribution. In this model, stress Y and the scale mixture of the strength of the gamma distribution, [Formula: see text], are modeled as independent random variables. The proposed model is compared with the traditional model through a simulation study. Additionally, the asymptotic distribution is used to construct an asymptotic confidence interval for [Formula: see text]. Finally, we use two real datasets to demonstrate the practical applicability of our proposed method.

Hierarchical Bayesian modeling for calibration and uncertainty quantification of constitutive material models: Application to a uniaxial steel material model

Constitutive material models play an integral role in finite element (FE) modeling of structural components and systems and are governed by a set of parameters. The values of the material model parameters are important for the FE model to be able to predict the real structural behavior as closely as possible. Realistic cyclic material constitutive models well calibrated to experimental data are critical when predicting the nonlinear dynamic response of structural systems to extreme loads (e.g., earthquakes, high winds, blast). Hence, this study focuses on the inference and uncertainty quantification of material model parameters from an experimental dataset consisting of data from experiments on multiple specimens of the same kind, which is a topic of great interest to the structural modeling and simulation community. Traditionally, Bayesian inference is performed by pooling the data from all the specimens, or by considering the data from a single specimen, disregarding the inherent variability (aleatory uncertainty) among specimens of the same kind. Consequently, traditional approaches fail to capture the population distribution of material model parameters, which represents the aleatory specimen-to-specimen variability that is essential for conducting reliability analysis and uncertainty propagation studies. Moreover, these approaches cannot accurately simulate the response of new specimens. To overcome these limitations, this study introduces a novel application of hierarchical Bayesian inference for calibrating material model parameters. By simultaneously considering experimental data from multiple specimens of the same kind, hierarchical Bayesian modeling quantifies both epistemic uncertainty and aleatory specimen-to-specimen variability. In this study, the parameters of the well-known Giuffré-Menegotto-Pinto (GMP) uniaxial material model for reinforcing steel are calibrated and validated based on a dataset of experimental cyclic tests conducted on thirty-six steel specimens, or coupons, originating from three different manufacturing mills, complying to two different manufacturing standards, and subjected to different strain histories. These strain histories are similar to those experienced during seismic events by reinforcing steel bars within reinforced concrete structures. The material model parameters are calibrated using both the traditional and the hierarchical Bayesian approach and the results are compared. Based on the experimental dataset employed in this study, a joint probability distribution of the GMP material model parameters that quantifies both epistemic (due to finite sample size) and aleatory (coupon-to-coupon) variability or uncertainty is provided which can be used for uncertainty propagation in reliability and risk analyses of nonlinear dynamic systems.

Importance sampling for counting statistics in one-dimensional systems.

In this paper, we consider the problem of numerical investigation of the counting statistics for a class of one-dimensional systems. Importance sampling, the cornerstone technique usually implemented for such problems, critically hinges on selecting an appropriate biased distribution. While an exponential tilt in the observable stands as the conventional choice for various problems, its efficiency in the context of counting statistics may be significantly hindered by the genuine discreteness of the observable. To address this challenge, we propose an alternative strategy, which we call importance sampling with the local tilt. We demonstrate the efficiency of the proposed approach through the analysis of three prototypical examples: a set of independent Gaussian random variables, Dyson gas, and symmetric simple exclusion process with a steplike initial condition.

Modeling uncertainty: the impact of noise in T cell differentiation

BackgroundThe regulatory mechanisms guiding CD4 T cell differentiation are complex and are further influenced by intrinsic cell variability along with that of microenvironmental cues, such as cytokine and nutrient availability.ObjectiveThis study aims to expand our understanding of CD4 T cell differentiation by examining the influence of intrinsic noise on cell fate.MethodologyA model based on a complex regulatory network of early signaling events involved in CD4 T cell activation and differentiation was described in terms of a set of stochastic differential equation to assess the effect of noise intensity on differentiation efficiency to the Th1, Th2, Th17, Treg, and TFH effector phenotypes under defined cytokine and nutrient conditions.ResultsThe increase of noise intensity decreases differentiation efficiencies. In a microenvironment of Th1-inducing cytokines and optimal nutrient conditions, noise levels of 3%, 5% and 10% render Th1 differentiation efficiencies of 0.87, 0.76 and 0.62, respectively, underscoring the sensitivity of the network to random variations. Further increments of noise reveal that the network is relatively stable until noise levels of 20%, where the resulting cell phenotypes becomes heterogeneous. Notably, Treg differentiation showed the highest robustness to noise perturbations. A combined Th1-Th2 cytokine environment with optimal nutrient levels induces a dominant Th1 phenotype; however, removal of glutamine shifts the balance towards the Th2 phenotype at all noise levels, with an efficiency similar to that obtained under Th2-only cytokine conditions. Similarly, combinations of Th1/Treg and Treg/Th17-inducing cytokines along with the removal of either tryptophan or oxygen shift the dominant Th1 and Treg phenotypes towards Treg and Th17 respectively. Model results are consistent with differentiation efficiency patterns obtained under well-controlled experimental settings reported in the literature.ConclusionThe stochastic CD4 T cell mathematical model presented here demonstrates a noise-dependent modulation of T cell differentiation induced by cytokines and nutrient availability. Modeling results can be explained by the network topology, which assures that the system will arrive at stable states of cell functionality despite variable levels of biological intrinsic noise. Moreover, the model provides insights into the robustness of the T cell differentiation process.

Blocked Gibbs sampler for hierarchical Dirichlet processes

– Posterior computation in hierarchical Dirichlet process (HDP) mixture models is an active area of research in nonparametric Bayes inference of grouped data. Existing literature almost exclusively focuses on the Chinese restaurant franchise (CRF) analogy of the marginal distribution of the parameters, which can mix poorly and has a quadratic complexity with the sample size. A recently developed slice sampler allows for efficient blocked updates of the parameters, but is shown to be statistically unstable in our article. We develop a blocked Gibbs sampler that employs a truncated approximation of the underlying random measures to sample from the posterior distribution of HDP, which produces statistically stable results, is highly scalable with respect to sample size, and is shown to have good mixing. The heart of the construction is to endow the shared concentration parameter with an appropriately chosen gamma prior that allows us to break the dependence of the shared mixing proportions and permits independent updates of certain log-concave random variables in a block. En route, we develop an efficient rejection sampler for these random variables leveraging piece-wise tangent-line approximations. Supplementary materials, which include substantive additional details and code, are available online.

A Reliability-Based Design Approach for the Flexural Resistance of Compression Yielded Fibre-Reinforced Polymer (FRP)-Reinforced Concrete Beams

Fibre-reinforced polymer (FRP) reinforcement has been employed as an alternative to conventional steel reinforcement in concrete structures, which is attributed to its excellent strength and corrosion resistance. However, one drawback is that FRP reinforcements are brittle and affect the ductility of concrete structures. One of the recent effective techniques proposed to overcome ductility issues is the compression yielding (CY) concept. The CY mechanism allows the structure to fail differently than the conventional FRP-reinforced concrete structure. Thus, the existing design recommendations as per the current codes for FRP-reinforced concrete structures are not appropriate. Hence, reliability studies are crucial for the development of a functional CY beam design in order to emphasise the structure’s lifetime performance and to guarantee safety requirements. In this study, a reliability-based design approach is developed for a compression-yielded FRP-reinforced concrete beam (CY beam) using load and resistance factor design (LRFD). Firstly, the flexural failure modes of CY beams are discussed. The uncertainties involved in the development of the probabilistic model for the CY beam are defined. A case study is consequently conducted for a CY beam with random variables that are associated with the statistical characteristics of material properties and load. The reliability analysis method employed in this research is the Hasofer–Lind method. The results suggest the importance of choosing appropriate design variables and stochastic parameters for CY blocks that contribute to a higher level of reliability. The reliability index and resistance factors of a CY beam are then evaluated using the Monte Carlo Simulation computational method. The reliability index value of 3.336 is obtained from the simulation, which indicates that the CY beam demonstrates ductile behaviour. The results not only demonstrate ductile behaviour but also contribute to a possible reduction in material costs and a substantial safety margin. When compared to conventional FRP-reinforced concrete beams, for different load ratios, CY beams showed higher resistance and better reliability levels.

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Physiological Auricular and Cornual Asymmetries of the Sanmartinero Creole Bovine

Asymmetric studies can indicate disturbances in the developmental process. Fluctuating asymmetry (FA) is considered an indicator of stress. The Sanmartinero (SM) creole bovine is native to the department of Meta (Colombian Orinoquia) and its adaptation process has allowed it to live in extreme tropical environments. The aim of this cross-sectional and descriptive study was to present the current state of the knowledge of asymmetries in some cephalic characters of the SM creole bovine. A total of 94 animals were studied (18 uncastrated males and 76 females) from three different farms, with an age range of 0.5–10 years. For each animal, two measurements of the ear (width and length) and two measurements of the horn (perimeter and length) were obtained in vivo. The degree of asymmetry was calculated as (R − L)/(R + L). Bilateral differences pointed towards a fluctuating asymmetry (e.g., a random variation in the trait that is expected to be perfectly symmetrical) biased towards right for ear width and horn perimeter, and towards left for ear and horn length. Since the development of these structures—ears and horns—is under the control of the same set of genes, the fluctuating asymmetry could constitute a reflection of a normal condition.

Using rejection sampling probability of acceptance as a measure of independence

This paper proposes a new association statistic for determining whether random variables are statistically independent. The proposed association statistic can also be used to examine the strength of both linear and non-linear dependency between variables. This statistic is derived by examining how the conditional probabilities of events differ from their corresponding marginal probabilities. The new statistic is defined in terms of the probability of acceptance commonly associated with the Accept-Reject algorithm and has a very simple formula. The simulated density of the probability of acceptance can be used to measure the degree of uncertainty of the estimated values of the new association statistic. The results from simulations as well as examples that employ real data indicate that this new association statistic is very powerful in detecting linear and non-linear associations between two random variables. Supplementary materials for this article are available online.

Limit distribution of partial transposition of block random matrices

It is well known that, under some assumptions, the limit distribution of block random matrices and their partial transposition converges to the distributions of random variables in some non-commutative probability spaces. Using free probability theory, we obtain the relation between the free cumulants of the corresponding random variables. As an application, we can derive a new family of co-completely positive and [Formula: see text]-positive maps by using the Wishart ensemble.

Runtime Analysis of Single- and Multi-Objective Evolutionary Algorithms for Chance Constrained Optimization Problems with Normally Distributed Random Variables.

Chance constrained optimization problems allow to model problems where constraints involving stochastic components should only be violated with a small probability. Evolutionary algorithms have been applied to this scenario and shown to achieve high quality results. With this paper, we contribute to the theoretical understanding of evolutionary algorithms for chance constrained optimization. We study the scenario of stochastic components that are independent and normally distributed. Considering the simple single-objective (1+1) EA, we show that imposing an additional uniform constraint already leads to local optima for very restricted scenarios and an exponential optimization time. We therefore introduce a multi-objective formulation of the problem which trades off the expected cost and its variance. We show that multi-objective evolutionary algorithms are highly effective when using this formulation and obtain a set of solutions that contains an optimal solution for any possible confidence level imposed on the constraint. Furthermore, we prove that this approach can also be used to compute a set of optimal solutions for the chance constrained minimum spanning tree problem. In order to deal with potentially exponentially many trade-offs in the multi-objective formulation, we propose and analyze improved convex multi-objective approaches. Experimental investigations on instances of the NP-hard stochastic minimum weight dominating set problem confirm the benefit of the multi-objective and the improved convex multi-objective approach in practice.

Stabilisation of the swing pattern of an anisotropic simple pendulum

The suppression of the effects of anisotropy on a pendulum by use of a rotating mount was initially envisaged by Léon Foucault, based on his observations of the vibrations of a rod clamped in a lathe. However, the method seems to never have been tried due to the practical difficulties involved. We report a computational study of the stabilisation of the swing pattern of a simple pendulum, showing anisotropic behaviour in a static configuration, by rotation of the system mount. When the mount is static, for most initial conditions the swing patterns quickly evolves into unstable, complex Lissajous-like patterns. When the pendulum mount is rotated faster than the pendulum frequency effects of anisotropy are suppressed, and the swing pattern stabilises to that of an isotropic 3D simple pendulum. Suppression of mount anisotropy influence occurs for relatively low rotation rates. We also study swing evolution in the presence of random variations in the orientation of the mount principal axes. The use of computational techniques confirms Foucault’s original observations and hypothesis and provides an interesting avenue for students to engage meaningfully with this historically important and inspiring experiment in a novel and challenging manner.

Reverse Electrodialysis with Continuous Random Variation in Nanochannel Shape: Salinity Gradient-Driven Power Generation.

The shape of nanochannels plays a crucial role in the ion selectivity and overall performance of reverse electrodialysis (RED) systems. However, current research on two-dimensional nanochannel shapes is largely limited to a few fixed asymmetric forms. This study explores the impact of randomly shaped nanochannels using dimensionless methods, controlling their randomness by varying their length and shape amplitude. The research systematically compares how alterations in the nanochannel length and shape amplitude influence various system performance parameters. Our findings indicate that increasing the nanochannel length can significantly enhance the system performance. While drastic changes in the nanochannel shape amplitude positively affect the system performance, the most significant improvements arise from the interplay between the nanochannel length and shape amplitude. This compounding effect creates a local optimum, resulting in peak system performance. Within the range of dimensionless lengths from 0 to 30, the system reaches its optimal performance at a dimensionless length of approximately 25. Additionally, we explored two other influencing factors: the nanochannel surface charge density and the concentration gradient of the solution across the nanochannel. Optimal performance is observed when the nanochannel has a high surface charge density and a low concentration gradient, particularly with random shapes. This study advances the theoretical understanding of RED systems in two-dimensional nanochannels, guiding research towards practical operational conditions.

A general law of the iterated logarithm for non-additive probabilities

Motivated by some interesting problems in mathematical economics, quantum mechanics and finance, non-additive probabilities have been used to describe the phenomena which are generally non-additive. In this paper, we further study the law of the iterated logarithm (LIL) for non-additive probabilities, based on existing results. Under the framework of sublinear expectation initiated by Peng, we give two convergence results of Vn≔∑i=1nXinϕ(n) under some reasonable assumptions, where {Xi}i=1∞ is a sequence of random variables and ϕ is a positive nondecreasing function. From these, a general LIL for non-additive probabilities is proved for negatively dependent and non-identically distributed random variables. It turns out that our result is a natural extension of the Kolmogorov LIL and the Hartman–Wintner LIL. Theorem 1 and Theorem 2 in this paper can be seen an extension of Theorem 1 in Chen and Hu (2014).

Reliability based optimisation of composite plates under aeroelastic constraints via adapted surrogate modeling and genetic algorithms

Composite materials are vital in aerospace for their exceptional strength-to-weight ratio. This study delves into reliability-based optimisation of composite plates within aeroelastic constraints, employing an efficient global optimisation method and genetic algorithms. Initial analysis focuses on aeroelastic responses such as limit flutter speed, gust response, and static loads, emphasizing maximum strain assessment. To tackle optimisation challenges of composite stacking sequences, a homogenization technique with lamination parameters is applied. We then formulate a constrained optimisation problem to minimize gust response while meeting flutter and maximum strain constraints. Surrogate models based on conditioned Gaussian Processes are developed for each aeroelastic response, facilitating optimisation within the composite design space. These models, with potential for local refinement, expedite optimal solution identification. Further, we integrate reliability-based optimisation into the framework to determine a robust stacking sequence using genetic algorithms, accounting for random fiber orientation variations. This holistic approach integrates aeroelastic analysis, constrained optimisation, surrogate modeling, and reliability-based optimisation, proving effective in designing reliable, efficient composite structures for aerospace, thus enhancing performance and safety.

Representation of random variables as Lebesgue integrals

Representation of random variables as Lebesgue integrals

Event-Triggered Reduced-Order Filtering for Continuous Semi-Markov Jump Systems With Imperfect Measurements.

This article conducts the issue of event-triggered reduced-order filtering for continuous-time semi-Markov jump systems with imperfect measurements as well as randomly occurring uncertainties (ROUs). Specifically, the sojourn-time-dependent transition probability matrix (TPM) is presumed to be polytopic and a quantizer is introduced to quantize output signals aiming to reflect the reality. Both ROUs and sensor failures are generated by individual random variables belonging to be mutually independent Bernoulli-distributed white sequences. First, sufficient conditions for the existence of the event-triggered reduced-order filter are obtained by utilizing the dissipativity-based technique to ensure the asymptotical stability with a strictly dissipative performance of the filtering error system. The time-varying TPM is then fractionalized, which enhances the results as stated. Furthermore, the required reduced-order filter parameters are obtained by introducing slack symmetric matrix as well as cone complementarity linearization algorithm. The effectiveness of the suggested event-triggered reduced-order filter design method is shown through simulation results.

A New Concept for Reference Change Values-Regression to the Population Mean.

Reference change values (RCV) are used to indicate a change in analyte concentration that is unlikely to be due to random variation in the patient or the measurement. Current theory describes RCV relative to a first measurement result (X1). We investigate an alternative view predicting the starting point for RCV calculations from X1 and its location in the reference interval. Data for serum sodium, calcium, and total protein from the European Biological Variation study and from routine clinical collections were analyzed for the effect of the position of X1 within the reference interval on the following result from the same patient. A model to describe the effect was determined, and an equation to predict the RCV for a sample in a population was developed. For all data sets, the midpoints of the RCVs were dependent on the position of X1 in the population. Values for X1 below the population mean were more likely to be followed by a higher result, and X1 results above the mean were more likely to be followed by lower results. A model using population mean, reference interval dispersion, and result diagnostic variation provided a good fit with the data sets, and the derived equation predicted the changes seen. We have demonstrated that the position of X1 within the reference interval creates an asymmetrical RCV. This can be described as a regression to the population mean. Adding this concept to the theory of RCVs will be an important consideration in many cases.