Skew-normal Distribution Research Articles

ClrDV: a differential variability test for RNA-Seq data based on the skew-normal distribution.

Pathological conditions may result in certain genes having expression variance that differs markedly from that of the control. Finding such genes from gene expression data can provide invaluable candidates for therapeutic intervention. Under the dominant paradigm for modeling RNA-Seq gene counts using the negative binomial model, tests of differential variability are challenging to develop, owing to dependence of the variance on the mean. Here, we describe clrDV, a statistical method for detecting genes that show differential variability between two populations. We present the skew-normal distribution for modeling gene-wise null distribution of centered log-ratio transformation of compositional RNA-seq data. Simulation results show that clrDV has false discovery rate and probability of Type II error that are on par with or superior to existing methodologies. In addition, its run time is faster than its closest competitors, and remains relatively constant for increasing sample size per group. Analysis of a large neurodegenerative disease RNA-Seq dataset using clrDV successfully recovers multiple gene candidates that have been reported to be associated with Alzheimer's disease.

Stochastic representations and probabilistic characteristics of multivariate skew-elliptical distributions

The family of multivariate skew-normal distributions has many interesting properties. It is shown here that these hold for a general class of skew-elliptical distributions. For this class, several stochastic representations are established and then their probabilistic properties, such as characteristic function, moments, quadratic forms as well as transformation properties, are investigated.

Higher-order expansions of sample range from skew-normal distribution

Higher-order expansions of sample range from skew-normal distribution

Application of Asymmetric-GARCH Type Models to The Kenyan Exchange Rates

Modelling and forecasting the volatility of a financial time series has become essential in many economic and financial applications like portfolio optimization and risk management. The symmetric-GARCH type models can capture volatility and leptokurtosis. However, the models fail to capture leverage effects, volatility clustering, and the thick tail property of high-frequency financial time series. The main objective of this study was to apply the asymmetric-GARCH type models to Kenyan exchange to overcome the shortcomings of symmetric-GARCH type models. The study compared the asymmetric Conditional Heteroskedasticity class of models: EGARCH, TGARCH, APARCH, GJR-GARCH, and IGARCH. Secondary data on the exchange rate from January 1993 to June 2021 were obtained from the Central Bank of Kenya website. The best fit model is determined based on parsimony of the Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), Log-Likelihood criterion, and minimisation of prediction production errors (Mean error [ME] and Root Mean Absolute error [RMAE]). The optimal variance equation for the exchange rates data was APARCH (1,1) - ARMA (3,0) model with a skewed normal distribution (AIC = -4.6871, BIC = -4.5860). Volatility clustering was present in exchange rate data with evidence of the leverage effect. Estimated Kenya’s exchange rate volatility narrows over time, indicating sustained exchange rate stability.

Some Theoretical and Computational Aspects of the Truncated Multivariate Skew-Normal/Independent Distributions

In this article, we derive a closed-form expression for computing the probabilities of p-dimensional rectangles by means of a multivariate skew-normal distribution. We use a stochastic representation of the multivariate skew-normal/independent distributions to derive expressions that relate their probability density functions to the expected values of positive random variables. We also obtain an analogous expression for probabilities of p-dimensional rectangles for these distributions. Based on this, we propose a procedure based on Monte Carlo integration to evaluate the probabilities of p-dimensional rectangles through multivariate skew-normal/independent distributions. We use these findings to evaluate the probability density functions of a truncated version of this class of distributions, for which we also suggest a scheme to generate random vectors by using a stochastic representation involving a truncated multivariate skew-normal random vector. Finally, we derive distributional properties involving affine transformations and marginalization. We illustrate graphically several of our methodologies and results derived in this article.

Tangency portfolio weights under a skew-normal model in small and large dimensions

In this paper, we investigate the distributional properties of the estimated tangency portfolio (TP) weights assuming that the asset returns follow a matrix variate closed skew-normal distribution. We establish a stochastic representation of the linear combination of the estimated TP weights that fully characterizes its distribution. Using the stochastic representation we derive the mean and variance of the estimated weights of TP which are of key importance in portfolio analysis. Furthermore, we provide the asymptotic distribution of the linear combination of the estimated TP weights under the high-dimensional asymptotic regime, i.e., the dimension of the portfolio p and the sample size n tend to infinity such that p / n → c ∈ ( 0 , 1 ) . A good performance of the theoretical findings is documented in the simulation study. In an empirical study, we apply the theoretical results to real data of the stocks included in the S&P 500 index.

Likelihood Based Inference and Bias Reduction in the Modified Skew-t-Normal Distribution

In this paper, likelihood-based inference and bias correction based on Firth’s approach are developed in the modified skew-t-normal (MStN) distribution. The latter model exhibits a greater flexibility than the modified skew-normal (MSN) distribution since it is able to model heavily skewed data and thick tails. In addition, the tails are controlled by the shape parameter and the degrees of freedom. We provide the density of this new distribution and present some of its more important properties including a general expression for the moments. The Fisher’s information matrix together with the observed matrix associated with the log-likelihood are also given. Furthermore, the non-singularity of the Fisher’s information matrix for the MStN model is demonstrated when the shape parameter is zero. As the MStN model presents an inferential problem in the shape parameter, Firth’s method for bias reduction was applied for the scalar case and for the location and scale case.

Multivariate Asymmetric Distributions of Copula Related Random Variables

It is known that normal distribution plays an important role in analysing symmetric data. However, this symmetric assumption may not hold in many real word and in such cases, asymmetric distribution, including skew normal distribution, are known as the best alternative. Constructing asymmetric distributions is carried out using the conditional/selection approach of several independent variable conditioning on other set of variables and this approach does not work well when the independence between variablesviolated. In this work we construct an asymmetric distribution when variables are dependent using a copula. Specifically, we consider the random vectors X and Y are connected using a copula function CX,Y and we study the selection distribution Z = (X|Y ∈ T ).We present some special cases of our proposed distribution, among them, multivariate skew-normal distribution. Some properties such as moments and moment generating function are investigated. Also, numerical analysis including simulation study as well asa real data set analysis are presented for illustration.

Modeling Dependence in Survival Times using Log-Skew Normal Shared Frailties

In survival studies, event times under a common influence are often grouped together in clusters. The association between and within these clusters can be studied using frailty models where randomness in the data or heterogeneity arising due to unknown covariates is described using a frailty variable. In shared frailty models, frailty value is common or shared for all observations within a cluster while it is conditionally independent for different clusters. In this article, we consider a model whose baseline distribution is Weibull and shared frailties follow log skew-normal distribution. This distribution increases flexibility of the model as it allows the frailty term to be positively or negatively skewed and estimation of skewness parameter enables us to comment on dependence structure of the random component. A simulation study is performed and Bayesian estimates of treatment effects, variance and skewness of frailty term are obtained using Metropolis-Hastings algorithm. It is shown that while bias and expected loss for estimates of all parameters reduce as dataset size increases, frailty parameters are more efficiently estimated when the random component is considered to be skewed. The model is also applied to two real-life datasets where positive and negative skewness is observed in the frailty term.

Investigations of the Temperature Field and Cracking Risk in Early Age Massive Concrete in the # 0 Segment of a Box Girder Bridge

Massive high-strength concrete is often used for the # 0 segment of continuous box girders in modern large-span bridges. Thus, the release of substantial hydration heat results in thermal cracking during construction, which greatly affects bridge safety and durability. To predict the risk of structural cracking, it is important to clarify the distribution of the temperature gradient associated with the hydration heat in the # 0 segment and develop an accurate and simple method for calculations to assess of cracking. In this study, based on an actual bridge project, the temperature variation of the # 0 segment caused by hydration heat were measured, and the temperature distribution and development were investigated. The finite element method (FEM) was utilized to simulate the temperature field of the # 0 segment. Three thermodynamic methods for the assessment of cracking were compared and analyzed by practical calculations, and the influence of different parameters on the cracking risk was investigated. Measurements showed that the temperature field of the # 0 segment had a skewed unimodal normal distribution in time and a U-shaped distribution of low and high temperatures in outer and inner positions, respectively. The temperature distributions associated with the hydration heat were discontinuous in time for the root section and obeyed different but correlated Gaussian functions along the width of different pouring layers of the intermediate crossbeam at the # 0 segment. Additionally, the results of the FEM simulation were very consistent with the measurements, which verified the accuracy of the FEM simulation. The cracking assessment methods in EC2 and GB 50496 were used to exactly evaluate the cracking of massive concrete structures. On the basis of the results of the study, it was recommended to reduce casting temperature, delay removal of formwork and control curing temperature above 15°C to curb cracking.

Analytical formulation for explaining the variations in traffic states: A fundamental diagram modeling perspective with stochastic parameters

Despite the simplicity and practicality of (deterministic) fundamental diagram models in highway traffic flow theory, the wide scattering effect observed in empirical data remains highly controversial, particularly for explaining traffic state variations. Owing to the analytical properties of the fundamental diagram modeling approach, in this study, we proposed an analytical and quantitative method for analyzing traffic state variations. We investigated the scattering effect in the fundamental diagram and proposed two stochastic fundamental diagram (SFD) models with lognormal and skew-normal distributions to explain the variations in traffic states. The first SFD model assumes that the scattering effect results from stochasticity in both the free-flow speed and the speed at critical density. Both random variables were assumed to follow the lognormal distribution. In the second SFD model, an integrated error term that was assumed to follow the skew-normal distribution over different density ranges was appended to the deterministic fundamental diagram. The properties of these two SFD models were analyzed and compared, and the parameters in these SFD models were calibrated using real-world loop detector data. The observed scatters from the empirical data were reproduced well by the simulated fundamental diagram model, indicating the validity of the proposed SFD models for explaining traffic state variations. Using these two analytical SFD models, we can analyze the stochastic capacity of freeways with closed forms. More importantly, the sources of stochasticity in freeway capacity can be traced in terms of randomly distributed parameters in fundamental diagram models.

Low-pressure steam dropwise condensation on durable PFA-coated horizontal tube: Droplet dynamics in active region

The performance of low-pressure steam condensers is vital to the thermal energy utilization efficiency of the power cycle, refrigeration, heat pump, and seawater desalination systems. Dropwise condensation demonstrates better heat transfer performance than traditional filmwise condensation, and its performance is determined by the condensate droplet dynamic behaviors on hydrophobic surfaces. However, the droplet dynamics behavior on the horizontal tube has not been adequately investigated. In this study, a durable PFA-coated copper tube is achieved to sustain stable dropwise condensation at the vapor pressure ranging from 5 to 100 kPa. By optical observation of condensation on the horizontally oriented tube, the droplet dynamic characteristics and the influence of vapor pressure on the droplet dynamics are examined. We show that an ‘active droplet region’ (ADR), which is characterized by frequent droplet shedding and surface renewal promoting more active nucleation, growth, and departure of droplets, exists along the tube circumference. As a result of the varying droplet dynamics, a skew-normal distribution of sliding droplet numbers presents along the tube circumference. The droplets departing within ADR account for 80% of the total departure droplets. Then, the effect of steam pressure on the droplet dynamics behavior in ADR is studied systematically. The average sweeping frequency of droplets under steam pressure of 100 kPa increases by -122% more than that under steam pressure of 5 kPa. The growth rate of droplets and migration speed improve with steam pressure. Thus, the evolution rate of the transient size distribution becomes rapid as the vapor pressure increases. Taking vapor pressure and circumferential angle into consideration, a semi-empirical model for the prediction of condensate droplet departure radius on a horizontal tube is proposed, of which the standard deviation can be reduced to less than 15%. These findings and insights into droplet dynamic behaviors on PFA-coated hydrophobic horizontal tube are of great significance for the novel surface design strategy for the enhancement of dropwise condensation heat transfer.

Study on Bayesian Skew-Normal Linear Mixed Model and Its Application in Fire Insurance.

Fire insurance is a crucial component of property insurance, and its rating depends on the forecast of insurance loss claim data. Fire insurance loss claim data have complicated characteristics such as skewness and heavy tail. The traditional linear mixed model is commonly difficult to accurately describe the distribution of loss. Therefore, it is crucial to establish a scientific and reasonable distribution model of fire insurance loss claim data. In this study, the random effects and random errors in the linear mixed model are firstly assumed to obey the skew-normal distribution. Then, a skew-normal linear mixed model is established using the Bayesian MCMC method based on a set of U.S. property insurance loss claims data. Comparative analysis is conducted with the linear mixed model of logarithmic transformation. Afterward, a Bayesian skew-normal linear mixed model for Chinese fire insurance loss claims data is designed. The posterior distribution of claim data parameters and related parameter estimation are employed with the R language JAGS package to obtain the predicted and simulated loss claim values. Finally, the optimization model in this study is used to determine the insurance rate. The results demonstrate that the model established by the Bayesian MCMC method can overcome data skewness, and the fitting and correlation with the sample data are better than the log-normal linear mixed model. Hence, it can be concluded that the distribution model proposed in this paper is reasonable for describing insurance claims. This study innovates a new approach for calculating the insurance premium rate and expands the application of the Bayesian method in the fire insurance field.

Local and global measures of the shear moduli of jammed disk packings.

Strain-controlled isotropic compression gives rise to jammed packings of repulsive, frictionless disks with either positive or negative global shear moduli. We carry out computational studies to understand the contributions of the negative shear moduli to the mechanical response of jammed disk packings. We first decompose the ensemble-averaged, global shear modulus as 〈G〉=(1-F_{-})〈G_{+}〉+F_{-}〈G_{-}〉, where F_{-} is the fraction of jammed packings with negative shear moduli and 〈G_{+}〉 and 〈G_{-}〉 are the average values from packings with positive and negative moduli, respectively. We show that 〈G_{+}〉 and 〈|G_{-}|〉 obey different power-law scaling relations above and below pN^{2}∼1. For pN^{2}>1, both 〈G_{+}〉N and 〈|G_{-}|〉N∼(pN^{2})^{β}, where β∼0.5 for repulsive linear spring interactions. Despite this, 〈G〉N∼(pN^{2})^{β^{'}} with β^{'}≳0.5 due to the contributions from packings with negative shear moduli. We show further that the probability distribution of global shear moduli P(G) collapses at fixed pN^{2} and different values of p and N. We calculate analytically that P(G) is a Γ distribution in the pN^{2}≪1 limit. As pN^{2} increases, the skewness of P(G) decreases and P(G) becomes a skew-normal distribution with negative skewness in the pN^{2}≫1 limit. We also partition jammed disk packings into subsystems using Delaunay triangulation of the disk centers to calculate local shear moduli. We show that the local shear moduli defined from groups of adjacent triangles can be negative even when G>0. The spatial correlation function of local shear moduli C(r[over ⃗]) displays weak correlations for pn_{sub}^{2}<10^{-2}, where n_{sub} is the number of particles within each subsystem. However, C(r[over ⃗]) begins to develop long-ranged spatial correlations with fourfold angular symmetry for pn_{sub}^{2}≳10^{-2}.

A novel hypothesis testing method of skew-normal distribution based on the fourth-order moment property

A novel method for testing skew-normal distribution is proposed, and this method is called the fourth-order moment test (FOMT). It is a statistic constructed based on the fourth-order moment of the standard skew-normal distribution. We compared FOMT with the Kolmogorov-Smirnov test (KS test), the Anderson-Darling test (AD test), AS test, and W test, both of which generate random numbers of skew-normal distribution by stochastic representing and compares the probability of the Type I error for tests. The true distributions of different alternative hypotheses are simulated by Monte Carlo (MC), and the power functions of the tests are calculated. Monte Carlo simulations show that FOMT has a significant advantage in power function. A real data application is analyzed by the proposed methodology.

Jam density and stopbar location estimation with trajectory data at signalized intersections

Jam density, or its reciprocal jam spacing, is a parameter difficult to estimate. In fact, most traffic signal control and traffic state estimation studies published until date generally assume a given value, with no estimation from data whatsoever. Nevertheless, estimating jam density is crucial, since any deviation from its true value will propagate to queue lengths and volume estimates. In the possession of only trajectory data, its estimation is even harder since not all vehicles are observed. In this paper, we first define the data generating process of the jam spacing parameter using sparse trajectory data. Then, we propose several estimators to estimate jam spacing and its variance. We use as measurements the distances between successive vehicle stops. The first estimator is a Maximum-Likelihood estimator (MLE) of a Geometrically Skewed Normal (GSN) distribution, to be used whenever there is lane information. The second estimator is a MLE of a shifted GSN with Normal Stopbar (SGSN-NSB) to be used when observed distances are measured from the stopbar estimate. In addition, we propose two least-squares counterparts, LSE and LSE-SB, based on least absolute remainders. Finally, we assess and compare the bias and efficiency of the estimators in both synthetic and real-world data, obtaining satisfactory results. MLE estimators are shown to outperform their LSE counterparts in both situations.

A federated learning differential privacy algorithm for non-Gaussian heterogeneous data

Multi-center heterogeneous data are a hot topic in federated learning. The data of clients and centers do not follow a normal distribution, posing significant challenges to learning. Based on the assumption that the client data have a multivariate skewed normal distribution, we improve the DP-Fed-mv-PPCA model. We use a Bayesian framework to construct prior distributions of local parameters and use expectation maximization and pseudo-Newton algorithms to obtain robust parameter estimates. Then, the clipping algorithm and differential privacy algorithm are used to solve the problem in which the model parameters do not have a display solution and achieve privacy guarantee. Furthermore, we verified the effectiveness of our model using synthetic and actual data from the Internet of vehicles.

Abstract 861: Modelling the role of EGFR mutation in the initiation of never-smoker lung cancer

Abstract Genetic mutations associated with never-smoker lung cancer frequently appear in normal lung tissue, in both cancer and non-cancer patients. It is unclear what circumstances are required for a mutation to fix in healthy tissue and eventually lead to a tumor. To investigate this, alveolar type-II murine cells were grown in a 3D nutrient system, forming spheroidal clusters. The volumes of these clusters were measured at 7 and 14 days and a mathematical model was devised to describe their growth. In the absence of an EGFR mutation, variations in growth between clusters can be explained by a skew-normal distribution of initial division rates. These rates decline exponentially as the clusters age, and the decline rate is the same across clusters. When cells possess an epidermal growth factor receptor (EGFR) mutation or are grown in the presence of interleukin-1 beta (il-1b) growth factor, clusters form more complex shapes, which we believe may be driven by branching processes and cell-cell interactions. Accordingly, more complex models are needed to explain their development. This work lays the groundwork for new, parameterized models of mutational spread in lung tissue, and a deeper understanding of the role of mutation in carcinogenesis. Citation Format: Helena M. Coggan, Clare E. Weeden, Mohit Dalwadi, Charles Swanton, Philip Pearce, Karen M. Page. Modelling the role of EGFR mutation in the initiation of never-smoker lung cancer [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 861.

New Insights on the Multivariate Skew Exponential Power Distribution

ABSTRACT The multivariate exponential power is a useful distribution for modeling departures from normality in data by means of a tail weight scalar parameter that regulates the non-normality of the model. The incorporation of a shape asymmetry vector into the model serves to account for potential asymmetries and gives rise to the multivariate skew exponential power distribution. This work is aimed at revisiting the skew exponential power distribution taking as a starting point its formulation as a scale mixture of skew-normal distributions. The paper provides some highlights and theoretical insights on the role played by its parameters to assess two complementary aspects of the multivariate non-normality such as directional asymmetry and tail weight behavior regardless of the asymmetry. The intuition behind both issues relies on well-known mathematical ideas about skewness maximization and convex transform stochastic orderings.

Finite mixture of semiparametric multivariate skew-normal distributions

Although the normal distribution is the most frequently used distribution for modeling a given data in many applications due to its desirable theoretical properties and computational convenience, it is often not suitable to model the data having asymmetric or heavy tailed distributions. The skew-normal distribution is an important alternative to the normal distribution as it can cover not only the normal distribution but also some asymmetric distributions. However, it cannot well approximate heavy tailed distributions. In this paper, we propose a semiparametric skew-normal distribution which contains skew-normal distributions using the nonparametric scale mixture of skew-normal distributions and apply the proposed model to finite mixture models so that we can obtain more efficient and insightful knowledge in the model-based cluster analysis. We provide a feasible algorithm to compute all parametric and nonparametric components in the proposed model. Numerical examples to show the applicability of the proposed model are also presented.