Skew-normal Distribution Research Articles

Histological image color normalization using a skewed normal distribution mixed model.

Color variation between histological images may influence the performance of computer-aided histological image analysis. Therefore, among the most essential and challenging tasks in histological image analysis are the reduction of the color variation between images and the preservation of the histological information contained in the images. In recent years, many methods have been introduced with respect to the color normalization of histological images. In this study, we introduce a new clustering method referred to as the skewed normal distribution mixed model clustering algorithm. Realizing that the color distribution of hue values approximates the combination of several skewed normal distributions, we propose to use the skewed normal distribution mixture model to analyze the hue distribution. The proposed skewed normal distribution mixture model clustering algorithm includes saturation-weighted hue histograms because it takes into account the saturation and hue information of a particular histogram image, which can diminish the influence of achromatic pixels. Finally, we conducted extensive experiments based on three data sets and compared them with commonly used color normalization methods. The experiments show that the proposed algorithm has better performance in stain separation and color normalization compared to other methods.

Testing the skewness of skew-normal distribution by Bayes factors

Likelihood ratio test is based on the ratio of the maximum values of likelihood functions on the whole parameter space and the null space, respectively. However, the likelihood ratio test is sometimes intractable such as the skew-normal distribution because of the irregular likelihood behavior. In this paper, we investigate the testing problem concerning the skew parameter based on Bayes factor, which is just the ratio of the integrals of the likelihoods. Like Wilks phenomenon of the classical likelihood ratio, the asymptotic chi-square distributions of some linear functions of the logarithms of Bayes factors are derived. Then selecting appropriate prior, we can construct tests with asymptotically nominal significance level. At the same time, we give the local power of the tests, which is the same as the power of classical likelihood ratio tests. We also provide an exact test based on the Bayes factor with specific prior, which is competitive with existing tests. Simulation studies show that our tests are effective.

Solar panel scale inversion with higher-order statistical moments of echo laser pulse waveform

The space environment is becoming increasingly complicated; therefore, precise detection and identification of space targets are critical to preserving space security. Compared to optical and radar imaging, obtaining space target information using laser echo waveform is more efficient for detection. Based on the skew-normal distribution decomposition, the connection between sub-echoes and target scale following the decomposition of the space target pulse laser echo waveform is explored, and an inversion approach is suggested to identify the scale information of the solar panel utilizing the intersection of skewness and kurtosis contours in the coordinate system of the variables to be solved, based on the high-order moment characteristics of waveforms such as skewness and kurtosis. Using the proposed method, we realized the scale inversion of a 60-cm solar panel of a cube satellite inclined at 45 deg, with the turntable and the detection system placed 80 m apart. The results show that the skewness and kurtosis of the decomposed echo can represent the target size information, and the approach used here can successfully extract the solar panel scale information of a typical satellite, providing methods and data support for space target detection and classification.

A New Family of Distributions Based on Proportional Hazards

In this article, we introduce a new family of symmetric-asymmetric distributions based on skew distributions and on the family of order statistics with proportional hazards. This new family of distributions is able to fit both unimodal and bimodal asymmetric data. Furthermore, it contains, as special cases, the symmetric distribution and the “skew-symmetric” family, and therefore the skew-normal distribution. Another interesting feature of the family is that the parameter controlling the distributional shape in bimodal cases takes values in the interval (0, 1); this is an advantage for computing maximum likelihood estimates of model parameters, which is performed by numerical methods. The practical utility of the proposed distribution is illustrated in two real data applications.

Using (ECM) Algorithm to Estimate the Missing Values and Make Comparison between (MLE) and (GA) Algorithm for Estimating Parameters of Multivariate Skew Normal Distribution (MSN)

The estimation of statistical parameters for multivariate data leads to waste in the information if the missing values are neglected, which will subsequently lead to inaccurate estimates. Therefore, the incomplete data must be estimated using one of the statistical estimation methods to obtain accurate results and thus obtaining good estimates for the parameters.The aim of this paper is to estimate the missing values for the multivariate skew normal distribution function using the Expectation Conditional Maximization (ECM) algorithm. After estimating the missing values, the parameters are estimated using Maximum Likelihood Estimation (MLE) with the Newton-Raphson algorithm, as well as using the Genetic Algorithm (GA). Using simulation, the Mean Squared Error (MSE) was calculated to find out which method is the best for estimation by comparing the two methods using different sample sizes (400, 600, and 800). The (GA) that is based on the (ECM) algorithm to estimate the missing values proved to be better and more efficient than the (MLE) method in terms of the results.

The physical and mechanical properties for flexible biomass particles using computer vision

The combustion and fluidization behavior of biomass depend on the physical properties (size, morphology, and density) and mechanical performances (elastic modulus, Poisson's ratio, tensile strength and failure strain), but their quantitative models have rarely been focused in previous researchers. Hence, a static image measurement for particle physical properties is studied. Combining the uniaxial tension and digital image correlation technology, the dynamic image measurement method for the mechanical properties is proposed. The results indicate that the average roundness, rectangularity, and sphericity of present biomass particles are 0.2, 0.4, and 0.16, respectively. The equivalent diameter and density obey the skewed normal distribution. The tensile strength and failure stress are sensitive to stretching rate, fiber size and orientation. The distribution intervals of elastic modulus and Poisson's ratio are 30–600 MPa and 0.25–0.307, respectively. The stress–strain curves obtained from imaging experiments agree well with the result of finite element method. This study provides the operating parameters for the numerical simulation of particles in the fluidized bed and combustor. Furthermore, the computer vision measurement method can be extended to the investigations of fossil fuels.

Heteroscedastic partially linear model under skew-normal distribution with application in ragweed pollen concentration

We introduce a new class of heteroscedastic partially linear model (PLM) with skew-normal distribution. Maximum likelihood estimation of the model parameters by the ECM algorithm (Expectation/Conditional Maximization) as well as influence diagnostics for the new model are investigated. In addition, a Likelihood Ratio test for assessing the homogeneity of the scale parameter is presented. Simulation studies for assessing the performance of the ECM algorithm and the Likelihood Ratio test statistics for homogeneity of variance are developed. Also, a study for misspecification of the structure function is considered. Finally, an application of the new heteroscedastic PLM to a real data set on ragweed pollen concentration is presented to show that it provides a better fit than the classic homocedastic PLM. We hope that the proposed model may attract applications in different areas of knowledge.

On Moments of Folded and Doubly Truncated Multivariate Extended Skew-Normal Distributions

ABSTRACT This article develops recurrence relations for integrals that relate the density of multivariate extended skew-normal (ESN) distribution, including the well-known skew-normal (SN) distribution introduced by Azzalini and Dalla-Valle and the popular multivariate normal distribution. These recursions offer a fast computation of arbitrary order product moments of the multivariate truncated extended skew-normal and multivariate folded extended skew-normal distributions with the product moments as a byproduct. In addition to the recurrence approach, we realized that any arbitrary moment of the truncated multivariate extended skew-normal distribution can be computed using a corresponding moment of a truncated multivariate normal distribution, pointing the way to a faster algorithm since a less number of integrals is required for its computation which results much simpler to evaluate. Since there are several methods available to calculate the first two moments of a multivariate truncated normal distribution, we propose an optimized method that offers a better performance in terms of time and accuracy, in addition to consider extreme cases in which other methods fail. Finally, we present an application in finance where multivariate tail conditional expectation (MTCE) for SN distributed data is calculated using analytical expressions involving normal left-truncated moments. The R MomTrunc package provides these new efficient methods for practitioners. Supplementary files for this article are available online.

PAR(1) model analysis: a web-based shiny application for analysing periodic autoregressive models

In this paper, a web-based shiny application called the ‘PAR(1) Model Analysis’— that allows the modelling, estimation and prediction of a periodic autoregressive time series with scale mixtures of skew-normal innovations, a general and quite flexible class of error distributions—is presented. The class of scale mixtures of skew-normal distributions is often used for statistical procedures of analysing symmetrical and asymmetrical data. The formulation of the scale of a mixture of skew-normal periodic autoregressive models and the estimating methods, which will be applied in the web application, is briefly explained. The embedded tools in the web application and their applications on data analysis are fully described, and finally, a real data example is completely analysed by using this app to illustrate its handling.

Bayesian censored piecewise regression mixture models with skewness

ABSTRACT This paper presents censored mixture regression models with piecewise growth curves for assessing longitudinal data that exhibit multiphasic features. Such features may include censoring, skewness, measurement errors in covariates, and mixtures of unobserved subpopulations. In the process of describing those features, identification of differential effects of predictors on a response variable for a heterogeneous population (subpopulations) has recently been highly sought. Regression mixture models are key methods for assessing differential effects of predictors. In this article, we extend regression mixture models with normal distribution to incorporate (i) skew-normal distribution, (ii) left-censoring, (iii) measurement errors, and (iv) piecewise growth mixture modeling for describing multiphasic trajectories over time where the observed observations come from a mixture of unobserved subgroups. The proposed methods are illustrated using real data from an AIDS clinical study and a Bayesian approach.

Optimization of the algal species Chlorella miniata growth: Mathematical modelling and evaluation of temperature and light intensity effects

Growth of Chlorella miniata , a green microalga was investigated during this study under various temperature and light intensity values with the purpose of determining growth rate changes of the microalgae with cultivation parameters, experiments were carried out using airlift photobioreactors with a study volume of 6 L. Culturing conditions were between 66 and 385 μmol photon m −2 s −1 and between 14 and 30 °C for light intensity and ambient temperature, respectively. Acquired data were then used to test various mathematical models for coherency with experimental results. Aiba Model for light intensity and Skewed Normal Distribution Model for temperature parameters performed superior compared to the rest of the mathematical models used during the study. Utilizing both mathematical models a novel model was deduced to express effects of both light intensity and temperature parameters in combination on algal growth. Then the developed model was used to calculate the optimum growth condition of the species. The proposed mathematical model showed good coherency with experimental data and an average relative error of 1.97% for both temperature and light intensity effects on algal growth. The theoretical optimum temperature and light intensity for the maximum specific growth rate were calculated to be 22.43 °C and 291.5 μmol photon m −2 s −1 respectively. • The theoretical optimum temperature for growth was calculated to be 22.43 °C. • The theoretical optimum light intensity for growth was calculated to be 291.5 μmol photon m −2 s −1 . • Aiba model was selected for describing light intensity growth relationship. • Skewed Normal Distribution Model was selected for temperature growth relationship.

Study of European style options under Itô-McKean Brownian motion with Azzalini skew-normal distribution

In this article, we deal European style option, with arbitrary payoff which includes both put and call options, on an asset whose price evolves as It?-McKean skew Brownian motion with Azzalini skew-normal distribution. Initially, we investigate a condition which leads the It?-McKean skew Brownian motion to be a martingale. Next, we price the option and show that if the payoff function is convex then so is the price function. After this, we show if the payoff is finite then the price function satisfies a partial differential equation with respect to time. Further, we provide a necessary and sufficient condition for the price function to satisfy Feymann-Kac type equation. Next, we study Black-Scholes type equation and give expressions for the delta hedge. Finally, we study the particular case of an European call option in order to compare some of our results with the existing literature. Our results can be used to investigate the optimal exercise boundary, discrete time hedging strategies etc. of the option.

Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-[formula omitted] distribution

In this paper, we compute doubly truncated moments for the selection elliptical class of distributions, including some multivariate asymmetric versions of well-known elliptical distributions, such as the normal, Student’s t, slash, among others. We address the moments for doubly truncated members of this family, establishing neat formulation for high-order moments and its first two moments. We establish sufficient and necessary conditions for the existence of these truncated moments. Further, we propose optimized methods to handle the extreme setting of the parameters, partitions with almost zero volume or no truncation, which are validated with numerical studies. All results have been particularized to the unified skew-t distribution, a complex multivariate asymmetric heavy-tailed distribution which includes the extended skew-t, extended skew-normal, skew-t, and skew-normal distributions as particular and limiting cases.

Survivability of Suddenly Loaded Arrays of Micropillars.

When a multicomponent system is suddenly loaded, its capability of bearing the load depends not only on the strength of components but also on how a load released by a failed component is distributed among the remaining intact ones. Specifically, we consider an array of pillars which are located on a flat substrate and subjected to an impulsive and compressive load. Immediately after the loading, the pillars whose strengths are below the load magnitude crash. Then, loads released by these crashed pillars are transferred to and assimilated by the intact ones according to a load-sharing rule which reflects the mechanical properties of the pillars and the substrate. A sequence of bursts involving crashes and load transfers either destroys all the pillars or drives the array to a stable configuration when a smaller number of pillars sustain the applied load. By employing a fibre bundle model framework, we numerically study how the array integrity depends on sudden loading amplitudes, randomly distributed pillar strength thresholds and varying ranges of load transfer. Based on the simulation, we estimate the survivability of arrays of pillars defined as the probability of sustaining the applied load despite numerous damaged pillars. It is found that the resulting survival functions are accurately fitted by the family of complementary cumulative skew-normal distributions.

An adaptive polynomial skewed-normal transformation model for distribution reconstruction and reliability evaluation with rare events

Evaluating the failure probabilities with rare events remains a challenging task, especially when dealing with complex limit state functions with high-dimensional random inputs. To handle this problem, an adaptive polynomial skewed-normal transformation (A-PSNT) model is proposed in this paper for evaluating the probability distribution of the limit state function and failure probability. First, the polynomial transformation models on the basis of normal distribution are introduced, where the limitations are also pointed out. Then, the skewed-normal distribution is presented, which serves as the basis of the proposed polynomial transformation model. The PSNT with a given order can be initiated, whereby the unknown coefficients can be determined by using the probability weighted moments (PWMs)-matching technique. Further, a tail error criterion is proposed to select the most appropriate order for the PSNT model, which makes the proposed method an adaptive one. To handle the low- to high-dimensional random inputs, two low-discrepancy sampling methods are employed to generate samples for unbiasedly estimating the PWMs of the limit state function. Once the probability distribution of the limit state function is reconstructed by the proposed A-PSNT model, the failure probability, especially for the rare one, can be determined accordingly. The proposed method is validated through numerical examples involving both static and dynamic, low- to high-dimensional problems, where some classical reliability analysis methods are also adopted for comparisons. The results show that the proposed A-PSNT model not only can effectively reconstruct the probability distribution, but also is able to accurately predict the failure probability with rare events.

Variable selection in finite mixture of location and mean regression models using skew-normal distribution

In this article, a variable selection method for the finite mixture of location regression (FMLR) and the finite mixture of mean regression (FMMR) models with a skew-normal error are discussed. The univariate skew-normal distribution was introduced by Sahu et al. will be used in this work, which is attractive because estimation of the skewness parameter does not present the same degree of difficulty as in the case with Azzalini one and, moreover, it allows easy implementation of the EM algorithm. A penalized likelihood approach for variable selection in FMLR and FMMR models was introduced in this article. With a data-adaptive method for selecting tuning parameters, we establish the theoretical properties of our procedure, including consistency in variable selection, the oracle property in estimation. The EM algorithm facilitated by Gauss–Newton method for efficient numerical computations are developed. Simulation studies and a real data set are used to illustrate the proposed methodologies.

An overview on the progeny of the skew-normal family— A personal perspective

In the last two decades or so, much work has been dedicated to the portion of distribution theory stemming from the skew-normal distribution and its ramification. This contribution presents an outline of the theme, without attempting a detailed review, which would be unfeasible, given the amount of available material. The aim is to present a panoramic view of the theme, leaving out the fine details, with rather more emphasis on the evolution of the underlying ideas and on the breath of the overall developments, as for range of specific directions considered.

A heterogeneous Bayesian regression model for skewed spatial data

After displaying skewness, spatial data can cause nonstationarity. This paper develops a hierarchical skew-Gaussian process capable of simultaneously handling skewness and nonstationarity. At the first level of the hierarchy, we specify a multivariate skew-normal distribution for the realizations of the process over a fixed finite set. At the second level, it is extended to a process across the domain based on the referenced locations. We applied Markov chain Monte Carlo algorithms for Bayesian inference. The proposed methodology is illustrated by some simulation experiments and by applying it to rainfall data modelling.

Experimental investigation on statistical characteristics of cell voltage distribution for a PEMFC stack under dynamic driving cycle

The proton exchange membrane fuel cell (PEMFC) stack consists of individual cells in series. Its operating life is subjected to performance of the weakest cell because of the short-board effect, thus voltage uniformity during dynamic long-running process is significant to its durability. In this work, based on a 1044 h aging experiment on a 6.55 kW PEMFC stack under dynamic driving cycle, voltage uniformity is analyzed. In a single cycle, voltage uniformity becomes worse with the increase of loading current and there are some local maxima of voltage coefficient variation (Cv) at the moment of loading or unloading step. Cv value at higher current is greater and increases faster with cycles. At the end of experiment, Cv at 135 A is more than 6%. Besides, skewness (Sk) is used to evaluate the skew direction and degree of cell voltage data in a cycle. In most cycles, Sk values at 34.22 A are above 0 and Sk values at 59.70 A and 135 A are below 0. After the Box-Cox transformation, which is used to improve symmetry of data and reduce Sk, the cell voltage data have passed the verifications of normal fitting, probability-probability plot and quantile-quantile plot. Therefore, it is found that cell voltage data tend to obey skewed normal distribution, which is of positive significance for improving voltage uniformity and durability of PEMFC stack.

Bivariate Distributions Underlying Responses to Ordinal Variables

The association between two ordinal variables can be expressed with a polychoric correlation coefficient. This coefficient is conventionally based on the assumption that responses to ordinal variables are generated by two underlying continuous latent variables with a bivariate normal distribution. When the underlying bivariate normality assumption is violated, the estimated polychoric correlation coefficient may be biased. In such a case, we may consider other distributions. In this paper, we aimed to provide an illustration of fitting various bivariate distributions to empirical ordinal data and examining how estimates of the polychoric correlation may vary under different distributional assumptions. Results suggested that the bivariate normal and skew-normal distributions rarely hold in the empirical datasets. In contrast, mixtures of bivariate normal distributions were often not rejected.