Modeling functional data with spatially heterogeneous shape characteristics

We propose a novel class of models for functional data exhibiting skewness or other shape characteristics that vary with spatial or temporal location. We use copulas so that the marginal distributions and the dependence structure can be modeled independently. Dependence is modeled with a Gaussian or t-copula, so that there is an underlying latent Gaussian process. We model the marginal distributions using the skew t family. The mean, variance, and shape parameters are modeled nonparametrically as functions of location. A computationally tractable inferential framework for estimating heterogeneous asymmetric or heavy-tailed marginal distributions is introduced. This framework provides a new set of tools for increasingly complex data collected in medical and public health studies. Our methods were motivated by and are illustrated with a state-of-the-art study of neuronal tracts in multiple sclerosis patients and healthy controls. Using the tools we have developed, we were able to find those locations along the tract most affected by the disease. However, our methods are general and highly relevant to many functional data sets. In addition to the application to one-dimensional tract profiles illustrated here, higher-dimensional extensions of the methodology could have direct applications to other biological data including functional and structural magnetic resonance imaging (MRI).     

Aspects of the mineral nutrition of some native British plants — inter-site variation

Fifteen common native British plants were each sampled at a range of sites in Great Britain and green tissues analysed for several inorganic nutrients. Sampling criteria are discussed. The inter-site variation of each element within a species is assessed as a frequency distribution of raw data. Sample values of parameters including arithmetic mean, variance (coefficient of variation), skewness and kurtosis are presented. Their stability is assessed from nitrogen in sub-samples of Pteridium. This suggested sample sizes were adequate but some distributions had sufficient kurtosis to affect the variance. These parameters showed distinctions between macro- and micro-elements and between species. Some mean values sharply distinguished between species and may help to assess current theories of strategy and adaptation but a wider range of species is needed to clarify trends. Coefficients of variation are discussed and were relatively low for a nation-wide survey after allowing for sampling constraints. Coefficients of skewness and kurtosis showed two-thirds of the sample distributions were non-normal. Ecological aspects of the distributions are discussed and are relevant to studies along environmental gradients. Published hypotheses linking positive skewness to stress in the field are considered and two other postulates discussed. Distribution bounds such as those confining 95% of the values are discussed in relation to possible critical levels of nutrients.Nomenclature follows Clapham et al. (1981), Excursion flora of the British Isles. 3rd ed. University Press, Cambridge, except Chamaenerion.     

Scalability Properties of the S-Distribution

Previous work has shown that the S-distribution is a valuable tool for data analysis and for the classification of continuous and discrete distribution functions. The distribution has four parameters: one determines its location, one is related to the variance, and two control its shape. The distributional structure allows symmetry as well as skewness to the right or the left. This offers great flexibility and, among other analyses, facilitates the simultaneous investigation of random variables whose distributions differ in shape. The present paper demonstrates that mean, variance, and quantiles of any S-distribution can be computed with simple algebraic operations from corresponding properties of a standard S-distribution, sSd , which is characterized by only the two shape parameters. This scalability is comparable with the use of z-scores when dealing with normal distributions.     

Circular distributions based on nonnegative trigonometric sums

A new family of distributions for circular random variables is proposed. It is based on nonnegative trigonometric sums and can be used to model data sets which present skewness and/or multimodality. In this family of distributions, the trigonometric moments are easily expressed in terms of the parameters of the distribution. The proposed family is applied to two data sets, one related with the directions taken by ants and the other with the directions taken by turtles, to compare their goodness of fit versus common distributions used in the literature.     

Estimation of transformation parameters for microarray data

MOTIVATION AND RESULTS: Durbin et al. (2002), Huber et al. (2002) and Munson (2001) independently introduced a family of transformations (the generalized-log family) which stabilizes the variance of microarray data up to the first order. We introduce a method for estimating the transformation parameter in tandem with a linear model based on the procedure outlined in Box and Cox (1964). We also discuss means of finding transformations within the generalized-log family which are optimal under other criteria, such as minimum residual skewness and minimum mean-variance dependency. AVAILABILITY: R and Matlab code and test data are available from the authors on request.     

Differentiation and frequency distributions of body weights in plants and animals

The basic and simplest system that one can consider in ecology is a group of individuals of equal age and representing one species, that is, a cohort. This paper is an attempt to show that analysis of such a system may be of great importance to understanding basic ecological problems, such as, intraspecific competition and the dynamics of a single population. It is easy to observe that in even-aged populations individuals differ in weights. A close look can show that weight distributions in even-aged populations may have different skewness. Most common are distributions with coefficients of skewness greater than zero. Sometimes weight distributions are symmetrical or with skewness coefficients less than zero. In a cohort of growing individuals the coefficient of skewness changes with time: most often starting from zero (symmetrical distribution), it increases in time; sometimes after an initial increase it can decrease in the final stage of growth, which is related to an increased mortality of individuals. The rate of change in skewness, and the skewness itself depend on the density of individuals in a cohort and on food conditions. They are greater at higher densities and increase with deteriorating food conditions. Weight distributions are symmetrical at low densities and optimal food conditions. The differences in individual weights measured by variance of weight distributions or coefficient of variation follow the same pattern, but observed changes with time, density and food conditions are not so clear. These conclusions rest upon the review of numerous papers concerning both plants and animals, which is presented in this paper. In the past, the properties of weight distributions in even-aged populations were explained not by interactions between individuals, but rather as a natural outcome of the growth process of non-interacting individuals. The exponential equation of growth, with relative growth rate having a normal distribution in populations, was used to support this hypothesis. Obtained weight distributions were of positive skewness; however, this model, which in fact is able to describe the growth process only in its initial stage, cannot explain the changes of skewness of weight distributions with density and food conditions. A model has been developed which includes competitive interactions among members of even-aged populations to explain observed properties of weight distributions in them. The basic assumption is that intraspecific competition leads to uneven partitioning of resources, which are the object of competition. Functions describing resource partitioning among individuals are included into the model.(ABSTRACT TRUNCATED AT 400 WORDS)     

Why the beta-function cannot be used to estimate skewness of species responses

Abstract. The beta-function (β-function) has been suggested for testing the significance of the skewness of species responses along a gradient. However, the location of the optimum and skewness are correlated so that these parameters cannot be estimated independently. The only way for an independent estimation is to let the endpoints of the response curve vary. In that case they would no longer define the range of species occurrence. However, non-linear estimation of endpoints often leads to overwhelming problems in model fitting. Therefore, the beta-function is not suitable to test the shape of species response curves. Hierarchic models proposed by Huisman et al. (1993) seem to be superior to generalized additive models or third-degree polynomials and seem to be the best alternative to study the skewness of responses.     

Models for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums

Johnson and Wehrly (1978, Journal of the American Statistical Association 73, 602-606) and Wehrly and Johnson (1980, Biometrika 67, 255-256) show one way to construct the joint distribution of a circular and a linear random variable, or the joint distribution of a pair of circular random variables from their marginal distributions and the density of a circular random variable, which in this article is referred to as joining circular density. To construct flexible models, it is necessary that the joining circular density be able to present multimodality and/or skewness in order to model different dependence patterns. Fernández-Durán (2004, Biometrics 60, 499-503) constructed circular distributions based on nonnegative trigonometric sums that can present multimodality and/or skewness. Furthermore, they can be conveniently used as a model for circular-linear or circular-circular joint distributions. In the current work, joint distributions for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums are presented and applied to two data sets, one for circular-linear data related to the air pollution patterns in Mexico City and the other for circular-circular data related to the pair of dihedral angles between consecutive amino acids in a protein.     

Commingling analysis of obesity in twins

Evidence is presented for multiple components in the distribution of human fatness across several large twin samples, after removing age effects and allowing for residual skewness in component distributions. The upper component distributions corresponded to overweight or obesity in samples of middle-aged or older individuals. A bivariate analysis demonstrated that, while monozygotic co-twins appeared to be drawn from the same component distributions (normal or overweight), the twin correlations varied across components, with the lowest correlation in the overweight group. While these analyses cannot provide a definitive test of competing genetic and environmental hypotheses, this approach is useful for generating hypotheses about the causes of obesity. When combined with other published literature, our results suggest that the genetic background largely determines the propensity to become obese. Whether a predisposed person becomes obese and the extent of obesity depend on environmental exposures that are largely independent of early family experience. Both genes and environment appear to be important in obesity, but it appears that some genotypes may be much more sensitive to the environment than are others.     

Repeatability of Quantitative Sodium Magnetic Resonance Imaging for Estimating Pseudo-Intracellular Sodium Concentration and Pseudo-Extracellular Volume Fraction in Brain at 3 T

The purpose of this study is to assess the repeatability of the quantification of pseudo-intracellular sodium concentration (C₁) and pseudo-extracellular volume fraction (α) estimated in brain in vivo using sodium magnetic resonance (MRI) at 3 T. Eleven healthy subjects were scanned twice, with two sodium MRI acquisitions (with and without fluid suppression by inversion recovery), and two double inversion recovery (DIR) proton MRI. DIR MRIs were used to create masks of gray and white matter (GM, WM), that were subsequently applied to the C₁ and α maps calculated from sodium MRI and a tissue three-compartment model, in order to measure the distributions of these two parameters in GM, WM or full brain (GM+WM) separately. The mean, median, mode, standard deviation (std), skewness and kurtosis of the C₁ and α distributions in whole GM, WM and full brain were calculated for each subject, averaged over all data, and used as parameters for the repeatability assessment. The coefficient of variation (CV) was calculated as a measure of reliability for the detection of intra-subject changes in C₁ and αfor each parameter, while intraclass correlation (ICC) was used as a measure of repeatability. It was found that the CV of most of the parameters was around 10–20% (except for C₁ kurtosis which is about 40%) for C₁ and α measurements, and that ICC was moderate to very good (0.4 to 0.9) for C₁ parameters and for some of the α parameters (mainly skewness and kurtosis). In conclusion, the proposed method could allow to reliably detect changes of 50% and above of the different measurement parameters of C₁ and αin neuropathologies (multiple sclerosis, tumor, stroke, Alzheimer’s disease) compared to healthy subjects, and that skewness and kurtosis of the distributions of C₁ and αseem to be the more sensitive parameters to these changes.     

The role of local versus biogeographical processes in influencing diversity and body‐size variation in mammal assemblages

Our objective was to estimate and analyze the body‐size distribution parameters of terrestrial mammal assemblages at different spatial scales, and to determine whether these parameters are controlled by local ecological processes or by larger‐scale ones. Based on 93 local assemblages, plus the complete mammal assemblage from three continents (Africa, North, and South America), we estimated three key distribution parameters (diversity/size slope, skewness, and modal size) and compared the values to those expected if size distributions are mainly controlled by local interactions. Mammal diversity decreased much faster as body size increased than predicted by fractal niche theory, both at continental and at local scales, with continental distributions showing steeper slopes than the localities within them. South America showed a steeper slope (after controlling for species diversity), compared to Africa and North America, at local and continental scales. We also found that skewness and modal body size can show strikingly different correlations with predictor variables, such as species richness and median size, depending on the use of untransformed versus log‐transformed data, due to changes in the distribution density generated by log‐transformation. The main differences in slope, skewness, and modal size between local and continental scales appear to arise from the same biogeographical process, where small‐sized species increase in diversity much faster (due to higher spatial turnover rates) than large‐sized species. This process, which can operate even in the absence of competitive saturation at local scales, generates continental assemblages with steeper slopes, smaller modal sizes, and higher right skewness (toward small‐sized species) compared to local communities. In addition, historical factors can also affect the size distribution slopes, which are significantly steeper, in South American mammal assemblages (probably due to stronger megafauna extinction events in South America) than those in North America and Africa.     

Influence of Magnetic Field in Three-Dimensional Flow of Couple Stress Nanofluid over a Nonlinearly Stretching Surface with Convective Condition

This article investigates the magnetohydrodynamic (MHD) three-dimensional flow of couple stress nanofluid subject to the convective boundary condition. Flow is generated due to a nonlinear stretching of the surface in two lateral directions. Temperature and nanoparticles concentration distributions are studied through the Brownian motion and thermophoresis effects. Couple stress fluid is considered electrically conducting through a non-uniform applied magnetic field. Mathematical formulation is developed via boundary layer approach. Nonlinear ordinary differential systems are constructed by employing suitable transformations. The resulting systems have been solved for the convergent series solutions of velocities, temperature and nanoparticles concentration profiles. Graphs are sketched to see the effects of different interesting flow parameters on the temperature and nanoparticles concentration distributions. Numerical values are computed to analyze the values of skin-friction coefficients and Nusselt number.     

Stand Diameter Distribution Modelling and Prediction Based on Richards Function

The objective of this study was to introduce application of the Richards equation on modelling and prediction of stand diameter distribution. The long-term repeated measurement data sets, consisted of 309 diameter frequency distributions from Chinese fir (Cunninghamia lanceolata) plantations in the southern China, were used. Also, 150 stands were used as fitting data, the other 159 stands were used for testing. Nonlinear regression method (NRM) or maximum likelihood estimates method (MLEM) were applied to estimate the parameters of models, and the parameter prediction method (PPM) and parameter recovery method (PRM) were used to predict the diameter distributions of unknown stands. Four main conclusions were obtained: (1) R distribution presented a more accurate simulation than three-parametric Weibull function; (2) the parameters p, q and r of R distribution proved to be its scale, location and shape parameters, and have a deep relationship with stand characteristics, which means the parameters of R distribution have good theoretical interpretation; (3) the ordinate of inflection point of R distribution has significant relativity with its skewness and kurtosis, and the fitted main distribution range for the cumulative diameter distribution of Chinese fir plantations was 0.4∼0.6; (4) the goodness-of-fit test showed diameter distributions of unknown stands can be well estimated by applying R distribution based on PRM or the combination of PPM and PRM under the condition that only quadratic mean DBH or plus stand age are known, and the non-rejection rates were near 80%, which are higher than the 72.33% non-rejection rate of three-parametric Weibull function based on the combination of PPM and PRM.     

A general approach for two-stage analysis of multilevel clustered non-Gaussian data

In this article, we propose a two-stage approach to modeling multilevel clustered non-Gaussian data with sufficiently large numbers of continuous measures per cluster. Such data are common in biological and medical studies utilizing monitoring or image-processing equipment. We consider a general class of hierarchical models that generalizes the model in the global two-stage (GTS) method for nonlinear mixed effects models by using any square-root-n-consistent and asymptotically normal estimators from stage 1 as pseudodata in the stage 2 model, and by extending the stage 2 model to accommodate random effects from multiple levels of clustering. The second-stage model is a standard linear mixed effects model with normal random effects, but the cluster-specific distributions, conditional on random effects, can be non-Gaussian. This methodology provides a flexible framework for modeling not only a location parameter but also other characteristics of conditional distributions that may be of specific interest. For estimation of the population parameters, we propose a conditional restricted maximum likelihood (CREML) approach and establish the asymptotic properties of the CREML estimators. The proposed general approach is illustrated using quartiles as cluster-specific parameters estimated in the first stage, and applied to the data example from a collagen fibril development study. We demonstrate using simulations that in samples with small numbers of independent clusters, the CREML estimators may perform better than conditional maximum likelihood estimators, which are a direct extension of the estimators from the GTS method.     

Detection of rare major genes in lipid levels

Summary A statistical test of polygenic inheritance (TPI) against the alternative of a rare major gene is presented. It is designed for a random sample of quantitative observations on index cases and siblings of those index cases (probands) selected on the basis of the observed measurements of these probands. The test focuses on an increase of the variance of siblings of probands over its value under polygenic inheritance, such an increase being expected in the presence of a major gene producing a shift of the quantitative observations. Certain data on lipids are then analyzed by this test. A major gene can tentatively be confirmed for triglycerides but not for cholesterol. In addition, the values of all index cases are subjected to an analysis of a mixture of normal distributions (NOCOMP computer program), resulting in a significant second component for triglycerides but not for cholesterol. For both TPI and NOCOMP, the exponent in a power transformation is estimated by maximum likelihood simultaneously with all other parameters, so that these analysis methods are robust against a wide range of skewness in the data, which is demonstrated by manipulation of the observations and their reanalyis.     

Bayesian analysis of multivariate mixed models for a prospective cohort study using skew‐elliptical distributions

Classical multivariate mixed models that acknowledge the correlation of patients through the incorporation of normal error terms are widely used in cohort studies. Violation of the normality assumption can make the statistical inference vague. In this paper, we propose a Bayesian parametric approach by relaxing this assumption and substituting some flexible distributions in fitting multivariate mixed models. This strategy allows for the skewness and the heavy tails of error‐term distributions and thus makes inferences robust to the violation. This approach uses flexible skew‐elliptical distributions, including skewed, fat, or thin‐tailed distributions, and imposes the normal model as a special case. We use real data obtained from a prospective cohort study on the low back pain to illustrate the usefulness of our proposed approach.     

Preference patterns for skewed gambles in rhesus monkeys

While standard models of risky choice account for the first and second statistical moments of reward outcome distributions (mean and variance, respectively), they often ignore the third moment, skewness. Determining a decision-maker''s attitude about skewness is useful because it can help constrain process models of the mental steps involved in risky choice. We measured three rhesus monkeys’ preferences for gambles whose outcome distributions had almost identical means and variances but differed in skewness. We tested five distributions of skewness: strong negative, weak negative, normal, weak positive and strong positive. Monkeys preferred positively skewed gambles to negatively skewed ones and preferred strongly skewed and normal (i.e. unskewed) gambles to weakly skewed ones. This pattern of preferences cannot be explained solely by monotonic deformations of the utility curve or any other popular single account, but can be accounted for by multiple interacting factors.     

Competition and skewness in plantations

Data published recently on the stem diameters in experimental Pinus Radiata plantations, show a skewness which is initially zero, first becomes negative and later reverses direction, becomes positive and increases indefinitely. This and other behaviour are explained using a zone-of-influence model based entirely upon competition between neighbouring trees. Negative skewness can be identified with the early stages of competition when only the largest trees compete. The model also generates bimodal distributions when competition is intense, as observed experimentally in annual plants. Further modes are generated as competition is increased further.     

Computational modeling of flow-induced shear stresses within 3D salt-leached porous scaffolds imaged via micro-CT

Flow-induced shear stresses have been found to be a stimulatory factor in pre-osteoblastic cells seeded in 3D porous scaffolds and cultured under continuous flow perfusion. However, due to the complex internal structure of porous scaffolds, analytical estimation of the local shear forces is impractical. The primary goal of this work is to investigate the shear stress distributions within Poly(l-lactic acid) scaffolds via computation. Scaffolds used in this study are prepared via salt leeching with various geometric characteristics (80–95% porosity and 215–402.5 μm average pore size). High resolution micro-computed tomography is used to obtain their 3D structure. Flow of osteogenic media through the scaffolds is modeled via lattice Boltzmann method. It is found that the surface stress distributions within the scaffolds are characterized by long tails to the right (a positive skewness). Their shape is not strongly dependent on the scaffold manufacturing parameters, but the magnitudes of the stresses are. Correlations are prepared for the estimation of the average surface shear stress experienced by the cells within the scaffolds and of the probability density function of the surface stresses. Though the manufacturing technique does not appear to affect the shape of the shear stress distributions, presence of manufacturing defects is found to be significant: defects create areas of high flow and high stress along their periphery. The results of this study are applicable to other polymer systems provided that they are manufactured by a similar salt leeching technique, while the imaging/modeling approach is applicable to all scaffolds relevant to tissue engineering.