Statistical analysis of shape of objects based on landmark data.

Two objects with homologous landmarks are said to be of the same shape if the configurations of landmarks of one object can be exactly matched with that of the other by translation, rotation/reflection, and scaling. The observations on an object are coordinates of its landmarks with reference to a set of orthogonal coordinate axes in an appropriate dimensional space. The origin, choice of units, and orientation of the coordinate axes with respect to an object may be different from object to object. In such a case, how do we quantify the shape of an object, find the mean and variation of shape in a population of objects, compare the mean shapes in two or more different populations, and discriminate between objects belonging to two or more different shape distributions. We develop some methods that are invariant to translation, rotation, and scaling of the observations on each object and thereby provide generalizations of multivariate methods for shape analysis.

Distinct desmocollin isoforms occur in the same desmosomes and show reciprocally graded distributions in bovine nasal epidermis.

The adhesive core of the desmosome is composed of cadherin-like glycoproteins of two families, desmocollins and desmogleins. Three isoforms of each are expressed in a tissue-specific and developmentally regulated pattern. In bovine nasal epidermis, the three desmocollin (Dsc) isoforms are expressed in overlapping domains; Dsc3 expression is strongest in the basal layer, while Dsc2 and Dsc1 are strongly expressed in the suprabasal layers. Herein we have investigated whether different isoforms are assembled into the same or distinct desmosomes by performing double immunogold labeling using isoform-specific antibodies directed against Dsc1 and Dsc3. The results show that individual desmosomes harbor both isoforms in regions where their expression territories overlap. Quantification showed that the ratio of the proteins in each desmosome altered gradually from basal to immediately suprabasal and upper suprabasal layers, labeling for Dsc1 increasing and Dsc3 decreasing. Thus desmosomes are constantly modified as cells move up the epidermis, with continuing turnover of the desmosomal glycoproteins. Statistical analysis of the quantitative data showed a possible relationship between the distributions of the two isoforms. This gradual change in desmosomal composition may constitute a vertical adhesive gradient within the epidermis, having important consequences for cell positioning and differentiation.

Bayesian estimation of g-and-k distributions using MCMC

In this paper we investigate a Bayesian procedure for the estimation of a flexible generalised distribution, notably the MacGillivray adaptation of the g-and-κ distribution. This distribution, described through its inverse cdf or quantile function, generalises the standard normal through extra parameters which together describe skewness and kurtosis. The standard quantile-based methods for estimating the parameters of generalised distributions are often arbitrary and do not rely on computation of the likelihood. MCMC, however, provides a simulation-based alternative for obtaining the maximum likelihood estimates of parameters of these distributions or for deriving posterior estimates of the parameters through a Bayesian framework. In this paper we adopt the latter approach, The proposed methodology is illustrated through an application in which the parameter of interest is slightly skewed.

Comparison of the distributions of classical and adaptively aligned EEG power spectra

The distributions of eyes-closed resting electroencephalography (EEG) power spectra and their residuals were described and compared using classically averaged and adaptively aligned averaged spectra. Four minutes of eyes-closed resting EEG was available from 69 participants. Spectra were calculated with 0.5-Hz resolution and were analyzed at this level. It was shown that power in the individual 0.5 Hz frequency bins can be considered normally distributed when as few as three or four 2-second epochs of EEG are used in the average. A similar result holds for the residuals. Power at the peak Alpha frequency has quite different statistical behaviour to power at other frequencies and it is considered that power at peak Alpha represents a relatively individuated process that is best measured through aligned averaging. Previous analyses of contrasts in upper and lower alpha bands may be explained in terms of the variability or distribution of the peak Alpha frequency itself.

Bayesian invariant measurements of generalisation for discrete distributions

Neural network learning rules can be viewed as statistical estimators. They should be studied in Bayesian framework even if they are not Bayesian estimators. Generalisation should be measured by the divergence between the true distribution and the estimated distribution. Information divergences are invariant measurements of the divergence between two distributions. The posterior average information divergence is used to measure the generalisation ability of a network. The optimal estimators for multinomial distributions with Dirichlet priors are studied in detail. This confirms that the definition is compatible with intuition. The results also show that many commonly used methods can be put under this unified framework, by assume special priors and special divergences.

Typical kernel size and number of sparse random matrices over Galois fields: a statistical physics approach

Using methods of statistical physics, we study the average number and kernel size of general sparse random matrices over GF(q), with a given connectivity profile, in the thermodynamical limit of large matrices. We introduce a mapping of GF(q) matrices onto spin systems using the representation of the cyclic group of order q as the q-th complex roots of unity. This representation facilitates the derivation of the average kernel size of random matrices using the replica approach, under the replica symmetric ansatz, resulting in saddle point equations for general connectivity distributions. Numerical solutions are then obtained for particular cases by population dynamics. Similar techniques also allow us to obtain an expression for the exact and average number of random matrices for any general connectivity profile. We present numerical results for particular distributions.

Statistical mechanics of distribution networks

This thesis presents an analysis of the stability of complex distribution networks. We present a stability analysis against cascading failures. We propose a spin [binary] model, based on concepts of statistical mechanics. We test macroscopic properties of distribution networks with respect to various topological structures and distributions of microparameters. The equilibrium properties of the systems are obtained in a statistical mechanics framework by application of the replica method. We demonstrate the validity of our approach by comparing it with Monte Carlo simulations. We analyse the network properties in terms of phase diagrams and found both qualitative and quantitative dependence of the network properties on the network structure and macroparameters. The structure of the phase diagrams points at the existence of phase transition and the presence of stable and metastable states in the system. We also present an analysis of robustness against overloading in the distribution networks. We propose a model that describes a distribution process in a network. The model incorporates the currents between any connected hubs in the network, local constraints in the form of Kirchoff's law and a global optimizational criterion. The flow of currents in the system is driven by the consumption. We study two principal types of model: infinite and finite link capacity. The key properties are the distributions of currents in the system. We again use a statistical mechanics framework to describe the currents in the system in terms of macroscopic parameters. In order to obtain observable properties we apply the replica method. We are able to assess the criticality of the level of demand with respect to the available resources and the architecture of the network. Furthermore, the parts of the system, where critical currents may emerge, can be identified. This, in turn, provides us with the characteristic description of the spread of the overloading in the systems.

Some Properties of Mittag-Leffler Functions and Matrix-Variate Analogues: A Statistical Perspective

Mathematical Subject Classification 2010:26A33, 33E99, 15A52, 62E15.

Forecasting Responses of Seagrass Distributions to Changing Water Quality Using Monitoring Data

Extensive data sets on water quality and seagrass distributions in Florida Bay have been assembled under complementary, but independent, monitoring programs. This paper presents the landscape-scale results from these monitoring programs and outlines a method for exploring the relationships between two such data sets. Seagrass species occurrence and abundance data were used to define eight benthic habitat classes from 677 sampling locations in Florida Bay. Water quality data from 28 monitoring stations spread across the Bay were used to construct a discriminant function model that assigned a probability of a given benthic habitat class occurring for a given combination of water quality variables. Mean salinity, salinity variability, the amount of light reaching the benthos, sediment depth, and mean nutrient concentrations were important predictor variables in the discriminant function model. Using a cross-validated classification scheme, this discriminant function identified the most likely benthic habitat type as the actual habitat type in most cases. The model predicted that the distribution of benthic habitat types in Florida Bay would likely change if water quality and water delivery were changed by human engineering of freshwater discharge from the Everglades. Specifically, an increase in the seasonal delivery of freshwater to Florida Bay should cause an expansion of seagrass beds dominated by Ruppia maritima and Halodule wrightii at the expense of the Thalassia testudinum-dominated community that now occurs in northeast Florida Bay. These statistical techniques should prove useful for predicting landscape-scale changes in community composition in diverse systems where communities are in quasi-equilibrium with environmental drivers.

Experimental deuteron momentum distributions with reduced final state interactions

This dissertation presents a study of the D( e, e′p)n reaction carried out at the Thomas Jefferson National Accelerator Facility (Jefferson Lab) for a set of fixed values of four-momentum transfer Q 2 = 2.1 and 0.8 (GeV/c)2 and for missing momenta pm ranging from pm = 0.03 to pm = 0.65 GeV/c. The analysis resulted in the determination of absolute D(e,e′ p)n cross sections as a function of the recoiling neutron momentum and it's scattering angle with respect to the momentum transfer [vector] q. The angular distribution was compared to various modern theoretical predictions that also included final state interactions. The data confirmed the theoretical prediction of a strong anisotropy of final state interaction contributions at Q2 of 2.1 (GeV/c)2 while at the lower Q2 value, the anisotropy was much less pronounced. At Q2 of 0.8 (GeV/c)2, theories show a large disagreement with the experimental results. The experimental momentum distribution of the bound proton inside the deuteron has been determined for the first time at a set of fixed neutron recoil angles. The momentum distribution is directly related to the ground state wave function of the deuteron in momentum space. The high momentum part of this wave function plays a crucial role in understanding the short-range part of the nucleon-nucleon force. At Q2 = 2.1 (GeV/c)2, the momentum distribution determined at small neutron recoil angles is much less affected by FSI compared to a recoil angle of 75°. In contrast, at Q2 = 0.8 (GeV/c)2 there seems to be no region with reduced FSI for larger missing momenta. Besides the statistical errors, systematic errors of about 5–6 % were included in the final results in order to account for normalization uncertainties and uncertainties in the determi- nation of kinematic veriables. The measurements were carried out using an electron beam energy of 2.8 and 4.7 GeV with beam currents between 10 to 100 &mgr; A. The scattered electrons and the ejected protons originated from a 15cm long liquid deuterium target, and were detected in conicidence with the two high resolution spectrometers of Hall A at Jefferson Lab.^

THE PHYSICS OF IDEAS: INFERRING THE MECHANICS OF OPINION FORMATION FROM MACROSCOPIC STATISTICAL PATTERNS

In a microscopic setting, humans behave in rich and unexpected ways. In a macroscopic setting, however, distinctive patterns of group behavior emerge, leading statistical physicists to search for an underlying mechanism. The aim of this dissertation is to analyze the macroscopic patterns of competing ideas in order to discern the mechanics of how group opinions form at the microscopic level. First, we explore the competition of answers in online Q&A (question and answer) boards. We find that a simple individual-level model can capture important features of user behavior, especially as the number of answers to a question grows. Our model further suggests that the wisdom of crowds may be constrained by information overload, in which users are unable to thoroughly evaluate each answer and therefore tend to use heuristics to pick what they believe is the best answer. Next, we explore models of opinion spread among voters to explain observed universal statistical patterns such as rescaled vote distributions and logarithmic vote correlations. We introduce a simple model that can explain both properties, as well as why it takes so long for large groups to reach consensus. An important feature of the model that facilitates agreement with data is that individuals become more stubborn (unwilling to change their opinion) over time. Finally, we explore potential underlying mechanisms for opinion formation in juries, by comparing data to various types of models. We find that different null hypotheses in which jurors do not interact when reaching a decision are in strong disagreement with data compared to a simple interaction model. These findings provide conceptual and mechanistic support for previous work that has found mutual influence can play a large role in group decisions. In addition, by matching our models to data, we are able to infer the time scales over which individuals change their opinions for different jury contexts. We find that these values increase as a function of the trial time, suggesting that jurors and judicial panels exhibit a kind of stubbornness similar to what we include in our model of voting behavior.

An alternative methodology for fitting selectivity curves to pre-defined distributions

A non-linear least-squares methodology for simultaneously estimating parameters of selectivity curves with a pre-defined functional form, across size classes and mesh sizes, using catch size frequency distributions, was developed based on the model of Kirkwood and Walker [Kirkwood, G.P., Walker, T.L, 1986. Gill net selectivities for gummy shark, Mustelus antarcticus Gunther, taken in south-eastern Australian waters. Aust. J. Mar. Freshw. Res. 37, 689-697] and [Wulff, A., 1986. Mathematical model for selectivity of gill nets. Arch. Fish Wiss. 37, 101-106]. Observed catches of fish of size class I in mesh m are modeled as a function of the estimated numbers of fish of that size class in the population and the corresponding selectivities. A comparison was made with the maximum likelihood methodology of [Kirkwood, G.P., Walker, T.I., 1986. Gill net selectivities for gummy shark, Mustelus antarcticus Gunther, taken in south-eastern Australian waters. Aust. J. Mar. Freshw. Res. 37, 689-697] and [Wulff, A., 1986. Mathematical model for selectivity of gill nets. Arch. Fish Wiss; 37, 101-106], using simulated catch data with known selectivity curve parameters, and two published data sets. The estimated parameters and selectivity curves were generally consistent for both methods, with smaller standard errors for parameters estimated by non-linear least-squares. The proposed methodology is a useful and accessible alternative which can be used to model selectivity in situations where the parameters of a pre-defined model can be assumed to be functions of gear size; facilitating statistical evaluation of different models and of goodness of fit. (C) 1998 Elsevier Science B.V.

How well can models predict changes in species distributions? A 13-year-old otter model revisited

Species distribution and ecological niche models are increasingly used in biodiversity management and conservation. However, one thing that is important but rarely done is to follow up on the predictive performance of these models over time, to check if their predictions are fulfilled and maintain accuracy, or if they apply only to the set in which they were produced. In 2003, a distribution model of the Eurasian otter (Lutra lutra) in Spain was published, based on the results of a country-wide otter survey published in 1998. This model was built with logistic regression of otter presence-absence in UTM 10 km2 cells on a diverse set of environmental, human and spatial variables, selected according to statistical criteria. Here we evaluate this model against the results of the most recent otter survey, carried out a decade later and after a significant expansion of the otter distribution area in this country. Despite the time elapsed and the evident changes in this species’ distribution, the model maintained a good predictive capacity, considering both discrimination and calibration measures. Otter distribution did not expand randomly or simply towards vicinity areas,m but specifically towards the areas predicted as most favourable by the model based on data from 10 years before. This corroborates the utility of predictive distribution models, at least in the medium term and when they are made with robust methods and relevant predictor variables.

Statistical learning of random probability measures

The study of random probability measures is a lively research topic that has attracted interest from different fields in recent years. In this thesis, we consider random probability measures in the context of Bayesian nonparametrics, where the law of a random probability measure is used as prior distribution, and in the context of distributional data analysis, where the goal is to perform inference given avsample from the law of a random probability measure. The contributions contained in this thesis can be subdivided according to three different topics: (i) the use of almost surely discrete repulsive random measures (i.e., whose support points are well separated) for Bayesian model-based clustering, (ii) the proposal of new laws for collections of random probability measures for Bayesian density estimation of partially exchangeable data subdivided into different groups, and (iii) the study of principal component analysis and regression models for probability distributions seen as elements of the 2-Wasserstein space. Specifically, for point (i) above we propose an efficient Markov chain Monte Carlo algorithm for posterior inference, which sidesteps the need of split-merge reversible jump moves typically associated with poor performance, we propose a model for clustering high-dimensional data by introducing a novel class of anisotropic determinantal point processes, and study the distributional properties of the repulsive measures, shedding light on important theoretical results which enable more principled prior elicitation and more efficient posterior simulation algorithms. For point (ii) above, we consider several models suitable for clustering homogeneous populations, inducing spatial dependence across groups of data, extracting the characteristic traits common to all the data-groups, and propose a novel vector autoregressive model to study of growth curves of Singaporean kids. Finally, for point (iii), we propose a novel class of projected statistical methods for distributional data analysis for measures on the real line and on the unit-circle.