Maximum Likelihood Estimation of Mixture Densities for Binned and Truncated Multivariate Data

Binning and truncation of data are common in data analysis and machine learning. This paper addresses the problem of fitting mixture densities to multivariate binned and truncated data. The EM approach proposed by McLachlan and Jones (Biometrics, 44: 2, 571-578, 1988) for the univariate case is generalized to multivariate measurements. The multivariate solution requires the evaluation of multidimensional integrals over each bin at each iteration of the EM procedure. Naive implementation of the procedure can lead to computationally inefficient results. To reduce the computational cost a number of straightforward numerical techniques are proposed. Results on simulated data indicate that the proposed methods can achieve significant computational gains with no loss in the accuracy of the final parameter estimates. Furthermore, experimental results suggest that with a sufficient number of bins and data points it is possible to estimate the true underlying density almost as well as if the data were not binned. The paper concludes with a brief description of an application of this approach to diagnosis of iron deficiency anemia, in the context of binned and truncated bivariate measurements of volume and hemoglobin concentration from an individual's red blood cells.

Modelling High-Dimensional Data by Mixtures of Factor Analyzers

We focus on mixtures of factor analyzers from the perspective of a method for model-based density estimation from high-dimensional data, and hence for the clustering of such data. This approach enables a normal mixture model to be fitted to a sample of n data points of dimension p, where p is large relative to n. The number of free parameters is controlled through the dimension of the latent factor space. By working in this reduced space, it allows a model for each component-covariance matrix with complexity lying between that of the isotropic and full covariance structure models. We shall illustrate the use of mixtures of factor analyzers in a practical example that considers the clustering of cell lines on the basis of gene expressions from microarray experiments. (C) 2002 Elsevier Science B.V. All rights reserved.

Mean anisotropy of homogeneous Gaussian random fields and anisotropic norms of linear translation-invariant operators on multidimensional integer lattices

Sensitivity of output of a linear operator to its input can be quantified in various ways. In Control Theory, the input is usually interpreted as disturbance and the output is to be minimized in some sense. In stochastic worst-case design settings, the disturbance is considered random with imprecisely known probability distribution. The prior set of probability measures can be chosen so as to quantify how far the disturbance deviates from the white-noise hypothesis of Linear Quadratic Gaussian control. Such deviation can be measured by the minimal Kullback-Leibler informational divergence from the Gaussian distributions with zero mean and scalar covariance matrices. The resulting anisotropy functional is defined for finite power random vectors. Originally, anisotropy was introduced for directionally generic random vectors as the relative entropy of the normalized vector with respect to the uniform distribution on the unit sphere. The associated a-anisotropic norm of a matrix is then its maximum root mean square or average energy gain with respect to finite power or directionally generic inputs whose anisotropy is bounded above by a≥0. We give a systematic comparison of the anisotropy functionals and the associated norms. These are considered for unboundedly growing fragments of homogeneous Gaussian random fields on multidimensional integer lattice to yield mean anisotropy. Correspondingly, the anisotropic norms of finite matrices are extended to bounded linear translation invariant operators over such fields.

Rock fracture mean trace length estimation and confidence interval calculation using maximum likelihood methods

There has been a resurgence of interest in the mean trace length estimator of Pahl for window sampling of traces. The estimator has been dealt with by Mauldon and Zhang and Einstein in recent publications. The estimator is a very useful one in that it is non-parametric. However, despite some discussion regarding the statistical distribution of the estimator, none of the recent works or the original work by Pahl provide a rigorous basis for the determination a confidence interval for the estimator or a confidence region for the estimator and the corresponding estimator of trace spatial intensity in the sampling window. This paper shows, by consideration of a simplified version of the problem but without loss of generality, that the estimator is in fact the maximum likelihood estimator (MLE) and that it can be considered essentially unbiased. As the MLE, it possesses the least variance of all estimators and confidence intervals or regions should therefore be available through application of classical ML theory. It is shown that valid confidence intervals can in fact be determined. The results of the work and the calculations of the confidence intervals are illustrated by example. (C) 2003 Elsevier Science Ltd. All rights reserved.

Urban land use change analysis and modelling: a case study of Setubal-Sesimbra, Portugal

Dissertation submitted in partial fulfilment of the requirements for the Degree of Master of Science in Geospatial Technologies

On chaos, transient chaos and ghosts in single populations models with allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)(infinity) occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects. (C) 2011 Elsevier Ltd. All rights reserved.

Evaluation of Spatial interpolation techniques for mapping climate variables with low sample density: a case study using a new gridded dataset of Bangladesh

Dissertation submitted in partial fulfillment of the requirements for the Degree of Master of Science in Geospatial Technologies.

Validation of NASA rotor 67 with openFOAM's transonic density-based solver

Dissertação para obtenção do Grau de Mestre em Engenharia Mecânica

Relative information entropy in cosmology: The problem of information entanglement

The necessary information to distinguish a local inhomogeneous mass density field from its spatial average on a compact domain of the universe can be measured by relative information entropy. The Kullback-Leibler (KL) formula arises very naturally in this context, however, it provides a very complicated way to compute the mutual information between spatially separated but causally connected regions of the universe in a realistic, inhomogeneous model. To circumvent this issue, by considering a parametric extension of the KL measure, we develop a simple model to describe the mutual information which is entangled via the gravitational field equations. We show that the Tsallis relative entropy can be a good approximation in the case of small inhomogeneities, and for measuring the independent relative information inside the domain, we propose the R\'enyi relative entropy formula.

Contractive metrics for a Boltzmann equation for granular gases: diffusive equilibriaThe Patterson-Sullivan embedding and minimal volume entropy for outer space

We quantify the long-time behavior of a system of (partially) inelastic particles in a stochastic thermostat by means of the contractivity of a suitable metric in the set of probability measures. Existence, uniqueness, boundedness of moments and regularity of a steady state are derived from this basic property. The solutions of the kinetic model are proved to converge exponentially as t→ ∞ to this diffusive equilibrium in this distance metrizing the weak convergence of measures. Then, we prove a uniform bound in time on Sobolev norms of the solution, provided the initial data has a finite norm in the corresponding Sobolev space. These results are then combined, using interpolation inequalities, to obtain exponential convergence to the diffusive equilibrium in the strong L¹-norm, as well as various Sobolev norms.

Excreção urinária de 17-Cetoesteroides neutros no cavalo normal e no cavalo castrado

Total urinary neutral 17-steroids were determined in normal and in castrated horses. One liter of a 15-26 hours urine collection was hydrolysed by refluxing with 10% HC1 (v/v) for ten minutes and extracted with peroxyde-free ethyl ether. The extract was purified by washing with saturated NaHCO³ and KOH solutions. One half of the crude neutral fraction was fractionated with Girard's "T" reagent . The Zimmermann reaction was performed both in the ketonic and in the crude neutral extracts, using alcoholic 2.5N KOH and a 60 minutes period for the colour development in the dark. Optical density measuments were made in a grating Coleman Universal Spectrophotometer at 420 mµ and 520mµ; for the crude neutral fraction a colour correction equation was applied. The aliquot fraction used for colorimety was adjusted for keeping optical density measurements within the range 0.2 to 0.7. Androsterone (mp. 184-184.5°C) with an absorption maximum at 290.5 mµ (Beckman Model DU Spectrophotometer) was used as a reference standard. Table I, ilustrates the results obtained. At the 0.05 probability level there is a significant difference among castrated and normal group means (Fischer's "t" test.) when were used the data obtained from the ketonic fractions; in spite of the use of a colour correction applied for inespecific chromogens, the same results could not be obtained with the crude neutral fractions, Since Girard's reagent fractionation is generaly accepted as the best method for correcting the inespecific chromogen interference in the determination of the 17-ketosteroids by the Zimmermann reaction, we emphasize the value of the results obtained with the ketonic fractions. From these results it appears, as occurs in others mammals, that castrated horses show a lower level of urinary 17-ketosteroids excretion than the normal horses. The significance of the horse testis contribution for the neutral urinary steroid metabolites is discussed. Since horse urine has a low androgenic activity, the fractionation of the neutral 17-ketosteroids must be studied more accurately.

Theoretical estimates of consumable food and probability of acquiring food in larvae of Chrysomya putoria (Diptera: Calliphoridae)

An indirect estimate of consumable food and probability of acquiring food in a blowfly species, Chrysomya putoria, is presented. This alternative procedure combines three distinct models to estimate consumable food in the context of the exploitative competition experienced by immature individuals in blowfly populations. The relevant parameters are derived from data for pupal weight and survival and estimates of density-independent larval mortality in twenty different larval densities. As part of this procedure, the probability of acquiring food per unit of time and the time taken to exhaust the food supply are also calculated. The procedure employed here may be valuable for estimations in insects whose immature stages develop inside the food substrate, where it is difficult to partial out confounding effects such as separation of faeces. This procedure also has the advantage of taking into account the population dynamics of immatures living under crowded conditions, which are particularly characteristic of blowflies and other insects as well.

Fourth order pseudo maximum likelihood methods

We extend PML theory to account for information on the conditional moments up to order four, but without assuming a parametric model, to avoid a risk of misspecification of the conditional distribution. The key statistical tool is the quartic exponential family, which allows us to generalize the PML2 and QGPML1 methods proposed in Gourieroux et al. (1984) to PML4 and QGPML2 methods, respectively. An asymptotic theory is developed. The key numerical tool that we use is the Gauss-Freud integration scheme that solves a computational problem that has previously been raised in several fields. Simulation exercises demonstrate the feasibility and robustness of the methods [Authors]

Lumbar spine texture enhances 10-year fracture probability assessment.

We found that lumbar spine texture analysis using trabecular bone score (TBS) is a risk factor for MOF and a risk factor for death in a retrospective cohort study from a large clinical registry for the province of Manitoba, Canada. INTRODUCTION: FRAX® estimates the 10-year probability of major osteoporotic fracture (MOF) using clinical risk factors and femoral neck bone mineral density (BMD). Trabecular bone score (TBS), derived from texture in the spine dual X-ray absorptiometry (DXA) image, is related to bone microarchitecture and fracture risk independently of BMD. Our objective was to determine whether TBS provides information on MOF probability beyond that provided by the FRAX variables. METHODS: We included 33,352 women aged 40-100 years (mean 63 years) with baseline DXA measurements of lumbar spine TBS and femoral neck BMD. The association between TBS, the FRAX variables, and the risk of MOF or death was examined using an extension of the Poisson regression model. RESULTS: During the mean of 4.7 years, 1,754 women died and 1,872 sustained one or more MOF. For each standard deviation reduction in TBS, there was a 36 % increase in MOF risk (HR 1.36, 95 % CI 1.30-1.42, p < 0.001) and a 32 % increase in death (HR 1.32, 95 % CI 1.26-1.39, p < 0.001). When adjusted for significant clinical risk factors and femoral neck BMD, lumbar spine TBS was still a significant predictor of MOF (HR 1.18, 95 % CI 1.12-1.23) and death (HR 1.20, 95 % CI 1.14-1.26). Models for estimating MOF probability, accounting for competing mortality, showed that low TBS (10th percentile) increased risk by 1.5-1.6-fold compared with high TBS (90th percentile) across a broad range of ages and femoral neck T-scores. CONCLUSIONS: Lumbar spine TBS is able to predict incident MOF independent of FRAX clinical risk factors and femoral neck BMD even after accounting for the increased death hazard.

Have all the men and women with a fragility fracture a low bone mineral density? A Nationwide Swiss Survey : FM3

Background: The prevalence of a low bone mineral density (T-score <-1 SD) in postmenopausal women with a fragility fracture may vary from 70% to less than 50%. In one study (Siris ES. Arch Intern Med. 2004;164:1108-12), the prevalence of osteoporosis was very low at 6.4%. The corresponding values in men are rarely reported. Methods: In a nationwide Swiss survey, all consecutive patients aged 50+ presenting with one or more fractures to the emergency ward, were recruited by 8 participating hospitals (University Hospitals: Basel, Bern, and Lausanne; cantonal hospitals: Fribourg, Luzern, and St Gallen; regional hospitals: Estavayer and Riaz) between 2004 and 2006. Diagnostic workup was collected for descriptive analysis. Results: 3667 consecutive patients with a fragility fracture, 2797 women (73.8 ± 11.6 years) and 870 men (70.0 ± 12.1 years), were included. DXA measurement was performed in 1152 (44%) patients. The mean of the lowest T-score values was -2.34 SD in women and -2.16 SD in men. In the 908 women, the prevalence of osteoporosis and osteopenia according to the fracture type was: sacrum (100%, 0%), rib (100%, 0%), thoracic vertebral (78%, 22%), femur trochanter (67%, 26%), pelvis (66%, 32%), lumbar vertebral (63%, 28%), femoral neck (53%, 34%), femur shaft (50%, 50%), proximal humerus (50%, 34%), distal forearm (41%, 45%), tibia proximal (41%, 31%), malleolar lateral (28%, 46%), malleolar median (13%, 47%). The corresponding percentages in the 244 men were: distal forearm (70%, 19%), rib (63%, 11%), pelvis (60%, 20%), malleolar median (60%, 32%), femur trochanter (48%, 31%), thoracic vertebral (47%, 53%), lumbar vertebral (43%, 36%), proximal humerus (40%, 43%), femoral neck (28%, 55%), tibia proximal (26%, 36%), malleolar lateral (18%, 56%). Conclusion: The probability of underlying osteoporosis or osteopenia in men and women aged 50+ who experienced a fragility fracture was beyond 75% in fractures of the sacrum, pelvis, spine, femur, proximal humerus and distal forearm. The medial and lateral malleolar fractures had the lowest predictive value in women, not in men.