Mixture modelling for cluster analysis

Cluster analysis via a finite mixture model approach is considered. With this approach to clustering, the data can be partitioned into a specified number of clusters g by first fitting a mixture model with g components. An outright clustering of the data is then obtained by assigning an observation to the component to which it has the highest estimated posterior probability of belonging; that is, the ith cluster consists of those observations assigned to the ith component (i = 1,..., g). The focus is on the use of mixtures of normal components for the cluster analysis of data that can be regarded as being continuous. But attention is also given to the case of mixed data, where the observations consist of both continuous and discrete variables.

Biomechanics of the rostrum in crocodilians: A comparative analysis using finite-element modeling

This article reports the use of simple beam and finite-element models to investigate the relationship between rostral shape and biomechanical performance in living crocodilians under a range of loading conditions. Load cases corresponded to simple biting, lateral head shaking, and twist feeding behaviors. The six specimens were chosen to reflect, as far as possible, the full range of rostral shape in living crocodilians: a juvenile Caiman crocodilus, subadult Alligator mississippiensis and Crocodylus johnstoni, and adult Caiman crocodilus, Melanosuchus niger, and Paleosuchus palpebrosus. The simple beam models were generated using morphometric landmarks from each specimen. Three of the finite-element models, the A. mississippiensis, juvenile Caiman crocodilus, and the Crocodylus johnstoni, were based on CT scan data from respective specimens, but these data were not available for the other models and so these-the adult Caiman crocodilus, M. niger, and P. palpebrosus-were generated by morphing the juvenile Caiman crocodilus mesh with reference to three-dimensional linear distance measured from specimens. Comparison of the mechanical performance of the six finite-element models essentially matched results of the simple beam models: relatively tall skulls performed best under vertical loading and tall and wide skulls performed best under torsional loading. The widely held assumption that the platyrostral (dorsoventrally flattened) crocodilian skull is optimized for torsional loading was not supported by either simple beam theory models or finite-element modeling. Rather than being purely optimized against loads encountered while subduing and processing food, the shape of the crocodilian rostrum may be significantly affected by the hydrodynamic constraints of catching agile aquatic prey. This observation has important implications for our understanding of biomechanics in crocodilians and other aquatic reptiles.

Estimating the SCAN*PRO model of store sales: HB, FM or just OLS?

In this paper we investigate whether consideration of store-level heterogeneity in marketing mix effects improves the accuracy of the marketing mix elasticities, fit, and forecasting accuracy of the widely-applied SCAN*PRO model of store sales. Models with continuous and discrete representations of heterogeneity, estimated using hierarchical Bayes (HB) and finite mixture (FM) techniques, respectively, are empirically compared to the original model, which does not account for store-level heterogeneity in marketing mix effects, and is estimated using ordinary least squares (OLS). The empirical comparisons are conducted in two contexts: Dutch store-level scanner data for the shampoo product category, and an extensive simulation experiment. The simulation investigates how between- and within-segment variance in marketing mix effects, error variance, the number of weeks of data, and the number of stores impact the accuracy of marketing mix elasticities, model fit, and forecasting accuracy. Contrary to expectations, accommodating store-level heterogeneity does not improve the accuracy of marketing mix elasticities relative to the homogeneous SCAN*PRO model, suggesting that little may be lost by employing the original homogeneous SCAN*PRO model estimated using ordinary least squares. Improvements in fit and forecasting accuracy are also fairly modest. We pursue an explanation for this result since research in other contexts has shown clear advantages from assuming some type of heterogeneity in market response models. In an Afterthought section, we comment on the controversial nature of our result, distinguishing factors inherent to household-level data and associated models vs. general store-level data and associated models vs. the unique SCAN*PRO model specification.

A new creep test method for asphalt mixtures

A new creep test, Partial Triaxial Test (PTT), was developed to study the permanent deformation properties of asphalt mixtures. The PTT used two duplicate platens whose diameters were smaller than the diameter of the cylindrical asphalt mixtures specimen. One base platen was centrally placed under the specimen and another loading platen was centrally placed on the top surface of the specimen. Then the compressive repeated load was applied on the loading platen and the vertical deformation of the asphalt mixture was recorded in the PTTs. Triaxial repeated load permanent deformation tests (TRT) and PTTs were respectively conducted on AC20 and SMA13 asphalt mixtures at 40°C and 60°C so as to provide the parameters of the creep constitutive relations in the ABAQUS finite element models (FEMs) which were built to simulate the laboratory wheel tracking tests. The real laboratory wheel tracking tests were also conducted on AC20 and SMA13 asphalt mixtures at 40°C and 60°C. Then the calculated rutting depth from the FEMs were compared with the measured rutting depth of the laboratory wheeling tracking tests. Results indicated that PTT was able to characterize the permanent deformation of the asphalt mixtures in laboratory. The rutting depth calculated using the parameters estimated from PTTs' results was closer to and showed better matches with the measured rutting than the rutting depth calculated using the parameters estimated from TRTs' results. Main reason was that PTT could better simulate the changing confinement conditions of asphalt mixtures in the laboratory wheeling tracking tests than the TRT.

A Bayesian Spatial Mixture Model for FMRI Analysis

We develop, implement and study a new Bayesian spatial mixture model (BSMM). The proposed BSMM allows for spatial structure in the binary activation indicators through a latent thresholded Gaussian Markov random field. We develop a Gibbs (MCMC) sampler to perform posterior inference on the model parameters, which then allows us to assess the posterior probabilities of activation for each voxel. One purpose of this article is to compare the HJ model and the BSMM in terms of receiver operating characteristics (ROC) curves. Also we consider the accuracy of the spatial mixture model and the BSMM for estimation of the size of the activation region in terms of bias, variance and mean squared error. We perform a simulation study to examine the aforementioned characteristics under a variety of configurations of spatial mixture model and BSMM both as the size of the region changes and as the magnitude of activation changes.

Bayesian Inference in Large-scale Problems

Many modern applications fall into the category of "large-scale" statistical problems, in which both the number of observations n and the number of features or parameters p may be large. Many existing methods focus on point estimation, despite the continued relevance of uncertainty quantification in the sciences, where the number of parameters to estimate often exceeds the sample size, despite huge increases in the value of n typically seen in many fields. Thus, the tendency in some areas of industry to dispense with traditional statistical analysis on the basis that "n=all" is of little relevance outside of certain narrow applications. The main result of the Big Data revolution in most fields has instead been to make computation much harder without reducing the importance of uncertainty quantification. Bayesian methods excel at uncertainty quantification, but often scale poorly relative to alternatives. This conflict between the statistical advantages of Bayesian procedures and their substantial computational disadvantages is perhaps the greatest challenge facing modern Bayesian statistics, and is the primary motivation for the work presented here.

Two general strategies for scaling Bayesian inference are considered. The first is the development of methods that lend themselves to faster computation, and the second is design and characterization of computational algorithms that scale better in n or p. In the first instance, the focus is on joint inference outside of the standard problem of multivariate continuous data that has been a major focus of previous theoretical work in this area. In the second area, we pursue strategies for improving the speed of Markov chain Monte Carlo algorithms, and characterizing their performance in large-scale settings. Throughout, the focus is on rigorous theoretical evaluation combined with empirical demonstrations of performance and concordance with the theory.

One topic we consider is modeling the joint distribution of multivariate categorical data, often summarized in a contingency table. Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. In Chapter 2, we derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions.

Latent class models for the joint distribution of multivariate categorical, such as the PARAFAC decomposition, data play an important role in the analysis of population structure. In this context, the number of latent classes is interpreted as the number of genetically distinct subpopulations of an organism, an important factor in the analysis of evolutionary processes and conservation status. Existing methods focus on point estimates of the number of subpopulations, and lack robust uncertainty quantification. Moreover, whether the number of latent classes in these models is even an identified parameter is an open question. In Chapter 3, we show that when the model is properly specified, the correct number of subpopulations can be recovered almost surely. We then propose an alternative method for estimating the number of latent subpopulations that provides good quantification of uncertainty, and provide a simple procedure for verifying that the proposed method is consistent for the number of subpopulations. The performance of the model in estimating the number of subpopulations and other common population structure inference problems is assessed in simulations and a real data application.

In contingency table analysis, sparse data is frequently encountered for even modest numbers of variables, resulting in non-existence of maximum likelihood estimates. A common solution is to obtain regularized estimates of the parameters of a log-linear model. Bayesian methods provide a coherent approach to regularization, but are often computationally intensive. Conjugate priors ease computational demands, but the conjugate Diaconis--Ylvisaker priors for the parameters of log-linear models do not give rise to closed form credible regions, complicating posterior inference. In Chapter 4 we derive the optimal Gaussian approximation to the posterior for log-linear models with Diaconis--Ylvisaker priors, and provide convergence rate and finite-sample bounds for the Kullback-Leibler divergence between the exact posterior and the optimal Gaussian approximation. We demonstrate empirically in simulations and a real data application that the approximation is highly accurate, even in relatively small samples. The proposed approximation provides a computationally scalable and principled approach to regularized estimation and approximate Bayesian inference for log-linear models.

Another challenging and somewhat non-standard joint modeling problem is inference on tail dependence in stochastic processes. In applications where extreme dependence is of interest, data are almost always time-indexed. Existing methods for inference and modeling in this setting often cluster extreme events or choose window sizes with the goal of preserving temporal information. In Chapter 5, we propose an alternative paradigm for inference on tail dependence in stochastic processes with arbitrary temporal dependence structure in the extremes, based on the idea that the information on strength of tail dependence and the temporal structure in this dependence are both encoded in waiting times between exceedances of high thresholds. We construct a class of time-indexed stochastic processes with tail dependence obtained by endowing the support points in de Haan's spectral representation of max-stable processes with velocities and lifetimes. We extend Smith's model to these max-stable velocity processes and obtain the distribution of waiting times between extreme events at multiple locations. Motivated by this result, a new definition of tail dependence is proposed that is a function of the distribution of waiting times between threshold exceedances, and an inferential framework is constructed for estimating the strength of extremal dependence and quantifying uncertainty in this paradigm. The method is applied to climatological, financial, and electrophysiology data.

The remainder of this thesis focuses on posterior computation by Markov chain Monte Carlo. The Markov Chain Monte Carlo method is the dominant paradigm for posterior computation in Bayesian analysis. It has long been common to control computation time by making approximations to the Markov transition kernel. Comparatively little attention has been paid to convergence and estimation error in these approximating Markov Chains. In Chapter 6, we propose a framework for assessing when to use approximations in MCMC algorithms, and how much error in the transition kernel should be tolerated to obtain optimal estimation performance with respect to a specified loss function and computational budget. The results require only ergodicity of the exact kernel and control of the kernel approximation accuracy. The theoretical framework is applied to approximations based on random subsets of data, low-rank approximations of Gaussian processes, and a novel approximating Markov chain for discrete mixture models.

Data augmentation Gibbs samplers are arguably the most popular class of algorithm for approximately sampling from the posterior distribution for the parameters of generalized linear models. The truncated Normal and Polya-Gamma data augmentation samplers are standard examples for probit and logit links, respectively. Motivated by an important problem in quantitative advertising, in Chapter 7 we consider the application of these algorithms to modeling rare events. We show that when the sample size is large but the observed number of successes is small, these data augmentation samplers mix very slowly, with a spectral gap that converges to zero at a rate at least proportional to the reciprocal of the square root of the sample size up to a log factor. In simulation studies, moderate sample sizes result in high autocorrelations and small effective sample sizes. Similar empirical results are observed for related data augmentation samplers for multinomial logit and probit models. When applied to a real quantitative advertising dataset, the data augmentation samplers mix very poorly. Conversely, Hamiltonian Monte Carlo and a type of independence chain Metropolis algorithm show good mixing on the same dataset.

SIMULATING LARGE DEFORMATION OF INTRANEURAL GANGLION CYST USING FINITE ELEMENT METHOD

Intraneural Ganglion Cyst is disorder observed in the nerve injury, it is still unknown and very difficult to predict its propagation in the human body so many times it is referred as an unsolved history. The treatments for this disorder are to remove the cystic substance from the nerve by a surgery. However these treatments may result in neuropathic pain and recurrence of the cyst. The articular theory proposed by Spinner et al., (Spinner et al. 2003) considers the neurological deficit in Common Peroneal Nerve (CPN) branch of the sciatic nerve and adds that in addition to the treatment, ligation of articular branch results into foolproof eradication of the deficit. Mechanical modeling of the affected nerve cross section will reinforce the articular theory (Spinner et al. 2003). As the cyst propagates, it compresses the neighboring fascicles and the nerve cross section appears like a signet ring. Hence, in order to mechanically model the affected nerve cross section; computational methods capable of modeling excessively large deformations are required. Traditional FEM produces distorted elements while modeling such deformations, resulting into inaccuracies and premature termination of the analysis. The methods described in research report have the capability to simulate large deformation. The results obtained from this research shows significant deformation as compared to the deformation observed in the conventional finite element models. The report elaborates the neurological deficit followed by detail explanation of the Smoothed Particle Hydrodynamic approach. Finally, the results show the large deformation in stages and also the successful implementation of the SPH method for the large deformation of the biological organ like the Intra-neural ganglion cyst.

Assessing model credibility of CT-Based finite element model to predict bone strength

Osteoporosis is one of the major causes of mortality among the elderly. Nowadays, areal bone mineral density (aBMD) is used as diagnostic criteria for osteoporosis; however, this is a moderate predictor of the femur fracture risk and does not capture the effect of some anatomical and physiological properties on the bone strength estimation. Data from past research suggest that most fragility femur fractures occur in patients with aBMD values outside the pathological range. Subject-specific finite element models derived from computed tomography data are considered better tools to non-invasively assess hip fracture risk. In particular, the Bologna Biomechanical Computed Tomography (BBCT) is an In Silico methodology that uses a subject specific FE model to predict bone strength. Different studies demonstrated that the modeling pipeline can increase predictive accuracy of osteoporosis detection and assess the efficacy of new antiresorptive drugs. However, one critical aspect that must be properly addressed before using the technology in the clinical practice, is the assessment of the model credibility. The aim of this study was to define and perform verification and uncertainty quantification analyses on the BBCT methodology following the risk-based credibility assessment framework recently proposed in the VV-40 standard. The analyses focused on the main verification tests used in computational solid mechanics: force and moment equilibrium check, mesh convergence analyses, mesh quality metrics study, evaluation of the uncertainties associated to the definition of the boundary conditions and material properties mapping. Results of these analyses showed that the FE model is correctly implemented and solved. The operation that mostly affect the model results is the material properties mapping step. This work represents an important step that, together with the ongoing clinical validation activities, will contribute to demonstrate the credibility of the BBCT methodology.

Thermomechanical processing for recovery of desired < 001 > fiber texture in electric motor steels

A processing route has been developed for recovering the desired lambda fiber in iron-silicon electrical steel needed for superior magnetic properties in electric motor application. The lambda fiber texture is available in directionally solidified iron-silicon steel with the < 001 > columnar grains but was lost after heavy rolling and recrystallization required for motor laminations. Two steps of light rolling each followed by recrystallization were found to largely restore the desired fiber texture. This strengthening of the < 001 > fiber texture had been predicted on the basis of the strain-induced boundary migration (SIBM) mechanism during recrystallization of lightly rolled steel from existing grains of near the ideal orientation, due to postulated low stored energies. Taylor and finite element models supported the idea of the low stored energy of the lambda fiber grains. The models also showed that the lambda fiber grains, though unstable during rolling, only rotated away from their initial orientations quite slowly.

Influence of patient age and implantation technique on the probability of re-replacement of the homograft aortic valve

Background and aim of the study: Results of valve re-replacement (reoperation) in 898 patients undergoing aortic valve replacement with cryopreserved homograft valves between 1975 and 1998 are reported. The study aim was to provide estimates of unconditional probability of valve reoperation and cumulative incidence function (actual risk) of reoperation. Methods: Valves were implanted by subcoronary insertion (n = 500), inclusion cylinder (n = 46), and aortic root replacement (n = 352). Probability of reoperation was estimated by adopting a mixture model framework within which estimates were adjusted for two risk factors: patient age at initial replacement, and implantation technique. Results: For a patient aged 50 years, the probability of reoperation in his/her lifetime was estimated as 44% and 56% for non-root and root replacement techniques, respectively. For a patient aged 70 years, estimated probability of reoperation was 16% and 25%, respectively. Given that a reoperation is required, patients with non-root replacement have a higher hazard rate than those with root replacement (hazards ratio = 1.4), indicating that non-root replacement patients tend to undergo reoperation earlier before death than root replacement patients. Conclusion: Younger patient age and root versus non-root replacement are risk factors for reoperation. Valve durability is much less in younger patients, while root replacement patients appear more likely to live longer and hence are more likely to require reoperation.

Standard errors of fitted component means of normal mixtures

In this paper use consider the problem of providing standard errors of the component means in normal mixture models fitted to univariate or multivariate data by maximum likelihood via the EM algorithm. Two methods of estimation of the standard errors are considered: the standard information-based method and the computationally-intensive bootstrap method. They are compared empirically by their application to three real data sets and by a small-scale Monte Carlo experiment.

A general hazard model for lifetime data in the presence of cure rate

Historically, the cure rate model has been used for modeling time-to-event data within which a significant proportion of patients are assumed to be cured of illnesses, including breast cancer, non-Hodgkin lymphoma, leukemia, prostate cancer, melanoma, and head and neck cancer. Perhaps the most popular type of cure rate model is the mixture model introduced by Berkson and Gage [1]. In this model, it is assumed that a certain proportion of the patients are cured, in the sense that they do not present the event of interest during a long period of time and can found to be immune to the cause of failure under study. In this paper, we propose a general hazard model which accommodates comprehensive families of cure rate models as particular cases, including the model proposed by Berkson and Gage. The maximum-likelihood-estimation procedure is discussed. A simulation study analyzes the coverage probabilities of the asymptotic confidence intervals for the parameters. A real data set on children exposed to HIV by vertical transmission illustrates the methodology.

Fitting mixtures of Kent distributions to aid in joint set identification

When examining a rock mass, joint sets and their orientations can play a significant role with regard to how the rock mass will behave. To identify joint sets present in the rock mass, the orientation of individual fracture planer can be measured on exposed rock faces and the resulting data can be examined for heterogeneity. In this article, the expectation-maximization algorithm is used to lit mixtures of Kent component distributions to the fracture data to aid in the identification of joint sets. An additional uniform component is also included in the model to accommodate the noise present in the data.

Determinação do momento crítico de flambagem lateral com distorção em vigas mistas contínuas de aço e concreto com perfis de alma senoidal

As vigas mistas de aço e concreto estão sendo largamente utilizadas em construções de edifícios e pontes. Ao se combinar o aço com o concreto obtêm-se estruturas mais econômicas, uma vez que se tira proveito das melhores características de cada material. Nas regiões de momento negativo de uma viga mista contínua, a mesa inferior e parte da alma estão comprimidas, se a alma do perfil não tiver rigidez suficiente para evitar a flexão lateral, ela distorcerá gerando um deslocamento lateral e um giro na mesa comprimida, caracterizando um modo de flambagem denominado flambagem lateral com distorção (FLD). O procedimento de verificação à FLD da EN 1994-1-1:2004 originou o método de cálculo da ABNT NBR 8800:2008, entretanto a EN 1994-1-1:2004 não fornece expressão para o cálculo do momento crítico elástico, enquanto a ABNT NBR 8800:2008 prescreve uma formulação proposta por Roik, Hanswille e Kina (1990) desenvolvida para vigas mistas com perfis de alma plana. Embora as normas prescrevam um método de verificação à FLD para vigas mistas com perfis de alma plana, poucos estudos têm sido feitos sobre esse estado-limite. Além disso, tanto a ABNT NBR 8800:2008 quanto as normas internacionais não abordam perfis de alma senoidal. Neste trabalho, foram implementadas análises de flambagem elástica, com auxílio do software ANSYS 14.0 (2011), em modelos de elementos finitos que retratem o comportamento à FLD de vigas mistas de aço e concreto com perfis de alma plana e senoidal. Os modelos numéricos foram constituídos pelo perfil de aço, por uma mola rotacional que restringe parcialmente o giro da mesa superior e uma restrição ao deslocamento lateral, ao longo de todo o comprimento da viga. Os resultados numéricos são comparados com os obtidos pelas formulações de Roik, Hanswille e Kina (1990) e de Hanswille (2002), adaptadas para levar em consideração a corrugação da alma do perfil de aço. Para avaliação das formulações supracitadas e da consistência da modelagem numérica adotada, o momento crítico elástico foi determinado para vigas mistas com perfis de aço de alma plana. Como resultado, um método para o cálculo do momento crítico elástico de vigas mistas de alma senoidal é proposto.

Estudo numérico e experimental sobre o comportamento dinâmico de um edifício

Dissertação para obtenção do grau de Mestre em Engenharia Civil na Área de Especialização em Estruturas