A log-linear regression model for the beta-Birnbaum-Saunders distribution with censored data

The beta-Birnbaum-Saunders (Cordeiro and Lemonte, 2011) and Birnbaum-Saunders (Birnbaum and Saunders, 1969a) distributions have been used quite effectively to model failure times for materials subject to fatigue and lifetime data. We define the log-beta-Birnbaum-Saunders distribution by the logarithm of the beta-Birnbaum-Saunders distribution. Explicit expressions for its generating function and moments are derived. We propose a new log-beta-Birnbaum-Saunders regression model that can be applied to censored data and be used more effectively in survival analysis. We obtain the maximum likelihood estimates of the model parameters for censored data and investigate influence diagnostics. The new location-scale regression model is modified for the possibility that long-term survivors may be presented in the data. Its usefulness is illustrated by means of two real data sets. (C) 2011 Elsevier B.V. All rights reserved.

The Truncated Inflated Beta Distribution

The study of proportions is a common topic in many fields of study. The standard beta distribution or the inflated beta distribution may be a reasonable choice to fit a proportion in most situations. However, they do not fit well variables that do not assume values in the open interval (0, c), 0 < c < 1. For these variables, the authors introduce the truncated inflated beta distribution (TBEINF). This proposed distribution is a mixture of the beta distribution bounded in the open interval (c, 1) and the trinomial distribution. The authors present the moments of the distribution, its scoring vector, and Fisher information matrix, and discuss estimation of its parameters. The properties of the suggested estimators are studied using Monte Carlo simulation. In addition, the authors present an application of the TBEINF distribution for unemployment insurance data.

Mixture Cure Survival Models with Dependent Censoring

A number of authors have studies the mixture survival model to analyze survival data with nonnegligible cure fractions. A key assumption made by these authors is the independence between the survival time and the censoring time. To our knowledge, no one has studies the mixture cure model in the presence of dependent censoring. To account for such dependence, we propose a more general cure model which allows for dependent censoring. In particular, we derive the cure models from the perspective of competing risks and model the dependence between the censoring time and the survival time using a class of Archimedean copula models. Within this framework, we consider the parameter estimation, the cure detection, and the two-sample comparison of latency distribution in the presence of dependent censoring when a proportion of patients is deemed cured. Large sample results using the martingale theory are obtained. We applied the proposed methodologies to the SEER prostate cancer data.

Cement distribution, volume, and compliance in vertebroplasty: some answers from an anatomy-based nonlinear finite element study

STUDY DESIGN: The biomechanics of vertebral bodies augmented with real distributions of cement were investigated using nonlinear finite element (FE) analysis. OBJECTIVES: To compare stiffness, strength, and stress transfer of augmented versus nonaugmented osteoporotic vertebral bodies under compressive loading. Specifically, to examine how cement distribution, volume, and compliance affect these biomechanical variables. SUMMARY OF BACKGROUND DATA: Previous FE studies suggested that vertebroplasty might alter vertebral stress transfer, leading to adjacent vertebral failure. However, no FE study so far accounted for real cement distributions and bone damage accumulation. METHODS: Twelve vertebral bodies scanned with high-resolution pQCT and tested in compression were augmented with various volumes of cements and scanned again. Nonaugmented and augmented pQCT datasets were converted to FE models, with bone properties modeled with an elastic, plastic and damage constitutive law that was previously calibrated for the nonaugmented models. The cement-bone composite was modeled with a rule of mixture. The nonaugmented and augmented FE models were subjected to compression and their stiffness, strength, and stress map calculated for different cement compliances. RESULTS: Cement distribution dominated the stiffening and strengthening effects of augmentation. Models with cement connecting either the superior or inferior endplate (S/I fillings) were only up to 2 times stiffer than the nonaugmented models with minimal strengthening, whereas those with cement connecting both endplates (S + I fillings) were 1 to 8 times stiffer and 1 to 12 times stronger. Stress increases above and below the cement, which was higher for the S + I cases and was significantly reduced by increasing cement compliance. CONCLUSION: The developed FE approach, which accounts for real cement distributions and bone damage accumulation, provides a refined insight into the mechanics of augmented vertebral bodies. In particular, augmentation with compliant cement bridging both endplates would reduce stress transfer while providing sufficient strengthening.

Mixture modeling for joint analysis of survival, discrete, and continuous data

Mixture modeling is commonly used to model categorical latent variables that represent subpopulations in which population membership is unknown but can be inferred from the data. In relatively recent years, the potential of finite mixture models has been applied in time-to-event data. However, the commonly used survival mixture model assumes that the effects of the covariates involved in failure times differ across latent classes, but the covariate distribution is homogeneous. The aim of this dissertation is to develop a method to examine time-to-event data in the presence of unobserved heterogeneity under a framework of mixture modeling. A joint model is developed to incorporate the latent survival trajectory along with the observed information for the joint analysis of a time-to-event variable, its discrete and continuous covariates, and a latent class variable. It is assumed that the effects of covariates on survival times and the distribution of covariates vary across different latent classes. The unobservable survival trajectories are identified through estimating the probability that a subject belongs to a particular class based on observed information. We applied this method to a Hodgkin lymphoma study with long-term follow-up and observed four distinct latent classes in terms of long-term survival and distributions of prognostic factors. Our results from simulation studies and from the Hodgkin lymphoma study demonstrated the superiority of our joint model compared with the conventional survival model. This flexible inference method provides more accurate estimation and accommodates unobservable heterogeneity among individuals while taking involved interactions between covariates into consideration.^

The distribution of zinc, cadmium, copper and iron in seawater of the Iceland-Faroe Ridge area

During the International ICES Expedition "Overflow '73" a total of 174 samples from 18 stations were collected by R. V. "Meteor" in the waters of the Iceland-Faroe Ridge area. They were filtered on board ship (through 0.4 mym "Nuclepore" filters), then stored in 500 cm**3 quartz bottles (at -20 °C) and analyzed in air-filtered laboratories on land for zinc and cadmium by means of the differential pulse anodic stripping voltammetry technique and copper and iron by flameless atomic absorption spectrometry. The overall averages of 1.9 myg Zn l**-1, 0.07 myg Cd l**-1, 0.5 myg Cu l**-1 and 0.9 myg Fe l**-1 are in good agreement with recent "baseline" studies of open-ocean waters. The mixture of low salinity water masses from the North Iceland Shelf/Arctic Intermediate Waters seem to maintain distinctly lower concentration of Cd, Cu and Fe than the waters from the North Atlantic and the Norwegian Sea where quite similar mean values are found. There is only little evidence for the assumption that overflow events on the ridge are influencing the concentrations of dissolved metals in the near-bottom layers.

On the Statistical Distribution of Object-Oriented System Properties

The statistical distributions of different software properties have been thoroughly studied in the past, including software size, complexity and the number of defects. In the case of object-oriented systems, these distributions have been found to obey a power law, a common statistical distribution also found in many other fields. However, we have found that for some statistical properties, the behavior does not entirely follow a power law, but a mixture between a lognormal and a power law distribution. Our study is based on the Qualitas Corpus, a large compendium of diverse Java-based software projects. We have measured the Chidamber and Kemerer metrics suite for every file of every Java project in the corpus. Our results show that the range of high values for the different metrics follows a power law distribution, whereas the rest of the range follows a lognormal distribution. This is a pattern typical of so-called double Pareto distributions, also found in empirical studies for other software properties.

Distribution of planktonic foraminifers, d18O and d13C in Globigerinoides sacculifer as well as Cibicidoides kullenbergi at DSDP Holes 90-590A and 90-590B

High biogenic sedimentation rates in the late Neogene at DSDP Site 590 (1293 m) provide an exceptional opportunity to evaluate late Neogene (late Miocene to latest Pliocene) paleoceanography in waters transitional between temperate and warm-subtropical water masses. Oxygen and carbon isotope analyses and quantitative planktonic foraminiferal data have been used to interpret the late Neogene paleoceanographic evolution of this site. Faunal and isotopic data from Site 590 show a progression of paleoceanographic events between 6.7 and 4.3 Ma, during the latest Miocene and early Pliocene. First, a permanent depletion in both planktonic and benthic foraminiferal d13C, between 6.7 and 6.2 Ma, can be correlated to the globally recognized late Miocene carbon isotope shift. Second, a 0.5 per mil enrichment in benthic foraminiferal d18O between 5.6 and 4.7 Ma in the latest Miocene to early Pliocene corresponds to the latest Miocene oxygen isotopic enrichment at Site 284, located in temperate waters south of Site 590. This enrichment in d18O coincides with a time of cool surface waters, as is suggested by high frequencies of Neogloboquadrina pachyderma and low frequencies of the warmer-water planktonic foraminifers, as well as by an enrichment in planktonic foraminiferal d18O relative to the earlier Miocene. By 4.6 Ma, benthic foraminiferal d18O values become depleted and remain fairly stable until about 3.8 Ma. The early Pliocene (~4.3 to 3.2 Ma) is marked by a significant increase in biogenic sedimentation rates (37.7 to 83.3 m/m.y.). During this time, heaviest values in planktonic foraminiferal d18O are associated with a decrease in the gradient between surface and intermediate-water d13C and d18O, a 1.0 per mil depletion in the d13C of two species of planktonic foraminifers, and a mixture of warm and cool planktonic foraminiferal elements. These data suggest that localized upwelling at the Subtropical Divergence produced an increase in surface-water productivity during the early Pliocene. A two-step enrichment in benthic foraminiferal d18O occurs in the late Pliocene sequence at Site 590. A 0.3 per mil average enrichment at about 3.6 Ma is followed by a 0.5 per mil enrichment at 2.7 Ma. These two events can be correlated with the two-step isotopic enrichment associated with late Pliocene climatic instability and the initiation of Northern Hemisphere glaciation.

On a resampling approach for tests on the number of clusters with mixture model-based clustering of tissue samples

We consider the problem of assessing the number of clusters in a limited number of tissue samples containing gene expressions for possibly several thousands of genes. It is proposed to use a normal mixture model-based approach to the clustering of the tissue samples. One advantage of this approach is that the question on the number of clusters in the data can be formulated in terms of a test on the smallest number of components in the mixture model compatible with the data. This test can be carried out on the basis of the likelihood ratio test statistic, using resampling to assess its null distribution. The effectiveness of this approach is demonstrated on simulated data and on some microarray datasets, as considered previously in the bioinformatics literature. (C) 2004 Elsevier Inc. All rights reserved.

A simple implementation of a normal mixture approach to differential gene expression in multiclass microarrays

Motivation: An important problem in microarray experiments is the detection of genes that are differentially expressed in a given number of classes. We provide a straightforward and easily implemented method for estimating the posterior probability that an individual gene is null. The problem can be expressed in a two-component mixture framework, using an empirical Bayes approach. Current methods of implementing this approach either have some limitations due to the minimal assumptions made or with more specific assumptions are computationally intensive. Results: By converting to a z-score the value of the test statistic used to test the significance of each gene, we propose a simple two-component normal mixture that models adequately the distribution of this score. The usefulness of our approach is demonstrated on three real datasets.

Mixture density networks

Minimization of a sum-of-squares or cross-entropy error function leads to network outputs which approximate the conditional averages of the target data, conditioned on the input vector. For classifications problems, with a suitably chosen target coding scheme, these averages represent the posterior probabilities of class membership, and so can be regarded as optimal. For problems involving the prediction of continuous variables, however, the conditional averages provide only a very limited description of the properties of the target variables. This is particularly true for problems in which the mapping to be learned is multi-valued, as often arises in the solution of inverse problems, since the average of several correct target values is not necessarily itself a correct value. In order to obtain a complete description of the data, for the purposes of predicting the outputs corresponding to new input vectors, we must model the conditional probability distribution of the target data, again conditioned on the input vector. In this paper we introduce a new class of network models obtained by combining a conventional neural network with a mixture density model. The complete system is called a Mixture Density Network, and can in principle represent arbitrary conditional probability distributions in the same way that a conventional neural network can represent arbitrary functions. We demonstrate the effectiveness of Mixture Density Networks using both a toy problem and a problem involving robot inverse kinematics.

Regularisation of mixture density networks

Mixture Density Networks are a principled method to model conditional probability density functions which are non-Gaussian. This is achieved by modelling the conditional distribution for each pattern with a Gaussian Mixture Model for which the parameters are generated by a neural network. This thesis presents a novel method to introduce regularisation in this context for the special case where the mean and variance of the spherical Gaussian Kernels in the mixtures are fixed to predetermined values. Guidelines for how these parameters can be initialised are given, and it is shown how to apply the evidence framework to mixture density networks to achieve regularisation. This also provides an objective stopping criteria that can replace the `early stopping' methods that have previously been used. If the neural network used is an RBF network with fixed centres this opens up new opportunities for improved initialisation of the network weights, which are exploited to start training relatively close to the optimum. The new method is demonstrated on two data sets. The first is a simple synthetic data set while the second is a real life data set, namely satellite scatterometer data used to infer the wind speed and wind direction near the ocean surface. For both data sets the regularisation method performs well in comparison with earlier published results. Ideas on how the constraint on the kernels may be relaxed to allow fully adaptable kernels are presented.

Multi-valued control problems and mixture density network

We have proposed a novel robust inversion-based neurocontroller that searches for the optimal control law by sampling from the estimated Gaussian distribution of the inverse plant model. However, for problems involving the prediction of continuous variables, a Gaussian model approximation provides only a very limited description of the properties of the inverse model. This is usually the case for problems in which the mapping to be learned is multi-valued or involves hysteritic transfer characteristics. This often arises in the solution of inverse plant models. In order to obtain a complete description of the inverse model, a more general multicomponent distributions must be modeled. In this paper we test whether our proposed sampling approach can be used when considering an arbitrary conditional probability distributions. These arbitrary distributions will be modeled by a mixture density network. Importance sampling provides a structured and principled approach to constrain the complexity of the search space for the ideal control law. The effectiveness of the importance sampling from an arbitrary conditional probability distribution will be demonstrated using a simple single input single output static nonlinear system with hysteretic characteristics in the inverse plant model.

Regularisation of mixture density networks

Mixture Density Networks are a principled method to model conditional probability density functions which are non-Gaussian. This is achieved by modelling the conditional distribution for each pattern with a Gaussian Mixture Model for which the parameters are generated by a neural network. This thesis presents a novel method to introduce regularisation in this context for the special case where the mean and variance of the spherical Gaussian Kernels in the mixtures are fixed to predetermined values. Guidelines for how these parameters can be initialised are given, and it is shown how to apply the evidence framework to mixture density networks to achieve regularisation. This also provides an objective stopping criteria that can replace the `early stopping' methods that have previously been used. If the neural network used is an RBF network with fixed centres this opens up new opportunities for improved initialisation of the network weights, which are exploited to start training relatively close to the optimum. The new method is demonstrated on two data sets. The first is a simple synthetic data set while the second is a real life data set, namely satellite scatterometer data used to infer the wind speed and wind direction near the ocean surface. For both data sets the regularisation method performs well in comparison with earlier published results. Ideas on how the constraint on the kernels may be relaxed to allow fully adaptable kernels are presented.

Towards asphalt mixture morphology evaluation with the virtual specimen approach

The morphology of asphalt mixture can be defined as a set of parameters describing the geometrical characteristics of its constituent materials, their relative proportions as well as spatial arrangement in the mixture. The present study is carried out to investigate the effect of the morphology on its meso- and macro-mechanical response. An analysis approach is used for the meso-structural characterisation based on the X-ray computed tomography (CT) data. Image processing techniques are used to systematically vary the internal structure to obtain different morphology structures. A morphology framework is used to characterise the average mastic coating thickness around the main load carrying structure in the structures. The uniaxial tension simulation shows that the mixtures with the lowest coating thickness exhibit better inter-particle interaction with more continuous load distribution chains between adjacent aggregate particles, less stress concentrations and less strain localisation in the mastic phase.