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PubMed2006

Mixture modeling for the detection of subpopulations in a pharmacokinetic/pharmacodynamic analysis.

To be able to estimate accurately parameters entering a non-linear mixed effects model taking into account that one or more subpopulations of patients can exist rather than assuming that the entire population is best described by unimodal distributions for the random effects, we proposed a methodology based on the likelihood approximation using the Gauss-Hermite quadrature. The idea is to combine the estimation of the model parameters and the detection of homogeneous subgroups of patients in a given population using a Gaussian mixture for the distribution of the random effects. As the accuracy of the likelihood approximation is likely to govern the quality of the estimation of the different parameters entering the non-linear mixed effects model, we based this approximation on the use of an adjustable Gauss-Hermite quadrature. Moreover, to complete this methodology, we propose a strategy allowing the detection and explanation of heterogeneity based on the Kullback-Leibler test, which was used to estimate the number of components in the Gaussian mixture. In order to evaluate the capability of the method to take into account heterogeneity, this strategy was performed in a PK/PD analysis using the database and the structural model selected in a previous analysis. In this analysis, non-responders were found out using NONMEM [Beal and Sheiner. NONMEM Users Guides. NONMEM Project Group, University of California, San Francisio, 1992] in a population of diabetic patients treated with a once-a-day new formulation of an antidiabetic drug. The authors looked for a subpopulation of patients for whom the therapeutic effect would vanish. In this paper, we looked for subpopulations of patients exhibiting specificities with respect to different parameters entering the description of the effect. The results obtained with our approach are compared in terms of parameter estimation and heterogeneity detection to those obtained in the previous analysis.

Blood Glucose↗
PubMed2006

On the quantitative genetics of mixture characters.

Finite mixture models are helpful for uncovering heterogeneity due to hidden structure. Quantitative genetics issues of continuous characters having a finite mixture of Gaussian components as statistical distribution are explored in this article. The partition of variance in a mixture, the covariance between relatives under the supposition of an additive genetic model, and the offspring-parent regression are derived. Formulas for assessing the effect of mass selection operating on a mixture are given. Expressions for the genetic and phenotypic correlations between mixture and Gaussian traits and between two mixture traits are presented. It is found that, if there is heterogeneity in a population at the genetic or environmental level, then genetic parameters based on theory treating distributions as homogeneous can lead to misleading interpretations. Some peculiarities of mixture characters are: heritability depends on the mean values of the component distributions, the offspring-parent regression is nonlinear, and genetic or phenotypic correlations cannot be interpreted devoid of the mixture proportions and of the parameters of the distributions mixed.

Environment↗
PubMed2001

Maximum likelihood inference for left-censored HIV RNA data.

Left-censored data are characteristic of many bioassays due to inherent limit of detection and limit of quantitation (LOQ) in the assays. This paper examines how the left-censoring of plasma HIV RNA measurements, collected for the Hemophilia Growth and Development Study, affects the quantification of viral load and the assessment of its association with a continuous or dichotomous outcome. Data analyses using maximum likelihood estimation are compared to analyses where the LOQ or LOQ/2 value is substituted for the left-censored observations, and also to other methods like multiple imputation. A Gaussian distribution is assumed for the log-transformed plasma HIV RNA data, and simulations are used to explore the sensitivity of the results to changes in the model parameters. The robustness of the estimators is also investigated when the data are generated from a mixture of two Gaussian distributions. Maximum likelihood is in general the least biased method. However, multiple imputation assuming a censored Gaussian imputation model and substituting the censored values with the expectation of its conditional predictive distribution are also competitive to maximum likelihood, and may be appealing because of their simpler computational algorithms.

Adolescent↗
PubMed2006

Probe-level linear model fitting and mixture modeling results in high accuracy detection of differential gene expression.

BACKGROUND: The identification of differentially expressed genes (DEGs) from Affymetrix GeneChips arrays is currently done by first computing expression levels from the low-level probe intensities, then deriving significance by comparing these expression levels between conditions. The proposed PL-LM (Probe-Level Linear Model) method implements a linear model applied on the probe-level data to directly estimate the treatment effect. A finite mixture of Gaussian components is then used to identify DEGs using the coefficients estimated by the linear model. This approach can readily be applied to experimental design with or without replication. RESULTS: On a wholly defined dataset, the PL-LM method was able to identify 75% of the differentially expressed genes within 10% of false positives. This accuracy was achieved both using the three replicates per conditions available in the dataset and using only one replicate per condition. CONCLUSION: The method achieves, on this dataset, a higher accuracy than the best set of tools identified by the authors of the dataset, and does so using only one replicate per condition.

Algorithms↗
PubMed2002

Unsupervised continuous sleep analysis.

One aim of the EU-funded project SIESTA was to develop a new way of describing the human sleep-wake continuum with high temporal resolution, and independent of subjective rules, to serve as an alternative to traditional sleep scoring. Here, we report new findings obtained with a fully automatic, probabilistic sleep-analyzer using Hidden Markov Models (HMMs) based on data from a single electroencephalogram (EEG) channel. HMMs allow the analysis of non-stationary time series by modeling both the probability density functions of locally stationary data and the transition probabilities between these stable states. In the context of sleep analysis, the locally stable states can be thought of as sleep stages. The sleep-wake continuum was modeled as a mixture of three different processes by defining a three-state Gaussian Observation HMM (GOHMM). No class information from human scorers was used to train the model. The probabilities of being in any of the three states at each point in time roughly indicate the amount of wakefulness, deep sleep and rapid-eye-movement (REM) sleep with a one-second time resolution. Although it was not the aim to replicate the traditional Rechtschaffen and Kales (R&K) scoring, pseudo R&K hypnograms were constructed from the probability plots in order to compare the analyzer results with classical sleep stages by human experts. We expected that the analyzer would be able to classify data correctly from the "cornerstones" of human sleep (wakefulness, deep sleep, and REM sleep). Contrary to our previous efforts, we trained the HMMs on data from two different sleep laboratories separately, instead of generalizing data from diverse laboratories. While these stages could be detected with an accuracy of around 80% at the sleep laboratory for which we already had achieved the best results, there was no improvement from previous results by the training of a separate model in the other laboratory. This finding indicates clear laboratory effects in the signal characteristics, probably due to differences in hardware and filter settings. The presented approach, going beyond a mere replication of the traditional R&K standard, offers a continuous description of human sleep which is based on probabilistic principles. It provides a second-by-second quantification of the sleep-wake continuum and captures, although being entirely data-driven instead of rule-based, the three main processes in human sleep: wakefulness, deep sleep and REM sleep.

Adult↗
PubMed2006

Modeling the amplitude statistics of ultrasonic images.

In this paper, a new statistical model for representing the amplitude statistics of ultrasonic images is presented. The model is called the Rician inverse Gaussian (RiIG) distribution, due to the fact that it is constructed as a mixture of the Rice distribution and the Inverse Gaussian distribution. The probability density function (pdf) of the RiIG model is given in closed form as a function of three parameters. Some theoretical background on this new model is discussed, and an iterative algorithm for estimating its parameters from data is given. Then, the appropriateness of the RiIG distribution as a model for the amplitude statistics of medical ultrasound images is experimentally studied. It is shown that the new distribution can fit to the various shapes of local histograms of linearly scaled ultrasound data better than existing models. A log-likelihood cross-validation comparison of the predictive performance of the RiIG, the K, and the generalized Nakagami models turns out in favor of the new model. Furthermore, a maximum a posteriori (MAP) filter is developed based on the RiIG distribution. Experimental studies show that the RiIG MAP filter has excellent filtering performance in the sense that it smooths homogeneous regions, and at the same time preserves details.

Algorithms↗
PubMed2003

Singularities in mixture models and upper bounds of stochastic complexity.

A learning machine which is a mixture of several distributions, for example, a gaussian mixture or a mixture of experts, has a wide range of applications. However, such a machine is a non-identifiable statistical model with a lot of singularities in the parameter space, hence its generalization property is left unknown. Recently an algebraic geometrical method has been developed which enables us to treat such learning machines mathematically. Based on this method, this paper rigorously proves that a mixture learning machine has the smaller Bayesian stochastic complexity than regular statistical models. Since the generalization error of a learning machine is equal to the increase of the stochastic complexity, the result of this paper shows that the mixture model can attain the more precise prediction than regular statistical models if Bayesian estimation is applied in statistical inference.

Bayes Theorem↗
PubMed1996

An introduction to finite mixture distributions.

Finite mixture densities can be used to model data from populations known or suspected to contain a number of separate subpopulations. Most commonly used are mixture densities with Gaussian (univariate or multivariate) components, but mixtures with other types of component are also increasingly used to model, for example, survival times. This paper gives a general introduction to the topic which should help when considering the other more specialized papers in this issue.

Adult↗
PubMed2000

Probabilistic-based approach to optimal filtering

The signal-to-noise ratio maximizing approach in optimal filtering provides a robust tool to detect signals in the presence of colored noise. The method fails, however, when the data present a regimelike behavior. An approach is developed in this manuscript to recover local (in phase space) behavior in an intermittent regimelike behaving system. The method is first formulated in its general form within a Gaussian framework, given an estimate of the noise covariance, and demands that the signal corresponds to minimizing the noise probability distribution for any given value, i.e., on isosurfaces, of the data probability distribution. The extension to the non-Gaussian case is provided through the use of finite mixture models for data that show regimelike behavior. The method yields the correct signal when applied in a simplified manner to synthetic time series with and without regimes, compared to the signal-to-noise ratio approach, and helps identify the right frequency of the oscillation spells in the classical and variants of the Lorenz system.

Journal Article↗
PubMed2006

Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification.

We present a method for automated segmentation of the vasculature in retinal images. The method produces segmentations by classifying each image pixel as vessel or nonvessel, based on the pixel's feature vector. Feature vectors are composed of the pixel's intensity and two-dimensional Gabor wavelet transform responses taken at multiple scales. The Gabor wavelet is capable of tuning to specific frequencies, thus allowing noise filtering and vessel enhancement in a single step. We use a Bayesian classifier with class-conditional probability density functions (likelihoods) described as Gaussian mixtures, yielding a fast classification, while being able to model complex decision surfaces. The probability distributions are estimated based on a training set of labeled pixels obtained from manual segmentations. The method's performance is evaluated on publicly available DRIVE (Staal et al., 2004) and STARE (Hoover et al., 2000) databases of manually labeled images. On the DRIVE database, it achieves an area under the receiver operating characteristic curve of 0.9614, being slightly superior than that presented by state-of-the-art approaches. We are making our implementation available as open source MATLAB scripts for researchers interested in implementation details, evaluation, or development of methods.

Algorithms↗
PubMed2005

Clustering expressed genes on the basis of their association with a quantitative phenotype.

Cluster analyses of gene expression data are usually conducted based on their associations with the phenotype of a particular disease. Many disease traits have a clearly defined binary phenotype (presence or absence), so that genes can be clustered based on the differences of expression levels between the two contrasting phenotypic groups. For example, cluster analysis based on binary phenotype has been successfully used in tumour research. Some complex diseases have phenotypes that vary in a continuous manner and the method developed for a binary trait is not immediately applicable to a continuous trait. However, understanding the role of gene expression in these complex traits is of fundamental importance. Therefore, it is necessary to develop a new statistical method to cluster expressed genes based on their association with a quantitative trait phenotype. We developed a model-based clustering method to classify genes based on their association with a continuous phenotype. We used a linear model to describe the relationship between gene expression and the phenotypic value. The model effects of the linear model (linear regression coefficients) represent the strength of the association. We assumed that the model effects of each gene follow a mixture of several multivariate Gaussian distributions. Parameter estimation and cluster assignment were accomplished via an Expectation-Maximization (EM) algorithm. The method was verified by analysing two simulated datasets, and further demonstrated using real data generated in a microarray experiment for the study of gene expression associated with Alzheimer's disease.

Algorithms↗
PubMed1998

Soft vector quantization and the EM algorithm.

The relation between hard c-means (HCM), fuzzy c-means (FCM), fuzzy learning vector quantization (FLVQ), soft competition scheme (SCS) of [Yair et al. (1992)] and probabilistic Gaussian mixtures (GM) have been pointed out recently by [Bezdek and Pal (1995)]. We extend this relation to their training, showing that learning rules by these models to estimate the cluster centers can be seen as approximations to the expectation-maximization (EM) method as applied to Gaussian mixtures. HCM and unsupervised, LVQ use 1-of-c type competition. In FCM and FLVQ, membership is the -2/(m-1)th power of the distance. In SCS and GM, Gaussian function is used. If the Gaussian membership function is used, the weighted within-groups sum of squared errors used as the fuzzy objective function corresponds to the maximum likelihood estimate in Gaussian mixtures with equal priors and covariances. The fuzzy clustering method named fuzzy c-means alternating optimization procedure (FCM-AO) proposed to optimize the former is then equivalent to batch EM and SCS's update rule is a variant of the online version of EM. The advantages of the probabilistic framework are: (i) we no longer have spurious spread parameters that needs fine tuning as m in fuzzy vector quantization or beta in SCS; instead we have a variance term that has a sound interpretation and that can be estimated from the sample; (ii) EM guarantees that the likelihood does not decrease, thus it converges to the nearest local optimum; (iii) EM also allows us to estimate the underlying distance norm and the cluster priors which we could not with the other approaches. We compare Gaussian mixtures trained with EM with LVQ (HCM), SCS and FLVQ on the IRIS dataset and see that it is more accurate due to its being able to take into account the covariance information. We finally note that vector quantization is generally an intermediate step before finding a final output for which supervision may be possible. Thus, instead of an uncoupled approach where an unsupervised method is used first to find the cluster parameters followed by supervised training of the mapping based on the memberships, we advocate a coupled approach where the cluster parameters and mapping are trained supervised in a coupled way. The uncoupled approach ignores the error at the outputs which may not be ideal.

Journal Article↗
PubMed2006

Bayesian wavelet-based image deconvolution: a GEM algorithm exploiting a class of heavy-tailed priors.

Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The well-known sparsity of the wavelet coefficients of real-world images is modeled by heavy-tailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys, and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the "nonnegative garrote" thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary second-order iterative method. The result is a GEM algorithm of O(N log N) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to state-of-the art methods, demanding comparable (in some cases, much less) computational complexity.

Algorithms↗
PubMed2005

Modeling of binary adsorption on heterogeneous surfaces characterized by a quasi-gaussian adsorption energy distribution.

The integral equation (IE) approach coupled with a quasi-Gaussian adsorption energy distribution is used to model the adsorption of single gases and their binary mixture on a heterogeneous solid surface. The adsorbing surface is assumed to be characterized by two, generally different in width, quasi-Gaussian distribution functions, each of them related to a single component of the mixture. The influence of correlations between the distribution functions associated with different components on the corresponding adsorption isotherms and phase diagrams is discussed. In particular, it is demonstrated that a lack of microscopic correlations between the adsorption energies of the components may lead to the formation of an azeotropic mixture. The predictions of the theory are also compared with the results of the grand canonical Monte Carlo (GCMC) simulations carried out for the system studied.

Journal Article↗
PubMed2001

Binary Gaussian core model: fluid-fluid phase separation and interfacial properties.

Using a mean-field equation of state we calculate the density-concentration phase diagrams for a binary mixture of repulsive Gaussian core particles over a range of size ratios. A simple mean-field density functional (DFT) approach, equivalent to the random phase approximation, is used to calculate the surface tension and density profiles of the interface between the demixed fluid phases of the binary mixture. For certain coexisting states oscillations are found in the density profiles on both sides of the interface, i.e., approaching both bulk phases. The form of the oscillations is determined by the asymptotic decay of the bulk total pairwise correlations, and the onset of oscillations in the interfacial density profiles depends on the location of the crossover line (Fisher-Widom line) in the bulk phase diagram where the asymptotic decay changes from monotonic to damped oscillatory. For certain particle size ratios we find another crossover line that separates a region of the phase diagram where the longest-range decay of the pairwise correlations is damped oscillatory from a region where the longest-range decay is damped oscillatory but with a different wavelength. We argue that many of the predictions of the simple DFT approach should remain valid in more refined treatments.

Journal Article↗
PubMed2005

Structural group classification technique based on regional fMRI BOLD responses.

This paper presents a new multigroup classification method based on subtle differences in regional brain activity during the completion of a functional magnetic resonance imaging (fMRI) challenge paradigm. Classification is performed based on features derived from BOLD time intensity curves in selected regions of interest (ROI). For each ROI, a mean time intensity curve [called mean regional response (MRR)] is calculated from realigned and normalized datasets. The overall subject performance is characterized with a vector of features obtained using nonlinear modeling of all subject's MRRs with a mixture of time shifted Gaussian functions. The classification is performed in the reduced-dimension optimal discrimination space, obtained through canonical transformations of original feature space. In order to demonstrate feasibility of the proposed method, classification of three groups of subjects is presented. The three groups are defined as heavy marijuana smokers after 24 hours of abstinence, heavy marijuana smokers after 28 days of abstinence, and healthy nonusing controls. The proposed method can be useful as an analytic tool for the discrimination of different groups of subjects based on temporal features of functional magnetic resonance imaging activation.

Adult↗
PubMed2006

PDE-based image restoration: a hybrid model and color image denoising.

The paper is concerned with PDE-based image restoration. A new model is introduced by hybridizing a nonconvex variant of the total variation minimization (TVM) and the motion by mean curvature (MMC) in order to deal with the mixture of the impulse and Gaussian noises reliably. We suggest the essentially nondissipative (ENoD) difference schemes for the MMC component to eliminate the impulse noise with a minimum (ideally no) introduction of dissipation. The MMC-TVM hybrid model and the ENoD schemes are applied for both gray-scale and color images. For color image denoising, we consider the chromaticity-brightness decomposition with the chromaticity formulated in the angle domain. An incomplete Crank-Nicolson alternating direction implicit time-stepping procedure is adopted to solve those differential equations efficiently. Numerical experiments have shown that the new hybrid model and the numerical schemes can remove the mixture of the impulse and Gaussian noises, efficiently and reliably, preserving edges quite satisfactorily.

Algorithms↗
PubMedDate not supplied

A comparison of parametric and nonparametric approaches to ROC analysis of quantitative diagnostic tests.

Receiver operating characteristic (ROC) analysis, which yields indices of accuracy such as the area under the curve (AUC), is increasingly being used to evaluate the performances of diagnostic tests that produce results on continuous scales. Both parametric and nonparametric ROC approaches are available to assess the discriminant capacity of such tests, but there are no clear guidelines as to the merits of each, particularly with non-binormal data. Investigators may worry that when data are non-Gaussian, estimates of diagnostic accuracy based on a binormal model may be distorted. The authors conducted a Monte Carlo simulation study to compare the bias and sampling variability in the estimates of the AUCs derived from parametric and nonparametric procedures. Each approach was assessed in data sets generated from various configurations of pairs of overlapping distributions; these included the binormal model and non-binormal pairs of distributions where one or both pair members were mixtures of Gaussian (MG) distributions with different degrees of departures from binormality. The biases in the estimates of the AUCs were found to be very small for both parametric and nonparametric procedures. The two approaches yielded very close estimates of the AUCs and the corresponding sampling variability even when data were generated from non-binormal models. Thus, for a wide range of distributions, concern about bias or imprecision of the estimates of the AUC should not be a major factor in choosing between the nonparametric and parametric approaches.

Bias↗