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PubMed1994

Parametric estimation in a genetic mixture model with application to nuclear family data.

The apparent conflict between the biometrician and Mendelian genetics has been recently resolved by the introduction of a genetic mixed model to analyze continuous traits measured on human families and to elucidate the mechanism of underlying major genes. The mixed model formulated by Elston and Stewart (1971, Human Heredity 21, 523-542), extended by Morton and MacLean (1974, American Journal of Human Genetics 26, 489-503), and reviewed, with further extensions, by Boyle and Elston (1979, Biometrics 35, 55-68) has become an extremely useful tool of wide applicability in the field of genetic epidemiology. This model allows for segregation at a major locus, a polygenic effect, and a sibling environmental variation. The main concern of this paper is with estimating the model parameters by the method of maximum likelihood. The expectation-maximization (EM) algorithm is developed to derive the estimates iteratively. An approximation of the information matrix when using the EM algorithm is given. We illustrate the methodology by fitting the model to the arterial blood pressure data collected by Miall and Oldham (1955, Clinical Science 14, 459-487).

Biometry↗
PubMed2006

Comparative assessment of statistical brain MR image segmentation algorithms and their impact on partial volume correction in PET.

Magnetic resonance imaging (MRI)-guided partial volume effect correction (PVC) in brain positron emission tomography (PET) is now a well-established approach to compensate the large bias in the estimate of regional radioactivity concentration, especially for small structures. The accuracy of the algorithms developed so far is, however, largely dependent on the performance of segmentation methods partitioning MRI brain data into its main classes, namely gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF). A comparative evaluation of three brain MRI segmentation algorithms using simulated and clinical brain MR data was performed, and subsequently their impact on PVC in 18F-FDG and 18F-DOPA brain PET imaging was assessed. Two algorithms, the first is bundled in the Statistical Parametric Mapping (SPM2) package while the other is the Expectation Maximization Segmentation (EMS) algorithm, incorporate a priori probability images derived from MR images of a large number of subjects. The third, here referred to as the HBSA algorithm, is a histogram-based segmentation algorithm incorporating an Expectation Maximization approach to model a four-Gaussian mixture for both global and local histograms. Simulated under different combinations of noise and intensity non-uniformity, MR brain phantoms with known true volumes for the different brain classes were generated. The algorithms' performance was checked by calculating the kappa index assessing similarities with the "ground truth" as well as multiclass type I and type II errors including misclassification rates. The impact of image segmentation algorithms on PVC was then quantified using clinical data. The segmented tissues of patients' brain MRI were given as input to the region of interest (RoI)-based geometric transfer matrix (GTM) PVC algorithm, and quantitative comparisons were made. The results of digital MRI phantom studies suggest that the use of HBSA produces the best performance for WM classification. For GM classification, it is suggested to use the EMS. Segmentation performed on clinical MRI data show quite substantial differences, especially when lesions are present. For the particular case of PVC, SPM2 and EMS algorithms show very similar results and may be used interchangeably. The use of HBSA is not recommended for PVC. The partial volume corrected activities in some regions of the brain show quite large relative differences when performing paired analysis on 2 algorithms, implying a careful choice of the segmentation algorithm for GTM-based PVC.

Algorithms↗
PubMed2001

Estimation of linkage disequilibrium for loci with multiple alleles: basic approach and an application using data from bighorn sheep.

The great expansion of population genetic data using molecular techniques now allows examination of the extent of linkage disequilibrium for many pairs of loci, each locus often with multiple alleles. The expectation-maximization (EM) algorithm for generating maximum likelihood estimates of gametic frequencies from multiallelic genotypic data is described and applied. The EM algorithm is used in desert bighorn sheep where the population size, and consequently the sample size, is often small. We calculated haplotype frequencies for all pairwise combinations of five major histocompatibility loci and three microsatellite loci in 14 populations; the performance of the algorithm is discussed. Disequilibrium values are calculated and tested for statistical significance. High levels of disequilibrium are found between all pairs of major histocompatibility complex (MHC) loci and between MHC and a linked microsatellite locus.

Alberta↗
PubMed2001

Haplotyping and estimation of haplotype frequencies for closely linked biallelic multilocus genetic phenotypes including nuclear family information.

With the discovery of single nucleotide polymorphisms (SNP) along the genome, genotyping of large samples of biallelic multilocus genetic phenotypes for (fine) mapping of disease genes or for population studies has become standard practice. A genetic trait, however, is mainly caused by an underlying defective haplotype, and populations are best characterized by their haplotype frequencies. Therefore, it is essential to infer from the phase-unknown genetic phenotypes in a sample drawn from a population the haplotype frequencies in the population and the underlying haplotype pairs in the sample in order to find disease predisposing genes by some association or haplotype sharing algorithm. Haplotype frequencies and haplotype pairs are estimated via a maximum likelihood approach by a well-known expectation maximization (EM) algorithm, adapting it to a large number (up to 30) of biallelic loci (SNP), and including nuclear family information, if available, into the analysis. Parents are treated as an independent sample from the population. Their genotyped offspring reduces the number of potential haplotype pairs for both parents, resulting in a higher accuracy of the estimation, and may also reduce computation time. In a series of simulations our approach of including nuclear family information has been tested against both the EM algorithm without nuclear family information and an alternative approach using GENEHUNTER for the haplotyping of the families, using the locus-by-locus allele counts of the sample. Our new approach is more precise in haplotyping in cases of a high number of heterozygous loci, whereas for a moderate number of heterozygous positions in the sample all three different approaches gave the same perfect results.

Algorithms↗
PubMed2002

Dual-modality imaging of cancer with SPECT/CT.

Dual-modality imaging is an in vivo diagnostic technique that obtains structural and functional information directly from patient studies in a way that cannot be achieved with separate imaging systems alone. Dual-modality imaging systems are configured by combining computed tomography (CT) with radionuclide imaging (using positron emission tomography (PET) or single-photon emission computed tomography (SPECT)) on a single gantry which allows both functional and structural imaging to be performed during a single imaging session without having the patient leave the imaging system. A SPECT/CT system developed at UCSF is being used in a study to determine if dual-modality imaging offers advantages for assessment of patients with prostate cancer using (111)In-ProstaScint, a radiolabeled antibody for the prostate-specific membrane antigen. (111)In-ProstaScint images are reconstructed using an iterative maximum-likelihood expectation-maximization (ML-EM) algorithm with correction for photon attenuation using a patient-specific map of attenuation coefficients derived from CT. The ML-EM algorithm accounts for the dual-photon nature of the 111In-labeled radionuclide, and incorporates correction for the geometric response of the radionuclide collimator. The radionuclide image then can be coregistered and overlaid in color on a grayscale CT image for improved localization of the functional information from SPECT. Radionuclide images obtained with SPECT/CT and reconstructed using ML-EM with correction for photon attenuation and collimator response improve image quality in comparison to conventional radionuclide images obtained with filtered backprojection reconstruction. These results illustrate the potential advantages of dual-modality imaging for improving the quality and the localization of radionuclide uptake for staging disease, planning treatment, and monitoring therapeutic response in patients with cancer.

Algorithms↗
PubMed2001

Segmentation of brain MR images through a hidden Markov random field model and the expectation-maximization algorithm.

The finite mixture (FM) model is the most commonly used model for statistical segmentation of brain magnetic resonance (MR) images because of its simple mathematical form and the piecewise constant nature of ideal brain MR images. However, being a histogram-based model, the FM has an intrinsic limitation--no spatial information is taken into account. This causes the FM model to work only on well-defined images with low levels of noise; unfortunately, this is often not the the case due to artifacts such as partial volume effect and bias field distortion. Under these conditions, FM model-based methods produce unreliable results. In this paper, we propose a novel hidden Markov random field (HMRF) model, which is a stochastic process generated by a MRF whose state sequence cannot be observed directly but which can be indirectly estimated through observations. Mathematically, it can be shown that the FM model is a degenerate version of the HMRF model. The advantage of the HMRF model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites. Although MRF modeling has been employed in MR image segmentation by other researchers, most reported methods are limited to using MRF as a general prior in an FM model-based approach. To fit the HMRF model, an EM algorithm is used. We show that by incorporating both the HMRF model and the EM algorithm into a HMRF-EM framework, an accurate and robust segmentation can be achieved. More importantly, the HMRF-EM framework can easily be combined with other techniques. As an example, we show how the bias field correction algorithm of Guillemaud and Brady (1997) can be incorporated into this framework to achieve a three-dimensional fully automated approach for brain MR image segmentation.

Algorithms↗
PubMed1999

Maximum likelihood estimation of cerebral blood flow in dynamic susceptibility contrast MRI.

For quantification of cerebral blood flow (CBF) using dynamic susceptibility contrast magnetic resonance imaging (DSC-MRI), knowledge of the tissue response function is necessary. To obtain this, the tissue contrast passage measurement must be corrected for the arterial input. This study proposes an iterative maximum likelihood expectation maximization (ML-EM) algorithm for this correction, which takes into account the noise in T2- or T2*-weighted image sequences. The ML-EM algorithm does not assume a priori knowledge of the shape of the response function; it automatically corrects for arrival time offsets and inherently yields positive response values. The results on synthetic image sequences are presented, for which the recovered flow values and the response functions are in good agreement with their expectation values. The method is illustrated by calculating the gray and white matter flow in a clinical example.

Cerebrovascular Circulation↗
PubMed2007

A new estimation method for the semiparametric accelerated failure time mixture cure model.

The proportional hazard (PH) mixture cure model and the accelerated failure time (AFT) mixture cure model are usually used in analysing failure time data with long-term survivors. However, the semiparametric AFT mixture cure model has attracted less attention than the semiparametric PH mixture cure model because of the complexity of its estimation method. In this paper, we propose a new estimation method for the semiparametric AFT mixture cure model. This method employs the EM algorithm and the rank estimator of the AFT model to estimate the parameters of interest. The M-step in the EM algorithm, which incorporates the rank-like estimating equation, can be carried out easily using the linear programming method. To evaluate the performance of the proposed method, we conduct a simulation study. The results of the simulation study demonstrate that the proposed method performs better than the existing estimation method and the semiparametric AFT mixture cure model improves the identifiability of the parameters in comparison to the parametric AFT mixture cure model. To illustrate, we apply the model and the proposed method to a data set of failure times from bone marrow transplant patients.

Algorithms↗
PubMed2026

ADAMIXTURE: adaptive first-order optimization for biobank-scale genetic clustering.

MOTIVATION: Estimating genetic clusters from sequencing data is a fundamental task in population and medical genetics, enabling demographic inference and adjustment for population structure in association studies. ADMIXTURE, a widely used model-based clustering method, employs an accelerated Expectation-Maximization (EM) algorithm to infer population parameters; however, its computational demands scale poorly, limiting its usefulness for modern biobank-sized datasets. While recent EM acceleration strategies employing second-order quasi-Newton schemes preserve accuracy, they remain computationally intensive. Conversely, EM-free approaches that prioritize speed often compromise solution quality. RESULTS: We introduce ADAMIXTURE, a novel optimization framework that integrates the EM algorithm with Adaptive Moment Estimation (Adam). Unlike traditional acceleration methods, ADAMIXTURE utilizes first-order gradients with adaptive learning rates derived from raw and squared moments to approximate curvature information, bypassing the computational overhead of Hessian approximations. This approach surpasses the convergence efficiency of second-order methods while maintaining the low computational complexity of first-order updates. Across simulated and large-scale empirical datasets, ADAMIXTURE demonstrates substantial reductions in wall-clock runtime and enhanced scalability compared to state-of-the-art methods, while maintaining comparable or improved inference accuracy. Its GPU implementation runs in under 2 h on half a million samples and variants, a two order of magnitude speedup over current state-of-the-art. AVAILABILITY AND IMPLEMENTATION: Source code is available at: https://github.com/AI-sandbox/ADAMIXTURE.

Clustering Algorithms↗
PubMed2005

Generalized linear latent variable models for repeated measures of spatially correlated multivariate data.

Observations of multiple-response variables across space and over time occur often in environmental and ecological studies. Compared to purely spatial models for a single response variable in the exponential family of distributions, fewer statistical tools are available for multiple-response variables that are not necessarily Gaussian. An exception is a common-factor model developed for multivariate spatial data by Wang and Wall (2003, Biostatistics 4, 569-582). The purpose of this article is to extend this multivariate space-only model and develop a flexible class of generalized linear latent variable models for multivariate spatial-temporal data. For statistical inference, maximum likelihood estimates and their standard deviations are obtained using a Monte Carlo EM algorithm. We also use a novel way to automatically adjust the Monte Carlo sample size, which facilitates the convergence of the Monte Carlo EM algorithm. The methodology is illustrated by an ecological study of red pine trees in response to bark beetle challenges in a forest stand of Wisconsin.

Algorithms↗
PubMed1993

Metropolis sampling in pedigree analysis.

This paper reviews and develops the applications of Markov chain Monte Carlo methods in pedigree analysis, with particular stress on the Metropolis algorithm. In likelihood based genetic analyses, standard deterministic algorithms often fail because of the computational complexity of the observed pedigree data under a proposed genetic model. The new Monte Carlo methods permit approximate maximum likelihood estimation in the presence of such complexity. Monte Carlo implementation of the EM algorithm is the key to successful maximum likelihood analysis. Gibbs sampling and the Metropolis algorithm are alternative ways of defining Markov chains for performing the E step of the EM algorithm. Two applications illustrate the power and simplicity of the Metropolis algorithm. One of these applications involves a discrete model for variance component analysis of quantitative traits; the other application involves a Monte Carlo version of location scores for multipoint linkage analysis.

Algorithms↗
PubMed2004

Algorithms for inferring haplotypes.

Haplotype phase information in diploid organisms provides valuable information on human evolutionary history and may lead to the development of more efficient strategies to identify genetic variants that increase susceptibility to human diseases. Molecular haplotyping methods are labor-intensive, low-throughput, and very costly. Therefore, algorithms based on formal statistical theories were shown to be very effective and cost-efficient for haplotype reconstruction. This review covers 1) population-based haplotype inference methods: Clark's algorithm, expectation-maximization (EM) algorithm, coalescence-based algorithms (pseudo-Gibbs sampler and perfect/imperfect phylogeny), and partition-ligation algorithm implemented by a fully Bayesian model (Haplotyper) or by EM (PLEM); 2) family-based haplotype inference methods; 3) the handling of genotype scoring uncertainties (i.e., genotyping errors and raw two-dimensional genotype scatterplots) in inferring haplotypes; and 4) haplotype inference methods for pooled DNA samples. The advantages and limitations of each algorithm are discussed. By using simulations based on empirical data on the G6PD gene and TNFRSF5 gene, I demonstrate that different algorithms have different degrees of sensitivity to various extents of population diversities and genotyping error rates. Future development of statistical algorithms for addressing haplotype reconstruction will resort more and more to ideas based on combinatorial mathematics, graphical models, and machine learning, and they will have profound impacts on population genetics and genetic epidemiology with the advent of the human HapMap.

Algorithms↗
PubMed2004

HAPLORE: a program for haplotype reconstruction in general pedigrees without recombination.

MOTIVATION: Haplotype reconstruction is an essential step in genetic linkage and association studies. Although many methods have been developed to estimate haplotype frequencies and reconstruct haplotypes for a sample of unrelated individuals, haplotype reconstruction in large pedigrees with a large number of genetic markers remains a challenging problem. METHODS: We have developed an efficient computer program, HAPLORE (HAPLOtype REconstruction), to identify all haplotype sets that are compatible with the observed genotypes in a pedigree for tightly linked genetic markers. HAPLORE consists of three steps that can serve different needs in applications. In the first step, a set of logic rules is used to reduce the number of compatible haplotypes of each individual in the pedigree as much as possible. After this step, the haplotypes of all individuals in the pedigree can be completely or partially determined. These logic rules are applicable to completely linked markers and they can be used to impute missing data and check genotyping errors. In the second step, a haplotype-elimination algorithm similar to the genotype-elimination algorithms used in linkage analysis is applied to delete incompatible haplotypes derived from the first step. All superfluous haplotypes of the pedigree members will be excluded after this step. In the third step, the expectation-maximization (EM) algorithm combined with the partition and ligation technique is used to estimate haplotype frequencies based on the inferred haplotype configurations through the first two steps. Only compatible haplotype configurations with haplotypes having frequencies greater than a threshold are retained. RESULTS: We test the effectiveness and the efficiency of HAPLORE using both simulated and real datasets. Our results show that, the rule-based algorithm is very efficient for completely genotyped pedigree. In this case, almost all of the families have one unique haplotype configuration. In the presence of missing data, the number of compatible haplotypes can be substantially reduced by HAPLORE, and the program will provide all possible haplotype configurations of a pedigree under different circumstances, if such multiple configurations exist. These inferred haplotype configurations, as well as the haplotype frequencies estimated by the EM algorithm, can be used in genetic linkage and association studies. AVAILABILITY: The program can be downloaded from http://bioinformatics.med.yale.edu.

Algorithms↗
PubMed2001

Maximum likelihood methods for cure rate models with missing covariates.

We propose maximum likelihood methods for parameter estimation for a novel class of semiparametric survival models with a cure fraction, in which the covariates are allowed to be missing. We allow the covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one-dimensional conditional distributions. We propose a novel EM algorithm for maximum likelihood estimation and derive standard errors by using Louis's formula (Louis, 1982, Journal of the Royal Statistical Society, Series B 44, 226-233). Computational techniques using the Monte Carlo EM algorithm are discussed and implemented. A real data set involving a melanoma cancer clinical trial is examined in detail to demonstrate the methodology.

Algorithms↗
PubMedDate not supplied

EM-REML estimation of covariance parameters in Gaussian mixed models for longitudinal data analysis.

This paper presents procedures for implementing the EM algorithm to compute REML estimates of variance covariance components in Gaussian mixed models for longitudinal data analysis. The class of models considered includes random coefficient factors, stationary time processes and measurement errors. The EM algorithm allows separation of the computations pertaining to parameters involved in the random coefficient factors from those pertaining to the time processes and errors. The procedures are illustrated with Pothoff and Roy's data example on growth measurements taken on 11 girls and 16 boys at four ages. Several variants and extensions are discussed.

Journal Article↗
PubMed2000

The Bayesian evidence scheme for regularizing probability-density estimating neural networks.

Training probability-density estimating neural networks with the expectation-maximization (EM) algorithm aims to maximize the likelihood of the training set and therefore leads to overfitting for sparse data. In this article, a regularization method for mixture models with generalized linear kernel centers is proposed, which adopts the Bayesian evidence approach and optimizes the hyperparameters of the prior by type II maximum likelihood. This includes a marginalization over the parameters, which is done by Laplace approximation and requires the derivation of the Hessian of the log-likelihood function. The incorporation of this approach into the standard training scheme leads to a modified form of the EM algorithm, which includes a regularization term and adapts the hyperparameters on-line after each EM cycle. The article presents applications of this scheme to classification problems, the prediction of stochastic time series, and latent space models.

Acquired Immunodeficiency Syndrome↗
PubMed2003

Estimating a state-space model from point process observations.

A widely used signal processing paradigm is the state-space model. The state-space model is defined by two equations: an observation equation that describes how the hidden state or latent process is observed and a state equation that defines the evolution of the process through time. Inspired by neurophysiology experiments in which neural spiking activity is induced by an implicit (latent) stimulus, we develop an algorithm to estimate a state-space model observed through point process measurements. We represent the latent process modulating the neural spiking activity as a gaussian autoregressive model driven by an external stimulus. Given the latent process, neural spiking activity is characterized as a general point process defined by its conditional intensity function. We develop an approximate expectation-maximization (EM) algorithm to estimate the unobservable state-space process, its parameters, and the parameters of the point process. The EM algorithm combines a point process recursive nonlinear filter algorithm, the fixed interval smoothing algorithm, and the state-space covariance algorithm to compute the complete data log likelihood efficiently. We use a Kolmogorov-Smirnov test based on the time-rescaling theorem to evaluate agreement between the model and point process data. We illustrate the model with two simulated data examples: an ensemble of Poisson neurons driven by a common stimulus and a single neuron whose conditional intensity function is approximated as a local Bernoulli process.

Action Potentials↗
PubMed1998

Robust maximum likelihood training of heteroscedastic probabilistic neural networks.

We consider the probabilistic neural network (PNN) that is a mixture of Gaussian basis functions having different variances. Such a Gaussian heteroscedastic PNN is more economic, in terms of the number of kernel functions required, than the Gaussian mixture PNN of a common variance. The expectation-maximisation (EM) algorithm, although a powerful technique for constructing maximum likelihood (ML) homoscedastic PNNs, often encounters numerical difficulties when training heteroscedastic PNNs. We combine a robust statistical technique known as the Jack-knife with the EM algorithm to provide a robust ML training algorithm. An artificial-data case, the two-dimensional XOR problem, and a real-data case, success or failure prediction of UK private construction companies, are used to evaluate the performance of this robust learning algorithm.

Journal Article↗