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PubMed2005

Randomized algorithms for motif detection.

MOTIVATION: Motif detection for DNA sequences has many important applications in biological studies, e.g. locating binding sites regulatory signals, designing genetic probes etc. In this paper, we propose a randomized algorithm, design an improved EM algorithm and combine them to form a software tool. RESULTS: (1) We design a randomized algorithm for consensus pattern problem. We can show that with high probability, our randomized algorithm finds a pattern in polynomial time with cost error at most x l for each string, where l is the length of the motif and can be any positive number given by the user. (2) We design an improved EM algorithm that outperforms the original EM algorithm. (3) We develop a software tool, MotifDetector, that uses our randomized algorithm to find good seeds and uses the improved EM algorithm to do local search. We compare MotifDetector with Buhler and Tompa's PROJECTION which is considered to be the best known software for motif detection. Simulations show that MotifDetector is slower than PROJECTION when the pattern length is relatively small, and outperforms PROJECTION when the pattern length becomes large. AVAILABILITY: It is available for free at http://www.cs.cityu.edu.hk/~lwang/software/motif/index.html, subject to copyright restrictions.

Algorithms↗
PubMed2001

Comparison of parametric FBP and OS-EM reconstruction algorithm images for PET dynamic study.

An ordered subsets expectation maximization (OS-EM) algorithm is used for image reconstruction to suppress image noise and to make non-negative value images. We have applied OS-EM to a digital brain phantom and to human brain 18F-FDG PET kinetic studies to generate parametric images. A 45 min dynamic scan was performed starting injection of FDG with a 2D PET scanner. The images were reconstructed with OS-EM (6 iterations, 16 subsets) and with filtered backprojection (FBP), and K1, k2 and k3 images were created by the Marquardt non-linear least squares method based on the 3-parameter kinetic model. Although the OS-EM activity images correlated fairly well with those obtained by FBP, the pixel correlations were poor for the k2 and k3 parametric images, but the plots were scattered along the line of identity and the mean values for K1, k2 and k3 obtained by OS-EM were almost equal to those by FBP. The kinetic fitting error for OS-EM was no smaller than that for FBP. The results suggest that OS-EM is not necessarily superior to FBP for creating parametric images.

Algorithms↗
PubMed1994

An evaluation of the accelerated expectation maximization algorithms for single-photon emission tomography image reconstruction.

We previously reported that brain single-photon emission tomography (SPET) images could be improved by using an attenuation coefficient map constructed with transmission data and the iterative expectation maximization (EM) algorithm. However, the conventional EM algorithm (CEM) typically requires 30-80 iterations to provide acceptable results, limiting its clinical applicability. Several methods have been proposed to accelerate the EM algorithm. The purpose of this study was to search for a practical method for accelerating the EM algorithm. The methods investigated here include the accelerated EM algorithm (ACEM) using additive correction, ACEM using multiplicative correction, and Tanaka's filtered iterative reconstruction method (FIR). These methods were assessed by simulated SPET studies of a phantom incorporating nonuniform attenuation and by reference to clinical brain SPET data. In the simulation studies, the above methods were evaluated by using three parameters (root mean square error, log likelihood value, and contrast recovery coefficient); the results showed that FIR had an advantage over other methods in terms of all parameters. The results obtained using the clinical data demonstrated that FIR could reconstruct acceptable images in only five iterations. These results show that FIR offers significant advantages over CEM or other ACEMs, indicating that FIR can make the EM algorithm practical for clinical use in SPET.

Adult↗
PubMed1999

An EM-type Algorithm for Ordered Restriction Map Alignment.

Constructing restriction maps is one of the important steps towards the determination of DNA sequences. Recently, the single-molecule approaches to constructing restriction maps, such as Optical Mapping by D. Schwartz et al., have developed. In practice, with the single-molecule approach like Optical Mapping, the identification of the restriction sites is complicated by several error factors due to resolving power of biological experiments. The ordered restriction map alignment problem is a problem to estimate the actual restriction sites from many imprecise copies of map from single molecule. In this paper, we formulate the problem on the basis of the statistical maximum likelihood estimate, and propose a new efficient local search algorithm for this problem, by applying the Expectation-Maximization (EM) algorithm along with the concept of two-clustering. Our algorithm works well for a lot of sets of simulated data, some of which we believe more difficult than the actual cases.

Journal Article↗
PubMed2005

Estimation of linear mixed models with a mixture of distribution for the random effects.

The aim of this paper is to propose an algorithm to estimate linear mixed model when random effect distribution is a mixture of Gaussians. This heterogeneous linear mixed model relaxes the classical Gaussian assumption for the random effects and, when used for longitudinal data, can highlight distinct patterns of evolution. The observed likelihood is maximized using a Marquardt algorithm instead of the EM algorithm which is frequently used for mixture models. Indeed, the EM algorithm is computationally expensive and does not provide good convergence criteria nor direct estimates of the variance of the parameters. The proposed method also allows to classify subjects according to the estimated profiles by computing posterior probabilities of belonging to each component. The use of heterogeneous linear mixed model is illustrated through a study of the different patterns of cognitive evolution in the elderly. HETMIXLIN is a free Fortran90 program available on the web site: http://www.isped.u-bordeaux2.fr.

Aged↗
PubMed1989

Correction of nonuniform attenuation in cardiac SPECT imaging.

Correction for photon attenuation in cardiac SPECT imaging using a measured attenuation distribution with an iterative expectation maximization (EM) algorithm and an iterative Chang algorithm were compared with the conventional filtered backprojection and an iterative EM algorithm without attenuation correction. The attenuation distribution was determined from a transmission computed tomography study that was obtained using an external collimated sheet source. The attenuation of the emitting photons was modeled in the EM algorithm by an attenuated projector-backprojector that used the estimated attenuation distribution to calculate attenuation factors for each pixel along each projection and backprojection ray. Results from a heart-lung phantom study and a 201Tl patient study demonstrated that the iterative EM algorithm with attenuation correction provided improved image quality in terms of reduced streak artifacts and noise, and more accurate quantitative information in terms of improved radioactivity distribution uniformity where uniformity existed, and better anatomic object definition.

Algorithms↗
PubMed2003

Relaxed ordered-subset algorithm for penalized-likelihood image restoration.

The expectation-maximization (EM) algorithm for maximum-likelihood image recovery is guaranteed to converge, but it converges slowly. Its ordered-subset version (OS-EM) is used widely in tomographic image reconstruction because of its order-of-magnitude acceleration compared with the EM algorithm, but it does not guarantee convergence. Recently the ordered-subset, separable-paraboloidal-surrogate (OS-SPS) algorithm with relaxation has been shown to converge to the optimal point while providing fast convergence. We adapt the relaxed OS-SPS algorithm to the problem of image restoration. Because data acquisition in image restoration is different from that in tomography, we employ a different strategy for choosing subsets, using pixel locations rather than projection angles. Simulation results show that the relaxed OS-SPS algorithm can provide an order-of-magnitude acceleration over the EM algorithm for image restoration. This new algorithm now provides the speed and guaranteed convergence necessary for efficient image restoration.

Journal Article↗
PubMed2005

Weighted expectation maximization reconstruction algorithms for thermoacoustic tomography.

Thermoacoustic tomography (TAT) is an emerging imaging technique with potential for a wide range of biomedical imaging applications. In this correspondence, we propose an infinite family of weighted expectation maximization (EM) algorithms for reconstruction of images from temporally truncated TAT measurement data. The weighted EM algorithms are equivalent mathematically to the conventional EM algorithm, but are shown to propagate data inconsistencies in different ways. Using simulated and experimental TAT measurement data, we demonstrate that suitable choices of weighted EM algorithms can effectively mitigate image artifacts that are attributable to temporal truncation of the TAT data function.

Acoustics↗
PubMed1999

Improved learning algorithms for mixture of experts in multiclass classification.

Mixture of experts (ME) is a modular neural network architecture for supervised learning. A double-loop Expectation-Maximization (EM) algorithm has been introduced to the ME architecture for adjusting the parameters and the iteratively reweighted least squares (IRLS) algorithm is used to perform maximization in the inner loop [Jordan, M.I., Jacobs, R.A. (1994). Hierarchical mixture of experts and the EM algorithm, Neural Computation, 6(2), 181-214]. However, it is reported in literature that the IRLS algorithm is of instability and the ME architecture trained by the EM algorithm, where IRLS algorithm is used in the inner loop, often produces the poor performance in multiclass classification. In this paper, the reason of this instability is explored. We find out that due to an implicitly imposed incorrect assumption on parameter independence in multiclass classification, an incomplete Hessian matrix is used in that IRLS algorithm. Based on this finding, we apply the Newton-Raphson method to the inner loop of the EM algorithm in the case of multiclass classification, where the exact Hessian matrix is adopted. To tackle the expensive computation of the Hessian matrix and its inverse, we propose an approximation to the Newton-Raphson algorithm based on a so-called generalized Bernoulli density. The Newton-Raphson algorithm and its approximation have been applied to synthetic data, benchmark, and real-world multiclass classification tasks. For comparison, the IRLS algorithm and a quasi-Newton algorithm called BFGS have also been applied to the same tasks. Simulation results have shown that the use of the proposed learning algorithms avoids the instability problem and makes the ME architecture produce good performance in multiclass classification. In particular, our approximation algorithm leads to fast learning. In addition, the limitation of our approximation algorithm is also empirically investigated in this paper.

Journal Article↗
PubMed1994

Mixed Bayesian networks: a mixture of Gaussian distributions.

Mixed Bayesian networks are probabilistic models associated with a graphical representation, where the graph is directed and the random variables are discrete or continuous. We propose a comprehensive method for estimating the density functions of continuous variables, using a graph structure and a set of samples. The principle of the method is to learn the shape of densities from a sample of continuous variables. The densities are approximated by a mixture of Gaussian distributions. The estimation algorithm is a stochastic version of the Expectation Maximization algorithm (Stochastic EM algorithm). The inference algorithm corresponding to our model is a variant of junction three method, adapted to our specific case. The approach is illustrated by a simulated example from the domain of pharmacokinetics. Tests show that the true distributions seem sufficiently fitted for practical application.

Algorithms↗
PubMed1998

Estimating equations with incomplete categorical covariates in the Cox model.

Incomplete covariate data is a common occurrence in many studies in which the outcome is survival time. When a full likelihood is specified, a useful technique for obtaining parameter estimates is the EM algorithm. We propose a set of estimating equations to estimate the parameters of Cox's proportional hazards model when some covariate values are missing. These estimating equations can be solved by an algorithm similar to the EM algorithm. Because of the computational burden of finding a solution to these estimating equations, we propose obtaining parameter estimates via Monte Carlo methods. Asymptotic variances of the parameter estimates are also derived. We present a clinical trials example with three covariates, two of which have some missing values.

Algorithms↗
PubMedDate not supplied

Assessment of geometrical distortion and activity distribution after attenuation correction: A SPECT phantom study.

BACKGROUND: If single photon emission computed tomography (SPECT) images are reconstructed with filtered backprojection (FBP), not accounting for photon attenuation, artifacts can occur related to geometrical distortion and inaccurate estimation of regional distribution of radioactivity. By reconstructing the images with an iterative algorithm such as the maximum likelihood-expectation maximization (ML-EM) that incorporates the attenuation distribution information, it is possible to compensate for nonuniform attenuation. The aim of this study was to assess whether correction for nonuniform attenuation in SPECT can reduce the geometrical distortion and improve the activity quantitation. METHODS AND RESULTS: Three capillary sources containing the same amount of technetium 99m were imaged by a dual-headed SPECT system provided with two gadolinium 153 scanning transmission line sources, in nonuniform attenuation conditions. The images were reconstructed (1) with the use of FBP, (2) with the iterative ML-EM algorithm, and (3) with the iterative ML-EM algorithm incorporating attenuation maps. The geometrical distortion was estimated by comparing the spread that occurred in 2 orthogonal directions in the reconstructed transverse slices, expressed by full width at half maximum related to the x-axis and y-axis line spread functions. The accuracy of activity quantitation was analyzed by comparing the counts in regions of interest placed over the transverse slices of the 3 sources, located in different attenuating areas. The FBP-reconstructed slices showed a spread of image intensity toward the direction of minor attenuation; the source shape improved in the iterative ML-EM images, as well as in the iterative attenuation-corrected ML-EM images. The sources located deep in the phantom showed an apparent decrease in image intensity in both FBP and ML-EM images, which became less evident in the iterative attenuation-corrected ML-EM images. CONCLUSIONS: Image reconstruction with the iterative ML-EM algorithm, without the use of attenuation maps, can reduce geometrical distortion and eliminate streak artifacts, leading to an improvement in the object's shape and size, but does not reduce activity underestimation and inaccurate quantitation. In the iterative attenuation-corrected ML-EM images, there was a significant improvement in the accurate quantitation of activity distribution and a further reduction in geometrical distortion. In conclusion, nonisotropic attenuation correction with iterative ML-EM reduced the geometrical distortion of images and improved the accuracy of activity quantitation.

Abdomen↗
PubMed2004

[Evaluation of the number of sampling step angles in SPECT Images: comparison between FBP and OS-EM reconstruction algorithms].

We compared filtered back projection (FBP) and ordered subset expectation maximization (OS-EM) reconstruction algorithms to examine the effects of sampling step angle on SPECT image reconstruction. Image sampling was compared between conditions of constant total sampling time (16 min) and constant sampling time (40 sec) for each step with angles of 3, 5, 7.5, and 10 degrees. Generally, OS-EM was superior to FBP in terms of image reconstruction. Changes in the sampling step angles did not significantly affect the quality of the reconstructed images with OS-EM insofar as acquisition counts were sufficient for evaluation. On the other hand, the quality of FBP images deteriorated with larger angles such as 7.5 and 10 degrees, and it was difficult to obtain stable results. In conclusion, OS-EM improves the quality of images required for examination.

Algorithms↗
PubMed2003

Ordered subsets Bayesian tomographic reconstruction using 2-D smoothing splines as priors.

The ordered subsets expectation maximization (OS-EM) algorithm has enjoyed considerable interest for accelerating the well-known EM algorithm for emission tomography. The OS principle has also been applied to several regularized EM algorithms, such as nonquadratic convex minimization-based maximum a posteriori (MAP) algorithms. However, most of these methods have not been as practical as OS-EM due to their complex optimization methods and difficulties in hyperparameter estimation. We note here that, by relaxing the requirement of imposing sharp edges and using instead useful quadratic spline priors, solutions are much easier to compute, and hyperparameter calculation becomes less of a problem. In this work, we use two-dimensional smoothing splines as priors and apply a method of iterated conditional modes for the optimization. In this case, step sizes or line-search algorithms necessary for gradient-based descent methods are avoided. We also accelerate the resulting algorithm using the OS approach and propose a principled way of scaling smoothing parameters to retain the strength of smoothing for different subset numbers. Our experimental results show that the OS approach applied to our quadratic MAP algorithms provides a considerable acceleration while retaining the advantages of quadratic spline priors.

Algorithms↗
PubMed2005

Weighted expectation maximization reconstruction algorithms with application to gated megavoltage tomography.

We propose and investigate weighted expectation maximization (EM) algorithms for image reconstruction in x-ray tomography. The development of the algorithms is motivated by the respiratory-gated megavoltage tomography problem, in which the acquired asymmetric cone-beam projections are limited in number and unevenly sampled over view angle. In these cases, images reconstructed by use of the conventional EM algorithm can contain ring- and streak-like artefacts that are attributable to a combination of data inconsistencies and truncation of the projection data. By use of computer-simulated and clinical gated fan-beam megavoltage projection data, we demonstrate that the proposed weighted EM algorithms effectively mitigate such image artefacts.

Algorithms↗
PubMed1991

Counting algorithms for linkage: correction to Morton and Collins.

In a recent paper, Morton & Collins (1990) claimed: (1) that the Lander-Green algorithm for genetic linkage analysis is not the EM algorithm for finding the maximum likelihood map; and (2) that a proposed alternative algorithm does have these properties. Here, we show that these assertions are both incorrect: the Lander-Green algorithm is an EM algorithm, while the Morton-Collins algorithm is not. We note that Morton and Collins concur with these conclusions.

Algorithms↗
PubMed2005

The analysis of incomplete cost data due to dropout.

Incomplete data due to premature withdrawal (dropout) constitute a serious problem in prospective economic evaluations that has received only little attention to date. The aim of this simulation study was to investigate how standard methods for dealing with incomplete data perform when applied to cost data with various distributions and various types of dropout. Selected methods included the product-limit estimator of Lin et al. the expectation maximisation (EM-) algorithm, several types of multiple imputation (MI) and various simple methods like complete case analysis and mean imputation. Almost all methods were unbiased in the case of dropout completely at random (DCAR), but only the product-limit estimator, the EM-algorithm and the MI approaches provided adequate estimates of the standard error (SE). The best estimates of the mean and SE for dropout at random (DAR) were provided by the bootstrap EM-algorithm, MI regression and MI Monte Carlo Markov chain. These methods were able to deal with skewed cost data in combination with DAR and only became biased when costs also included the costs of expensive events. None of the methods were able to deal adequately with informative dropout. In conclusion, the EM-algorithm with bootstrap, MI regression and MI MCMC are robust to the multivariate normal assumption and are the preferred methods for the analysis of incomplete cost data when the assumption of DCAR is not justified.

Algorithms↗
PubMed2004

Maximum-likelihood estimation of haplotype frequencies in nuclear families.

The importance of haplotype analysis in the context of association fine mapping of disease genes has grown steadily over the last years. Since experimental methods to determine haplotypes on a large scale are not available, phase has to be inferred statistically. For individual genotype data, several reconstruction techniques and many implementations of the expectation-maximization (EM) algorithm for haplotype frequency estimation exist. Recent research work has shown that incorporating available genotype information of related individuals largely increases the precision of haplotype frequency estimates. We, therefore, implemented a highly flexible program written in C, called FAMHAP, which calculates maximum likelihood estimates (MLEs) of haplotype frequencies from general nuclear families with an arbitrary number of children via the EM-algorithm for up to 20 SNPs. For more loci, we have implemented a locus-iterative mode of the EM-algorithm, which gives reliable approximations of the MLEs for up to 63 SNP loci, or less when multi-allelic markers are incorporated into the analysis. Missing genotypes can be handled as well. The program is able to distinguish cases (haplotypes transmitted to the first affected child of a family) from pseudo-controls (non-transmitted haplotypes with respect to the child). We tested the performance of FAMHAP and the accuracy of the obtained haplotype frequencies on a variety of simulated data sets. The implementation proved to work well when many markers were considered and no significant differences between the estimates obtained with the usual EM-algorithm and those obtained in its locus-iterative mode were observed. We conclude from the simulations that the accuracy of haplotype frequency estimation and reconstruction in nuclear families is very reliable in general and robust against missing genotypes.

Algorithms↗