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PubMed2003

A constrained EM algorithm for principal component analysis.

We propose a constrained EM algorithm for principal component analysis (PCA) using a coupled probability model derived from single-standard factor analysis models with isotropic noise structure. The single probabilistic PCA, especially for the case where there is no noise, can find only a vector set that is a linear superposition of principal components and requires postprocessing, such as diagonalization of symmetric matrices. By contrast, the proposed algorithm finds the actual principal components, which are sorted in descending order of eigenvalue size and require no additional calculation or postprocessing. The method is easily applied to kernel PCA. It is also shown that the new EM algorithm is derived from a generalized least-squares formulation.

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PubMed2004

[Simultaneous spatial resolution correction in SPECT reconstruction using OS-EM algorithm].

Single photon emission computed tomography (SPECT) is widely used for Nuclear Medicine. The low spatial resolution is mainly due to the limited collimator resolution. We have developed ordered subsets-expectation maximization (OS-EM) algorithm incorporating three dimensional spatial resolution correction (3D-SRC;horizontal and vertical direction) for distance-dependent blurring. In this paper we evaluate the fundamental properties of OS-EM algorithm including distance-dependent resolution correction using some phantoms (cubic phantom, cardiac and brain phantoms) and 12 patients performed 99mTc-tetrofosmin myocardial SPECT. In cubic phantom, the resolution in transaxial, coronal and sagital images are significantly improved by 3D-SRC. In short axial images of cardiac phantom, 3D-SRC shows more excellent images than no correction. In brain phantom, the blurring at the edge of the brain structure is improved by using OS-EM algorithm with 3D-SRC. In patients, the resolution in transaxial and short axial images is significant improved. These results suggest that this method improves the resolution in SPECT images and 3D-SRC is especially available in SPECT images of other axes in addition to transaxis. This method has advantage that both reconstruction and resolution correction are simultaneously performed. In the near future, we think that OS-EM algorithm incorporating attenuation, scatter and resolution correction will be used for quantitative SPECT images.

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PubMed2000

Asymptotic convergence rate of the EM algorithm for gaussian mixtures

It is well known that the convergence rate of the expectation-maximization (EM) algorithm can be faster than those of convention first-order iterative algorithms when the overlap in the given mixture is small. But this argument has not been mathematically proved yet. This article studies this problem asymptotically in the setting of gaussian mixtures under the theoretical framework of Xu and Jordan (1996). It has been proved that the asymptotic convergence rate of the EM algorithm for gaussian mixtures locally around the true solution Theta* is o(e(0. 5-epsilon)(Theta*)), where epsilon > 0 is an arbitrarily small number, o(x) means that it is a higher-order infinitesimal as x --> 0, and e(Theta*) is a measure of the average overlap of gaussians in the mixture. In other words, the large sample local convergence rate for the EM algorithm tends to be asymptotically superlinear when e(Theta*) tends to zero.

Journal Article↗
PubMed2000

Interest of the ordered subsets expectation maximization (OS-EM) algorithm in pinhole single-photon emission tomography reconstruction: a phantom study.

Pinhole single-photon emission tomography (SPET) has been proposed to improve the trade-off between sensitivity and resolution for small organs located in close proximity to the pinhole aperture. This technique is hampered by artefacts in the non-central slices. These artefacts are caused by truncation and by the fact that the pinhole SPET data collected in a circular orbit do not contain sufficient information for exact reconstruction. The ordered subsets expectation maximization (OS-EM) algorithm is a potential solution to these problems. In this study a three-dimensional OS-EM algorithm was implemented for data acquired on a single-head gamma camera equipped with a pinhole collimator (PH OS-EM). The aim of this study was to compare the PH OS-EM algorithm with the filtered back-projection algorithm of Feldkamp, Davis and Kress (FDK) and with the conventional parallel-hole geometry as a whole, using a line source phantom, Picker's thyroid phantom and a phantom mimicking the human cervical column. Correction for the angular dependency of the sensitivity in the pinhole geometry was based on a uniform flood acquisition. The projection data were shifted according to the measured centre of rotation. No correction was made for attenuation, scatter or distance-dependent camera resolution. The resolution measured with the line source phantom showed a significant improvement with PH OS-EM as compared with FDK, especially in the axial direction. Using Picker's thyroid phantom, one iteration with eight subsets was sufficient to obtain images with similar noise levels in uniform regions of interest to those obtained with the FDK algorithm. With these parameters the reconstruction time was 2.5 times longer than for the FDK method. Furthermore, there was a reduction in the artefacts caused by the circular orbit SPET acquisition. The images obtained from the phantom mimicking the human cervical column indicated that the improvement in image quality with PH OS-EM is relevant for future clinical use and that the improvements obtained using the OS-EM algorithm are more significant for the pinhole geometry than for the conventional parallel-hole geometry. We conclude that PH OS-EM is a practical and promising alternative for pinhole SPET reconstruction.

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PubMed1997

EM algorithms without missing data.

Most problems in computational statistics involve optimization of an objective function such as a loglikelihood, a sum of squares, or a log posterior function. The EM algorithm is one of the most effective algorithms for maximization because it iteratively transfers maximization from a complex function to a simple, surrogate function. This theoretical perspective clarifies the operation of the EM algorithm and suggests novel generalizations. Besides simplifying maximization, optimization transfer usually leads to highly stable algorithms with well-understood local and global convergence properties. Although convergence can be excruciatingly slow, various devices exist for accelerating it. Beginning with the EM algorithm, we review in this paper several optimization transfer algorithms of substantial utility in medical statistics.

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PubMed2001

An EM algorithm for estimating SPECT emission and transmission parameters from emissions data only.

A maximum-likelihood (ML) expectation-maximization (EM) algorithm (called EM-IntraSPECT) is presented for simultaneously estimating single photon emission computed tomography (SPECT) emission and attenuation parameters from emission data alone. The algorithm uses the activity within the patient as transmission tomography sources, with which attenuation coefficients can be estimated. For this initial study, EM-IntraSPECT was tested on computer-simulated attenuation and emission maps representing a simplified human thorax as well as on SPECT data obtained from a physical phantom. Two evaluations were performed. First, to corroborate the idea of reconstructing attenuation parameters from emission data, attenuation parameters (mu) were estimated with the emission intensities (lambda) fixed at their true values. Accurate reconstructions of attenuation parameters were obtained. Second, emission parameters lambda and attenuation parameters mu were simultaneously estimated from the emission data alone. In this case there was crosstalk between estimates of lambda and mu and final estimates of lambda and mu depended on initial values. Estimates degraded significantly as the support extended out farther from the body, and an explanation for this is proposed. In the EM-IntraSPECT reconstructed attenuation images, the lungs, spine, and soft tissue were readily distinguished and had approximately correct shapes and sizes. As compared with standard EM reconstruction assuming a fix uniform attenuation map, EM-IntraSPECT provided more uniform estimates of cardiac activity in the physical phantom study and in the simulation study with tight support, but less uniform estimates with a broad support. The new EM algorithm derived here has additional applications, including reconstructing emission and transmission projection data under a unified statistical model.

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PubMed1997

The EM algorithm and medical studies: a historical link.

Anderson Gray McKendrick's 1926 paper, 'Applications of mathematics to medical problems', was the earliest reference cited in Dempster et al.'s 1977 paper that defined and popularized the EM algorithm. McKendrick's paper was prominently featured by Joseph Oscar Irwin in his 1962 inaugural address as the President of the Royal Statistical Society (in the UK), entitled 'The place of mathematics in medical and biological statistics'. The link of McKendrick's work to the EM algorithm is due to an improvement made by Irwin on a novel method McKendrick used for estimating an infection rate when the observed data do not distinguish between those individuals who are not susceptible to the infection and those who are susceptible, but do not develop symptoms. This article examines this link, along the way illustrating the central ideas underlying the EM algorithm as well as its properties; the examination also suggests a profiling strategy for speeding up EM, which may be worthy of general investigation. McKendrick's data on an epidemic of cholera are used for illustration and to compare EM with Irwin's method as well as the Newton-Raphson algorithm. Issues beyond computation are also discussed whenever appropriate.

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PubMed2005

The full EM algorithm for the MLEs of QTL effects and positions and their estimated variances in multiple-interval mapping.

The advent of complete genetic linkage maps of DNA markers has made systematic studies of mapping quantitative trait loci (QTL) in experimental organisms feasible. The method of multiple-interval mapping provides an appropriate way for mapping QTL using genetic markers. However, efficient algorithms for the computation involved remain to be developed. In this article, a full EM algorithm for the simultaneous computation of the MLEs of QTL effects and positions is developed. EM-based formulas are derived for computing the observed Fisher information matrix. The full EM algorithm is compared with an ECM algorithm developed by Kao and Zeng (1997, Biometrics 53, 653-665). The validity of the inverted observed Fisher information matrix as an estimate of the variance matrix of the MLEs is demonstrated by a simulation study.

Algorithms↗
PubMed2003

Optimisation of the OS-EM algorithm and comparison with FBP for image reconstruction on a dual-head camera: a phantom and a clinical 18F-FDG study.

Iterative reconstruction algorithms, such as the ordered subsets expectation maximisation (OS-EM), are a promising alternative to filtered backprojection (FBP). The aims of this study were first to optimise the OS-EM algorithm in terms of iteration number and to study the usefulness of post-filtering, and second to compare OS-EM and FBP for image reconstruction on a fluorine-18 fluorodeoxyglucose ((18)F-FDG) dual-head camera (DHC). These two goals were addressed using phantom acquisitions. The performances of these algorithms were also studied in patient acquisitions performed on a DHC and a PET on the same day. Phantom experiments were performed on a DHC using a Jaszczak phantom containing six spheres filled with (18)F-FDG, two background levels (0.95, 6.80 kBq/ml) and three object contrasts (5.9, 3.7, 2.7). The reconstruction algorithms were FBP with a Gaussian filter (FWHM 0.5-2 pixel width) and OS-EM using 8-128 equivalent iterations (equivalent to the ML-EM algorithm) with and without Gaussian post-filtering [OS-EM (iterations, pixel width)]. Contrast recovery coefficient (CRC) and noise characteristics were assessed. Twenty-two patients (21 male, one female; age 55+/-15 years) with lung cancer underwent, on the same day, PET (1 h post injection of 37 MBq/kg (18)F-FDG) and DHC acquisitions (3 h post injection). DHC data were reconstructed using six methods: FBP (1), OS-EM (16), (40), (40,1), (64) and (64,1). These sets were evaluated by two observers and compared to PET reconstructed with OS-EM (16). The number of detected lesions and the visual quality were assessed. A marked improvement in CRC was observed with OS-EM as compared with FBP when more than 24 iterations were used. The CRC increased markedly from 8 to 40 iterations and then reached a plateau. The noise was stable until 40 iterations and then increased. The best compromise was obtained for OS-EM (32) and OS-EM (40,1). For the patient study, OS-EM provided images of better visual quality, but with no significant difference in detection sensitivity. OS-EM was superior to FBP in terms of contrast recovery and noise level. The optimal compromise between contrast recovery and noise was obtained for OS-EM (32) and (40,1) on the phantom study. The clinical study showed that OS-EM yielded images of better visual quality but with no improvement in terms of detection of lung cancer.

Algorithms↗
PubMedDate not supplied

The PX-EM algorithm for fast stable fitting of Henderson's mixed model.

This paper presents procedures for implementing the PX-EM algorithm of Liu, Rubin and Wu to compute REML estimates of variance covariance components in Henderson's linear mixed models. The class of models considered encompasses several correlated random factors having the same vector length e.g., as in random regression models for longitudinal data analysis and in sire-maternal grandsire models for genetic evaluation. Numerical examples are presented to illustrate the procedures. Much better results in terms of convergence characteristics (number of iterations and time required for convergence) are obtained for PX-EM relative to the basic EM algorithm in the random regression.

Journal Article↗
PubMed2005

Statistical inference in evolutionary models of DNA sequences via the EM algorithm.

We describe statistical inference in continuous time Markov processes of DNA sequences related by a phylogenetic tree. The maximum likelihood estimator can be found by the expectation maximization (EM) algorithm and an expression for the information matrix is also derived. We provide explicit analytical solutions for the EM algorithm and information matrix.

Journal Article↗
PubMed1994

Noise properties of the EM algorithm: II. Monte Carlo simulations.

In an earlier paper we derived a theoretical formulation for estimating the statistical properties of images reconstructed using the iterative ML-EM algorithm. To gain insight into this complex problem, two levels of approximation were considered in the theory. These techniques revealed the dependence of the variance and covariance of the reconstructed image noise on the source distribution, imaging system transfer function, and iteration number. In this paper a Monte Carlo approach was taken to study the noise properties of the ML-EM algorithm and to test the predictions of the theory. The study also served to evaluate the approximations used in the theory. Simulated data from phantoms were used in the Monte Carlo experiments. The ML-EM statistical properties were calculated from sample averages of a large number of images with different noise realizations. The agreement between the more exact form of the theoretical formulation and the Monte Carlo formulation was better than 10% in most cases examined, and for many situations the agreement was within the expected error of the Monte Carlo experiments. Results from the studies provide valuable information about the noise characteristics of ML-EM reconstructed images. Furthermore, the studies demonstrate the power of the theoretical and Monte Carlo approaches for investigating noise properties of statistical reconstruction algorithms.

Algorithms↗
PubMed1988

Fitting mixture models to grouped and truncated data via the EM algorithm.

The fitting of finite mixture models via the EM algorithm is considered for data which are available only in grouped form and which may also be truncated. A practical example is presented where a mixture of two doubly truncated log-normal distributions is adopted to model the distribution of the volume of red blood cells in cows during recovery from anemia.

Algorithms↗
PubMed2003

Recognition of translation initiation sites of eukaryotic genes based on an EM algorithm.

With the rapid increase of DNA databases of human and other eukaryotic model organisms, a large great number of genes need to be distinguished from the DNA databases. Exact recognition of translation initiation sites (TISs) of eukaryotic genes is very important to understand the translation initiation process, predict the detailed structure of eukaryotic genes, and annotate uncharacterized sequences. The problem has not been solved satisfactorily, especially for recognizing TISs of the eukaryotic genes with shorter first exons. It is an important task for extracting new features and finding new powerful algorithms for recognizing TISs of eukaryotic genes. In this paper, the important characteristics of shorter flanking fragments around TISs are extracted and an expectation-maximization (EM) algorithm based on incomplete data is used to recognize TISs of eukaryotic genes. The accuracy is up to 87.8% over a six-fold cross-validation test. The result shows that the identification variables are effectively extracted and the EM algorithm is a powerful tool to predict the TISs of eukaryotic genes. The algorithm also can be applied to other classification or clustering tasks in bioinformatics.

Algorithms↗
PubMed2005

A version of the EM algorithm for proportional hazard model with random effects.

Proportional hazard models with multivariate random effects (frailties) acting multiplicatively on the baseline hazard have recently become a topic of an intensive research. One of the main practical problems related to the models is the estimation of parameters. To this aim, several approaches based on the EM algorithm have been proposed. The major difference between these approaches is the method of the computation of conditional expectations required at the E-step. In this paper an alternative implementation of the EM algorithm is proposed, in which the expected values are computed with the use of the Laplace approximation. The method is computationally less demanding than the approaches developed previously. Its performance is assessed based on a simulation study and compared to a non-EM based estimation approach proposed by Ripatti and Palmgren (2000).

Algorithms↗
PubMed2006

An extended EM algorithm for joint feature extraction and classification in brain-computer interfaces.

For many electroencephalogram (EEG)-based brain-computer interfaces (BCIs), a tedious and time-consuming training process is needed to set parameters. In BCI Competition 2005, reducing the training process was explicitly proposed as a task. Furthermore, an effective BCI system needs to be adaptive to dynamic variations of brain signals; that is, its parameters need to be adjusted online. In this article, we introduce an extended expectation maximization (EM) algorithm, where the extraction and classification of common spatial pattern (CSP) features are performed jointly and iteratively. In each iteration, the training data set is updated using all or part of the test data and the labels predicted in the previous iteration. Based on the updated training data set, the CSP features are reextracted and classified using a standard EM algorithm. Since the training data set is updated frequently, the initial training data set can be small (semi-supervised case) or null (unsupervised case). During the above iterations, the parameters of the Bayes classifier and the CSP transformation matrix are also updated concurrently. In online situations, we can still run the training process to adjust the system parameters using unlabeled data while a subject is using the BCI system. The effectiveness of the algorithm depends on the robustness of CSP feature to noise and iteration convergence, which are discussed in this article. Our proposed approach has been applied to data set IVa of BCI Competition 2005. The data analysis results show that we can obtain satisfying prediction accuracy using our algorithm in the semisupervised and unsupervised cases. The convergence of the algorithm and robustness of CSP feature are also demonstrated in our data analysis.

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PubMed2000

Mapping a quantitative trait locus via the EM algorithm and Bayesian classification.

Mapping a locus controlling a quantitative genetic trait (e.g., blood pressure) to a specific genomic region is of considerable interest. Data on the quantitative trait under consideration and several codominant genetic markers with known genomic locations are collected from members of families and statistically analyzed to draw inferences on the genomic position of the trait locus. The vector of parameters of interest comprises the pairwise recombination fractions, theta, between the putative quantitative trait locus and the marker loci. One of the major complications in estimating theta for a quantitative trait in humans is the lack of haplotype information on members of families. The purpose of this study was to devise a computationally simple and efficient method of estimation of theta in the absence of haplotype information. We have proposed a two-stage estimation procedure using the expectation-maximization (EM) algorithm. In the first stage, parameters of the QTL are estimated based on data of a sample of unrelated individuals. From estimates thus obtained, we have used a Bayes' rule to infer QTL genotypes of parents in families. Finally, in the second stage of the procedure, we have proposed an EM algorithm for obtaining the maximum likelihood estimate of theta based on data of informative families (which are identified upon inferring parental QTL genotypes performed in the first stage). We have shown, using simulated data, that the proposed procedure is cost-effective, computationally simple, and statistically efficient. As expected, analysis of data on multiple markers jointly is more efficient than the analysis based on single markers.

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