The probability distribution of parameters learned with the EM algorithm

On convergence and parameter selection of the EM and DA-EM algorithms for Gaussian mixtures

The ability to theoretically model the propagation of photon noise through PET and SPECT tomographic reconstruction algorithms is crucial in evaluating the reconstructed image quality as a function of parameters of the algorithm. In a previous approach for the important case of the iterative ML - EM (maximum-likelihood - expectation-maximization) algorithm, judicious linearizations were used to model theoretically the propagation of a mean image and a covariance matrix from one iteration to the next. Our analysis extends this approach to the case of MAP (maximum a posteriori) - EM algorithms, where the EM approach incorporates prior terms. We analyse in detail two cases: a MAP - EM algorithm incorporating an independent gamma prior, and a one-step-late (OSL) version of a MAP - EM algorithm incorporating a multivariate Gaussian prior, for which familiar smoothing priors are special cases. To validate our theoretical analyses, we use a Monte Carlo methodology to compare, at each iteration, theoretical estimates of mean and covariance with sample estimates, and show that the theory works well in practical situations where the noise and bias in the reconstructed images do not assume extreme values.

Many diseases show dichotomous phenotypic variation but do not follow a simple Mendelian pattern of inheritance. Variances of these binary diseases are presumably controlled by multiple loci and environmental variants. A least-squares method has been developed for mapping such complex disease loci by treating the binary phenotypes (0 and 1) as if they were continuous. However, the least-squares method is not recommended because of its ad hoc nature. Maximum Likelihood (ML) and Bayesian methods have also been developed for binary disease mapping by incorporating the discrete nature of the phenotypic distribution. In the ML analysis, the likelihood function is usually maximized using some complicated maximization algorithms (e.g. the Newton-Raphson or the simplex algorithm). Under the threshold model of binary disease, we develop an Expectation Maximization (EM) algorithm to solve for the maximum likelihood estimates (MLEs). The new EM algorithm is developed by treating both the unobserved genotype and the disease liability as missing values. As a result, the EM iteration equations have the same form as the normal equation system in linear regression. The EM algorithm is further modified to take into account sexual dimorphism in the linkage maps. Applying the EM-implemented ML method to a four-way-cross mouse family, we detected two regions on the fourth chromosome that have evidence of QTLs controlling the segregation of fibrosarcoma, a form of connective tissue cancer. The two QTLs explain 50-60% of the variance in the disease liability. We also applied a Bayesian method previously developed (modified to take into account sex-specific maps) to this data set and detected one additional QTL on chromosome 13 that explains another 26% of the variance of the disease liability. All the QTLs detected primarily show dominance effects.

Alzheimer’s disease is the most common form of dementia and the early detection is essential to prevent its proliferation. Real data available has been of paramount importance in order to achieve progress in the automatic detection despite presenting two major challenges: Multi-source observations containing Magnetic resonance (MRI), Positron emission tomography (PET) and Cerebrospinal fluid data (CSF); and also missing values within all these sources. Most machine learning techniques perform this predictive task by using a single data modality. Nevertheless, the integration of all these sources of evidence could possibly bring a higher performance at different stages of disease progression. The Expectation Maximization (EM) algorithm has been successfully employed to handle missing values, but it is not designed for typical Machine Learning scenarios where an imputation model is created over training data and subsequently applied on a testing set. In this work, we propose EMreg-KNN, a novel supervised and multi-source imputation algorithm. Based on the EM algorithm, EMreg-KNN builds a regression ensemble model for the imputation of future data thus allowing the further utilization of any vector-based Machine Learning method to automatically assess the Alzheimer’s disease diagnosis. Using the ADNI database, the proposed method achieves significant improvements on F1, AUC and Accuracy measures over classical imputation methods for this database using four classification algorithms. Considering these classifiers in four different classification scenarios, our algorithm is experimentally superior in terms of the F measure, in nearly 82% of the cases under evaluation.

Left Ventricular Diastolic Dysfunction (LVDD) is a decompensatory change in the relaxation properties of the heart, the risk for which increases with age. Currently, physicians use a decision-tree-like algorithm to distinguish between discrete LVDD levels. This approach, based on cut-off thresholds, can potentially lead to information loss and possibly to misdiagnosis. This paper aims to explore an alternative diagnostic method to determine LVDD risk level, taking into account a wide variety of attributes available in patient records, without pre-setting cut-off thresholds. Using a large dataset derived from the Baltimore Longitude Study of Aging (BLSA), and adjusting the data for age and gender, we employ the Chi Square test and the information gain criterion to identify attributes that correlate well with the physician-assigned grades; such attributes are referred to as distinguishing attributes. We then apply the expectation maximization (EM) algorithm, as well as the K-Means, in order to cluster records that are represented using distinguishing attributes. While clusters resulting from the K-Means are not stable, three stable and tightly-formed clusters, which are obtained from the EM algorithm, roughly correspond to the physician-assigned categories. Based on the results from the EM algorithm, we can compute a patient's probability to have low, high or no risk for LVDD, and use this probability as a basis for defining a risk score to determine the patient's LVDD severity.

Objective: Data being missed is an unavoidable problem in cohort studies. This paper compares the imputation effect of eight common missing data imputation methods involved in cutting longitudinal data through simulation study to provide a valuable reference for the treatment of missing data in longitudinal studies. Methods: The simulation study is based on R language software and generates missing longitudinal data by the Monte Carlo method. By comparing the average absolute deviation, average relative deviation, and TypeⅠerror from the regression analysis of different imputation methods, the imputation effect of varying imputation methods on missing longitudinal data and the influence on subsequent multivariate analysis are evaluated. Results: The mean imputation, k nearest neighbor (KNN), regression imputation, and random forest all have a similar imputation effect, which is also steady. However, the hot deck is inferior to the above imputation methods. K-means clustering and expectation maximization (EM) algorithm are among the worst and unstable. Mean imputation, EM algorithm, random forest, KNN, and regression imputation can control TypeⅠerror. Still, multiple imputations, hot deck, and K-means clustering cannot effectively manage the TypeⅠerror. Conclusions: For missing data in longitudinal studies, mean imputation, KNN, regression imputation, and random forest can be used as better imputation methods under the mechanism of missing at random. When the missing ratio is not too large, multiple imputations and hot deck can also perform well, but K-means clustering and EM algorithm are not recommended.

Methodology for PET system modeling using image-space techniques in the expectation maximization (EM) algorithm is presented. The approach, applicable to both list-mode data and projection data, is of particular significance to EM algorithm implementations which otherwise only use basic system models (such as those which calculate the system matrix elements on the fly). A basic version of the proposed technique can be implemented using image-space convolution, in order to include resolution effects into the system matrix, so that the EM algorithm gradually recovers the modeled resolution with each update. The improved system modeling (achieved by inclusion of two convolutions per iteration) results in both enhanced resolution and lower noise, and there is often no need for regularization-other than to limit the number of iterations. Tests have been performed with simulated list-mode data and also with measured projection data from a GE Advance PET scanner, for both [/sup 18/F]-FDG and [/sup 124/I]-NaI. The method demonstrates improved image quality in all cases when compared to the conventional FBP and EM methods presently used for clinical data (which do not include resolution modeling). The benefits of this approach for /sup 124/I (which has a low positron yield and a large positron range, usually resulting in noisier and poorer resolution images) are particularly noticeable.

In this work, an expectation maximization (EM) algorithm is derived for maximum likelihood estimation of the autocorrelation function of a linear distortion channel as well as the level of additive noise, under the assumption that the source signal comes from a mixture of Gaussian sources. To facilitate parameter initialization in the EM algorithm, a correlation-matching based estimation algorithm is developed for the channel autocorrelation function. The proposed EM algorithm was evaluated on speech-derived simulated data of multiple autoregressive Gaussian sources and real speech of isolated digits under signal-to-noise ratios (SNRs) of 20 dB down to 0 dB. The algorithm is shown to produce convergent estimation results as well as estimates of signal statistics that lead to significantly improved classification accuracy under additive and convolutive noise conditions.

We are interested in an estimate of the usual residents in the Netherlands. Capture-recapture estimation with three registers enables us to estimate the size of the total population, of which the usual residents are a part. However, usual residence cannot be used as a covariate because it is not available in one of the registers. We approach this as a missing data problem. There are different methods available to handle missing data. In this manuscript we use Expectation Maximization (EM) algorithm and Predictive Mean Matching (PMM). The EM algorithm is often used in categorical data analysis, but PMM has the advantage of flexibility in the choice for a specific part of the observed data used for the imputation of the missing data. Four scenarios have been identified where the missing data are completed via either the EM algorithm or PMM imputation, resulting in different population size estimates for usual residence. It was found that the different scenarios lead to different population size estimates. Even small changes in the completed data lead to different population size estimates. In this study PMM imputation performs best according flexibility and it is theoretically better motivated.

Due to the lack of information about the causes of outages, the estimation of the interactions between component failures that capture the general outage propagation patterns is a typical parameter estimation problem with incomplete data. In this paper, we estimate these interactions by the expectation maximization (EM) algorithm. The proposed method is validated with simulated cascading failure data from the AC OPA model on the IEEE 118-bus system. The EM algorithm can accurately estimate the interactions and identify the key links and key components using only a small number of the original cascades from a detailed cascading blackout model, which is critical for online cascading failure analysis and decision making. Compared with AC OPA simulation, the highly probabilistic interaction model simulation based on the proposed interaction estimation method can achieve a speed-up of 100.61.

Within the context of emission tomography, we study volumetric reconstruction methods based on the Expectation Maximization (EM) algorithm. We show, for the first time, the equivalence of the standard implementation of the EM‐based reconstruction with an implementation based on hardware‐accelerated volume rendering for nearest‐neighbor (NN) interpolation. This equivalence suggests that higher‐order kernels should be used with caution and do not necessarily lead to better performance. We also show that the EM algorithm can easily be adapted for different lattices, the body‐centered cubic (BCC) one in particular. For validation purposes, we use the 3D version of the Shepp‐Logan synthetic phantom, for which we derive closed‐form analytical expressions of the projection data. The experimental results show the theoretically‐predicted optimality of NN interpolation in combination with the EM algorithm, for both the noiseless and the noisy case. Moreover, reconstruction on the BCC lattice leads to superior accuracy, more compact data representation, and better noise reduction compared to the Cartesian one. Finally, we show the usefulness of the proposed method for optical projection tomography of a mouse embryo.

This research article attempts to classify the distinguish authorship problem of three existing Tamil scholars during the Indian independence period, namely, Mahakavi Bharathiar (MB), Subramaniya Iyer (SI) and T.V. Kalyanasundaranar (TVK), using expectation maximisation (EM) algorithm method. The three trendy scholars, as stated above, had written a good number of articles on the India's Freedom Movement during the pre-independence period and published in the magazine India, in the year 1906, is considered for this study. The database consists of 36 articles of blocks of MB, 31 blocks of articles of TVK, 38 blocks of articles of MB, each block having many sentences and 19 function words parameters were used. In the above point of view, this study attempts to identify three authorship of three authors using their stylistic traits and classification of articles of MB, TVK and SI using EM algorithm model. EM algorithm has been used newly as a modern tool and attracted broad research in various disciplines including text mining. Application of EM algorithm has increased considerably in areas of pattern recognition and classification problems in the field of text mining over the last decade.

Joint modeling of survival and longitudinal data has been studied extensively in the recent literature. The likelihood approach is one of the most popular estimation methods employed within the joint modeling framework. Typically, the parameters are estimated using maximum likelihood, with computation performed by the expectation maximization (EM) algorithm. However, one drawback of this approach is that standard error (SE) estimates are not automatically produced when using the EM algorithm. Many different procedures have been proposed to obtain the asymptotic covariance matrix for the parameters when the number of parameters is typically small. In the joint modeling context, however, there may be an infinite-dimensional parameter, the baseline hazard function, which greatly complicates the problem, so that the existing methods cannot be readily applied. The profile likelihood and the bootstrap methods overcome the difficulty to some extent; however, they can be computationally intensive. In this paper, we propose two new methods for SE estimation using the EM algorithm that allow for more efficient computation of the SE of a subset of parametric components in a semiparametric or high-dimensional parametric model. The precision and computation time are evaluated through a thorough simulation study. We conclude with an application of our SE estimation method to analyze an HIV clinical trial dataset.

Missing data in large data analysis has affected further analysis conducted on dataset. To fill in missing data, Nearest Neighbour Method (NNM) and Expectation Maximization (EM) algorithm are the two most widely used methods. Thus, this research aims to compare both methods by imputing missing data of air quality in five monitoring stations (CA0030, CA0039, CA0042, CA0049, CA0050) in Sabah, Malaysia. PM10 (particulate matter with aerodynamic size below 10 microns) dataset in the range from 2003–2007 (Part A) and 2008–2012 (Part B) are used in this research. To make performance evaluation possible, missing data is introduced in the datasets at 5 different levels (5%, 10%, 15%, 25% and 40%). The missing data is imputed by using both NNM and EM algorithm. The performance of both data imputation methods is evaluated using performance indicators (RMSE, MAE, IOA, COD) and regression analysis. Based on performance indicators and regression analysis, NNM performs better compared to EM in imputing data for stations CA0039, CA0042 and CA0049. This may be due to air quality data missing at random (MAR). However, this is not the case for CA0050 and part B of CA0030. This may be due to fluctuation that could not be detected by NNM. Accuracy evaluation using Mean Absolute Percentage Error (MAPE) shows that NNM is more accurate imputation method for most of the cases.

This paper proposes a method for obtaining nonlinear models of distribution grid based on available measurements from the power grid. We formulate a maximum likelihood estimation (MLE) problem that estimates unknown line parameters—specifically, the impedance between nodes—using measured voltage magnitudes and net active and reactive power injections at each node. The nonlinear model for the distribution grid uses a nonlinear approximation of the DistFlow model, which includes line losses and is parameterized by the unknown line impedances. We solve the resulting MLE problem using an expectation maximization (EM) algorithm, tailored for the nonlinear model, and provide a numerically robust implementation. The proposed method is demonstrated on the IEEE 37-node test network, and we compare it with the state-of-the-art methods. The proposed method achieves a 70% reduction in voltage error and an error for state variables that is more than 10000 times smaller. A final comparison uses data from a real network, and the proposed method achieves parameter estimates with errors 100 times smaller than competing methods.