Negative binomial maximum likelihood expectation maximization (NB-MLEM) algorithm for reconstruction of pre-corrected PET data

The iterative maximum-likelihood expectation-maximization (ML-EM) algorithm is an excellent algorithm for image reconstruction and usually provides better images than the filtered backprojection (FBP) algorithm. However, a windowed FBP algorithm can outperform the ML-EM in certain occasions, when the least-squared difference from the true image, that is, the least-squared error (LSE), is used as the comparison criterion. Computer simulations were carried out for the two algorithms. For a given data set the best reconstruction (compared to the true image) from each algorithm was first obtained, and the two reconstructions are compared. The stopping iteration number of the ML-EM algorithm and the parameters of the windowed FBP algorithm were determined, so that they produced an image that was closest to the true image. However, to use the LSE criterion to compare algorithms, one must know the true image. How to select the optimal parameters when the true image is unknown is a practical open problem. For noisy Poisson projections, computer simulation results indicate that the ML-EM images are better than the regular FBP images, and the windowed FBP algorithm images are better than the ML-EM images. For the noiseless projections, the FBP algorithms outperform the ML-EM algorithm. The computer simulations reveal that the windowed FBP algorithm can provide a reconstruction that is closer to the true image than the ML-EM algorithm.

Modeling over-dispersed crash data with a long tail: Examining the accuracy of the dispersion parameter in Negative Binomial models

If X 1, X 2, … is a sequence of independent Bernoulli random variables, the number of successes in the first n trials has a binomial distribution and the number of failures before the rth success has a negative binomial distribution. From both the binomial and the negative binomial distributions, the Poisson distribution is obtainable as a limit. Moreover, gamma distributions (integer shape parameters) are limits of negative binomial distributions, and the normal distribution is a limit of negative binomial, Poisson, and gamma distributions. These basic facts from elementary probability have natural extensions to two dimensions because there is a unique natural bivariate Bernoulli distribution. In this article, such extensions yielding a family of bivariate distributions are obtained and studied.

<p>Modeling seismicity data is challenging and it remains a subject of ongoing research. Assumptions about the distribution of earthquake numbers play an important role in seismic hazard and risk analysis. The most common distribution that has been widely used in modeling earthquake numbers is the Poisson distribution because of its simplicity and easy to use. However, the heterogeneity in earthquake data and temporal dependencies that are often present in many real earthquake sequences make the use of the Poisson distribution inadequate. So, we propose the use of a Hidden Markov model (HMM) with state-specific Negative Binomial distributions in which some states are allowed to approach the Poisson distribution. A HMM is a generalization of a mixture model where the different unobservable (hidden) states are related through a Markov process rather than being independent of each other. We parameterize the Negative Binomial distribution in terms of the mean and dispersion (clustering) parameter. Maximum likelihood estimates of the models’ parameters are obtained through an Expectation-Maximization algorithm (EM-algorithm).</p><p>We apply the model to real earthquake data. We have selected the area of Killini Western Greece to test the proposed hypothesis. The area of Killini has been selected based on the fact that in a time window of 17 years three clusters of seismicity associated with strong mainshocks are included in the catalog. Application of the model to the data resulted in three states, representing different levels of seismicity (low, medium, high). The state that corresponds to the low seismicity level approaches the Poisson distribution while the other two states (medium and high) are following the Negative Binomial distribution. This result complies with the nature of the data. The variation within each state that is introduced to the model by the Negative Binomial distribution is greater in the states of medium and high seismicity. </p>

The question of how to characterize the bacterial density in a body of water when data are available as counts from a number of small-volume samples was examined for cases where either the Poisson or negative binomial probability distributions could be used to describe the bacteriological data. The suitability of the Poisson distribution when replicate analyses were performed under carefully controlled conditions and of the negative binomial distribution for samples collected from different locations and over time were illustrated by two examples. In cases where the negative binomial distribution was appropriate, a procedure was given for characterizing the variability by dividing the bacterial counts into homogeneous groups. The usefulness of this procedure was illustrated for the second example based on survey data for Lake Erie. A further illustration of the difference between results based on the Poisson and negative binomial distributions was given by calculating the probability of obtaining all samples sterile, assuming various bacterial densities and sample sizes.

Iterative algorithms such as maximum likelihood expectation maximization (ML-EM) algorithm are rapidly becoming the standard for image reconstruction in emission computed tomography. The maximum likelihood approach provides images with superior noise characteristics compared to conventional filtered backprojection algorithm. A major drawback of the iterative image reconstruction methods is their high computational cost. In this paper, we develop a new algorithm called the improved ordered subset expectation maximization (IOS-EM) algorithm. This algorithm modifies the number of projections in each subset and the step size (i.e., the relaxation factor) for each iteration in order to recover various frequency components in early iteration steps. In the method presented in this paper, the number of projections in a subset increases and the step size decreases after each iteration. In addition, pixel data are grouped into subdivisions to accelerate image reconstruction. Experimental results show that the IOS-EM algorithm can provide high quality reconstructed images at a small number of iterations.

We investigated the performance of an ordered-subsets expectation-maximization (OSEM) algorithm for accelerated reconstruction in cone-beam SPECT. SPECT scans were performed using a Defrise phantom filled with 0.9/spl mu/Ci/ml of Tc-99m and a dual-head gamma camera equipped with one cone-beam (CBC, f=70 cm) and one parallel-beam collimator (PBC). Images were reconstructed using a fully-3D approach with resolution and attenuation modeling and an ordered-subsets version of a maximum-likelihood expectation-maximization algorithm (MLEM). Three grouping patterns of subsets were applied: consecutive, orthogonal, and uniform. In contrast to PBC SPECT, we observe that, in CBC SPECT, the reconstruction grouping pattern of the subsets is very important for the image quality obtained. Only when the projection data grouped into a subset were selected as uniformly as possible from all the acquired views, were the image quality and the noise in the images very close to results obtained using MLEM. However, we note that, for both CBC and PBC SPECT, the loglikelihood for a given iteration is practically the same for different grouping patterns of subsets.

Developing a Random Parameters Negative Binomial-Lindley Model to analyze highly over-dispersed crash count data

The enhanced depth discrimination of a confocal scanning optical microscope is produced by a pinhole aperture placed in front of the detector to reject out-of-focus light. Strictly confocal behavior is impractical because an infinitesimally small aperture would collect very little light and would result in images with a poor signal-to-noise ratio (SNR), while a finite-sized partially confocal aperture provides a better SNR but reduced depth discrimination. Reconstruction algorithms, such as the expectationmaximization algorithm for maximum likelihood, can be applied to partially confocal images in order to achieve better resolution, but because they are sensitive to noise in the data, there is a practical trade-off involved. With a small aperture, fewer iterations of the reconstruction algorithm are necessary to achieve the desired resolution, but the low a priori SNR will result in a noisy reconstruction, at least when no regularization is used. With a larger aperture the a priori SNR is larger but the resolution is lower, and more iterations of the algorithm are necessary to reach the desired resolution; at some point the a posteriori SNR is lower than the a priori value. We present a theoretical analysis of the SNR-toresolution trade-off partially confocal imaging, and we present two studies that use the expectationmaximization algorithm as a postprocessor; these studies show that a for a given task there is an optimum aperture size, departures from which result in a lower a posteriori SNR.

The foliage in a fully-leafed deciduous forest canopy is clumped. Consequently, theory indicates that the probability of beam penetration will be estimated more accurately with a model based on the negative binomial distribution than with a model based on the Poisson distribution, incorporating an assumption of a spherical leaf inclination angle distribution. Flux densities of photosynthetically active radiation (PAR) were measured in and above a deciduous forest canopy and were computed with the canopy radiative transfer models based on the negative binomial and Poisson distributions. These radiation values were used to compute canopy photosynthesis and stomatal conductance. Canopy photosynthesis and stomatal conductance, based on the negative binomial model, overestimated values computed from measured PAR profiles by 8 and 9%. respectively. The canopy photosynthesis and stomatal conductance values computed with the spherical Poisson model under-estimated measured values by 17 and 10%, respectively. Thus, the negative binomial radiative transfer model improves estimates of canopy photosynthesis and, to a lesser extent, stomatal conductance, inside a deciduous forest.

BackgroundRNA sequencing (RNA-seq) is a powerful tool for genome-wide expression profiling of biological samples with the advantage of high-throughput and high resolution. There are many existing algorithms nowadays for quantifying expression levels and detecting differential gene expression, but none of them takes the misaligned reads that are mapped to non-exonic regions into account. We developed a novel algorithm, XBSeq, where a statistical model was established based on the assumption that observed signals are the convolution of true expression signals and sequencing noises. The mapped reads in non-exonic regions are considered as sequencing noises, which follows a Poisson distribution. Given measureable observed and noise signals from RNA-seq data, true expression signals, assuming governed by the negative binomial distribution, can be delineated and thus the accurate detection of differential expressed genes.ResultsWe implemented our novel XBSeq algorithm and evaluated it by using a set of simulated expression datasets under different conditions, using a combination of negative binomial and Poisson distributions with parameters derived from real RNA-seq data. We compared the performance of our method with other commonly used differential expression analysis algorithms. We also evaluated the changes in true and false positive rates with variations in biological replicates, differential fold changes, and expression levels in non-exonic regions. We also tested the algorithm on a set of real RNA-seq data where the common and different detection results from different algorithms were reported.ConclusionsIn this paper, we proposed a novel XBSeq, a differential expression analysis algorithm for RNA-seq data that takes non-exonic mapped reads into consideration. When background noise is at baseline level, the performance of XBSeq and DESeq are mostly equivalent. However, our method surpasses DESeq and other algorithms with the increase of non-exonic mapped reads. Only in very low read count condition XBSeq had a slightly higher false discovery rate, which may be improved by adjusting the background noise effect in this situation. Taken together, by considering non-exonic mapped reads, XBSeq can provide accurate expression measurement and thus detect differential expressed genes even in noisy conditions.

Effectiveness of sampling plans by attributes based on mixture distributions characterising microbial clustering in food

Depth-of-interaction enhanced Compton camera using pixelated LYSO(Ce) scintillator arrays and dual-ended SiPMs.

Methodologic Challenges in the Analysis of Count Data in Radiology Health Services Research

Estimating Retail Demand with Poisson Mixtures and Out-of-Sample Likelihood