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Results for “penalized likelihood maximization”

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PubMed1994

Fisher's method of scoring in statistical image reconstruction: comparison of Jacobi and Gauss-Seidel iterative schemes.

Many algorithms for medical image reconstruction adopt versions of the expectation-maximization (EM) algorithm. In this approach, parameter estimates are obtained which maximize a complete data likelihood or penalized likelihood, in each iteration. Implicitly (and sometimes explicitly) penalized algorithms require smoothing of the current reconstruction in the image domain as part of their iteration scheme. In this paper, we discuss alternatives to EM which adapt Fisher's method of scoring (FS) and other methods for direct maximization of the incomplete data likelihood. Jacobi and Gauss-Seidel methods for non-linear optimization provide efficient algorithms applying FS in tomography. One approach uses smoothed projection data in its iterations. We investigate the convergence of Jacobi and Gauss-Seidel algorithms with clinical tomographic projection data.

Algorithms↗

PubMed2006

Penalized-likelihood sinogram restoration for computed tomography.

We formulate computed tomography (CT) sinogram preprocessing as a statistical restoration problem in which the goal is to obtain the best estimate of the line integrals needed for reconstruction from the set of noisy, degraded measurements. CT measurement data are degraded by a number of factors-including beam hardening and off-focal radiation-that produce artifacts in reconstructed images unless properly corrected. Currently, such effects are addressed by a sequence of sinogram-preprocessing steps, including deconvolution corrections for off-focal radiation, that have the potential to amplify noise. Noise itself is generally mitigated through apodization of the reconstruction kernel, which effectively ignores the measurement statistics, although in high-noise situations adaptive filtering methods that loosely model data statistics are sometimes applied. As an alternative, we present a general imaging model relating the degraded measurements to the sinogram of ideal line integrals and propose to estimate these line integrals by iteratively optimizing a statistically based objective function. We consider three different strategies for estimating the set of ideal line integrals, one based on direct estimation of ideal "monochromatic" line integrals that have been corrected for single-material beam hardening, one based on estimation of ideal "polychromatic" line integrals that can be readily mapped to monochromatic line integrals, and one based on estimation of ideal transmitted intensities, from which ideal, monochromatic line integrals can be readily estimated. The first two approaches involve maximization of a penalized Poisson-likelihood objective function while the third involves minimization of a quadratic penalized weighted least squares (PWLS) objective applied in the transmitted intensity domain. We find that at low exposure levels typical of those being considered for screening CT, the Poisson-likelihood based approaches outperform the PWLS objective as well as a standard approach based on adaptive filtering followed by deconvolution. At higher exposure levels, the approaches all perform similarly.

Algorithms↗

PubMed2004

Graphical modeling of the joint distribution of alleles at associated loci.

Pairwise linkage disequilibrium, haplotype blocks, and recombination hotspots provide only a partial description of the patterns of dependences and independences between the allelic states at proximal loci. On the gross scale, where recombination and spatial relationships dominate, the associations can be reasonably described in these terms. However, on the fine scale of current high-density maps, the mutation process is also important and creates associations between loci that are independent of the physical ordering and that can not be summarized with pairwise measures of association. Graphical modeling provides a standard statistical framework for characterizing precisely these sorts of complex stochastic data. Although graphical models are often used in situations in which assumptions lead naturally to specific models, it is less well known that estimation of graphical models is also a developed field. We show how decomposable graphical models can be fitted to dense genetic data. The objective function is the maximized log likelihood for the model penalized by a multiple of the model's degrees of freedom. We also describe how this can be modified to incorporate prior information of locus position. Simulated annealing is used to find good solutions. Part of the appeal of this approach is that categorical phenotypes can be included in the same analysis and association with polymorphisms can be assessed jointly with the interlocus associations. We illustrate our method with genotypic data from 25 loci in the ELAC2 gene. The results contain third- and fourth-order locus interactions and show that, at this density of markers, linkage disequilibrium is not a simple function of physical distance. Graphical models provide more flexibility to express these features of the joint distribution of alleles than do monotonic functions connecting physical and genetic maps.

Alleles↗

PubMed2025

A robust transfer learning approach for high-dimensional linear regression to support integration of multi-source gene expression data.

Transfer learning aims to integrate useful information from multi-source datasets to improve the learning performance of target data. This can be effectively applied in genomics when we learn the gene associations in a target tissue, and data from other tissues can be integrated. However, heavy-tail distribution and outliers are common in genomics data, which poses challenges to the effectiveness of current transfer learning approaches. In this paper, we study the transfer learning problem under high-dimensional linear models with t-distributed error (Trans-PtLR), which aims to improve the estimation and prediction of target data by borrowing information from useful source data and offering robustness to accommodate complex data with heavy tails and outliers. In the oracle case with known transferable source datasets, a transfer learning algorithm based on penalized maximum likelihood and expectation-maximization algorithm is established. To avoid including non-informative sources, we propose to select the transferable sources based on cross-validation. Extensive simulation experiments as well as an application demonstrate that Trans-PtLR demonstrates robustness and better performance of estimation and prediction when heavy-tail and outliers exist compared to transfer learning for linear regression model with normal error distribution. Data integration, Variable selection, T distribution, Expectation maximization algorithm, Genotype-Tissue Expression, Cross validation.

Linear Models↗

PubMed1995

Time-dependent effects of fixed covariates in Cox regression.

A nonparametric modification is proposed for Cox's proportional hazards model (Cox, 1972, Journal of the Royal Statistical Society, Series B 34, 187-220), where the covariates are fixed, but their effects are allowed to vary in time. Parameters are introduced for the covariate effects at the (uncensored) survival times. Estimates are obtained by maximizing the penalized partial log-likelihood that arises if a penalty function of the first order differences of consecutive parameters is subtracted from the partial log-likelihood. The choice of the smoothing parameter, which determines the weight of the penalty, is based on Akaike's Information Criterion. The methods are illustrated in a set of ovarian cancer data and in a set of kidney transplantation data.

Antineoplastic Agents↗

PubMed2001

Modelling bivariate ordinal responses smoothly with examples from ophthalmology and genetics.

A non-parametric implementation of the bivariate Dale model (BDM) is presented as an extension of the generalized additive model (GAM) of Hastie and Tibshirani. The original BDM is an example of a bivariate generalized linear model. In this paper smoothing is introduced on the marginal as well as on the association level. Our non-parametric procedure can be used as a diagnostic tool for identifying parametric transformations of the covariates in the linear BDM, hence it also provides a kind of goodness-of-fit test for a bivariate generalized linear model. Cubic smoothing spline functions for the covariates are estimated by maximizing a penalized version of the log-likelihood. The method is applied to two studies. The first study is the classical Wisconsin Epidemiologic Study of Diabetic Retinopathy. The second study is a twin study, where the association between the elements of twin pairs is of primary interest. The results show that smoothing on the association level can give a significant improvement to the model fit.

Adolescent↗

PubMed2005

A fast image reconstruction algorithm based on penalized-likelihood estimate.

Statistical iterative methods for image reconstruction like maximum likelihood expectation maximization (ML-EM) are more robust and flexible than analytical inversion methods and allow for accurately modeling the counting statistics and the photon transport during acquisition. They are rapidly becoming the standard for image reconstruction in emission computed tomography. The maximum likelihood approach provides images with superior noise characteristics compared to the conventional filtered back projection algorithm. But a major drawback of the statistical iterative image reconstruction is its high computational cost. In this paper, a fast algorithm is proposed as a modified OS-EM (MOS-EM) using a penalized function, which is applied to the least squares merit function to accelerate image reconstruction and to achieve better convergence. The experimental results show that the algorithm can provide high quality reconstructed images with a small number of iterations.

Algorithms↗

PubMed2003

Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms.

We present two types of globally convergent relaxed ordered subsets (OS) algorithms for penalized-likelihood image reconstruction in emission tomography: modified block sequential regularized expectation-maximization (BSREM) and relaxed OS separable paraboloidal surrogates (OS-SPS). The global convergence proof of the existing BSREM (De Pierro and Yamagishi, 2001) required a few a posteriori assumptions. By modifying the scaling functions of BSREM, we are able to prove the convergence of the modified BSREM under realistic assumptions. Our modification also makes stepsize selection more convenient. In addition, we introduce relaxation into the OS-SPS algorithm (Erdoğan and Fessler, 1999) that otherwise would converge to a limit cycle. We prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed OS-SPS is a special case; main results of the proofs are from (Nedić and Bertsekas, 2001) and (Correa and Lemaréchal, 1993). Simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms.

Algorithms↗

PubMed2006

Convergent incremental optimization transfer algorithms: application to tomography.

No convergent ordered subsets (OS) type image reconstruction algorithms for transmission tomography have been proposed to date. In contrast, in emission tomography, there are two known families of convergent OS algorithms: methods that use relaxation parameters, and methods based on the incremental expectation-maximization (EM) approach. This paper generalizes the incremental EM approach by introducing a general framework, "incremental optimization transfer." The proposed algorithms accelerate convergence speeds and ensure global convergence without requiring relaxation parameters. The general optimization transfer framework allows the use of a very broad family of surrogate functions, enabling the development of new algorithms. This paper provides the first convergent OS-type algorithm for (nonconcave) penalized-likelihood (PL) transmission image reconstruction by using separable paraboloidal surrogates (SPS) which yield closed-form maximization steps. We found it is very effective to achieve fast convergence rates by starting with an OS algorithm with a large number of subsets and switching to the new "transmission incremental optimization transfer (TRIOT)" algorithm. Results show that TRIOT is faster in increasing the PL objective than nonincremental ordinary SPS and even OS-SPS yet is convergent.

Algorithms↗

PubMed2004

Object dependency of resolution in reconstruction algorithms with interiteration filtering applied to PET data.

In this paper, we study the resolution properties of those algorithms where a filtering step is applied after every iteration. As concrete examples we take filtered preconditioned gradient descent algorithms for the Poisson log likelihood for PET emission data. For nonlinear estimators, resolution can be characterized in terms of the linearized local impulse response (LLIR). We provide analytic approximations for the LLIR for the class of algorithms mentioned above. Our expressions clearly show that when interiteration filtering (with linear filters) is used, the resolution properties are, in most cases, spatially varying, object dependent and asymmetric. These nonuniformities are solely due to the interaction between the filtering step and the Poisson noise model. This situation is similar to penalized likelihood reconstructions as studied previously in the literature. In contrast, nonregularized and postfiltered maximum-likelihood expectation maximization (MLEM) produce images with nearly "perfect" uniform resolution when convergence is reached. We use the analytic expressions for the LLIR to propose three different approaches to obtain nearly object independent and uniform resolution. Two of them are based on calculating filter coefficients on a pixel basis, whereas the third one chooses an appropriate preconditioner. These three approaches are tested on simulated data for the filtered MLEM algorithm or the filtered separable paraboloidal surrogates algorithm. The evaluation confirms that images obtained using our proposed regularization methods have nearly object independent and uniform resolution.

Algorithms↗

PubMed2006

Image reconstruction for positron emission tomography using fuzzy nonlinear anisotropic diffusion penalty.

Iterative algorithms such as maximum likelihood-expectation maximization (ML-EM) become the standard for the reconstruction in emission computed tomography. However, such algorithms are sensitive to noise artifacts so that the reconstruction begins to degrade when the number of iterations reaches a certain value. In this paper, we have investigated a new iterative algorithm for penalized-likelihood image reconstruction that uses the fuzzy nonlinear anisotropic diffusion (AD) as a penalty function. The proposed algorithm does not suffer from the same problem as that of ML-EM algorithm, and it converges to a low noisy solution even if the iteration number is high. The fuzzy reasoning instead of a nonnegative monotonically decreasing function was used to calculate the diffusion coefficients which control the whole diffusion. Thus, the diffusion strength is controlled by fuzzy rules expressed in a linguistic form. The proposed method makes use of the advantages of fuzzy set theory in dealing with uncertain problems and nonlinear AD techniques in removing the noise as well as preserving the edges. Quantitative analysis shows that the proposed reconstruction algorithm is suitable to produce better reconstructed images when compared with ML-EM, ordered subsets EM (OS-EM), Gaussian-MAP, MRP, TV-EM reconstructed images.

Algorithms↗

PubMed2000

Estimation of multivariate frailty models using penalized partial likelihood.

There exists a growing literature on the estimation of gamma distributed multiplicative shared frailty models. There is, however, often a need to model more complicated frailty structures, but attempts to extend gamma frailties run into complications. Motivated by hip replacement data with a more complicated dependence structure, we propose a model based on multiplicative frailties with a multivariate log-normal joint distribution. We give a justification and an estimation procedure for this generally structured frailty model, which is a generalization of the one presented by McGilchrist (1993, Biometrics 49, 221-225). The estimation is based on Laplace approximation of the likelihood function. This leads to estimating equations based on a penalized fixed effects partial likelihood, where the marginal distribution of the frailty terms determines the penalty term. The tuning parameters of the penalty function, i.e., the frailty variances, are estimated by maximizing an approximate profile likelihood. The performance of the approximation is evaluated by simulation, and the frailty model is fitted to the hip replacement data.

Biometry↗

PubMed2003

Platelets: a multiscale approach for recovering edges and surfaces in photon-limited medical imaging.

The nonparametric multiscale platelet algorithms presented in this paper, unlike traditional wavelet-based methods, are both well suited to photon-limited medical imaging applications involving Poisson data and capable of better approximating edge contours. This paper introduces platelets, localized functions at various scales, locations, and orientations that produce piece-wise linear image approximations, and a new multiscale image decomposition based on these functions. Platelets are well suited for approximating images consisting of smooth regions separated by smooth boundaries. For smoothness measured in certain Hölder classes, it is shown that the error of m-term platelet approximations can decay significantly faster than that of m-term approximations in terms of sinusoids, wavelets, or wedgelets. This suggests that platelets may outperform existing techniques for image denoising and reconstruction. Fast, platelet-based, maximum penalized likelihood methods for photon-limited image denoising, deblurring and tomographic reconstruction problems are developed. Because platelet decompositions of Poisson distributed images are tractable and computationally efficient, existing image reconstruction methods based on expectation-maximization type algorithms can be easily enhanced with platelet techniques. Experimental results suggest that platelet-based methods can outperform standard reconstruction methods currently in use in confocal microscopy, image restoration, and emission tomography.

Algorithms↗

PubMed2002

Maximum likelihood inference for multivariate frailty models using an automated Monte Carlo EM algorithm.

We present a maximum likelihood estimation procedure for the multivariate frailty model. The estimation is based on a Monte Carlo EM algorithm. The expectation step is approximated by averaging over random samples drawn from the posterior distribution of the frailties using rejection sampling. The maximization step reduces to a standard partial likelihood maximization. We also propose a simple rule based on the relative change in the parameter estimates to decide on sample size in each iteration and a stopping time for the algorithm. An important new concept is acquiring absolute convergence of the algorithm through sample size determination and an efficient sampling technique. The method is illustrated using a rat carcinogenesis dataset and data on vase lifetimes of cut roses. The estimation results are compared with approximate inference based on penalized partial likelihood using these two examples. Unlike the penalized partial likelihood estimation, the proposed full maximum likelihood estimation method accounts for all the uncertainty while estimating standard errors for the parameters.

Algorithms↗

PubMed2006

Estimating haplotype effects on dichotomous outcome for unphased genotype data using a weighted penalized log-likelihood approach.

OBJECTIVE: To develop a method to estimate haplotype effects on dichotomous outcomes when phase is unknown, that can also estimate reliable effects of rare haplotypes. METHODS: In short, the method uses a logistic regression approach, with weights attached to all possible haplotype combinations of an individual. An EM-algorithm was used: in the E-step the weights are estimated, and the M-step consists of maximizing the joint log-likelihood. When rare haplotypes were present, a penalty function was introduced. We compared four different penalties. To investigate statistical properties of our method, we performed a simulation study for different scenarios. The evaluation criteria are the mean bias of the parameter estimates, the root of the mean squared error, the coverage probability, power, Type I error rate and the false discovery rate. RESULTS: For the unpenalized approach, mean bias was small, coverage probabilities were approximately 95%, power ranged from 15.2 to 44.7% depending on haplotype frequency, and Type I error rate was around 5%. All penalty functions reduced the standard errors of the rare haplotypes, but introduced bias. This trade-off decreased power. CONCLUSION: The unpenalized weighted log-likelihood approach performs well. A penalty function can help to estimate an effect for rare haplotypes.

Algorithms↗

PubMed1999

A theoretical study of the contrast recovery and variance of MAP reconstructions from PET data.

We examine the spatial resolution and variance properties of PET images reconstructed using maximum a posteriori (MAP) or penalized-likelihood methods. Resolution is characterized by the contrast recovery coefficient (CRC) of the local impulse response. Simplified approximate expressions are derived for the local impulse response CRC's and variances for each voxel. Using these results we propose a practical scheme for selecting spatially variant smoothing parameters to optimize lesion detectability through maximization of the local CRC-to-noise ratio in the reconstructed image.

Analysis of Variance↗

PubMed2004

Emission image reconstruction for randoms-precorrected PET allowing negative sinogram values.

Most positron emission tomography (PET) emission scans are corrected for accidental coincidence (AC) events by real-time subtraction of delayed-window coincidences, leaving only the randoms-precorrected data available for image reconstruction. The real-time randoms precorrection compensates in mean for AC events but destroys the Poisson statistics. The exact log-likelihood for randoms-precorrected data is inconvenient, so practical approximations are needed for maximum likelihood or penalized-likelihood image reconstruction. Conventional approximations involve setting negative sinogram values to zero, which can induce positive systematic biases, particularly for scans with low counts per ray. We propose new likelihood approximations that allow negative sinogram values without requiring zero-thresholding. With negative sinogram values, the log-likelihood functions can be nonconcave, complicating maximization; nevertheless, we develop monotonic algorithms for the new models by modifying the separable paraboloidal surrogates and the maximum-likelihood expectation-maximization (ML-EM) methods. These algorithms ascend to local maximizers of the objective function. Analysis and simulation results show that the new shifted Poisson (SP) model is nearly free of systematic bias yet keeps low variance. Despite its simpler implementation, the new SP performs comparably to the saddle-point model which has shown the best performance (as to systematic bias and variance) in randoms-precorrected PET emission reconstruction.

Algorithms↗

PubMed1994

A Bayesian approach to subvoxel tissue classification in NMR microscopic images of trabecular bone.

NMR microscopy is currently being used as an investigational tool for the evaluation of micromorphometric parameters of trabecular bone as a possible means to assess its strength. Since, typically, the image voxel size is not significantly smaller than individual trabecular elements, partial volume blurring can be a major complication for accurate tissue classification. In this paper, a Bayesian segmentation technique is reported that achieves improved subvoxel tissue classification. Each voxel is subdivided either into eight subvoxels twice the original resolution, or up to four subvoxels along the transaxial direction and the subvoxels optimally classified as volume blurring, the likelihood for the number of marrow subvoxels in each voxel can be computed on the basis of its measured signal. To resolve the ambiguity of the location of the marrow subvoxels, a Gibbs distribution is introduced to model the interaction between the subvoxels. Neighboring subvoxel pairs with the same tissue label are encouraged, and pairs with distinct labels are penalized. The segmentation is achieved by maximizing the a posteriori probability of the label image using the block ICM (iterative conditional mode) algorithm. The potential of the proposed technique is demonstrated in real and synthetic NMR microscopic images.

Algorithms↗