Estimation of haplotype associated with several quantitative phenotypes based on maximization of area under a receiver operating characteristic (ROC) curve
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Value (mathematics)
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The problem of estimating the parameters which determine a mixture density is reviewed as well as maximum likelihood estimation for it. A particular iterative procedure for numerically approximating maximum likelihood estimates for mixture density problems is considered. This EM algorithm, is a specialization to the mixture density context of a general algorithm of the same name used to approximate maximum likelihood estimates for incomplete data problems. The formulation and theoretical and practical properties of the EM algorithm for mixture densities are discussed focussing in particular on mixtures of densities from exponential families.
Maximum density
Density estimation
Restricted maximum likelihood
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Expectation-maximization is a broadly applicable approach to the iterative computation of maximum likelihood estimates. Each iteration of expectation-maximization method consists of two steps: the expectation step and the maximization step. Expectation-maximization method is useful in a variety of problems where the maximum likelihood estimates are very difficult to find. The basic idea of expectation-maximization method is to relate incomplete data problems to complete data problems where estimation by maximum likelihood method is much simpler.
Maximization
Restricted maximum likelihood
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Maximization
Restricted maximum likelihood
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This paper describes a simulation study of (small sample) maximum likelihood estimation for mixtures of distributions. Maximum likelihood (ML) leads to a multidimensional maximization problem for which a modified Newton method and the EM algorithm are used. The moment and Kabir method are applied to provide initial points. For twenty-seven different mixture distributions a number of samples is generated by means of a pseudo-random number generator. Frequently the supplied initial points are not admissible. Although an ad hoc method for initial points behaves well, still no estimates are obtained for a number of samples.
Maximization
Restricted maximum likelihood
Sample (material)
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Images produced in emission tomography with the expectation-maximization algorithm have been observed to become more noisy and to have large distortions near edges as iterations proceed and the images converge towards the maximum-likelihood estimate. It is our conclusion that these artifacts are fundamental to reconstructions based on maximum-likelihood estimation as it has been applied usually; they are not due to the use of the expectation-maximization algorithm, which is but one numerical approach for finding the maximum-likelihood estimate. In this paper, we develop a mathematical approach for suppressing both the noise and edge artifacts by modifying the maximum-likelihood approach to include constraints which the estimate must satisfy.
Maximization
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Abstract A new method for analysis of electron microscope autoradiographs is described which is based on the maximum‐likelihood method of statistics for estimating the intensities of radioactivity in organelle structures. We adopted a Poisson statistical model to describe the autoradiographic grain distributions that we prove results from the underlying Poisson nature of the radioactive decays as well as the additive errors introduced during the formation of grains. Within the model, an interative procedure derived from the expectation‐maximization algorithm of mathematical statistics is used to generate the maximum‐likelihood estimates. The algorithm has the properties that at every stage of the iteration process the likelihood of the data increases; and for all initial nonzero starting points the algorithm converges to the maximum‐likelihood estimates of the organelle intensities. The maximum‐likelihood approach differs from the mask‐analysis method, and other published quantitative algorithms in the following ways: (1) In deriving estimates of the radioactivity intensities the maximum‐likelihood algorithm requires that we obtain the actual locations of the grains as well as the micrograph geometries; each micrograph is digitized so that both the grain locations as well as the geometries of the organelle structures can be used. (2) The maximum‐likelihood algorithm iteratively computes the minimum‐meansquared‐error estimate of the underlying emission locations that resulted in the observed grain distributions, from which intensity estimates are generated; this algorithm does not minimize a chi‐squared error statistic. (3) The maximum‐likelihood approach is based on a Poisson model and is therefore valid for low‐count experiments; there are no minimum constraints on data collection for any single organelle compartment. (4) The maximum‐likelihood algorithm requires the form of the point‐spread function describing the emission spread; a probability matrix based on the use of overlay masks is not required. (5) The maximum‐likelihood algorithm does not change for different organelle geometries; arbitrary geometries are incorporated by maximizing the likelihood‐function subject to the geometry constraints. We have performed a preliminary evaluation of the quantitative accuracy of the maximum‐likelihood and mask‐analysis algorithms. Based on two different phantoms in which we compared the squared error resulting from the two algorithms, we find that the new maximum‐likelihood approach provides substantially improved estimates of the radioactivity intensities of the phantoms.
Statistic
Restricted maximum likelihood
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Abstract In this paper the strategy of the expectation maximization algorithm is developed in order to solve the problem of maximization of the likelihood which becomes a complex problem when it is expected to estimate multiple parameters at the same time. All this will be achieved with the introduction of basic concepts of statistical estimators and will be verified by simulations carried out in the MATLAB software.
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Expectation maximization algorithm (EM) is used to create estimator with the same qualities of maximum likelihood Estimator taking into consideration the existence of two types of data, Data viewing (observed data) and hidden data (missing data), in this research the estimating parameters of factor analysis model (maximum likelihood method) has been done by using expectation maximization algorithm and applied factor analysis (maximum likelihood method) on data for patients infected with breast cancer, and found from the results importance all of the variables in breast cancer variables except first variable (level of education) and fifth variable (hormone treatment used).
Restricted maximum likelihood
Maximization
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The application of the space alternating, expectation-maximization (SAGE) method is discussed for a problem in coherent-speckle imaging (or two-dimensional spectrum estimation). For this algorithm, convergence of the likelihood is demonstrated to be significantly faster than that of an expectation-maximization (EM) based algorithm that has been previously proposed. In addition, the utilization of the SAGE method for penalized maximum-likelihood estimation is demonstrated.
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Speckle noise
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Generalized maximum likelihood sequence detection and estimation (GMLSDE) is developed in this paper based on the expectation and maximization (EM) algorithm. The GMLSDE couples the estimation of channel parameters and data detection in the framework of the maximum likelihood (ML) criterion and unifies many MLSD/MLSDE structure receivers for different channel models. The GMLSDE clarifies the relation among channel model, receiver structure and degree of optimality. The per-survivor processing (PSP) and per-branch processing (PBP) methods emerge naturally from the EM aspect of the GMLSDE as well.
Maximization
Sequence (biology)
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