Maximum Entropy Filter Research Articles

State estimation comparison for a high-dimensional nonlinear system by particle-based filtering methods

The sequential filtering scheme provides a suitable framework for estimating and tracking geophysical states of systems as new data become available online. Mathematical foundations of sequential Bayesian filtering are reviewed with emphasis on practical issues for both particle filters and Kalman-based filters. In this study, we further investigate the study of Kim (2005) such that the sequential Importance resampling method (SIR), Ensemble Kalman Filter (EnKF), and the Maximum Entropy Filter (MEF) are tested in a relatively high dimensional ocean model that conceptually represents the Atlantic thermohaline circulation. The model exhibits large-amplitude transitions between strong (thermo-dominated) and weak (salinity-dominated) circulations that represent climate states between ice-age and normal climate.The performance of the particle-based schemes is compared with the convergent results from SIR based on measurement errors, observation locations, and particle sizes in various sets of twin experiments. The sensitivity analysis shows strength and weakness of each filtering method when applied to multimodal non-linear systems. As the number of particles is increased, SIR achieves the convergent results that are mathematically optimal solutions. EnKF shows suboptimal results regardless of sample sizes, and MEF achieves the optimal solution even with a small sample size. Both EnKF, and MEF produces robust results with a relatively small sample size or increased measurement locations. Small measurement errors or short intervals of observations (or, more frequent observations) significantly improve the performances of SIR and EnKF, and MEF still show robust results even with a relatively small sample size or sparse measurement locations when the system experiences the transition between one region to the other region.

Automatic Detection of Exudates from Digital Color Fundus Images

retinopathy is a widespread disease that may cause blindness. Early diagnosis and treatment will reduce its side effects and protect the eye. In this paper, a new algorithm for exudates detection is proposed. In the preprocessing step, the green channel of the color image is used, and then median filter followed by Contrast Limited Adaptive Histogram Equalization (CLAHE) is applied. The K-means clustering technique is used to select exudates objects. Optic disc is localized using maximum entropy filter and morphological closing. It is demonstrated that combining the K-means with CLAHE of the median filtered image results in 99.39% correct exudates. Experimental results show a reliable and accurate method for segmenting exudates from color retinal images. Performance of the proposed method is evaluated using a set of 52 images from a publicly available dataset STARE. KeywordsRetinopathy, Exudates detection, Entropy filter, K- means clustering

Maximum likelihood parameter estimation for a high-dimensional system by particle-based filters

Particle-based filters are applied to estimate poorly-known parameters in an oceanic thermohaline circulation model. The model is a stochastic partial differential equation that conceptually represents the Atlantic meridional overturning circulation. It exhibits rapid transitions between strong (temperature-dominated) and weak (salinity-dominated) circulations, and its stability depends on a stochastic term in the equation.Accurate estimation of the variance of the random term is critical in characterizing the circulation system, and the Maximum Likelihood Estimation is utilized via particle-based filters to estimate the parameter. The performance of different types of particle-based filters such as the Sequential Importance Resampling filter, Ensemble Kalman filter, and Maximum entropy filter are compared by the results of parameter estimation of the intermediate ocean model. The results show that all the three filters correctly estimate the stochastic parameter and, more importantly, suggest that particle-based filters are a promising tool for parameter estimation for high-dimensional systems.

Beyond Gaussian Statistical Modeling in Geophysical Data Assimilation

AbstractThis review discusses recent advances in geophysical data assimilation beyond Gaussian statistical modeling, in the fields of meteorology, oceanography, as well as atmospheric chemistry. The non-Gaussian features are stressed rather than the nonlinearity of the dynamical models, although both aspects are entangled. Ideas recently proposed to deal with these non-Gaussian issues, in order to improve the state or parameter estimation, are emphasized.The general Bayesian solution to the estimation problem and the techniques to solve it are first presented, as well as the obstacles that hinder their use in high-dimensional and complex systems. Approximations to the Bayesian solution relying on Gaussian, or on second-order moment closure, have been wholly adopted in geophysical data assimilation (e.g., Kalman filters and quadratic variational solutions). Yet, nonlinear and non-Gaussian effects remain. They essentially originate in the nonlinear models and in the non-Gaussian priors. How these effects are handled within algorithms based on Gaussian assumptions is then described. Statistical tools that can diagnose them and measure deviations from Gaussianity are recalled.The following advanced techniques that seek to handle the estimation problem beyond Gaussianity are reviewed: maximum entropy filter, Gaussian anamorphosis, non-Gaussian priors, particle filter with an ensemble Kalman filter as a proposal distribution, maximum entropy on the mean, or strictly Bayesian inferences for large linear models, etc. Several ideas are illustrated with recent or original examples that possess some features of high-dimensional systems. Many of the new approaches are well understood only in special cases and have difficulties that remain to be circumvented. Some of the suggested approaches are quite promising, and sometimes already successful for moderately large though specific geophysical applications. Hints are given as to where progress might come from.

Spectral restoration in HREELS

The spectral resolution of HREELS is determined by both instrumental and sample-related factors. Despite substantial progress in electron optic design, these factors are still a limitation in assigning the vibrational modes of complex molecules and in determining linewidths. To the extent that all spectral features are broadened in the same way and within signal to noise limitations, currently available spectral restoration methods may provide a reliable solution. In principle, the measured elastic peak provides a template for deconvolution of these factors from the observed lineshapes. We compare results of a direct deconvolution method, in which division in the Fourier domain is followed by a linearised maximum entropy filter, with iterative methods based on maximum likelihood, maximum entropy and Bayesian principles involving only multiplication (i.e. convolution) in the Fourier domain. These methods are applicable both to spectra of modest resolution (intrinsic linewidths > > instrumental linewidth) and to high resolution spectra (linewidths ≈ instrumental resolution) in which spectral lineshapes are measurably different. The asymmetry and tailing of the elastic peak, which are critical to restoration of low frequency features, are determined by a number of processes such as multiple scattering, the thermal population of vibrational states, and low energy continuum excitations (e–h pairs, phonons) of the substrate. These factors, as well as practical considerations, are discussed in relation to measuring data for optimal spectral restoration.

Geometry of the Kimura-Georgiou parametrization of modelling filters

We discuss questions concerning the geometry of the Kimura-Georgiou para-metrizalion of the set ( + (n) of degree n positive real transfer functions with the first n coefficients in the Laurent expansion about infinity prescribed. For example, one interesting question which has been raised is whether this set is star-shaped about the maximum entropy solution. This, of course, would be implied by convexity and would imply that,( plus;(n) is diffeomorphic to euclidean n-space. All three of these geometric properties would be of interest, for example, when using geometric or optimization techniques to construct an n-dimensional modelling filter with variations about the maximum entropy filter. Our first main result, which also lends support to conjectures concerning convexity and star-shapedness, is that. ( plus;(n) is in fact diffeomorphic to euclidean n-space. Our proof makes use of certain results from differential topology. There are of course several intimate relations between positive reality and stability properties of both real and complex polynomials. On this basis we observe that the convexity of the parametrization implies the convexity of the set of real Schur polynomials of degree n and is implied by the convexity of the set of complex Schur polynomials of degree n. From this it immediately follows that ( plus;(n) is not convex for n ≥ 3 and that. ( plus;(n) is convex and hence star-shaped. The case n = 2 is especially interesting since the real Schur region is convex whereas the complex is not. Another of our main results is to refine the above observations to settle this question in general, showing that for n ≥ 3 there is an open set of Schur parameters for which, ( plus;(n) fails to be star-shaped about the maximum entropy filter.