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An Improved Unbiased Particle Filter

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arXiv 2023年

作者： Jasra, Ajay Maama, Mohamed Ombao, Hernando Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia Statistics Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this paper we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. We assume that, for numerical reasons, one has to time-discretize the diffusion process which typically leads to filtering that is subject to discretization bias. The approach in [16] establishes that when only having access to the time discretized diffusion it is possible to remove the discretization bias with an estimator of finite variance. We improve on the method in [16] by introducing a modified estimator based on the recent work of [17]. We show that this new estimator is unbiased and has finite variance. Moreover, we conjecture and verify in numerical simulations that substantial gains are obtained. That is, for a given mean square error (MSE) and a particular class of multi-dimensional diffusion, the cost to achieve the said MSE falls. Copyright © 2023, The Authors. All rights reserved.

关键词： Mean square error

Learning nonlinear integral operators via Recurrent Neural Networks and its application in solving Integro-differential Equations

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arXiv 2023年

作者： Bassi, Hardeep Zhu, Yuanran Liang, Senwei Yin, Jia Reeves, Cian C. Vlček, Vojtěch Yang, Chao Department of Applied Mathematics University of California Merced Merced95340 United States Applied Mathematics and Computational Research Division Lawerence Berkeley National Laboratory Berkeley94720 United States Department of Physics University of California Santa Barbara Santa Barbara93117 United States Department of Chemistry and Biochemistry University of California Santa Barbara Santa Barbara93117 United States Department of Materials University of California Santa Barbara Santa Barbara93117 United States

In this paper, we propose using LSTM-RNNs (Long Short-Term Memory-Recurrent Neural Networks) to learn and represent nonlinear integral operators that appear in nonlinear integro-differential equations (IDEs). The LSTM-RNN representation of the nonlinear integral operator allows us to turn a system of nonlinear integro-differential equations into a system of ordinary differential equations for which many efficient solvers are available. Furthermore, because the use of LSTM-RNN representation of the nonlinear integral operator in an IDE eliminates the need to perform a numerical integration in each numerical time evolution step, the overall temporal cost of the LSTM-RNN-based IDE solver can be reduced to O(nT) from O(n2T) if a nT-step trajectory is to be computed. We illustrate the efficiency and robustness of this LSTM-RNN-based numerical IDE solver with a model problem. Additionally, we highlight the generalizability of the learned integral operator by applying it to IDEs driven by different external forces. As a practical application, we show how this methodology can effectively solve the Dyson's equation for quantum many-body systems. © 2023, CC BY.

关键词： Integrodifferential equations

Characterizing the SLOPE Trade-off: A Variational Perspective and the Donoho-Tanner Limit

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arXiv 2021年

作者： Bu, Zhiqi Klusowski, Jason M. Rush, Cynthia Su, Weijie J. Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania United States Department of Operations Research and Financial Engineering Princeton University United States Department of Statistics Columbia University United States Department of Statistics and Data Science University of Pennsylvania United States

Sorted l1 regularization has been incorporated into many methods for solving high-dimensional statistical estimation problems, including the SLOPE estimator in linear regression. In this paper, we study how this relatively new regularization technique improves variable selection by characterizing the optimal SLOPE trade-off between the false discovery proportion (FDP) and true positive proportion (TPP) or, equivalently, between measures of type I error and power. Assuming a regime of linear sparsity and working under Gaussian random designs, we obtain an upper bound on the optimal trade-off for SLOPE, showing its capability of breaking the Donoho-Tanner power limit. To put it into perspective, this limit is the highest possible power that the Lasso, which is perhaps the most popular l1-based method, can achieve even with arbitrarily strong effect sizes. Next, we derive a tight lower bound that delineates the fundamental limit of sorted l1 regularization in optimally trading the FDP off for the TPP. Finally, we show that on any problem instance, SLOPE with a certain regularization sequence outperforms the Lasso, in the sense of having a smaller FDP, larger TPP and smaller l2 estimation risk simultaneously. Our proofs are based on a novel technique that reduces a calculus of variations problem to a class of infinite-dimensional convex optimization problems and a very recent result from approximate message passing theory. Copyright © 2021, The Authors. All rights reserved.

关键词： Message passing

Biofluid mechanics of special organs and the issue of system control: Sixth international bio-fluid mechanics symposium and workshop, March 28-30, 2008 Pasadena, California

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Biofluid mechanics of special organs and the issue of system...

作者： Zamir, Mair Moore Jr., James E. Fujioka, Hideki Gaver III, Donald P. Department of Applied Mathematics University of Western Ontario 1151 Richmond Street London ON N6A 3K7 Canada Department of Medical Biophysics University of Western Ontario London ON N6A 3K7 Canada Department of Biomedical Engineering Texas A and M University College Station TX United States Center for Computational Science Tulane University New Orleans LA United States Department of Biomedical Engineering Tulane University New Orleans LA United States

In the field of fluid flow within the human body, focus has been placed on the transportation of blood in the systemic circulation since the discovery of that system;but, other fluids and fluid flow phenomena pervade the body. Some of the most fascinating fluid flow phenomena within the human body involve fluids other than blood and a service other than transport-the lymphatic and pulmonary systems are two striking examples. While transport is still involved in both cases, this is not the only service which they provide and blood is not the only fluid involved. In both systems, filtration, extraction, enrichment, and in general some "treatment" of the fluid itself is the primary function. The study of the systemic circulation has also been conventionally limited to treating the system as if it were an open-loop system governed by the laws of fluid mechanics alone, independent of physiological controls and regulations. This implies that system failures can be explained fully in terms of the laws of fluid mechanics, which of course is not the case. In this paper we examine the clinical implications of these issues and of the special biofluid mechanics issues involved in the lymphatic and pulmonary systems. © 2010 Biomedical Engineering Society.

关键词： Flow of fluids

Enhancing Electrocardiography Data Classification Confidence: A Robust Gaussian Process Approach (MuyGPs)

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arXiv 2024年

作者： Nnyaba, Ukamaka V. Shemtaga, Hewan M. Collins, David W. Muyskens, Amanda L. Priest, Benjamin W. Billor, Nedret Department of Mathematics and Statistics Auburn University AuburnAL36849 United States Department of Mathematics and Statistics Auburn University AuburnAL36849 United States Computational Engineering Division Lawrence Livermore National Laboratory LivermoreCA94550 United States Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94550 United States

Analyzing electrocardiography (ECG) data is essential for diagnosing and monitoring various heart diseases. The clinical adoption of automated methods requires accurate confidence measurements, which are largely absent from existing classification methods. In this paper, we present a robust Gaussian Process classification hyperparameter training model (MuyGPs) for discerning normal heartbeat signals from the signals affected by different arrhythmias and myocardial infarction. We compare the performance of MuyGPs with traditional Gaussian process classifier as well as conventional machine learning models, such as, Random Forest, Extra Trees, k-Nearest Neighbors and Convolutional Neural Network. Comparing these models reveals MuyGPs as the most performant model for making confident predictions on individual patient ECGs. Furthermore, we explore the posterior distribution obtained from the Gaussian process to interpret the prediction and quantify uncertainty. In addition, we provide a guideline on obtaining the prediction confidence of the machine learning models and quantitatively compare the uncertainty measures of these models. Particularly, we identify a class of less-accurate (ambiguous) signals for further diagnosis by an expert. © 2024, CC BY.

关键词： Convolutional neural networks

Construction and Validation of an Obstetric Risk Classification Engine to Support Intensive Care Unit (ICU) Admission Using Machine Learning

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SSRN

SSRN 2022年

作者： Soares, Fabiano Miguel Rose, Lívia Ohana da Rocha Oak Cecatti, José Guilherme Luz, Adriana Gomes Awe, Oluwafunmilola Deborah Esmi, Estevão Laureano de Carvalho Pacagnella, Rodolfo Department of Obstetrics and Gynecology Faculty of Medical Sciences State University of Campinas SP Campinas Brazil Department of Applied and Computational Mathematics Institute of Mathematics Statistics and Scientific Computing State University of Campinas SP Campinas Brazil Campus Universitário R. Alexander Flemming 101. Barão Geraldo Campinas SP. 13083-872 Brazil

Introduction: Maternal mortality remains a global problem, despite the efforts of many international organizations such as the World Health Organization to reduce it. In recent times, scientific evidence useful in the reduction of maternal mortality has become well established, which include implementation of bundles in emergency, and provision of Intensive Care Units (ICU) for obstetric complications. This is a major factor in the management of pregnant women, however, there is a need for optimization of the Intensive Care Unit (ICU) when the need arises. We aimed to develop a clinical support tool to guide decisions on which obstetric population required ICU ***: This study describes the development and validation of an obstetric risk classification engine among patients with maternal morbidity We used a supervised machine learning technique in a multicentric database of pregnant population including 9,550 women with Severe Maternal Morbidity. A decision tree model and two regression tree models were tested (Random Forest, Gradient boosting Machine -GBM, and Extreme Gradient Boosting -XGBoost) and after application, the accuracy, sensitivity and specificity were compared and the differences between the predictions were analyzed using Wilcoxon ***: The XGBoost algorithm showed the highest efficiency with an accuracy of 85%, sensitivity of 42%, specificity of 97%, AUC on the ROC curve of 86.7% and the estimated prevalence of ICU use (11.6%, in contrast to the actual prevalence of ICU, 21.52%), indicating the need to optimize the use of intensive ***: The Risk engine developed showed positive results, hence, can serve as a reinforcement factor in the management of women with complications so that those who need intensive care can be correctly identified, thereby optimizing the use of intensive beds. © 2022, The Authors. All rights reserved.

关键词： Intensive care units

Stable reconstruction of simple riemannian manifolds from unknown interior sources

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arXiv 2021年

作者： de Hoop, Maarten V. Ilmavirta, Joonas Lassas, Matti Saksala, Teemu Department of Computational and Applied Mathematics Rice University 6100 Main MS-134 HoustonTX77005-1892 United States Unit of Computing Sciences Tampere University Kalevantie 4 FI-33014 Finland FI-00014 Finland Department of Mathematics North Carolina State University 2311 Stinson Drive RaleighNC27695-8205 United States

Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a Riemannian manifold with boundary? With a finite set of sources we can only hope to get an approximate reconstruction, and we indeed provide a discrete metric approximation to the manifold with explicit data-driven error bounds when the manifold is simple. This is the geometrization of a seismological inverse problem where we measure the arrival times on the surface of waves from an unknown number of unknown interior microseismic events at unknown times. The closeness of two metric spaces with a marked boundary is measured by a labeled Gromov–Hausdorff distance. If measurements are done for infinite time and spatially dense sources, our construction produces the true Riemannian manifold and the finite-time approximations converge to it in the metric sense. Copyright © 2021, The Authors. All rights reserved.

关键词： Geometry

A Simple Approach for Local and Global Variable Importance in Nonlinear Regression Models

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arXiv 2023年

作者： Winn-Nuñez, Emily T. Griffin, Maryclare Crawford, Lorin Division of Applied Mathematics Brown University ProvidenceRI United States Department of Mathematics and Statistics University of Massachusetts Amherst AmherstMA United States Microsoft Research New England CambridgeMA United States Department of Biostatistics Brown University ProvidenceRI United States Center for Computational Molecular Biology Brown University ProvidenceRI United States

The ability to interpret machine learning models has become increasingly important as their usage in data science continues to rise. Most current interpretability methods are optimized to work on either (i) a global scale, where the goal is to rank features based on their contributions to overall variation in an observed population, or (ii) the local level, which aims to detail on how important a feature is to a particular individual in the data set. In this work, a new operator is proposed called the "GlObal And Local Score" (GOALS): a simple post hoc approach to simultaneously assess local and global feature variable importance in nonlinear models. Motivated by problems in biomedicine, the approach is demonstrated using Gaussian process regression where the task of understanding how genetic markers are associated with disease progression both within individuals and across populations is of high interest. Detailed simulations and real data analyses illustrate the flexible and efficient utility of GOALS over state-of-the-art variable importance strategies. Copyright © 2023, The Authors. All rights reserved.

关键词： Population statistics

Prior variances and depth un-biased estimators in EEG focal source imaging

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arXiv 2017年

作者： Koulouri, A. Rimpiläinen, V. Brookes, M. Kaipio, J.P. Institute for Computational and Applied Mathematics University of Münster Münster Germany Institute for Biomagnetism and Biosignalanalysis University of Münster Münster Germany Department of Electrical and Electronic Engineering Imperial College London London United Kingdom Department of Mathematics University of Auckland Auckland New Zealand University of Münster Einsteinstrasse 62 Münster Germany

In electroencephalography (EEG) source imaging, the inverse source estimates are depth biased in such a way that their maxima are often close to the sensors. This depth bias can be quantified by inspecting the statistics (mean and covariance) of these estimates. In this paper, we find weighting factors within a Bayesian framework for the used `1/`2 sparsity prior that the resulting maximum a posterior (MAP) estimates do not favour any particular source location. Due to the lack of an analytical expression for the MAP estimate when this sparsity prior is used, we solve the weights indirectly. First, we calculate the Gaussian prior variances that lead to depth un-biased maximum a posterior (MAP) estimates. Subsequently, we approximate the corresponding weight factors in the sparsity prior based on the solved Gaussian prior variances. Finally, we reconstruct focal source configurations using the sparsity prior with the proposed weights and two other commonly used choices of weights that can be found in literature. Copyright © 2017, The Authors. All rights reserved.

关键词： Electrophysiology