arXiv subjects

Lan Jiang

Publications and source records attributed to Lan Jiang.

24 records · Page 2Linked to original sources

arXiv2016

Tracking multiple moving objects in images using Markov Chain Monte Carlo

A new Bayesian state and parameter learning algorithm for multiple target tracking (MTT) models with image observations is proposed. Specifically, a Markov chain Monte Carlo algorithm is designed to sample from the posterior distribution of the unknown number of targets, their birth and death times, states and model parameters, which constitutes the complete solution to the tracking problem. The conventional approach is to pre-process the images to extract point observations and then perform tracking. We model the image generation process directly to avoid potential loss of information when extracting point observations. Numerical examples show that our algorithm has improved tracking performance over commonly used techniques, for both synthetic examples and real florescent microscopy data, especially in the case of dim targets with overlapping illuminated regions.

stat.AP↗

arXiv2015

GAIL---Guaranteed Automatic Integration Library in MATLAB: Documentation for Version 2.1

Automatic and adaptive approximation, optimization, or integration of functions in a cone with guarantee of accuracy is a relatively new paradigm. Our purpose is to create an open-source MATLAB package, Guaranteed Automatic Integration Library (GAIL), following the philosophy of reproducible research and sustainable practices of robust scientific software development. For our conviction that true scholarship in computational sciences are characterized by reliable reproducibility, we employ the best practices in mathematical research and software engineering known to us and available in MATLAB. This document describes the key features of functions in GAIL, which includes one-dimensional function approximation and minimization using linear splines, one-dimensional numerical integration using trapezoidal rule, and last but not least, mean estimation and multidimensional integration by Monte Carlo methods or Quasi Monte Carlo methods.

cs.MS↗

arXiv2014

Guaranteed Monte Carlo Methods for Bernoulli Random Variables

Simple Monte Carlo is a versatile computational method with a convergence rate of $O(n^{-1/2})$. It can be used to estimate the means of random variables whose distributions are unknown. Bernoulli random variables, $Y$, are widely used to model success (failure) of complex systems. Here $Y=1$ denotes a success (failure), and $p=\mathbb{E}(Y)$ denotes the probability of that success (failure). Another application of Bernoulli random variables is $Y=\mathbb{1}_{R}(\boldsymbol{X})$, where then $p$ is the probability of $\boldsymbol{X}$ lying in the region $R$. This article explores how estimate $p$ to a prescribed absolute error tolerance, $\varepsilon$, with a high level of confidence, $1-α$. The proposed algorithm automatically determines the number of samples of $Y$ needed to reach the prescribed error tolerance with the specified confidence level by using Hoeffding's inequality. The algorithm described here has been implemented in MATLAB and is part of the Guaranteed Automatic Integration Library (GAIL).

math.NA↗

arXiv2014

Bayesian tracking and parameter learning for non-linear multiple target tracking models

We propose a new Bayesian tracking and parameter learning algorithm for non-linear non-Gaussian multiple target tracking (MTT) models. We design a Markov chain Monte Carlo (MCMC) algorithm to sample from the posterior distribution of the target states, birth and death times, and association of observations to targets, which constitutes the solution to the tracking problem, as well as the model parameters. In the numerical section, we present performance comparisons with several competing techniques and demonstrate significant performance improvements in all cases.

stat.AP↗

arXiv2013

Guaranteed Conservative Fixed Width Confidence Intervals Via Monte Carlo Sampling

Monte Carlo methods are used to approximate the means, $μ$, of random variables $Y$, whose distributions are not known explicitly. The key idea is that the average of a random sample, $Y_1, ..., Y_n$, tends to $μ$ as $n$ tends to infinity. This article explores how one can reliably construct a confidence interval for $μ$ with a prescribed half-width (or error tolerance) $\varepsilon$. Our proposed two-stage algorithm assumes that the kurtosis of $Y$ does not exceed some user-specified bound. An initial independent and identically distributed (IID) sample is used to confidently estimate the variance of $Y$. A Berry-Esseen inequality then makes it possible to determine the size of the IID sample required to construct the desired confidence interval for $μ$. We discuss the important case where $Y=f(\vX)$ and $\vX$ is a random $d$-vector with probability density function $ρ$. In this case $μ$ can be interpreted as the integral $\int_{\reals^d} f(\vx) ρ(\vx) \dif \vx$, and the Monte Carlo method becomes a method for multidimensional cubature.

math.ST↗

arXiv2012

Estimating the Static Parameters in Linear Gaussian Multiple Target Tracking Models

We present both offline and online maximum likelihood estimation (MLE) techniques for inferring the static parameters of a multiple target tracking (MTT) model with linear Gaussian dynamics. We present the batch and online versions of the expectation-maximisation (EM) algorithm for short and long data sets respectively, and we show how Monte Carlo approximations of these methods can be implemented. Performance is assessed in numerical examples using simulated data for various scenarios and a comparison with a Bayesian estimation procedure is also provided.

stat.AP↗