DOCSLIB.ORG

Sequential Monte Carlo Techniques and Bayesian Filtering : Applications in Tracking

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Sequential Monte Carlo Techniques and Bayesian Filtering : Applications in Tracking 2 Sequential Monte Carlo Techniques and Bayesian Filtering : Applications in Tracking Samarjit Das Abstract—Visual tracking is a very important issue in various applications involving robotics and/or computer vision. The basic idea is to estimate a hidden state sequence from noisy observations under a given state space model. The state space model comprises of a system model and an observation model. The system models that we are going to discuss here has the markov property and the corresponding state space model is basically known as hidden markov model. Due to non-linearities in the system and the observation model, bayesian filtering ( also known as particle filtering ) is an ideal candidate for the purpose of tracking. Particle filtering is basically a non-linear, non-gaussian analog of kalman filter. It is based on sequential Monte Carlo techniques. In this article we are going to discuss Monte Carlo simulation techniques to solve numerical problems, especially to compute posterior probability distribution which involves complicated integrals. Then we are going to extend these ideas to sequential Monte Carlo techniques and develop algorithms leading to particle filtering. We demonstrate applications of particle filtering in areas such as target tracking in a wireless sensor network. Index Terms—Monte Carlo simulation, sequential Monte Carlo techniques, particle filtering, visual tracking ✦ 1 MONTE CARLO METHODS here δ(.) denotes a dirac delta function. Thus fˆX (x) is the discretized PMF (probability mass function) approx- Monte Carlo methods (also referred to as Monte Carlo imation of the continuous pdf fX . Now the integral can simulation) are a class of computational algorithms that be written as, rely on repeated random sampling to compute their results. These methods are quite often used to simu- b late physical and mathematical systems. They are most I = ψ(x)dx Za suitable for computation by computers as they rely on b ψ(x) repeated computation and random or pseudo random = f (x)dx Z f (x) X numbers. Monte Carlo methods tend to be used when a X it is infeasible or impossible to compute an exact re- ψ(X) I = EfX [ ] sult with deterministic algorithm. A simple example is fX (X) numerical computation of complicated integrals using where E [.] denotes the expected value w.r.t the Monte Carlo simulation. fX distribution fX . Now, we can estimate the expectation Consider the following case. Say we want to compute, and thus the integral using the discretized version of ψ(X) b the pdf. Or, Iˆ ≈ E [ ] which ca be computed as fˆX fX (X) I = ψ(x)dx follows, Za b N ψ x ψ(x) 1 where ( ) be a function with a very complicated closed Iˆ = ( δ(x − xi))dx form expression making it impossible to be integrated Z fX(x) N a Xi=1 in a closed form. We can solve this problem numerically N i using Monte Carlo technique. First, consider a random ˆ 1 ψ(x ) Or, I = i (1) N fX(x ) variable X distributed over the interval [a,b] with a Xi=1 probability density function (pdf) fX(x). The closed form expression of fX is known and we can draw samples Using the law of large numbers it can be shown that ˆ w.r.t. fX (.). We draw N (sufficiently large) samples from I → I as N → ∞. Thus by drawing a large number of the pdf, sample in the interval [a,b] we are able to numerically approximate the integral I using Monte Carlo technique. i x ∼ fX (x), i=1, ..., N The process of sampling from the distribution fX can be N extended to importance sampling technique by which we 1 i fˆX (x) , δ(x − x ) can estimate the properties of a particular distribution , N Xi=1 while only having samples from a different distribution (known as importance density or importance function) • This article was submitted as a term paper for the course ComS 577, Fall rather than the distribution of interest. This will be 2008 at Iowa State University, Ames, IA 50010. discussed in detail while developing the particle filtering algorithm. 3 posterior distribution. But in many practical problems the posterior is highly non-gaussian ( quite often mul- timodal ) together with non-linear φ(.) and g(.). Under such circumstances, sequential Monte Carlo technique based particle filtering algorithm gives us a way to solve this posterior estimation problem. In the next few sections, we shall develop the particle filter starting with Monte Carlo sampling for pdf approximation. We also discuss sequential Monte Carlo method (especially sequential importance sampling) and how it leads to Fig. 1. This figure shows the hidden markov model structure of particle filtering. the state space. The state sequence xt is a markov process and the observations yt, t = 1, 2, ... are conditionally independent given the state at t 3 DERIVATION OF THE PARTICLE FILTER 2 BAYESIAN FILTERING PROBLEM Similar to eq (4), say we are trying to compute the ex- Consider a state-space model with t =0,...,T , pectation w.r.t the joint posterior distribution p(x1:t|y1:t) xt = φ(xt−1)+ nt (2) Ep(x1:t|y1:t)[f(X1:t)] = f(x1:t)p(x1:t|y1:t)dx1:t =? yt = g(xt)+ vt (3) Z (5) where xt and yt denote the state and the observation at the current instant respectively. In real-life applications, In order to solve this problem, we have to look into for example, the state can be the position of a target two very important issues. a) Do we have a closed while the observation is noisy the sensor data about the form expression for p(x t|y t) ? b) Can we sample from current position and our goal could be to extract the true 1: 1: p(x t|y t) ? Depending on these two issues we can use ‘hidden’ state information using the observations and 1: 1: three different methods two solve this problem namely, the state dynamical model (the system model, eq (2)). simple Monte Carlo sampling, importance sampling and The functions φ(.) and g(.) can be any linear or non- bayesian importance sampling. The method involving linear function. The following things can be assumed bayesian importance is our main focus as it exactly is to be known. The initial state distribution p(x ), the 0 what bayesian filtering/particle does. These methods are state transition density p(x |x ) and the observation t t−1 discussed below. likelihood p(yt|xt). Here, p(.) is used to denote proba- bility density function. The transition density and the observation likelihood can be determined form the know 3.1 Simple Monte Carlo Sampling distributions of i.i.d noise sequences nt and vt. The state space is assumed to follow a hidden markov model Consider the case when we can sample from the joint (HMM). In other words, the state sequence xt is a posterior distribution p(x1:t|y1:t). In that case we can markov process and the observations yt, t = 1, 2, ... are approximate the distribution as a discretized version (as conditionally independent given the state at t. The states done in Sec. 1) in terms of the samples drawn from that are hidden (i.e. not observable); all we can know about distribution. Or, the system is through the observations known at each in- i stant. The graphical model of the HMM is shown in Fig. x1:t ∼ p(x1:t|y1:t), i=1, ..., N N 1. Under HMM assumption, p(xt|xt−1,past)= p(xt|xt−1) 1 pˆ(x |y ) ≈ δ(x − xi ) (6) and p(yt|xt,past)= p(yt|xt) e.g. p(yt|xt,yt−1)= p(yt|xt). 1:t 1:t N 1:t 1:t The goal of particle filtering (or bayesian filtering) is Xi=1 to estimate the joint posterior state distribution at time t Now, we can approximate the expectation in the equa- i.e. p(x |y ) or quite often its marginal p(x |y ). Here, 1:t 1:t t 1:t tion (5) as follows, x1:t = {x1, x2, ..., xt}. Finally, we would like to compute Ep(x1:t|y1:t)[f(X1:t)] ≈ Epˆ(x1:t|y1:t)[f(X1:t)] It = Ep(x |y )(f(Xt)) = f(xt)p(xt|y1:t)dxt t 1:t Z N 1 (4) = f(xi ) N 1:t Xi=1 When the posterior can be assumed to be gaussian and function φ(.) and g(.) to be linear the same problem can This is the most ideal case. In reality, almost always be recursively solved using Kalman filter [1]. To tackle we cannot sample from the posterior but we might have non-linearity of φ and g(.), the Extended Kalman filter [1] a closed form expression for the distribution. This leads can be used, but it still assumes the gaussianity of the to the second method known as importance sampling. 4 3.2 Importance Sampling form expression for the posterior p(x1:t|y1:t). But it can Importance sampling can be used to estimate the prop- be expressed in the following manner, erties of a particular distribution, while only having p(x1:t,y1:t) p(x1:t|y1:t) = thus samples from a different distribution (known as impor- p(y1:t) tance density or importance function) rather than the p(x1:t|y1:t) ∝ p(x1:t,y1:t) (8) distribution of interest. Thus when we have a closed It turns out that we can at least compute p(x1:t,y1:t) form expression for p(x1:t|y1:t) but cannot sample from it, we use an importance density to draw the samples.
Load more

Recommended publications
  • Kalman and Particle Filtering
    Abstract: The Kalman and Particle filters are algorithms that recursively update an estimate of the state and find the innovations driving a stochastic process given a sequence of observations. The Kalman filter accomplishes this goal by linear projections, while the Particle filter does so by a sequential Monte Carlo method. With the state estimates, we can forecast and smooth the stochastic process. With the innovations, we can estimate the parameters of the model. The article discusses how to set a dynamic model in a state-space form, derives the Kalman and Particle filters, and explains how to use them for estimation. Kalman and Particle Filtering The Kalman and Particle filters are algorithms that recursively update an estimate of the state and find the innovations driving a stochastic process given a sequence of observations. The Kalman filter accomplishes this goal by linear projections, while the Particle filter does so by a sequential Monte Carlo method. Since both filters start with a state-space representation of the stochastic processes of interest, section 1 presents the state-space form of a dynamic model. Then, section 2 intro- duces the Kalman filter and section 3 develops the Particle filter. For extended expositions of this material, see Doucet, de Freitas, and Gordon (2001), Durbin and Koopman (2001), and Ljungqvist and Sargent (2004). 1. The state-space representation of a dynamic model A large class of dynamic models can be represented by a state-space form: Xt+1 = ϕ (Xt,Wt+1; γ) (1) Yt = g (Xt,Vt; γ) . (2) This representation handles a stochastic process by finding three objects: a vector that l describes the position of the system (a state, Xt X R ) and two functions, one mapping ∈ ⊂ 1 the state today into the state tomorrow (the transition equation, (1)) and one mapping the state into observables, Yt (the measurement equation, (2)).
    [Show full text]
  • A Fourier-Wavelet Monte Carlo Method for Fractal Random Fields
    JOURNAL OF COMPUTATIONAL PHYSICS 132, 384±408 (1997) ARTICLE NO. CP965647 A Fourier±Wavelet Monte Carlo Method for Fractal Random Fields Frank W. Elliott Jr., David J. Horntrop, and Andrew J. Majda Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012 Received August 2, 1996; revised December 23, 1996 2 2H k[v(x) 2 v(y)] l 5 CHux 2 yu , (1.1) A new hierarchical method for the Monte Carlo simulation of random ®elds called the Fourier±wavelet method is developed and where 0 , H , 1 is the Hurst exponent and k?l denotes applied to isotropic Gaussian random ®elds with power law spectral the expected value. density functions. This technique is based upon the orthogonal Here we develop a new Monte Carlo method based upon decomposition of the Fourier stochastic integral representation of the ®eld using wavelets. The Meyer wavelet is used here because a wavelet expansion of the Fourier space representation of its rapid decay properties allow for a very compact representation the fractal random ®elds in (1.1). This method is capable of the ®eld. The Fourier±wavelet method is shown to be straightfor- of generating a velocity ®eld with the Kolmogoroff spec- ward to implement, given the nature of the necessary precomputa- trum (H 5 Ad in (1.1)) over many (10 to 15) decades of tions and the run-time calculations, and yields comparable results scaling behavior comparable to the physical space multi- with scaling behavior over as many decades as the physical space multiwavelet methods developed recently by two of the authors.
    [Show full text]
  • Applying Particle Filtering in Both Aggregated and Age-Structured Population Compartmental Models of Pre-Vaccination Measles
    bioRxiv preprint doi: https://doi.org/10.1101/340661; this version posted June 6, 2018. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. Applying particle filtering in both aggregated and age-structured population compartmental models of pre-vaccination measles Xiaoyan Li1*, Alexander Doroshenko2, Nathaniel D. Osgood1 1 Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada 2 Department of Medicine, Division of Preventive Medicine, University of Alberta, Edmonton, Alberta, Canada * [email protected] Abstract Measles is a highly transmissible disease and is one of the leading causes of death among young children under 5 globally. While the use of ongoing surveillance data and { recently { dynamic models offer insight on measles dynamics, both suffer notable shortcomings when applied to measles outbreak prediction. In this paper, we apply the Sequential Monte Carlo approach of particle filtering, incorporating reported measles incidence for Saskatchewan during the pre-vaccination era, using an adaptation of a previously contributed measles compartmental model. To secure further insight, we also perform particle filtering on an age structured adaptation of the model in which the population is divided into two interacting age groups { children and adults. The results indicate that, when used with a suitable dynamic model, particle filtering can offer high predictive capacity for measles dynamics and outbreak occurrence in a low vaccination context. We have investigated five particle filtering models in this project. Based on the most competitive model as evaluated by predictive accuracy, we have performed prediction and outbreak classification analysis.
    [Show full text]
  • Fourier, Wavelet and Monte Carlo Methods in Computational Finance
    Fourier, Wavelet and Monte Carlo Methods in Computational Finance Kees Oosterlee1;2 1CWI, Amsterdam 2Delft University of Technology, the Netherlands AANMPDE-9-16, 7/7/2016 Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 1 / 51 Joint work with Fang Fang, Marjon Ruijter, Luis Ortiz, Shashi Jain, Alvaro Leitao, Fei Cong, Qian Feng Agenda Derivatives pricing, Feynman-Kac Theorem Fourier methods Basics of COS method; Basics of SWIFT method; Options with early-exercise features COS method for Bermudan options Monte Carlo method BSDEs, BCOS method (very briefly) Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 1 / 51 Agenda Derivatives pricing, Feynman-Kac Theorem Fourier methods Basics of COS method; Basics of SWIFT method; Options with early-exercise features COS method for Bermudan options Monte Carlo method BSDEs, BCOS method (very briefly) Joint work with Fang Fang, Marjon Ruijter, Luis Ortiz, Shashi Jain, Alvaro Leitao, Fei Cong, Qian Feng Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 1 / 51 Feynman-Kac theorem: Z T v(t; x) = E g(s; Xs )ds + h(XT ) ; t where Xs is the solution to the FSDE dXs = µ(Xs )ds + σ(Xs )d!s ; Xt = x: Feynman-Kac Theorem The linear partial differential equation: @v(t; x) + Lv(t; x) + g(t; x) = 0; v(T ; x) = h(x); @t with operator 1 Lv(t; x) = µ(x)Dv(t; x) + σ2(x)D2v(t; x): 2 Kees Oosterlee (CWI, TU Delft) Comp. Finance AANMPDE-9-16 2 / 51 Feynman-Kac Theorem The linear partial differential equation: @v(t; x) + Lv(t; x) + g(t; x) = 0; v(T ; x) = h(x); @t with operator 1 Lv(t; x) = µ(x)Dv(t; x) + σ2(x)D2v(t; x): 2 Feynman-Kac theorem: Z T v(t; x) = E g(s; Xs )ds + h(XT ) ; t where Xs is the solution to the FSDE dXs = µ(Xs )ds + σ(Xs )d!s ; Xt = x: Kees Oosterlee (CWI, TU Delft) Comp.
    [Show full text]
  • Efficient Monte Carlo Methods for Estimating Failure Probabilities
    Reliability Engineering and System Safety 165 (2017) 376–394 Contents lists available at ScienceDirect Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress ffi E cient Monte Carlo methods for estimating failure probabilities MARK ⁎ Andres Albana, Hardik A. Darjia,b, Atsuki Imamurac, Marvin K. Nakayamac, a Mathematical Sciences Dept., New Jersey Institute of Technology, Newark, NJ 07102, USA b Mechanical Engineering Dept., New Jersey Institute of Technology, Newark, NJ 07102, USA c Computer Science Dept., New Jersey Institute of Technology, Newark, NJ 07102, USA ARTICLE INFO ABSTRACT Keywords: We develop efficient Monte Carlo methods for estimating the failure probability of a system. An example of the Probabilistic safety assessment problem comes from an approach for probabilistic safety assessment of nuclear power plants known as risk- Risk analysis informed safety-margin characterization, but it also arises in other contexts, e.g., structural reliability, Structural reliability catastrophe modeling, and finance. We estimate the failure probability using different combinations of Uncertainty simulation methodologies, including stratified sampling (SS), (replicated) Latin hypercube sampling (LHS), Monte Carlo and conditional Monte Carlo (CMC). We prove theorems establishing that the combination SS+LHS (resp., SS Variance reduction Confidence intervals +CMC+LHS) has smaller asymptotic variance than SS (resp., SS+LHS). We also devise asymptotically valid (as Nuclear regulation the overall sample size grows large) upper confidence bounds for the failure probability for the methods Risk-informed safety-margin characterization considered. The confidence bounds may be employed to perform an asymptotically valid probabilistic safety assessment. We present numerical results demonstrating that the combination SS+CMC+LHS can result in substantial variance reductions compared to stratified sampling alone.
    [Show full text]
  • Dynamic Detection of Change Points in Long Time Series
    AISM (2007) 59: 349–366 DOI 10.1007/s10463-006-0053-9 Nicolas Chopin Dynamic detection of change points in long time series Received: 23 March 2005 / Revised: 8 September 2005 / Published online: 17 June 2006 © The Institute of Statistical Mathematics, Tokyo 2006 Abstract We consider the problem of detecting change points (structural changes) in long sequences of data, whether in a sequential fashion or not, and without assuming prior knowledge of the number of these change points. We reformulate this problem as the Bayesian filtering and smoothing of a non standard state space model. Towards this goal, we build a hybrid algorithm that relies on particle filter- ing and Markov chain Monte Carlo ideas. The approach is illustrated by a GARCH change point model. Keywords Change point models · GARCH models · Markov chain Monte Carlo · Particle filter · Sequential Monte Carlo · State state models 1 Introduction The assumption that an observed time series follows the same fixed stationary model over a very long period is rarely realistic. In economic applications for instance, common sense suggests that the behaviour of economic agents may change abruptly under the effect of economic policy, political events, etc. For example, Mikosch and St˘aric˘a (2003, 2004) point out that GARCH models fit very poorly too long sequences of financial data, say 20 years of daily log-returns of some speculative asset. Despite this, these models remain highly popular, thanks to their forecast ability (at least on short to medium-sized time series) and their elegant simplicity (which facilitates economic interpretation). Against the common trend of build- ing more and more sophisticated stationary models that may spuriously provide a better fit for such long sequences, the aforementioned authors argue that GARCH models remain a good ‘local’ approximation of the behaviour of financial data, N.
    [Show full text]
  • Bayesian Filtering: from Kalman Filters to Particle Filters, and Beyond ZHE CHEN
    MANUSCRIPT 1 Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond ZHE CHEN Abstract— In this self-contained survey/review paper, we system- IV Bayesian Optimal Filtering 9 atically investigate the roots of Bayesian filtering as well as its rich IV-AOptimalFiltering..................... 10 leaves in the literature. Stochastic filtering theory is briefly reviewed IV-BKalmanFiltering..................... 11 with emphasis on nonlinear and non-Gaussian filtering. Following IV-COptimumNonlinearFiltering.............. 13 the Bayesian statistics, different Bayesian filtering techniques are de- IV-C.1Finite-dimensionalFilters............ 13 veloped given different scenarios. Under linear quadratic Gaussian circumstance, the celebrated Kalman filter can be derived within the Bayesian framework. Optimal/suboptimal nonlinear filtering tech- V Numerical Approximation Methods 14 niques are extensively investigated. In particular, we focus our at- V-A Gaussian/Laplace Approximation ............ 14 tention on the Bayesian filtering approach based on sequential Monte V-BIterativeQuadrature................... 14 Carlo sampling, the so-called particle filters. Many variants of the V-C Mulitgrid Method and Point-Mass Approximation . 14 particle filter as well as their features (strengths and weaknesses) are V-D Moment Approximation ................. 15 discussed. Related theoretical and practical issues are addressed in V-E Gaussian Sum Approximation . ............. 16 detail. In addition, some other (new) directions on Bayesian filtering V-F Deterministic
    [Show full text]
  • Monte Carlo Smoothing for Nonlinear Time Series
    Monte Carlo Smoothing for Nonlinear Time Series Simon J. GODSILL, Arnaud DOUCET, and Mike WEST We develop methods for performing smoothing computations in general state-space models. The methods rely on a particle representation of the filtering distributions, and their evolution through time using sequential importance sampling and resampling ideas. In particular, novel techniques are presented for generation of sample realizations of historical state sequences. This is carried out in a forward-filtering backward-smoothing procedure that can be viewed as the nonlinear, non-Gaussian counterpart of standard Kalman filter-based simulation smoothers in the linear Gaussian case. Convergence in the mean squared error sense of the smoothed trajectories is proved, showing the validity of our proposed method. The methods are tested in a substantial application for the processing of speech signals represented by a time-varying autoregression and parameterized in terms of time-varying partial correlation coefficients, comparing the results of our algorithm with those from a simple smoother based on the filtered trajectories. KEY WORDS: Bayesian inference; Non-Gaussian time series; Nonlinear time series; Particle filter; Sequential Monte Carlo; State- space model. 1. INTRODUCTION and In this article we develop Monte Carlo methods for smooth- g(yt+1|xt+1)p(xt+1|y1:t ) p(xt+1|y1:t+1) = . ing in general state-space models. To fix notation, consider p(yt+1|y1:t ) the standard Markovian state-space model (West and Harrison 1997) Similarly, smoothing can be performed recursively backward in time using the smoothing formula xt+1 ∼ f(xt+1|xt ) (state evolution density), p(xt |y1:t )f (xt+1|xt ) p(x |y ) = p(x + |y ) dx + .
    [Show full text]
  • Time Series Analysis 5
    Warm-up: Recursive Least Squares Kalman Filter Nonlinear State Space Models Particle Filtering Time Series Analysis 5. State space models and Kalman filtering Andrew Lesniewski Baruch College New York Fall 2019 A. Lesniewski Time Series Analysis Warm-up: Recursive Least Squares Kalman Filter Nonlinear State Space Models Particle Filtering Outline 1 Warm-up: Recursive Least Squares 2 Kalman Filter 3 Nonlinear State Space Models 4 Particle Filtering A. Lesniewski Time Series Analysis Warm-up: Recursive Least Squares Kalman Filter Nonlinear State Space Models Particle Filtering OLS regression As a motivation for the reminder of this lecture, we consider the standard linear model Y = X Tβ + "; (1) where Y 2 R, X 2 Rk , and " 2 R is noise (this includes the model with an intercept as a special case in which the first component of X is assumed to be 1). Given n observations x1;:::; xn and y1;:::; yn of X and Y , respectively, the ordinary least square least (OLS) regression leads to the following estimated value of the coefficient β: T −1 T βbn = (Xn Xn) Xn Yn: (2) The matrices X and Y above are defined as 0 T1 0 1 x1 y1 X = B . C 2 (R) Y = B . C 2 Rn; @ . A Matn;k and n @ . A (3) T xn yn respectively. A. Lesniewski Time Series Analysis Warm-up: Recursive Least Squares Kalman Filter Nonlinear State Space Models Particle Filtering Recursive least squares Suppose now that X and Y consists of a streaming set of data, and each new observation leads to an updated value of the estimated β.
    [Show full text]
  • Basic Statistics and Monte-Carlo Method -2
    Applied Statistical Mechanics Lecture Note - 10 Basic Statistics and Monte-Carlo Method -2 고려대학교 화공생명공학과 강정원 Table of Contents 1. General Monte Carlo Method 2. Variance Reduction Techniques 3. Metropolis Monte Carlo Simulation 1.1 Introduction Monte Carlo Method Any method that uses random numbers Random sampling the population Application • Science and engineering • Management and finance For given subject, various techniques and error analysis will be presented Subject : evaluation of definite integral b I = ρ(x)dx a 1.1 Introduction Monte Carlo method can be used to compute integral of any dimension d (d-fold integrals) Error comparison of d-fold integrals Simpson’s rule,… E ∝ N −1/ d − Monte Carlo method E ∝ N 1/ 2 purely statistical, not rely on the dimension ! Monte Carlo method WINS, when d >> 3 1.2 Hit-or-Miss Method Evaluation of a definite integral b I = ρ(x)dx a h X X X ≥ ρ X h (x) for any x X Probability that a random point reside inside X O the area O O I N' O O r = ≈ O O (b − a)h N a b N : Total number of points N’ : points that reside inside the region N' I ≈ (b − a)h N 1.2 Hit-or-Miss Method Start Set N : large integer N’ = 0 h X X X X X Choose a point x in [a,b] = − + X Loop x (b a)u1 a O N times O O y = hu O O Choose a point y in [0,h] 2 O O a b if [x,y] reside inside then N’ = N’+1 I = (b-a) h (N’/N) End 1.2 Hit-or-Miss Method Error Analysis of the Hit-or-Miss Method It is important to know how accurate the result of simulations are The rule of 3σ’s Identifying Random Variable N = 1 X X n N n=1 From
    [Show full text]
  • Adaptive Motion Model for Human Tracking Using Particle Filter
    2010 International Conference on Pattern Recognition Adaptive Motion Model for Human Tracking Using Particle Filter Mohammad Hossein Ghaeminia1, Amir Hossein Shabani2, and Shahryar Baradaran Shokouhi1 1Iran Univ. of Science & Technology, Tehran, Iran 2University of Waterloo, ON, Canada [email protected] [email protected] [email protected] Abstract is periodically adapted by an efficient learning procedure (Figure 1). The core of learning the motion This paper presents a novel approach to model the model is the parameter estimation for which the complex motion of human using a probabilistic sequence of velocity and acceleration are innovatively autoregressive moving average model. The analyzed to be modeled by a Gaussian Mixture Model parameters of the model are adaptively tuned during (GMM). The non-negative matrix factorization is then the course of tracking by utilizing the main varying used for dimensionality reduction to take care of high components of the pdf of the target’s acceleration and variations during abrupt changes [3]. Utilizing this velocity. This motion model, along with the color adaptive motion model along with a color histogram histogram as the measurement model, has been as measurement model in the PF framework provided incorporated in the particle filtering framework for us an appropriate approach for human tracking in the human tracking. The proposed method is evaluated by real world scenario of PETS benchmark [4]. PETS benchmark in which the targets have non- The rest of this paper is organized as the following. smooth motion and suddenly change their motion Section 2 overviews the related works. Section 3 direction. Our method competes with the state-of-the- explains the particle filter and the probabilistic ARMA art techniques for human tracking in the real world model.
    [Show full text]
  • A Tutorial on Quantile Estimation Via Monte Carlo
    A Tutorial on Quantile Estimation via Monte Carlo Hui Dong and Marvin K. Nakayama Abstract Quantiles are frequently used to assess risk in a wide spectrum of applica- tion areas, such as finance, nuclear engineering, and service industries. This tutorial discusses Monte Carlo simulation methods for estimating a quantile, also known as a percentile or value-at-risk, where p of a distribution’s mass lies below its p-quantile. We describe a general approach that is often followed to construct quantile estimators, and show how it applies when employing naive Monte Carlo or variance-reduction techniques. We review some large-sample properties of quantile estimators. We also describe procedures for building a confidence interval for a quantile, which provides a measure of the sampling error. 1 Introduction Numerous application settings have adopted quantiles as a way of measuring risk. For a fixed constant 0 < p < 1, the p-quantile of a continuous random variable is a constant x such that p of the distribution’s mass lies below x. For example, the median is the 0:5-quantile. In finance, a quantile is called a value-at-risk, and risk managers commonly employ p-quantiles for p ≈ 1 (e.g., p = 0:99 or p = 0:999) to help determine capital levels needed to be able to cover future large losses with high probability; e.g., see [33]. Nuclear engineers use 0:95-quantiles in probabilistic safety assessments (PSAs) of nuclear power plants. PSAs are often performed with Monte Carlo, and the U.S. Nuclear Regulatory Commission (NRC) further requires that a PSA accounts for the Hui Dong Amazon.com Corporate LLC∗, Seattle, WA 98109, USA e-mail: [email protected] ∗This work is not related to Amazon, regardless of the affiliation.
    [Show full text]