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Methods of Monte Carlo Simulation II
Methods of Monte Carlo Simulation II Ulm University Institute of Stochastics Lecture Notes Dr. Tim Brereton Summer Term 2014 Ulm, 2014 2 Contents 1 SomeSimpleStochasticProcesses 7 1.1 StochasticProcesses . 7 1.2 RandomWalks .......................... 7 1.2.1 BernoulliProcesses . 7 1.2.2 RandomWalks ...................... 10 1.2.3 ProbabilitiesofRandomWalks . 13 1.2.4 Distribution of Xn .................... 13 1.2.5 FirstPassageTime . 14 2 Estimators 17 2.1 Bias, Variance, the Central Limit Theorem and Mean Square Error................................ 19 2.2 Non-AsymptoticErrorBounds. 22 2.3 Big O and Little o Notation ................... 23 3 Markov Chains 25 3.1 SimulatingMarkovChains . 28 3.1.1 Drawing from a Discrete Uniform Distribution . 28 3.1.2 Drawing From A Discrete Distribution on a Small State Space ........................... 28 3.1.3 SimulatingaMarkovChain . 28 3.2 Communication .......................... 29 3.3 TheStrongMarkovProperty . 30 3.4 RecurrenceandTransience . 31 3.4.1 RecurrenceofRandomWalks . 33 3.5 InvariantDistributions . 34 3.6 LimitingDistribution. 36 3.7 Reversibility............................ 37 4 The Poisson Process 39 4.1 Point Processes on [0, )..................... 39 ∞ 3 4 CONTENTS 4.2 PoissonProcess .......................... 41 4.2.1 Order Statistics and the Distribution of Arrival Times 44 4.2.2 DistributionofArrivalTimes . 45 4.3 SimulatingPoissonProcesses. 46 4.3.1 Using the Infinitesimal Definition to Simulate Approx- imately .......................... 46 4.3.2 SimulatingtheArrivalTimes . 47 4.3.3 SimulatingtheInter-ArrivalTimes . 48 4.4 InhomogenousPoissonProcesses. 48 4.5 Simulating an Inhomogenous Poisson Process . 49 4.5.1 Acceptance-Rejection. 49 4.5.2 Infinitesimal Approach (Approximate) . 50 4.6 CompoundPoissonProcesses . 51 5 ContinuousTimeMarkovChains 53 5.1 TransitionFunction. 53 5.2 InfinitesimalGenerator . 54 5.3 ContinuousTimeMarkovChains . -
12 : Conditional Random Fields 1 Hidden Markov Model
10-708: Probabilistic Graphical Models 10-708, Spring 2014 12 : Conditional Random Fields Lecturer: Eric P. Xing Scribes: Qin Gao, Siheng Chen 1 Hidden Markov Model 1.1 General parametric form In hidden Markov model (HMM), we have three sets of parameters, j i transition probability matrix A : p(yt = 1jyt−1 = 1) = ai;j; initialprobabilities : p(y1) ∼ Multinomial(π1; π2; :::; πM ); i emission probabilities : p(xtjyt) ∼ Multinomial(bi;1; bi;2; :::; bi;K ): 1.2 Inference k k The inference can be done with forward algorithm which computes αt ≡ µt−1!t(k) = P (x1; :::; xt−1; xt; yt = 1) recursively by k k X i αt = p(xtjyt = 1) αt−1ai;k; (1) i k k and the backward algorithm which computes βt ≡ µt t+1(k) = P (xt+1; :::; xT jyt = 1) recursively by k X i i βt = ak;ip(xt+1jyt+1 = 1)βt+1: (2) i Another key quantity is the conditional probability of any hidden state given the entire sequence, which can be computed by the dot product of forward message and backward message by, i i i i X i;j γt = p(yt = 1jx1:T ) / αtβt = ξt ; (3) j where we define, i;j i j ξt = p(yt = 1; yt−1 = 1; x1:T ); i j / µt−1!t(yt = 1)µt t+1(yt+1 = 1)p(xt+1jyt+1)p(yt+1jyt); i j i = αtβt+1ai;jp(xt+1jyt+1 = 1): The implementation in Matlab can be vectorized by using, i Bt(i) = p(xtjyt = 1); j i A(i; j) = p(yt+1 = 1jyt = 1): 1 2 12 : Conditional Random Fields The relation of those quantities can be simply written in pseudocode as, T αt = (A αt−1): ∗ Bt; βt = A(βt+1: ∗ Bt+1); T ξt = (αt(βt+1: ∗ Bt+1) ): ∗ A; γt = αt: ∗ βt: 1.3 Learning 1.3.1 Supervised Learning The supervised learning is trivial if only we know the true state path. -
Hyperwage Theory
Hyperwage Theory About the cover The cover is a glowing red-hot bark of a tree embedded with the greatest equations of nature that tremendously impacted the thought of mankind and the course of civilization. These are the Maxwell’s electromagnetic field equations, the Navier-Stokes Theorem, Euler’s identity, Newton’s second law of motion, Newton’s law of gravitation, the ideal gas equation of state, Stefan’s law, the Second Law of Thermodynamics, Einstein’s mass-energy equivalence, Einstein’s gravity tensor equation, Lorenz time dilation, Planck’s Law, Heisenberg’s Uncertainty Principle, Schrodinger’s equation, Shannon’s Theorem, and Feynman diagrams in quantum electrodynamics. Concept and design by Thads Bentulan Hyperwage Theory The Misadventures of the Street Strategist Volume 10 Thads Bentulan Street Strategist First Edition 2008 Street Strategist Publications Limited Hyperwage Theory The Misadventures of the Street Strategist Volume 10 by Thads Bentulan ISBN 978-988-17536-9-4 This book is a compilation of articles that first appeared in BusinessWorld, unless otherwise noted. While the author endeavored to credit sources whenever possible, he welcomes corrections for any unacknowledged material. No part of this book may be reproduced, stored, or utilized in any form or by any means, electronic or mechanical, including photocopying, or recording, or by any information storage and retrieval system, including internet websites, without prior written permission. Copyright 2005-2008 by Thads Bentulan [email protected] 08/24/09 -
The 7Th Workshop on MARKOV PROCESSES and RELATED TOPICS
The 7th Workshop on MARKOV PROCESSES AND RELATED TOPICS July 19 - 23, 2010 No.6 Lecture room on 3th floor, Jingshi Building (京京京师师师大大大厦厦厦) Beijing Normal University Organizers: Mu-Fa Chen, Zeng-Hu Li, Feng-Yu Wang Supported by National Natural Science Foundation of China (No. 10721091) Probability Group, Stochastic Research Center School of Mathematical Sciences, Beijing Normal University No.19, Xinjiekouwai St., Beijing 100875, China Phone and Fax: 86-10-58809447 E-mail: [email protected] Website: http://math.bnu.edu.cn/probab/Workshop2010 Schedule 1 July 19 July 20 July 21 July 22 July 23 Chairman Mu-Fa Chen Fu-Zhou Gong Shui Feng Jie Xiong Dayue Chen Opening Shuenn-Jyi Sheu Anyue Chen Leonid Mytnik Zengjing Chen 8:30{8:40 8:30{9:00 8:30{9:00 8:30{9:00 8:30{9:00 M. Fukushima Feng-Yu Wang Jiashan Tang Quansheng Liu Dong Han 8:40{9:30 9:00{9:30 9:00{9:30 9:00{9:30 9:00{9:30 Speaker Tea Break Zhen-Qing Chen Renming Song Xia Chen Hao Wang Zongxia Liang 10:00{10:30 10:00{10:30 10:00{10:30 10:00{10:305 10:00{10:30 Panki Kim Christian Leonard Yutao Ma Xiaowen Zhou Fubao Xi 10:30{11:00 10:30{11:00 10:30{11:00 10:30{11:00 10:30{11:00 Jinghai Shao Xu Zhang Liang-Hui Xia Hui He Liqun Niu 11:00{11:30 11:00{11:20 11:00{11:20 11:00{11:20 11:00{11:20 Lunch Chairman Feng-Yu Wang Shizan Fang Zengjing Chen Zeng-Hu Li Chii-Ruey Hwang Tusheng Zhang Dayue Chen Shizan Fang 14:30{15:00 14:30{15:00 14:30{15:00 14:30{15:00 Ivan Gentil Xiang-Dong Li Alok Goswami Yimin Xiao 15:00{15:30 15:00{15:30 15:00{15:30 15:00{15:30 Speaker Tea Break Fu-Zhou Gong Jinwen -
MCMC Learning (Slides)
Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions MCMC Learning Varun Kanade Elchanan Mossel UC Berkeley UC Berkeley August 30, 2013 Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Outline Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Uniform Distribution Learning • Unknown target function f : {−1; 1gn ! {−1; 1g from some class C • Uniform distribution over {−1; 1gn • Random Examples: Monotone Decision Trees [OS06] • Random Walk: DNF expressions [BMOS03] • Membership Query: DNF, TOP [J95] • Main Tool: Discrete Fourier Analysis X Y f (x) = f^(S)χS (x); χS (x) = xi S⊆[n] i2S • Can utilize sophisticated results: hypercontractivity, invariance, etc. • Connections to cryptography, hardness, de-randomization etc. • Unfortunately, too much of an idealization. In practice, variables are correlated. Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Markov Random Fields • Graph G = ([n]; E). Each node takes some value in finite set A. n • Distribution over A : (for φC non-negative, Z normalization constant) 1 Y Pr((σ ) ) = φ ((σ ) ) v v2[n] Z C v v2C clique C Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Markov Random Fields • MRFs widely used in vision, computational biology, biostatistics etc. • Extensive Algorithmic Theory for sampling from MRFs, recovering parameters and structures • Learning Question: Given f : An ! {−1; 1g. (How) Can we learn with respect to MRF distribution? • Can we utilize the structure of the MRF to aid in learning? Uniform Distribution Learning Markov Random Fields Harmonic Analysis Experiments and Questions Learning Model • Let M be a MRF with distribution π and f : An ! {−1; 1g the target function • Learning algorithm gets i.i.d. -
Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality
Superpositions and Products of Ornstein-Uhlenbeck Type Processes: Intermittency and Multifractality Nikolai N. Leonenko School of Mathematics Cardiff University The second conference on Ambit Fields and Related Topics Aarhus August 15 Abstract Superpositions of stationary processes of Ornstein-Uhlenbeck (supOU) type have been introduced by Barndorff-Nielsen. We consider the constructions producing processes with long-range dependence and infinitely divisible marginal distributions. We consider additive functionals of supOU processes that satisfy the property referred to as intermittency. We investigate the properties of multifractal products of supOU processes. We present the general conditions for the Lq convergence of cumulative processes and investigate their q-th order moments and R´enyifunctions. These functions are nonlinear, hence displaying the multifractality of the processes. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios. Acknowledgments Joint work with Denis Denisov, University of Manchester, UK Danijel Grahovac, University of Osijek, Croatia Alla Sikorskii, Michigan State University, USA Murad Taqqu, Boston University, USA OU type process I OU Type process is the unique strong solution of the SDE: dX (t) = −λX (t)dt + dZ(λt) where λ > 0, fZ(t)gt≥0 is a (non-decreasing, for this talk) L´eviprocess, and an initial condition X0 is taken to be independent of Z(t). Note, in general Zt doesn't have to be a non-decreasing L´evyprocess. I For properties of OU type processes and their generalizations see Mandrekar & Rudiger (2007), Barndorff-Nielsen (2001), Barndorff-Nielsen & Stelzer (2011). I The a.s. -
Markov Random Fields: – Geman, Geman, “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images”, IEEE PAMI 6, No
MarkovMarkov RandomRandom FieldsFields withwith ApplicationsApplications toto MM--repsreps ModelsModels Conglin Lu Medical Image Display and Analysis Group University of North Carolina, Chapel Hill MarkovMarkov RandomRandom FieldsFields withwith ApplicationsApplications toto MM--repsreps ModelsModels Outline: Background; Definition and properties of MRF; Computation; MRF m-reps models. MarkovMarkov RandomRandom FieldsFields Model a large collection of random variables with complex dependency relationships among them. MarkovMarkov RandomRandom FieldsFields • A model based approach; • Has been applied to a variety of problems: - Speech recognition - Natural language processing - Coding - Image analysis - Neural networks - Artificial intelligence • Usually used within the Bayesian framework. TheThe BayesianBayesian ParadigmParadigm X = space of the unknown variables, e.g. labels; Y = space of data (observations), e.g. intensity values; Given an observation y∈Y, want to make inference about x∈X. TheThe BayesianBayesian ParadigmParadigm Prior PX : probability distribution on X; Likelihood PY|X : conditional distribution of Y given X; Statistical inference is based on the posterior distribution PX|Y ∝ PX •PY|X . TheThe PriorPrior DistributionDistribution • Describes our assumption or knowledge about the model; • X is usually a high dimensional space. PX describes the joint distribution of a large number of random variables; • How do we define PX? MarkovMarkov RandomRandom FieldsFields withwith ApplicationsApplications toto MM--repsreps ModelsModels Outline: 9 Background; Definition and properties of MRF; Computation; MRF m-reps models. AssumptionsAssumptions •X = {Xs}s∈S, where each Xs is a random variable; S is an index set and is finite; • There is a common state space R:Xs∈R for all s ∈ S; | R | is finite; • Let Ω = {ω=(x , ..., x ): x ∈ , 1≤i≤N} be s1 sN si R the set of all possible configurations. -
Anomaly Detection Based on Wavelet Domain GARCH Random Field Modeling Amir Noiboar and Israel Cohen, Senior Member, IEEE
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 5, MAY 2007 1361 Anomaly Detection Based on Wavelet Domain GARCH Random Field Modeling Amir Noiboar and Israel Cohen, Senior Member, IEEE Abstract—One-dimensional Generalized Autoregressive Con- Markov noise. It is also claimed that objects in imagery create a ditional Heteroscedasticity (GARCH) model is widely used for response over several scales in a multiresolution representation modeling financial time series. Extending the GARCH model to of an image, and therefore, the wavelet transform can serve as multiple dimensions yields a novel clutter model which is capable of taking into account important characteristics of a wavelet-based a means for computing a feature set for input to a detector. In multiscale feature space, namely heavy-tailed distributions and [17], a multiscale wavelet representation is utilized to capture innovations clustering as well as spatial and scale correlations. We periodical patterns of various period lengths, which often ap- show that the multidimensional GARCH model generalizes the pear in natural clutter images. In [12], the orientation and scale casual Gauss Markov random field (GMRF) model, and we de- selectivity of the wavelet transform are related to the biological velop a multiscale matched subspace detector (MSD) for detecting anomalies in GARCH clutter. Experimental results demonstrate mechanisms of the human visual system and are utilized to that by using a multiscale MSD under GARCH clutter modeling, enhance mammographic features. rather than GMRF clutter modeling, a reduced false-alarm rate Statistical models for clutter and anomalies are usually can be achieved without compromising the detection rate. related to the Gaussian distribution due to its mathematical Index Terms—Anomaly detection, Gaussian Markov tractability. -
A Novel Approach for Markov Random Field with Intractable Normalising Constant on Large Lattices
A novel approach for Markov Random Field with intractable normalising constant on large lattices W. Zhu ∗ and Y. Fany February 19, 2018 Abstract The pseudo likelihood method of Besag (1974), has remained a popular method for estimating Markov random field on a very large lattice, despite various documented deficiencies. This is partly because it remains the only computationally tractable method for large lattices. We introduce a novel method to estimate Markov random fields defined on a regular lattice. The method takes advantage of conditional independence structures and recur- sively decomposes a large lattice into smaller sublattices. An approximation is made at each decomposition. Doing so completely avoids the need to com- pute the troublesome normalising constant. The computational complexity arXiv:1601.02410v1 [stat.ME] 11 Jan 2016 is O(N), where N is the the number of pixels in lattice, making it computa- tionally attractive for very large lattices. We show through simulation, that the proposed method performs well, even when compared to the methods using exact likelihoods. Keywords: Markov random field, normalizing constant, conditional indepen- dence, decomposition, Potts model. ∗School of Mathematics and Statistics, University of New South Wales, Sydney 2052 Australia. Communicating Author Wanchuang Zhu: Email [email protected]. ySchool of Mathematics and Statistics, University of New South Wales, Sydney 2052 Australia. Email [email protected]. 1 1 Introduction Markov random field (MRF) models have an important role in modelling spa- tially correlated datasets. They have been used extensively in image and texture analyses ( Nott and Ryden´ 1999, Hurn et al. 2003), image segmentation (Pal and Pal 1993, Van Leemput et al. -
Markov Random Fields and Stochastic Image Models
Markov Random Fields and Stochastic Image Models Charles A. Bouman School of Electrical and Computer Engineering Purdue University Phone: (317) 494-0340 Fax: (317) 494-3358 email [email protected] Available from: http://dynamo.ecn.purdue.edu/»bouman/ Tutorial Presented at: 1995 IEEE International Conference on Image Processing 23-26 October 1995 Washington, D.C. Special thanks to: Ken Sauer Suhail Saquib Department of Electrical School of Electrical and Computer Engineering Engineering University of Notre Dame Purdue University 1 Overview of Topics 1. Introduction (b) Non-Gaussian MRF's 2. The Bayesian Approach i. Quadratic functions ii. Non-Convex functions 3. Discrete Models iii. Continuous MAP estimation (a) Markov Chains iv. Convex functions (b) Markov Random Fields (MRF) (c) Parameter Estimation (c) Simulation i. Estimation of σ (d) Parameter estimation ii. Estimation of T and p parameters 4. Application of MRF's to Segmentation 6. Application to Tomography (a) The Model (a) Tomographic system and data models (b) Bayesian Estimation (b) MAP Optimization (c) MAP Optimization (c) Parameter estimation (d) Parameter Estimation 7. Multiscale Stochastic Models (e) Other Approaches (a) Continuous models 5. Continuous Models (b) Discrete models (a) Gaussian Random Process Models 8. High Level Image Models i. Autoregressive (AR) models ii. Simultaneous AR (SAR) models iii. Gaussian MRF's iv. Generalization to 2-D 2 References in Statistical Image Modeling 1. Overview references [100, 89, 50, 54, 162, 4, 44] 4. Simulation and Stochastic Optimization Methods [118, 80, 129, 100, 68, 141, 61, 76, 62, 63] 2. Type of Random Field Model 5. Computational Methods used with MRF Models (a) Discrete Models i. -
A Note on Probability Theory
A Note on Probability Theory Ying Nian Wu, Note for STATS 200A Contents 1 Probability 3 1.1 Why probability? . .3 1.2 Three canonical examples . .4 1.3 Long run frequency . .4 1.4 Basic language and notation . .4 1.5 Axioms . .5 1.6 Sigma algebra . .5 1.7 Why sigma-algebra . .6 2 Measure 6 2.1 What is measure? . .6 2.2 Lebesgue measure . .7 2.3 Law of large number . .7 2.4 Concentration of measure . .8 2.5 Lebesgue integral . .9 2.6 Simple functions . 10 2.7 Convergence theorems . 10 3 Univariate distribution and expectation 11 3.1 Discrete random variable, expectation, long run average . 11 3.2 Continuous random variable, basic event, discretization . 12 3.3 How to think about density . 13 3.4 Existence of probability density function . 13 3.5 Cumulative density . 14 3.6 Uniform distribution . 14 3.7 Inversion method . 14 3.8 Transformation . 15 3.9 Polar method for normal random variable . 16 3.10 Counting techniques . 17 3.11 Bernoulli . 17 3.12 Binomial . 18 3.13 Normal approximation . 18 3.14 Geometric . 22 3.15 Poisson process . 22 3.16 Survival analysis . 24 1 4 Joint distribution and covariance 25 4.1 Joint distribution . 25 4.2 Expectation, variance, covariance . 26 4.3 Correlation as cosine of angle . 27 4.4 Correlation as the strength of regression . 28 4.5 Least squares derivation of regression . 28 4.6 Regression in terms of projections . 29 4.7 Independence and uncorrelated . 29 4.8 Multivariate statistics . 30 4.9 Multivariate normal . -
A Review on Statistical Inference Methods for Discrete Markov Random Fields Julien Stoehr, Richard Everitt, Matthew T
A review on statistical inference methods for discrete Markov random fields Julien Stoehr, Richard Everitt, Matthew T. Moores To cite this version: Julien Stoehr, Richard Everitt, Matthew T. Moores. A review on statistical inference methods for discrete Markov random fields. 2017. hal-01462078v2 HAL Id: hal-01462078 https://hal.archives-ouvertes.fr/hal-01462078v2 Preprint submitted on 11 Apr 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. A review on statistical inference methods for discrete Markov random fields Julien Stoehr1 1School of Mathematical Sciences & Insight Centre for Data Analytics, University College Dublin, Ireland Abstract Developing satisfactory methodology for the analysis of Markov random field is a very challenging task. Indeed, due to the Markovian dependence structure, the normalizing constant of the fields cannot be computed using standard analytical or numerical methods. This forms a central issue for any statistical approach as the likelihood is an integral part of the procedure. Furthermore, such unobserved fields cannot be integrated out and the likelihood evaluation becomes a doubly intractable problem. This report gives an overview of some of the methods used in the literature to analyse such observed or unobserved random fields.