DOCSLIB.ORG

Markov Random Fields

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Departement of Mathematics Markov Random Fields An extension from a temporal to a spatial setting Cornelia Richter STA381: Stochastic Simulation Seminar 03 June 2021 Departement of Mathematics Outline I 1. Spatial Models for Lattice Data 2. Conditionally Autoregressive Models 3. Graphs and Gaussian Markov Random Fields 4. Random Walk Models: Intrinsic Gaussian Markov Random Fields 5. Extension: A way of predicting in statistical models for random fields 6. Summary 7. Outlook 03 June 2021 Markov Random Fields Page 2 Departement of Mathematics Motivation – very important branch of probability theory – promising results in both theory and application 03 June 2021 Markov Random Fields Page 3 Departement of Mathematics Spatial Models for Lattice Data I – so far: Markov property based on time (continuous and discrete) – now: Markov property based on space (naturally continuous) ! typically called lattice or areal data 03 June 2021 Markov Random Fields Page 4 Departement of Mathematics Spatial Models for Lattice Data II Main difference between time series and lattice data: – time series: temporal direction forecasting (prediction) is very natural concept – lattice data: prediction very unusual, ! focus on smoothing (filtering) ! used to separate signal from noise or to fill in missing values 03 June 2021 Markov Random Fields Page 5 Departement of Mathematics Spatial Models for Lattice Data: Example – Partition of lower 48 US states into counties ! set of representative location of the block defines a lattice ! specific lattice has a varying density because of different county sizes 03 June 2021 Markov Random Fields Page 6 Departement of Mathematics Example Lattice Data US Counties I Figure: 48 lower US States 03 June 2021 Markov Random Fields Page 7 Departement of Mathematics Example Lattice Data US Counties II Figure: 3082 US Counties 03 June 2021 Markov Random Fields Page 8 Departement of Mathematics Example Lattice Data US Counties III Figure: Histogram of number of neighbors 03 June 2021 Markov Random Fields Page 9 Departement of Mathematics Conditionally Autoregressive Models I – basic idea: construct densities based on neighbors – random vector Y, density fY (y) ! f (yi j yj ; i 6= j); i = 1;::: n – approach: specify the joint density through n full conditional densities f (yi j y−i ) – assume Gaussian full conditionals, likely to get a valid joint density 03 June 2021 Markov Random Fields Page 10 Departement of Mathematics Conditionally Autoregressive Models II simple model for location i, given by P 2 Yi j yj ; j 6= i ∼ N( i;j6=i bij yj ; τi ); i = 1; :::; n ! with Brooks Lemma we conclude −1 Y ∼ Nn(0; (I − B) T), 2 B = bij , bii = 0, T = diag(τi ) 03 June 2021 Markov Random Fields Page 11 Departement of Mathematics Conditionally Autoregressive Models II Continued – covariance matrix is symmetric bij bji ! we require 2 = 2 τi τj – (I − B) has to be positive definite 03 June 2021 Markov Random Fields Page 12 Departement of Mathematics Conditionally Autoregressive Models III Neighbors – i and j are neighbors, denoted by i ∼ j – temporal neighbors: previous/next value – spatial neighbors: sharing a common edge or boundary on a Graph 03 June 2021 Markov Random Fields Page 13 Departement of Mathematics Conditionally Autoregressive Models III Neighbors - Example I Figure: Neighbours: left - 4 neighbors of s, right - 8 neighbors of s 03 June 2021 Markov Random Fields Page 14 Departement of Mathematics Conditionally Autoregressive Models III Neighbors - Example II Figure: Neighbours: middle - von Neumann neighborhood, right - Moore neighborhood 03 June 2021 Markov Random Fields Page 15 Departement of Mathematics Sudden Infant Death Syndrome Dataset (SIDS) – sudden, unexplained death of a less than one year old child – SIDS dataset has been analysed a lot in literature, e.g. in Cressie (1993) – dataset: ! birth counts and SIDS cases ! periods: 1974 - 1978 and 1979 - 1984 ! two ethnicities ! 100 counties of North Carolina (NC) – our analysis: neighbors, fitting an conditionally autoregressive model, Morans test, kriging 03 June 2021 Markov Random Fields Page 16 Departement of Mathematics Application CAR: SIDS I Figure: SIDS Data - Area 03 June 2021 Markov Random Fields Page 17 Departement of Mathematics Application CAR: SIDS I Figure: Neighbors: SIDS Data 03 June 2021 Markov Random Fields Page 18 Departement of Mathematics Application CAR: SIDS II Figure: SIDS Data - Rates 03 June 2021 Markov Random Fields Page 19 Departement of Mathematics Application CAR: SIDS II Figure: SIDS Data - Rates without outlier 03 June 2021 Markov Random Fields Page 20 Departement of Mathematics Application CAR: SIDS III Figure: SIDS Data - Summary CAR Fit 03 June 2021 Markov Random Fields Page 21 Departement of Mathematics Application CAR: SIDS IV Figure: SIDS Data - Summary CAR Fit 03 June 2021 Markov Random Fields Page 22 Departement of Mathematics Application CAR: SIDS V Figure: SIDS Data - Summary CAR Fit 03 June 2021 Markov Random Fields Page 23 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Graphs I – graph G = (V; E), V...nodes, E... edges – graphs can be directed or undirected 03 June 2021 Markov Random Fields Page 24 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Graphs II Figure: Graphs: left - undirected graph, right - directed graph 03 June 2021 Markov Random Fields Page 25 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Definitions I Definition: Precision The Precision of a random variable is the inverse of the variance. In case of random vectors the inverse of a covariance function is called precision matrix. Note: High precision (i.e. low variance) implies a lot of knowledge about the variable. 03 June 2021 Markov Random Fields Page 26 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Definitions II Definition: Markov Random Field (MRF) Let X = (Xs; s 2 S) be a set of discrete random variables taking values in set V. X is a Markov random field with respect to the neighborhood system (Ns; s 2 S), if P(Xs = x j Xt ; t 6= s) = P(Xs = x j Xt ; t 2 Ns), x 2 V; s 2 S 03 June 2021 Markov Random Fields Page 27 Departement of Mathematics Graphs and (Gaussian) Markov Random Fields Connection ! a random field is a MRF if it satisfies the Markov property ! a MRF is a set of random variables having a Markov property described by an undirected graph ! X ? Y denotes X and Y are independent ! X ? Y j Z = z denotes conditional independence of X and Y 03 June 2021 Markov Random Fields Page 28 Departement of Mathematics Graphs and Gaussian Markov Random Fields Definitions III Definition: Gaussian Markov Random Field (GMRF) T A random vector Y = (Y1; :::; Yn) is a Gaussian Markov Random Field with respect to a labeled graph G = (V; E) with mean µ and a symmetric and positive definite precision matrix Q, if its density is given by −n=2 1=2 1 T f (y) = (2π) det(Q) exp(− 2 (y − µ) Q(y − µ)) Qij 6= 0 , i; j 2 E8i 6= j 03 June 2021 Markov Random Fields Page 29 Departement of Mathematics Graphs and Gaussian Markov Random Fields Properties Property Let Y be a GMRF with respect to G = (V; E), with mean µ and precision matrix Q, then 1. Yi ? Yj j Y−ij = y−ij , Qij = 0 1 P 2. E[Y j Y− = y− ] = µ − Q (y − µ ) i i i i Qii j:j∼i ij j j 3. Prec(Yi j Y−i = y−i ) = Qii Qij 4. Cor(Yi Yj j Y−ij = y−ij ) = −p , i 6= j Qii Qjj 03 June 2021 Markov Random Fields Page 30 Departement of Mathematics Simulation of Markov Random Fields: Binary Simulation I Figure: Binary Simulation: Threshold 0, high sequence 03 June 2021 Markov Random Fields Page 31 Departement of Mathematics Simulation of Markov Random Fields: Binary Simulation I Figure: Binary Simulation: Threshold 0, low sequence 03 June 2021 Markov Random Fields Page 32 Departement of Mathematics Simulation of Markov Random Fields: Binary Simulation II Figure: Binary Simulation: Threshold 1, high sequence 03 June 2021 Markov Random Fields Page 33 Departement of Mathematics Simulation of Markov Random Fields: Binary Simulation II Figure: Binary Simulation: Threshold 1, low sequence 03 June 2021 Markov Random Fields Page 34 Departement of Mathematics Simulation of Markov Random Fields: Circular Embedding Figure: Circular Embedding: high sequence 03 June 2021 Markov Random Fields Page 35 Departement of Mathematics Simulation of Markov Random Fields: Circular Embedding Figure: Circular Embedding: low sequence 03 June 2021 Markov Random Fields Page 36 Departement of Mathematics Random Walk Models: Intrinsic Gaussian Markov Random Fields I – improper densities not integrating to one often used as priors in Bayesian frameworks ! resulting posterior may be appropriate – consider e.g. a regression setting −1 relax condition of constant µ, Yi+1 − Yi ∼ N(0; κ ), instead of Yi+1 − Yi ≡ 0 – NOW locations of n random variables are equispaced observations on the transection 03 June 2021 Markov Random Fields Page 37 Departement of Mathematics Random Walk Models: Intrinsic Gaussian Markov Random Fields II iid −1 ∆Yi = Yi+1 − Yi , i = 1;:::; n − 1, ∆Yi ∼ N(0; κ ) n−1 n−1 n−1 2 κ P 2 2 1 T ) f (∆y) / κ exp(− 2 i=1 (∆yi )) = κ exp(− 2 y Qy) 03 June 2021 Markov Random Fields Page 38 Departement of Mathematics Random Walk Models: Intrinsic Gaussian Markov Random Fields II cont. 0 1 1 −1 0 ··· 0 B . C B −1 2 −1 .. C B C B . C T Q = κ B 0 .. .. .. 0 C = κD D, B C B . C B . .. −1 2 −1 C @ A 0 ··· 0 −1 1 0 −1 1 0 ··· 1 B . C D = B 0 .. .. .. C 2 R(n−1)xn @ . A . .. −1 1 03 June 2021 Markov Random Fields Page 39 Departement of Mathematics Random Walk Models: Intrinsic Gaussian Markov
Recommended publications
  • Methods of Monte Carlo Simulation II

    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 .
  • Superprocesses and Mckean-Vlasov Equations with Creation of Mass

    Superprocesses and Mckean-Vlasov Equations with Creation of Mass

    Sup erpro cesses and McKean-Vlasov equations with creation of mass L. Overb eck Department of Statistics, University of California, Berkeley, 367, Evans Hall Berkeley, CA 94720, y U.S.A. Abstract Weak solutions of McKean-Vlasov equations with creation of mass are given in terms of sup erpro cesses. The solutions can b e approxi- mated by a sequence of non-interacting sup erpro cesses or by the mean- eld of multityp e sup erpro cesses with mean- eld interaction. The lat- ter approximation is asso ciated with a propagation of chaos statement for weakly interacting multityp e sup erpro cesses. Running title: Sup erpro cesses and McKean-Vlasov equations . 1 Intro duction Sup erpro cesses are useful in solving nonlinear partial di erential equation of 1+ the typ e f = f , 2 0; 1], cf. [Dy]. Wenowchange the p oint of view and showhowtheyprovide sto chastic solutions of nonlinear partial di erential Supp orted byanFellowship of the Deutsche Forschungsgemeinschaft. y On leave from the Universitat Bonn, Institut fur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany. 1 equation of McKean-Vlasovtyp e, i.e. wewant to nd weak solutions of d d 2 X X @ @ @ + d x; + bx; : 1.1 = a x; t i t t t t t ij t @t @x @x @x i j i i=1 i;j =1 d Aweak solution = 2 C [0;T];MIR satis es s Z 2 t X X @ @ a f = f + f + d f + b f ds: s ij s t 0 i s s @x @x @x 0 i j i Equation 1.1 generalizes McKean-Vlasov equations of twodi erenttyp es.
  • 12 : Conditional Random Fields 1 Hidden Markov Model

    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.
  • Lecture 19: Polymer Models: Persistence and Self-Avoidance

    Lecture 19: Polymer Models: Persistence and Self-Avoidance

    Lecture 19: Polymer Models: Persistence and Self-Avoidance Scribe: Allison Ferguson (and Martin Z. Bazant) Martin Fisher School of Physics, Brandeis University April 14, 2005 Polymers are high molecular weight molecules formed by combining a large number of smaller molecules (monomers) in a regular pattern. Polymers in solution (i.e. uncrystallized) can be characterized as long flexible chains, and thus may be well described by random walk models of various complexity. 1 Simple Random Walk (IID Steps) Consider a Rayleigh random walk (i.e. a fixed step size a with a random angle θ). Recall (from Lecture 1) that the probability distribution function (PDF) of finding the walker at position R~ after N steps is given by −3R2 2 ~ e 2a N PN (R) ∼ d (1) (2πa2N/d) 2 Define RN = |R~| as the end-to-end distance of the polymer. Then from previous results we q √ 2 2 ¯ 2 know hRN i = Na and RN = hRN i = a N. We can define an additional quantity, the radius of gyration GN as the radius from the centre of mass (assuming the polymer has an approximately 1 2 spherical shape ). Hughes [1] gives an expression for this quantity in terms of hRN i as follows: 1 1 hG2 i = 1 − hR2 i (2) N 6 N 2 N As an example of the type of prediction which can be made using this type of simple model, consider the following experiment: Pull on both ends of a polymer and measure the force f required to stretch it out2. Note that the force will be given by f = −(∂F/∂R) where F = U − TS is the Helmholtz free energy (T is the temperature, U is the internal energy, and S is the entropy).
  • 8 Convergence of Loop-Erased Random Walk to SLE2 8.1 Comments on the Paper

    8 Convergence of Loop-Erased Random Walk to SLE2 8.1 Comments on the Paper

    8 Convergence of loop-erased random walk to SLE2 8.1 Comments on the paper The proof that the loop-erased random walk converges to SLE2 is in the paper \Con- formal invariance of planar loop-erased random walks and uniform spanning trees" by Lawler, Schramm and Werner. (arxiv.org: math.PR/0112234). This section contains some comments to give you some hope of being able to actually make sense of this paper. In the first sections of the paper, δ is used to denote the lattice spacing. The main theorem is that as δ 0, the loop erased random walk converges to SLE2. HOWEVER, beginning in section !2.2 the lattice spacing is 1, and in section 3.2 the notation δ reappears but means something completely different. (δ also appears in the proof of lemma 2.1 in section 2.1, but its meaning here is restricted to this proof. The δ in this proof has nothing to do with any δ's outside the proof.) After the statement of the main theorem, there is no mention of the lattice spacing going to zero. This is hidden in conditions on the \inner radius." For a domain D containing 0 the inner radius is rad0(D) = inf z : z = D (364) fj j 2 g Now fix a domain D containing the origin. Let be the conformal map of D onto U. We want to show that the image under of a LERW on a lattice with spacing δ converges to SLE2 in the unit disc. In the paper they do this in a slightly different way.
  • Pdf File of Second Edition, January 2018

    Pdf File of Second Edition, January 2018

    Probability on Graphs Random Processes on Graphs and Lattices Second Edition, 2018 GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge copyright Geoffrey Grimmett Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 56 Figures copyright Geoffrey Grimmett Contents Preface ix 1 Random Walks on Graphs 1 1.1 Random Walks and Reversible Markov Chains 1 1.2 Electrical Networks 3 1.3 FlowsandEnergy 8 1.4 RecurrenceandResistance 11 1.5 Polya's Theorem 14 1.6 GraphTheory 16 1.7 Exercises 18 2 Uniform Spanning Tree 21 2.1 De®nition 21 2.2 Wilson's Algorithm 23 2.3 Weak Limits on Lattices 28 2.4 Uniform Forest 31 2.5 Schramm±LownerEvolutionsÈ 32 2.6 Exercises 36 3 Percolation and Self-Avoiding Walks 39 3.1 PercolationandPhaseTransition 39 3.2 Self-Avoiding Walks 42 3.3 ConnectiveConstantoftheHexagonalLattice 45 3.4 CoupledPercolation 53 3.5 Oriented Percolation 53 3.6 Exercises 56 4 Association and In¯uence 59 4.1 Holley Inequality 59 4.2 FKG Inequality 62 4.3 BK Inequalitycopyright Geoffrey Grimmett63 vi Contents 4.4 HoeffdingInequality 65 4.5 In¯uenceforProductMeasures 67 4.6 ProofsofIn¯uenceTheorems 72 4.7 Russo'sFormulaandSharpThresholds 80 4.8 Exercises 83 5 Further Percolation 86 5.1 Subcritical Phase 86 5.2 Supercritical Phase 90 5.3 UniquenessoftheIn®niteCluster 96 5.4 Phase Transition 99 5.5 OpenPathsinAnnuli 103 5.6 The Critical Probability in Two Dimensions 107
  • Mean Field Methods for Classification with Gaussian Processes

    Mean Field Methods for Classification with Gaussian Processes

    Mean field methods for classification with Gaussian processes Manfred Opper Neural Computing Research Group Division of Electronic Engineering and Computer Science Aston University Birmingham B4 7ET, UK. opperm~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, S6lvegatan 14 A S-223 62 Lund, Sweden CONNECT, The Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, 2100 Copenhagen 0, Denmark winther~thep.lu.se Abstract We discuss the application of TAP mean field methods known from the Statistical Mechanics of disordered systems to Bayesian classifi­ cation models with Gaussian processes. In contrast to previous ap­ proaches, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given. 1 Modeling with Gaussian Processes Bayesian models which are based on Gaussian prior distributions on function spaces are promising non-parametric statistical tools. They have been recently introduced into the Neural Computation community (Neal 1996, Williams & Rasmussen 1996, Mackay 1997). To give their basic definition, we assume that the likelihood of the output or target variable T for a given input s E RN can be written in the form p(Tlh(s)) where h : RN --+ R is a priori assumed to be a Gaussian random field. If we assume fields with zero prior mean, the statistics of h is entirely defined by the second order correlations C(s, S') == E[h(s)h(S')], where E denotes expectations 310 M Opper and 0. Winther with respect to the prior. Interesting examples are C(s, s') (1) C(s, s') (2) The choice (1) can be motivated as a limit of a two-layered neural network with infinitely many hidden units with factorizable input-hidden weight priors (Williams 1997).
  • MCMC Learning (Slides)

    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.
  • Markov Random Fields: – Geman, Geman, “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images”, IEEE PAMI 6, No

    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

    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.
  • Probability on Graphs Random Processes on Graphs and Lattices

    Probability on Graphs Random Processes on Graphs and Lattices

    Probability on Graphs Random Processes on Graphs and Lattices GEOFFREY GRIMMETT Statistical Laboratory University of Cambridge c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Geoffrey Grimmett Statistical Laboratory Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB3 0WB United Kingdom 2000 MSC: (Primary) 60K35, 82B20, (Secondary) 05C80, 82B43, 82C22 With 44 Figures c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 Contents Preface ix 1 Random walks on graphs 1 1.1 RandomwalksandreversibleMarkovchains 1 1.2 Electrical networks 3 1.3 Flowsandenergy 8 1.4 Recurrenceandresistance 11 1.5 Polya's theorem 14 1.6 Graphtheory 16 1.7 Exercises 18 2 Uniform spanning tree 21 2.1 De®nition 21 2.2 Wilson's algorithm 23 2.3 Weak limits on lattices 28 2.4 Uniform forest 31 2.5 Schramm±LownerevolutionsÈ 32 2.6 Exercises 37 3 Percolation and self-avoiding walk 39 3.1 Percolationandphasetransition 39 3.2 Self-avoiding walks 42 3.3 Coupledpercolation 45 3.4 Orientedpercolation 45 3.5 Exercises 48 4 Association and in¯uence 50 4.1 Holley inequality 50 4.2 FKGinequality 53 4.3 BK inequality 54 4.4 Hoeffdinginequality 56 c G. R. Grimmett 1/4/10, 17/11/10, 5/7/12 vi Contents 4.5 In¯uenceforproductmeasures 58 4.6 Proofsofin¯uencetheorems 63 4.7 Russo'sformulaandsharpthresholds 75 4.8 Exercises 78 5 Further percolation 81 5.1 Subcritical phase 81 5.2 Supercritical phase 86 5.3 Uniquenessofthein®nitecluster 92 5.4 Phase transition 95 5.5 Openpathsinannuli 99 5.6 The critical probability in two dimensions 103 5.7 Cardy's formula 110 5.8 The
  • Arxiv:1504.02898V2 [Cond-Mat.Stat-Mech] 7 Jun 2015 Keywords: Percolation, Explosive Percolation, SLE, Ising Model, Earth Topography

    Arxiv:1504.02898V2 [Cond-Mat.Stat-Mech] 7 Jun 2015 Keywords: Percolation, Explosive Percolation, SLE, Ising Model, Earth Topography

    Recent advances in percolation theory and its applications Abbas Ali Saberi aDepartment of Physics, University of Tehran, P.O. Box 14395-547,Tehran, Iran bSchool of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM) P.O. Box 19395-5531, Tehran, Iran Abstract Percolation is the simplest fundamental model in statistical mechanics that exhibits phase transitions signaled by the emergence of a giant connected component. Despite its very simple rules, percolation theory has successfully been applied to describe a large variety of natural, technological and social systems. Percolation models serve as important universality classes in critical phenomena characterized by a set of critical exponents which correspond to a rich fractal and scaling structure of their geometric features. We will first outline the basic features of the ordinary model. Over the years a variety of percolation models has been introduced some of which with completely different scaling and universal properties from the original model with either continuous or discontinuous transitions depending on the control parameter, di- mensionality and the type of the underlying rules and networks. We will try to take a glimpse at a number of selective variations including Achlioptas process, half-restricted process and spanning cluster-avoiding process as examples of the so-called explosive per- colation. We will also introduce non-self-averaging percolation and discuss correlated percolation and bootstrap percolation with special emphasis on their recent progress. Directed percolation process will be also discussed as a prototype of systems displaying a nonequilibrium phase transition into an absorbing state. In the past decade, after the invention of stochastic L¨ownerevolution (SLE) by Oded Schramm, two-dimensional (2D) percolation has become a central problem in probability theory leading to the two recent Fields medals.