Monte Carlo Simulations
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Getting Started with Geostatistics
FedGIS Conference February 24–25, 2016 | Washington, DC Getting Started with Geostatistics Brett Rose Objectives • Using geography as an integrating factor. • Introduce a variety of core tools available in GIS. • Demonstrate the utility of spatial & Geo statistics for a range of applications. • Modeling relationships and driving decisions Outline • Spatial Statistics vs Geostatistics • Geostatistical Workflow • Exploratory Spatial Data Analysis • Interpolation methods • Deterministic vs Geostatistical •Geostatistical Analyst in ArcGIS • Demos • Class Exercise In ArcGIS Geostatistical Analyst • interactive ESDA • interactive modeling including variography • many Kriging models (6 + cokriging) • pre-processing of data - Decluster, detrend, transformation • model diagnostics and comparison Spatial Analyst • rich set of tools to perform cell-based (raster) analysis • kriging (2 models), IDW, nearest neighbor … Spatial Statistics • ESDA GP tools • analyzing the distribution of geographic features • identifying spatial patterns What are spatial statistics in a GIS environment? • Software-based tools, methods, and techniques developed specifically for use with geographic data. • Spatial statistics: – Describe and spatial distributions, spatial patterns, spatial processes model , and spatial relationships. – Incorporate space (area, length, proximity, orientation, and/or spatial relationships) directly into their mathematics. In many ways spatial statistics extenD what the eyes anD minD Do intuitively to assess spatial patterns, trenDs anD relationships. -
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)). -
Reporting Guidelines for Simulation-Based Research in Social Sciences
Reporting Guidelines for Simulation-based Research in Social Sciences Hazhir Rahmandad John D. Sterman Grado Department of Industrial and Sloan School of Management Systems Engineering Massachusetts Institute of Technology Virginia Tech [email protected] [email protected] Abstract Reproducibility of research is critical for the healthy growth and accumulation of reliable knowledge, and simulation-based research is no exception. However, studies show many simulation-based studies in the social sciences are not reproducible. Better standards for documenting simulation models and reporting results are needed to enhance the reproducibility of simulation-based research in the social sciences. We provide an initial set of Reporting Guidelines for Simulation-based Research (RGSR) in the social sciences, with a focus on common scenarios in system dynamics research. We discuss these guidelines separately for reporting models, reporting simulation experiments, and reporting optimization results. The guidelines are further divided into minimum and preferred requirements, distinguishing between factors that are indispensable for reproduction of research and those that enhance transparency. We also provide a few guidelines for improved visualization of research to reduce the costs of reproduction. Suggestions for enhancing the adoption of these guidelines are discussed at the end. Acknowledgments: We would like to thank Mohammad Jalali for providing excellent research assistance for this paper. We thank David Lane for his helpful suggestions. 1. Introduction and Motivation Reproducibility of research is central to the progress of science. Only when research results are independently reproducible can different research projects build on each other, verify the results reported by other researchers, and convince the public of the reliability of their results (Laine, Goodman et al. -
Serious Games, Debriefing, and Simulation/Gaming As a Discipline
41610.1177/104687811 0390784CrookallSimulation & Gaming © 2010 SAGE Publications Reprints and permission: http://www. sagepub.com/journalsPermissions.nav Editorial Simulation & Gaming 41(6) 898 –920 Serious Games, © 2010 SAGE Publications Reprints and permission: http://www. Debriefing, and sagepub.com/journalsPermissions.nav DOI: 10.1177/1046878110390784 Simulation/Gaming http://sg.sagepub.com as a Discipline David Crookall Abstract At the close of the 40th Anniversary Symposium of S&G, this editorial offers some thoughts on a few important themes related to simulation/gaming. These are develop- ment of the field, the notion of serious games, the importance of debriefing, the need for research, and the emergence of a discipline. I suggest that the serious gaming com- munity has much to offer the discipline of simulation/gaming and that debriefing is vital both for learning and for establishing simulation/gaming as a discipline. Keywords academic programs, acceptability, after action review, 40th anniversary, computers, debriefing, definitions, design, discipline, engagement, events, experiential learning, feedback, games, interdisciplinarity, journals, learning, periodicals, philosophy of simulation, practice, processing experience, professional associations, publication, research, review articles, scholarship, serious games, S&G, simulation, simulation/ gaming, terminology, theory, training The field of simulation/gaming is, to be sure, rather fuzzy, and sits uneasily in many areas. The 40th Anniversary Symposium was an opportunity -
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. -
Monte Carlo Simulation in Statistics with a View to Recent Ideas
(sfi)2 Statistics for Innovation Monte Carlo simulation in statistics, with a view to recent ideas Arnoldo Frigessi [email protected] THE JOURNAL OF CHEMICAL PHYSICS VOLUME 21, NUMBER 6 JUNE, 1953 Equation of State Calculations by Fast Computing Machines NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER, Los Alamos Scientific Laboratory, Los Alamos, New Mexico AND EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois (Received March 6, 1953) A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion. The Metropolis algorithm from1953 has been cited as among the top 10 algorithms having the "greatest influence on the development and practice of science and engineering in the 20th century." MANIAC The MANIAC II 1952 Multiplication took a milli- second. (Metropolis choose the name MANIAC in the hope of stopping the rash of ”silly” acronyms for machine names.) Some MCMC milestones 1953 Metropolis Heat bath, dynamic Monte Carlo 1970 Hastings Political Analysis, 10:3 (2002) Society for Political Methodology 1984 Geman & Geman Gibbs -
The Development of Military Nuclear Strategy And
The Development of Military Nuclear Strategy and Anglo-American Relations, 1939 – 1958 Submitted by: Geoffrey Charles Mallett Skinner to the University of Exeter as a thesis for the degree of Doctor of Philosophy in History, July 2018 This thesis is available for Library use on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. I certify that all material in this thesis which is not my own work has been identified and that no material has previously been submitted and approved for the award of a degree by this or any other University. (Signature) ……………………………………………………………………………… 1 Abstract There was no special governmental partnership between Britain and America during the Second World War in atomic affairs. A recalibration is required that updates and amends the existing historiography in this respect. The wartime atomic relations of those countries were cooperative at the level of science and resources, but rarely that of the state. As soon as it became apparent that fission weaponry would be the main basis of future military power, America decided to gain exclusive control over the weapon. Britain could not replicate American resources and no assistance was offered to it by its conventional ally. America then created its own, closed, nuclear system and well before the 1946 Atomic Energy Act, the event which is typically seen by historians as the explanation of the fracturing of wartime atomic relations. Immediately after 1945 there was insufficient systemic force to create change in the consistent American policy of atomic monopoly. As fusion bombs introduced a new magnitude of risk, and as the nuclear world expanded and deepened, the systemic pressures grew. -
An Introduction to Spatial Autocorrelation and Kriging
An Introduction to Spatial Autocorrelation and Kriging Matt Robinson and Sebastian Dietrich RenR 690 – Spring 2016 Tobler and Spatial Relationships • Tobler’s 1st Law of Geography: “Everything is related to everything else, but near things are more related than distant things.”1 • Simple but powerful concept • Patterns exist across space • Forms basic foundation for concepts related to spatial dependency Waldo R. Tobler (1) Tobler W., (1970) "A computer movie simulating urban growth in the Detroit region". Economic Geography, 46(2): 234-240. Spatial Autocorrelation (SAC) – What is it? • Autocorrelation: A variable is correlated with itself (literally!) • Spatial Autocorrelation: Values of a random variable, at paired points, are more or less similar as a function of the distance between them 2 Closer Points more similar = Positive Autocorrelation Closer Points less similar = Negative Autocorrelation (2) Legendre P. Spatial Autocorrelation: Trouble or New Paradigm? Ecology. 1993 Sep;74(6):1659–1673. What causes Spatial Autocorrelation? (1) Artifact of Experimental Design (sample sites not random) Parent Plant (2) Interaction of variables across space (see below) Univariate case – response variable is correlated with itself Eg. Plant abundance higher (clustered) close to other plants (seeds fall and germinate close to parent). Multivariate case – interactions of response and predictor variables due to inherent properties of the variables Eg. Location of seed germination function of wind and preferred soil conditions Mechanisms underlying patterns will depend on study system!! Why is it important? Presence of SAC can be good or bad (depends on your objectives) Good: If SAC exists, it may allow reliable estimation at nearby, non-sampled sites (interpolation). -
A Selected Bibliography of Publications By, and About, J
A Selected Bibliography of Publications by, and about, J. Robert Oppenheimer Nelson H. F. Beebe University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Tel: +1 801 581 5254 FAX: +1 801 581 4148 E-mail: [email protected], [email protected], [email protected] (Internet) WWW URL: http://www.math.utah.edu/~beebe/ 17 March 2021 Version 1.47 Title word cross-reference $1 [Duf46]. $12.95 [Edg91]. $13.50 [Tho03]. $14.00 [Hug07]. $15.95 [Hen81]. $16.00 [RS06]. $16.95 [RS06]. $17.50 [Hen81]. $2.50 [Opp28g]. $20.00 [Hen81, Jor80]. $24.95 [Fra01]. $25.00 [Ger06]. $26.95 [Wol05]. $27.95 [Ger06]. $29.95 [Goo09]. $30.00 [Kev03, Kle07]. $32.50 [Edg91]. $35 [Wol05]. $35.00 [Bed06]. $37.50 [Hug09, Pol07, Dys13]. $39.50 [Edg91]. $39.95 [Bad95]. $8.95 [Edg91]. α [Opp27a, Rut27]. γ [LO34]. -particles [Opp27a]. -rays [Rut27]. -Teilchen [Opp27a]. 0-226-79845-3 [Guy07, Hug09]. 0-8014-8661-0 [Tho03]. 0-8047-1713-3 [Edg91]. 0-8047-1714-1 [Edg91]. 0-8047-1721-4 [Edg91]. 0-8047-1722-2 [Edg91]. 0-9672617-3-2 [Bro06, Hug07]. 1 [Opp57f]. 109 [Con05, Mur05, Nas07, Sap05a, Wol05, Kru07]. 112 [FW07]. 1 2 14.99/$25.00 [Ber04a]. 16 [GHK+96]. 1890-1960 [McG02]. 1911 [Meh75]. 1945 [GHK+96, Gow81, Haw61, Bad95, Gol95a, Hew66, She82, HBP94]. 1945-47 [Hew66]. 1950 [Ano50]. 1954 [Ano01b, GM54, SZC54]. 1960s [Sch08a]. 1963 [Kuh63]. 1967 [Bet67a, Bet97, Pun67, RB67]. 1976 [Sag79a, Sag79b]. 1981 [Ano81]. 20 [Goe88]. 2005 [Dre07]. 20th [Opp65a, Anoxx, Kai02]. -
Master of Science in Geospatial Technologies Geostatistics
Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa MasterMaster ofof ScienceScience inin GeospatialGeospatial TechnologiesTechnologies GeostatisticsGeostatistics PredictionsPredictions withwith GeostatisticsGeostatistics CarlosCarlos AlbertoAlberto FelgueirasFelgueiras [email protected]@isegi.unl.pt 1 Master of Science in Geoespatial Technologies PredictionsPredictions withwith GeostatisticsGeostatistics Contents Stochastic Predictions Introduction Deterministic x Stochastic Methods Regionalized Variables Stationary Hypothesis Kriging Prediction – Simple, Ordinary and with Trend Kriging Prediction - Example Cokriging CrossValidation and Validation Problems with Geostatistic Estimators Advantages on using geostatistics 2 Master of Science in Geoespatial Technologies PredictionsPredictions withwith GeostatisticsGeostatistics o sample locations Deterministic x Stochastic Methods z*(u) = K + estimation locations • Deterministic •The z *(u) value is estimated as a Deterministic Variable. An unique value is associated to its spatial location. • No uncertainties are associated to the z*(u)? estimations + • Stochastic •The z *(u) value is considered a Random Variable that has a probability distribution function associated to its possible values • Uncertainties can be associated to the estimations • Important: Samples are realizations of cdf Random Variables 3 Master of Science in Geoespatial Technologies PredictionsPredictions withwith GeostatisticsGeostatistics • Random Variables and Random -
The Convergence of Markov Chain Monte Carlo Methods 3
The Convergence of Markov chain Monte Carlo Methods: From the Metropolis method to Hamiltonian Monte Carlo Michael Betancourt From its inception in the 1950s to the modern frontiers of applied statistics, Markov chain Monte Carlo has been one of the most ubiquitous and successful methods in statistical computing. The development of the method in that time has been fueled by not only increasingly difficult problems but also novel techniques adopted from physics. In this article I will review the history of Markov chain Monte Carlo from its inception with the Metropolis method to the contemporary state-of-the-art in Hamiltonian Monte Carlo. Along the way I will focus on the evolving interplay between the statistical and physical perspectives of the method. This particular conceptual emphasis, not to mention the brevity of the article, requires a necessarily incomplete treatment. A complementary, and entertaining, discussion of the method from the statistical perspective is given in Robert and Casella (2011). Similarly, a more thorough but still very readable review of the mathematics behind Markov chain Monte Carlo and its implementations is given in the excellent survey by Neal (1993). I will begin with a discussion of the mathematical relationship between physical and statistical computation before reviewing the historical introduction of Markov chain Monte Carlo and its first implementations. Then I will continue to the subsequent evolution of the method with increasing more sophisticated implementations, ultimately leading to the advent of Hamiltonian Monte Carlo. 1. FROM PHYSICS TO STATISTICS AND BACK AGAIN At the dawn of the twentieth-century, physics became increasingly focused on under- arXiv:1706.01520v2 [stat.ME] 10 Jan 2018 standing the equilibrium behavior of thermodynamic systems, especially ensembles of par- ticles. -
Conditional Simulation
Basics in Geostatistics 3 Geostatistical Monte-Carlo methods: Conditional simulation Hans Wackernagel MINES ParisTech NERSC • April 2013 http://hans.wackernagel.free.fr Basic concepts Geostatistics Hans Wackernagel (MINES ParisTech) Basics in Geostatistics 3 NERSC • April 2013 2 / 34 Concepts Geostatistical model The experimental variogram serves to analyze the spatial structure of a regionalized variable z(x). It is fitted with a variogram model which is the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x). Kriging: Best Linear Unbiased Estimation of point values (or spatial averages) at any location of a region. Conditional simulation: generate an ensemble of realizations of the random function, conditional upon data. Statistics not linearly related to data can be computed from this ensemble. Concepts Geostatistical model The experimental variogram serves to analyze the spatial structure of a regionalized variable z(x). It is fitted with a variogram model which is the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x). Kriging: Best Linear Unbiased Estimation of point values (or spatial averages) at any location of a region. Conditional simulation: generate an ensemble of realizations of the random function, conditional upon data. Statistics not linearly related to data can be computed from this ensemble. Concepts Geostatistical model The experimental variogram serves to analyze the spatial structure of a regionalized variable z(x). It is fitted with a variogram model which is the structure function of a random function. The regionalized variable (reality) is viewed as one realization of the random function Z(x).