Available online at ScienceDirect. Physics Procedia 57 (2014 ) 48 52
|
|
- Sheila Walters
- 6 years ago
- Views:
Transcription
1 Available olie at ScieceDirect Physics Procedia 57 (214 ) th Aual CSP Workshops o Recet Developmets i Computer Simulatio Studies i Codesed Matter Physics, CSP 214 Applicatio of flat histogram methods to extract the log imagiary-time behavior i diagrammatic Mote Carlo Nikolaos G. Diamatis a, Efstratios Maousakis a,b a Departmet of Physics, Uiversity of Athes, Paepistimioupolis, Zografos, Athes, Greece b Departmet of Physics ad Natioal High Magetic Field Laboratory, Florida State Uiversity, Tallahassee, FL , USA Abstract We demostrate that usig flat histogram methods we ca extract the imagiary-time behavior of the Gree s fuctio G from diagrammatic Mote Carlo simulatios very accurately eve whe G chages by may orders of magitude. c 214 The Authors. Published by by Elsevier B.V. B.V. This is a ope access article uder the CC BY-NC-ND licese ( Peer-review uder resposibility of the Orgaizig Committee of CSP 214 coferece. Peer-review uder resposibility of The Orgaizig Committee of CSP 214 coferece Keywords: Computatioal techiques, computer modellig ad simulatio, Mote Carlo method 1. Itroductio Flat histogram techiques (Berg ad Neuhaus (1991); Oliviera et al. (1996); Wad ad Ladau (21)) have bee show to be very importat methods i order to allow efficiet samplig of the etire phase space ad trasitios betwee cofiguratios i classical systems udergoig a first order phase trasitio. Troyer et al. (Troyer, Wessel ad Alet (23)) have applied flat histogram methods to simulatio of equilibrium quatum statistical mechaics ad showed that it also makes the quatum Mote Carlo algorithm efficiet i hadlig the tuelig problem i first order phase trasitios. Furthermore, followig this idea, Gull et al. (Gull et al. (211)) have applied the flat histogram method to the cotiuous-time quatum Mote Carlo approach to the quatum impurity solver eeded for all dyamical mea-field theory applicatios. The goal of the preset paper is to show that oe ca use flat histogram techiques i order to accurately extract the log imagiary-time behavior of the Gree s fuctio i quatum may-body systems. More precisely, here we apply the flat histogram method to the diagrammatic-mote Carlo (diag-mc) method i order to extract for large. To illustrate the idea we choose the Fröhlich polaro problem (Fröhlich, Pelzer ad Zieau (195)) where the diag-mc has bee previously fruitfully applied (Prokof ev ad Svistiov (1998); Michecko et al. (2)). Some results of the idea preseted i this paper have bee published (Diamatis ad Maousakis (213)) ad the reader is referred to that work because it cotais complemetary iformatio. 2. The problem The diagrammatic Mote Carlo techique (Prokof ev ad Svistiov (1998); Michecko et al. (2)) is a Markov process i a space defied by all the terms (or diagrams) which appear i perturbatio theory. Take for example the The Authors. Published by Elsevier B.V. This is a ope access article uder the CC BY-NC-ND licese ( Peer-review uder resposibility of The Orgaizig Committee of CSP 214 coferece doi:1.116/j.phpro
2 Nikolaos G. Diamatis ad Efstratios Maousakis / Physics Procedia 57 ( 214 ) e+8 1.2e+8 =16 1e+8 1e+7 =8 = 12 = 16 I () 1e+8 8e+7 6e+7 I ( ) 1e+6 1e+5 4e+7 1 2e Fig. 1. I () as a fuctio of for fixed. I () as a fuctio of for three differet values ofplotted o a logarithmic scale. perturbatio expasio of the imagiary-time sigle-particle Gree s fuctio, which ca be schematically writte as follows: = I (), I ()= I (λ) (), I ()=G (), (1) I (λ) ()= λ d x 1 d x 2...d x F (λ) ( x 1, x 2,..., x,), (2) where ca be the order of the diagram adλis a variable which labels the diagrams withi the same order. As the order of the expasio icreases, the umber of itegratio variables icreases i a similar maer. I diag-mc the radom walk makes a series of trasitios betwee states{,λ, R} {,λ, R }, where R=( x 1,..., x ). Through such a Markov process the etire series of terms is sampled. This process geerates a histogram of the umber of times N the order appears i the Markov process. Sice we ca compute a low order diagram aalytically, say for example the zeroth order I (), the absolute value of all other orders is computed as follows: I ()= N N I (). For illustratio of the method we will use a simplified versio (Diamatis ad Maousakis (213)) of the Fröhlich polaro problem (Fröhlich, Pelzer ad Zieau (195)) which has a exact solutio ad, this allows us to compare the Mote-Carlo results with exact results. Whe we refer to the th order i this case we mea that the umber of phoo-propagators cotaied i the diagrams is ; therefore, the order i perturbatio expasio is 2, because there are 2 vertices whe there are phoo propagators. I Fig. 1 the calculated I () as a fuctio of is show for a fixed value ofas calculated for this model. Notice that the fuctio I is a Gaussia-like distributio which peaks at a value of = max (). Fig. 1 shows I () o a logarithmic scale for=8, 12 ad 16. Notice that as a fuctio of, max () grows almost liearly with, the value of I () at the maximum grows dramatically with icreasig. Namely, as icreases higher ad higher order diagrams give the most sigificat cotributio. As a result, for large eough, for ay give limited umber of Mote Carlo steps, the umber of walks ladig i small values of becomes very small or o-existet. However, whe the umber of MC steps which lad i the state = is zero or very small, it leads to a fatal situatio i our attempt to calculate the absolute value of I (), because this is obtaied usig the formula (3) ad a very small N implies a large ucertaity i the absolute value of all I (). This is illustrated i Fig. 2 where the red poits with error bars are obtaied with the diag-mc process. Notice that the (3)
3 5 Nikolaos G. Diamatis ad Efstratios Maousakis / Physics Procedia 57 ( 214 ) I () 1.6e+6 1.4e+6 1.2e+6 1e+6 8e+5 DMC + Multicaoical DMC I ( ) 1e e+5 4e+5 1 2e Fig. 2. Demostratio of the applicatio of the multicaoical algorithm. See text for details. size of the error bars is much larger tha the size of the fluctuatios of the poits for successive values of. Namely, the poits which represet I () form a rather smooth curve. This seems uusual give the size of the error bars. This ca be explaied by the fact that the error is due to the poor estimatio of N which propagates through the formula give by Eq. 3. Note that usig the same umber of MC steps becomes impossible to calculate I () beyod this value of=12 because the ratio I max ()/I becomes much larger tha the umber of MC steps. 3. How to solve the problem Here, we will solve the problem discussed i the previous sectio by adoptig the flat histogram techiques which have bee applied to simulatio of classical statistical mechaics (Berg ad Neuhaus (1991); Wad ad Ladau (21)). We map the particular value of to the eergy level i stadard flat histogram methods for classical statistical mechaics ad the sum of the terms givig I () to the desity of states which correspods to the correspodig cofiguratios. The flat histogram method reormalizes the desity of states I () for each by kow factors (which ca be easily estimated) ad, the, samples a more-or-less flat histogram of such populatios. Next, we use the idea of the multicaoical algorithm (Berg ad Neuhaus (1991)) as follows: First, for a give fixed value ofwe carry out a iitial exploratory ru, where we fid that the distributio I of the values of peaks at some value of = max, which depeds o the chose value of. The curve labeled i Fig. 2 shows the result obtaied for the histogram usig M = 1 6 diag-mc steps. This distributio falls off rapidly for > max, ad, thus, we ca determie the maximum value c of visited by the Markov process. We choose a value of m safely greater tha c, such that the value of I m is practically zero. The, we modify the probabilities associated with a particular cofiguratio of the th order by dividig the origial probabilities by a factor f () = max(1, N ). Usig these modified probabilities we carry out aother set of M diag-mc steps which yields a ew histogram with populatios N (1) show by curve labeled by 1 i Fig. 2. I the ext step, we divide the probabilities associated with a particular cofiguratio of the th order by the factor f (1) = f () max(1, N (1) ). Usig these modified probabilities we carry out a ew set of M diag-mc steps which yields a ew histogram with populatios N (2) show by curve labeled by 2 i Fig. 2. We cotiue this process several times where the probabilities at the i th step are divided by a factor f (i) = f (i 1) max(1, N (i) where N (i) is the populatio of the th order at the i th step. The curves labeled 3,4,5 ad 6 i Fig. 2 are obtaied by repeatig this process four more times. Notice that already at the 6 th step the histogram is reasoably flat icludig for small. Whe, the histogram becomes more-or-less flat at some k th step, we begi a Markov process for a relatively large umber of MC steps, by dividig the origial probabilities by the factor f (k), ad, by re-weightig the observables by the biasig factor f (k) we determie I () ad.
4 Nikolaos G. Diamatis ad Efstratios Maousakis / Physics Procedia 57 ( 214 ) Multicaoical Wag-Ladau Fig. 3. The results obtaied usig the fixed- diag-mc with applicatio of the multicaoical algorithm to make the histogram of I () asa fuctio of flat. with applicatio of the WL algorithm to make the histogram of I () as a fuctio of flat. The results i both cases are compared with the exact results. Fig. 2 shows the results of applyig the multicaoical algorithm as discussed i the previous paragraph for the same umber of MC steps ad approximately the same amout of CPU time as i the calculatio with the straightforward applicatio of the diag-mc to obtai the red curve i Fig. 2. Notice the sigificat reductio of the error bars. Furthermore, the flat histogram approach allows us to calculate I () for almost ay, somethig which is ot possible usig simple diag-mc. I Fig. 3 we compare with the exact results the results for obtaied for various values ofusig the multicaoical method to make the histogram flat (Fig. 3) or the W-L algorithm (Fig. 3). Notice that the agreemet holds over a rage where chages by 14 orders of magitude. 4. Samplig the histogram of Istead of fixig, we ca divide the iterval ofi small itervals of fixed duratioδ ad we defie g l as the time average of iside the particular iterval l. I this case, what we regard as the state i the Markov process acquires a additioal label l ad becomes{l,,λ, R} (where is the order,λaparticular diagram of order, ad R=( x 1,.., x ), the itegratio variables. Thus, we ca also have trasitios betwee differet l values. I this case i order to calculate g l we create a listn l of the umber of times the Markov radom walker lads i the iterval l ad we ca calculate the absolute value of g l as g l = N l N g, where g = 1. Notice that g l /g ca be a very large umber, ad ifis large eough it ca be much larger tha the total umber of MC steps. I this case becausen is expected to be either or a very small umber (ad, thus, with large error), it makes sese i order to apply the flat histogram method, to map the eergy level i the classical MC ad the desity of states to the particular iterval l ad g l respectively. This way, the histogram of g l becomes flat. Fig. 3 compares the results obtaied without applicatio of the flat histogram method (red circles with error bars) with those obtaied usig the W-L method (blue squares, the size of the error i this case is smaller tha the symbol-size ad it omitted for clarity) to make the histogram of g l flat. We have limited the maximum value ofi order to make it possible to obtai results with the simple diag-mc method, i.e., to obtai a o-zero value ofn. Agai the reaso for the errors i the stadard diag-mc without applyig the flat histogram method, is the fact that (4)
5 52 Nikolaos G. Diamatis ad Efstratios Maousakis / Physics Procedia 57 ( 214 ) Wag-Ladau DMC without guidace Fig. 4. Compariso of the stadard diag-mc with diag-mc+wl ad with the exact results. See text for details. N is very small ad, by usig the expressio (4) its error propagates to all other values of l. This is show i Fig. 3 where the results of differet rus, each with the same umber of MC steps are compared. Notice that the various lies are almost parallel to each other startig with differet value at l=. 5. Coclusios We have combied the idea of flat histogram methods with diagrammatic MC quatum simulatio techique i order to extract the sigle Gree fuctio at log imagiary time. We have show that a) Simple applicatio of the stadard diag-mc, without some a priori kowledge about the behavior of at log, leads to either very large errors or it becomes impossible to estimate if is large eough. We cosiderto be large whe the rage of values of I () for various values of or the rage of values of i the iterval (,) ivolves may orders of magitude. b) To cure this problem, we used the idea of flat histogram methods: I the case where we eed to apply the diag- MC by keepigfixed, we made the histogram of I () flat for various values of. I the case where we allowedto vary, which eables us to samplealso, we made flat the histogram of as fuctio of. c) We fid that this is a crucial improvemet over the stadard diag-mc i order to extract the imagiary time behavior of the Gree s fuctio. This allows us to extract the low-eergy physics of may-body systems. Refereces B. A. Berg, ad T. Neuhaus, Phys. Lett. B, 267, 249 (1991). P. M. C. de Oliviera et al.,jphys.26, 677 (1996). F. Wag ad D. P. Ladau, Phys. Rev. Lett. 86, 25 (21). M. Troyer, S. Wessel, ad F. Alet, Phys. Rev. Lett. 9, 1221 (23). E. Gull, A. J. Millis, A. I. Lichtestei, A. N. Rubstov, M. Troyer, P. Werer, Rev. Mod. Phys. 83, 349 (211). G. Li, W. Werer, ad A. N. Rubstov, S. Base, M. Potthoff, Phys. Rev. B 8, (29). E. Gull, Ph.D. Thesis. H. Fröhlich, H. Pelzer, ad S. Zieau, Phil. Mag. 41, 221 (195). N. V. Prokof ev ad B. V. Svistuov, Phys. Rev. Lett. 81, 2514 (1998). A. S. Mishcheko, N. V. Prokof ev, A. Sakamoto, B. V. Svistuov, Phys. Rev. B, 62, 6317 (2). N. G. Diamatis ad E. Maousakis, Phys. Rev. E, 88, 4332 (213).
Flat-histogram methods in quantum Monte Carlo simulations: Application to the t-j model
Joural of Physics: Coferece Series PAPER OPEN ACCESS Flat-histogram methods i quatum Mote Carlo simulatios: Applicatio to the t-j model To cite this article: Nikolaos G. Diamatis ad Efstratios Maousakis
More informationDouble Stage Shrinkage Estimator of Two Parameters. Generalized Exponential Distribution
Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M.
More informationDiscrete Mathematics for CS Spring 2008 David Wagner Note 22
CS 70 Discrete Mathematics for CS Sprig 2008 David Wager Note 22 I.I.D. Radom Variables Estimatig the bias of a coi Questio: We wat to estimate the proportio p of Democrats i the US populatio, by takig
More informationStatistics 511 Additional Materials
Cofidece Itervals o mu Statistics 511 Additioal Materials This topic officially moves us from probability to statistics. We begi to discuss makig ifereces about the populatio. Oe way to differetiate probability
More informationOrthogonal Gaussian Filters for Signal Processing
Orthogoal Gaussia Filters for Sigal Processig Mark Mackezie ad Kiet Tieu Mechaical Egieerig Uiversity of Wollogog.S.W. Australia Abstract A Gaussia filter usig the Hermite orthoormal series of fuctios
More information1 Inferential Methods for Correlation and Regression Analysis
1 Iferetial Methods for Correlatio ad Regressio Aalysis I the chapter o Correlatio ad Regressio Aalysis tools for describig bivariate cotiuous data were itroduced. The sample Pearso Correlatio Coefficiet
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationThe improvement of the volume ratio measurement method in static expansion vacuum system
Available olie at www.sciecedirect.com Physics Procedia 32 (22 ) 492 497 8 th Iteratioal Vacuum Cogress The improvemet of the volume ratio measuremet method i static expasio vacuum system Yu Hogya*, Wag
More information6.3 Testing Series With Positive Terms
6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationA statistical method to determine sample size to estimate characteristic value of soil parameters
A statistical method to determie sample size to estimate characteristic value of soil parameters Y. Hojo, B. Setiawa 2 ad M. Suzuki 3 Abstract Sample size is a importat factor to be cosidered i determiig
More informationENGI 4421 Confidence Intervals (Two Samples) Page 12-01
ENGI 44 Cofidece Itervals (Two Samples) Page -0 Two Sample Cofidece Iterval for a Differece i Populatio Meas [Navidi sectios 5.4-5.7; Devore chapter 9] From the cetral limit theorem, we kow that, for sufficietly
More informationCHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS. 8.1 Random Sampling. 8.2 Some Important Statistics
CHAPTER 8 FUNDAMENTAL SAMPLING DISTRIBUTIONS AND DATA DESCRIPTIONS 8.1 Radom Samplig The basic idea of the statistical iferece is that we are allowed to draw ifereces or coclusios about a populatio based
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationFACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures
FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 5 STATISTICS II. Mea ad stadard error of sample data. Biomial distributio. Normal distributio 4. Samplig 5. Cofidece itervals
More informationECE 8527: Introduction to Machine Learning and Pattern Recognition Midterm # 1. Vaishali Amin Fall, 2015
ECE 8527: Itroductio to Machie Learig ad Patter Recogitio Midterm # 1 Vaishali Ami Fall, 2015 tue39624@temple.edu Problem No. 1: Cosider a two-class discrete distributio problem: ω 1 :{[0,0], [2,0], [2,2],
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2016 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationStat 421-SP2012 Interval Estimation Section
Stat 41-SP01 Iterval Estimatio Sectio 11.1-11. We ow uderstad (Chapter 10) how to fid poit estimators of a ukow parameter. o However, a poit estimate does ot provide ay iformatio about the ucertaity (possible
More informationRandom Variables, Sampling and Estimation
Chapter 1 Radom Variables, Samplig ad Estimatio 1.1 Itroductio This chapter will cover the most importat basic statistical theory you eed i order to uderstad the ecoometric material that will be comig
More informationChapter 2 The Monte Carlo Method
Chapter 2 The Mote Carlo Method The Mote Carlo Method stads for a broad class of computatioal algorithms that rely o radom sampligs. It is ofte used i physical ad mathematical problems ad is most useful
More informationMath 140 Introductory Statistics
8.2 Testig a Proportio Math 1 Itroductory Statistics Professor B. Abrego Lecture 15 Sectios 8.2 People ofte make decisios with data by comparig the results from a sample to some predetermied stadard. These
More informationOn an Application of Bayesian Estimation
O a Applicatio of ayesia Estimatio KIYOHARU TANAKA School of Sciece ad Egieerig, Kiki Uiversity, Kowakae, Higashi-Osaka, JAPAN Email: ktaaka@ifokidaiacjp EVGENIY GRECHNIKOV Departmet of Mathematics, auma
More informationConfidence Interval for Standard Deviation of Normal Distribution with Known Coefficients of Variation
Cofidece Iterval for tadard Deviatio of Normal Distributio with Kow Coefficiets of Variatio uparat Niwitpog Departmet of Applied tatistics, Faculty of Applied ciece Kig Mogkut s Uiversity of Techology
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationChapter 8: Estimating with Confidence
Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig
More informationA proposed discrete distribution for the statistical modeling of
It. Statistical Ist.: Proc. 58th World Statistical Cogress, 0, Dubli (Sessio CPS047) p.5059 A proposed discrete distributio for the statistical modelig of Likert data Kidd, Marti Cetre for Statistical
More informationWHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? ABSTRACT
WHAT IS THE PROBABILITY FUNCTION FOR LARGE TSUNAMI WAVES? Harold G. Loomis Hoolulu, HI ABSTRACT Most coastal locatios have few if ay records of tsuami wave heights obtaied over various time periods. Still
More informationRAINFALL PREDICTION BY WAVELET DECOMPOSITION
RAIFALL PREDICTIO BY WAVELET DECOMPOSITIO A. W. JAYAWARDEA Departmet of Civil Egieerig, The Uiversit of Hog Kog, Hog Kog, Chia P. C. XU Academ of Mathematics ad Sstem Scieces, Chiese Academ of Scieces,
More informationSome Properties of the Exact and Score Methods for Binomial Proportion and Sample Size Calculation
Some Properties of the Exact ad Score Methods for Biomial Proportio ad Sample Size Calculatio K. KRISHNAMOORTHY AND JIE PENG Departmet of Mathematics, Uiversity of Louisiaa at Lafayette Lafayette, LA 70504-1010,
More informationDS 100: Principles and Techniques of Data Science Date: April 13, Discussion #10
DS 00: Priciples ad Techiques of Data Sciece Date: April 3, 208 Name: Hypothesis Testig Discussio #0. Defie these terms below as they relate to hypothesis testig. a) Data Geeratio Model: Solutio: A set
More informationThe standard deviation of the mean
Physics 6C Fall 20 The stadard deviatio of the mea These otes provide some clarificatio o the distictio betwee the stadard deviatio ad the stadard deviatio of the mea.. The sample mea ad variace Cosider
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationMATH 320: Probability and Statistics 9. Estimation and Testing of Parameters. Readings: Pruim, Chapter 4
MATH 30: Probability ad Statistics 9. Estimatio ad Testig of Parameters Estimatio ad Testig of Parameters We have bee dealig situatios i which we have full kowledge of the distributio of a radom variable.
More informationEstimation of a population proportion March 23,
1 Social Studies 201 Notes for March 23, 2005 Estimatio of a populatio proportio Sectio 8.5, p. 521. For the most part, we have dealt with meas ad stadard deviatios this semester. This sectio of the otes
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More informationSTA Learning Objectives. Population Proportions. Module 10 Comparing Two Proportions. Upon completing this module, you should be able to:
STA 2023 Module 10 Comparig Two Proportios Learig Objectives Upo completig this module, you should be able to: 1. Perform large-sample ifereces (hypothesis test ad cofidece itervals) to compare two populatio
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationProbability, Expectation Value and Uncertainty
Chapter 1 Probability, Expectatio Value ad Ucertaity We have see that the physically observable properties of a quatum system are represeted by Hermitea operators (also referred to as observables ) such
More informationResampling Methods. X (1/2), i.e., Pr (X i m) = 1/2. We order the data: X (1) X (2) X (n). Define the sample median: ( n.
Jauary 1, 2019 Resamplig Methods Motivatio We have so may estimators with the property θ θ d N 0, σ 2 We ca also write θ a N θ, σ 2 /, where a meas approximately distributed as Oce we have a cosistet estimator
More informationChapter 13, Part A Analysis of Variance and Experimental Design
Slides Prepared by JOHN S. LOUCKS St. Edward s Uiversity Slide 1 Chapter 13, Part A Aalysis of Variace ad Eperimetal Desig Itroductio to Aalysis of Variace Aalysis of Variace: Testig for the Equality of
More informationMixtures of Gaussians and the EM Algorithm
Mixtures of Gaussias ad the EM Algorithm CSE 6363 Machie Learig Vassilis Athitsos Computer Sciece ad Egieerig Departmet Uiversity of Texas at Arligto 1 Gaussias A popular way to estimate probability desity
More informationChapter 22. Comparing Two Proportions. Copyright 2010, 2007, 2004 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010, 2007, 2004 Pearso Educatio, Ic. Comparig Two Proportios Read the first two paragraphs of pg 504. Comparisos betwee two percetages are much more commo
More information1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable
More informationµ and π p i.e. Point Estimation x And, more generally, the population proportion is approximately equal to a sample proportion
Poit Estimatio Poit estimatio is the rather simplistic (ad obvious) process of usig the kow value of a sample statistic as a approximatio to the ukow value of a populatio parameter. So we could for example
More informationAnalysis of Experimental Measurements
Aalysis of Experimetal Measuremets Thik carefully about the process of makig a measuremet. A measuremet is a compariso betwee some ukow physical quatity ad a stadard of that physical quatity. As a example,
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationKinetics of Complex Reactions
Kietics of Complex Reactios by Flick Colema Departmet of Chemistry Wellesley College Wellesley MA 28 wcolema@wellesley.edu Copyright Flick Colema 996. All rights reserved. You are welcome to use this documet
More information1 Review of Probability & Statistics
1 Review of Probability & Statistics a. I a group of 000 people, it has bee reported that there are: 61 smokers 670 over 5 960 people who imbibe (drik alcohol) 86 smokers who imbibe 90 imbibers over 5
More informationx a x a Lecture 2 Series (See Chapter 1 in Boas)
Lecture Series (See Chapter i Boas) A basic ad very powerful (if pedestria, recall we are lazy AD smart) way to solve ay differetial (or itegral) equatio is via a series expasio of the correspodig solutio
More informationGUIDELINES ON REPRESENTATIVE SAMPLING
DRUGS WORKING GROUP VALIDATION OF THE GUIDELINES ON REPRESENTATIVE SAMPLING DOCUMENT TYPE : REF. CODE: ISSUE NO: ISSUE DATE: VALIDATION REPORT DWG-SGL-001 002 08 DECEMBER 2012 Ref code: DWG-SGL-001 Issue
More informationStochastic Simulation
Stochastic Simulatio 1 Itroductio Readig Assigmet: Read Chapter 1 of text. We shall itroduce may of the key issues to be discussed i this course via a couple of model problems. Model Problem 1 (Jackso
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationBinomial Distribution
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 1 2 3 4 5 6 7 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Overview Example: coi tossed three times Defiitio Formula Recall that a r.v. is discrete if there are either a fiite umber of possible
More informationBasis for simulation techniques
Basis for simulatio techiques M. Veeraraghava, March 7, 004 Estimatio is based o a collectio of experimetal outcomes, x, x,, x, where each experimetal outcome is a value of a radom variable. x i. Defiitios
More informationCHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION
CHAPTER 4 BIVARIATE DISTRIBUTION EXTENSION 4. Itroductio Numerous bivariate discrete distributios have bee defied ad studied (see Mardia, 97 ad Kocherlakota ad Kocherlakota, 99) based o various methods
More informationLecture 1 Probability and Statistics
Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark
More informationGoodness-Of-Fit For The Generalized Exponential Distribution. Abstract
Goodess-Of-Fit For The Geeralized Expoetial Distributio By Amal S. Hassa stitute of Statistical Studies & Research Cairo Uiversity Abstract Recetly a ew distributio called geeralized expoetial or expoetiated
More informationSNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
More informationTopic 9: Sampling Distributions of Estimators
Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationQuadratic Functions. Before we start looking at polynomials, we should know some common terminology.
Quadratic Fuctios I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationRoberto s Notes on Series Chapter 2: Convergence tests Section 7. Alternating series
Roberto s Notes o Series Chapter 2: Covergece tests Sectio 7 Alteratig series What you eed to kow already: All basic covergece tests for evetually positive series. What you ca lear here: A test for series
More informationSimilarity Solutions to Unsteady Pseudoplastic. Flow Near a Moving Wall
Iteratioal Mathematical Forum, Vol. 9, 04, o. 3, 465-475 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/imf.04.48 Similarity Solutios to Usteady Pseudoplastic Flow Near a Movig Wall W. Robi Egieerig
More informationChapter 6. Sampling and Estimation
Samplig ad Estimatio - 34 Chapter 6. Samplig ad Estimatio 6.. Itroductio Frequetly the egieer is uable to completely characterize the etire populatio. She/he must be satisfied with examiig some subset
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 010 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a ukow mea µ = E(X) of a distributio by
More information3 Resampling Methods: The Jackknife
3 Resamplig Methods: The Jackkife 3.1 Itroductio I this sectio, much of the cotet is a summary of material from Efro ad Tibshirai (1993) ad Maly (2007). Here are several useful referece texts o resamplig
More informationModule 1 Fundamentals in statistics
Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly
More informationApproximate Confidence Interval for the Reciprocal of a Normal Mean with a Known Coefficient of Variation
Metodološki zvezki, Vol. 13, No., 016, 117-130 Approximate Cofidece Iterval for the Reciprocal of a Normal Mea with a Kow Coefficiet of Variatio Wararit Paichkitkosolkul 1 Abstract A approximate cofidece
More informationAAEC/ECON 5126 FINAL EXAM: SOLUTIONS
AAEC/ECON 5126 FINAL EXAM: SOLUTIONS SPRING 2015 / INSTRUCTOR: KLAUS MOELTNER This exam is ope-book, ope-otes, but please work strictly o your ow. Please make sure your ame is o every sheet you re hadig
More informationChapter 23: Inferences About Means
Chapter 23: Ifereces About Meas Eough Proportios! We ve spet the last two uits workig with proportios (or qualitative variables, at least) ow it s time to tur our attetios to quatitative variables. For
More informationChapter 9: Numerical Differentiation
178 Chapter 9: Numerical Differetiatio Numerical Differetiatio Formulatio of equatios for physical problems ofte ivolve derivatives (rate-of-chage quatities, such as velocity ad acceleratio). Numerical
More informationSequences. Notation. Convergence of a Sequence
Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it
More informationChapter 11 Output Analysis for a Single Model. Banks, Carson, Nelson & Nicol Discrete-Event System Simulation
Chapter Output Aalysis for a Sigle Model Baks, Carso, Nelso & Nicol Discrete-Evet System Simulatio Error Estimatio If {,, } are ot statistically idepedet, the S / is a biased estimator of the true variace.
More informationNCSS Statistical Software. Tolerance Intervals
Chapter 585 Itroductio This procedure calculates oe-, ad two-, sided tolerace itervals based o either a distributio-free (oparametric) method or a method based o a ormality assumptio (parametric). A two-sided
More information3/8/2016. Contents in latter part PATTERN RECOGNITION AND MACHINE LEARNING. Dynamical Systems. Dynamical Systems. Linear Dynamical Systems
Cotets i latter part PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 13: SEQUENTIAL DATA Liear Dyamical Systems What is differet from HMM? Kalma filter Its stregth ad limitatio Particle Filter Its simple
More informationStatistical Fundamentals and Control Charts
Statistical Fudametals ad Cotrol Charts 1. Statistical Process Cotrol Basics Chace causes of variatio uavoidable causes of variatios Assigable causes of variatio large variatios related to machies, materials,
More informationAP Statistics Review Ch. 8
AP Statistics Review Ch. 8 Name 1. Each figure below displays the samplig distributio of a statistic used to estimate a parameter. The true value of the populatio parameter is marked o each samplig distributio.
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationBIOSTATISTICS. Lecture 5 Interval Estimations for Mean and Proportion. dr. Petr Nazarov
Microarray Ceter BIOSTATISTICS Lecture 5 Iterval Estimatios for Mea ad Proportio dr. Petr Nazarov 15-03-013 petr.azarov@crp-sate.lu Lecture 5. Iterval estimatio for mea ad proportio OUTLINE Iterval estimatios
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationMOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND.
XI-1 (1074) MOST PEOPLE WOULD RATHER LIVE WITH A PROBLEM THEY CAN'T SOLVE, THAN ACCEPT A SOLUTION THEY CAN'T UNDERSTAND. R. E. D. WOOLSEY AND H. S. SWANSON XI-2 (1075) STATISTICAL DECISION MAKING Advaced
More informationEstimation for Complete Data
Estimatio for Complete Data complete data: there is o loss of iformatio durig study. complete idividual complete data= grouped data A complete idividual data is the oe i which the complete iformatio of
More informationChapter 22. Comparing Two Proportions. Copyright 2010 Pearson Education, Inc.
Chapter 22 Comparig Two Proportios Copyright 2010 Pearso Educatio, Ic. Comparig Two Proportios Comparisos betwee two percetages are much more commo tha questios about isolated percetages. Ad they are more
More informationSAMPLING LIPSCHITZ CONTINUOUS DENSITIES. 1. Introduction
SAMPLING LIPSCHITZ CONTINUOUS DENSITIES OLIVIER BINETTE Abstract. A simple ad efficiet algorithm for geeratig radom variates from the class of Lipschitz cotiuous desities is described. A MatLab implemetatio
More informationBasics of Probability Theory (for Theory of Computation courses)
Basics of Probability Theory (for Theory of Computatio courses) Oded Goldreich Departmet of Computer Sciece Weizma Istitute of Sciece Rehovot, Israel. oded.goldreich@weizma.ac.il November 24, 2008 Preface.
More informationThe axial dispersion model for tubular reactors at steady state can be described by the following equations: dc dz R n cn = 0 (1) (2) 1 d 2 c.
5.4 Applicatio of Perturbatio Methods to the Dispersio Model for Tubular Reactors The axial dispersio model for tubular reactors at steady state ca be described by the followig equatios: d c Pe dz z =
More informationRelations between the continuous and the discrete Lotka power function
Relatios betwee the cotiuous ad the discrete Lotka power fuctio by L. Egghe Limburgs Uiversitair Cetrum (LUC), Uiversitaire Campus, B-3590 Diepebeek, Belgium ad Uiversiteit Atwerpe (UA), Campus Drie Eike,
More informationCSE 527, Additional notes on MLE & EM
CSE 57 Lecture Notes: MLE & EM CSE 57, Additioal otes o MLE & EM Based o earlier otes by C. Grat & M. Narasimha Itroductio Last lecture we bega a examiatio of model based clusterig. This lecture will be
More informationInterval Estimation (Confidence Interval = C.I.): An interval estimate of some population parameter is an interval of the form (, ),
Cofidece Iterval Estimatio Problems Suppose we have a populatio with some ukow parameter(s). Example: Normal(,) ad are parameters. We eed to draw coclusios (make ifereces) about the ukow parameters. We
More informationBIOS 4110: Introduction to Biostatistics. Breheny. Lab #9
BIOS 4110: Itroductio to Biostatistics Brehey Lab #9 The Cetral Limit Theorem is very importat i the realm of statistics, ad today's lab will explore the applicatio of it i both categorical ad cotiuous
More informationTHE KALMAN FILTER RAUL ROJAS
THE KALMAN FILTER RAUL ROJAS Abstract. This paper provides a getle itroductio to the Kalma filter, a umerical method that ca be used for sesor fusio or for calculatio of trajectories. First, we cosider
More informationPower and Type II Error
Statistical Methods I (EXST 7005) Page 57 Power ad Type II Error Sice we do't actually kow the value of the true mea (or we would't be hypothesizig somethig else), we caot kow i practice the type II error
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry Daa-Picard Departmet of Applied Mathematics Jerusalem College of Techology
More informationMath 10A final exam, December 16, 2016
Please put away all books, calculators, cell phoes ad other devices. You may cosult a sigle two-sided sheet of otes. Please write carefully ad clearly, USING WORDS (ot just symbols). Remember that the
More informationG. R. Pasha Department of Statistics Bahauddin Zakariya University Multan, Pakistan
Deviatio of the Variaces of Classical Estimators ad Negative Iteger Momet Estimator from Miimum Variace Boud with Referece to Maxwell Distributio G. R. Pasha Departmet of Statistics Bahauddi Zakariya Uiversity
More informationComparing your lab results with the others by one-way ANOVA
Comparig your lab results with the others by oe-way ANOVA You may have developed a ew test method ad i your method validatio process you would like to check the method s ruggedess by coductig a simple
More information