Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading
|
|
- Bertina Singleton
- 6 years ago
- Views:
Transcription
1 Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding
2 Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method Klmn Filter 4d-Vritionl Method Applictions of dt ssimiltion in erth system science
3 Motivtion
4 DARC Wht is dt ssimiltion? Dt ssimiltion is the technique whereby observtionl dt re combined with output from numericl model to produce n optiml estimte of the evolving stte of the system
5 Why We Need Dt Assimiltion rnge of observtions rnge of techniques different errors dt gps quntities not mesured quntities linked
6 DARC
7 Some Uses of Dt Assimiltion Opertionl wether nd ocen forecsting Sesonl wether forecsting Lnd-surfce process Globl climte dtsets Plnning stellite mesurements Evlution of models nd observtions DARC
8 Preliminry Concepts
9 Wht We Wnt o Know x t tmos stte vector s t c surfce fluxes model prmeters X t x t, s t, c
10 Wht We Also Wnt o Know Errors in models Errors in observtions Wht observtions to mke
11 DAA ASSIMILAION SYSEM Dt Cche A Error Sttistics model observtions O DAS A Numericl Model F B
12 he Dt Assimiltion Process observtions forecsts compre reject djust estimtes of stte & prmeters
13 X observtion model trjectory t
14 DARC Dt Assimiltion: n nlogy Driving with your eyes closed: open eyes every 0 seconds nd correct trjectory
15 Bsic Concept of Dt Assimiltion Informtion is ccumulted in time into the model stte nd propgted to ll vribles
16 DARC Wht re the benefits of dt ssimiltion? Qulity control Combintion of dt Errors in dt nd in model Filling in dt poor regions Designing observtionl systems Mintining consistency Estimting unobserved quntities
17 DARC Methods of Dt Assimiltion Optiml interpoltion or pprox to it 3D vritionl method 3DVr 4D vritionl method 4DVr Klmn filter with pproximtions
18 ypes of Dt Assimiltion Sequentil Non-sequentil Intermittent Continous
19 Sequentil Intermittent Assimiltion obs obs obs obs obs obs nlysis model nlysis model nlysis
20 Sequentil Continuous Assimiltion obs obs obs obs obs
21 Non-sequentil Intermittent Assimiltion obs obs obs obs obs obs nlysis + model nlysis + model nlysis + model
22 Non-sequentil Continuous Assimiltion obs obs obs obs obs obs nlysis + model
23 Sttisticl Approch to Dt Assimiltion
24 DARC
25 Dt Assimiltion Mde Simple sclr cse
26 Lest Squres Method Minimum Vrince re two mesurements the 0, 0 > < > < > < > >< < + + ε ε σ ε σ ε ε ε ε ε t t
27 Estimte of the observtio ns : t s liner combintio n + he nlysis should be unbised : < > t +
28 Lest Squres Method Continued subject to the constrint : men squred error by minimizing its Estimte + + > + < > + >< < t t t σ σ ε ε σ
29 Lest Squres Method Continued σ σ σ + σ σ σ + σ σ σ + he precision of the nlysis is the sum of the precisions of the mesurements he nlysis therefore hs higher precision thn ny single mesurement if the sttistics re correct
30 Vritionl Approch 0 for which the vlue of is + J J σ σ J
31 Mximum Likelihood Estimte Obtin or ssume probbility distributions for the errors he best estimte of the stte is chosen to hve the gretest probbility, or mximum likelihood If errors normlly distributed,unbised nd uncorrelted, then sttes estimted by minimum vrince nd mximum likelihood re the sme
32 Mximum Likelihood Approch Bysin Derivtion A s s u m e w e hve lredy mde n observtio n the bckground forecst in dt ssimiltion hen the probbilit y distributi on of the truth for Gussin error sttistics is :, p exp σ the prior pdf
33 Mximum Likelihood Continued Byes' s formul for the posterior p d f given the observtio n i s : p p p p exp σ exp σ since p Mximize We get the i s the normlisin g fctor independen t sme posterior nswer p d f minimizing the cost Equivlenc e holds for multi - dimension l cse s o r ln to estimte o f the function for truth Gussin sttistics
34 Simple Sequentil Assimiltion error vrince i s : nlysis the nd, by : given i s weight optiml he the " innovtion " i s where Let b o b b b o b o b o b W W W W σ σ σ σ σ + +
35 Comments he nlysis is obtined by dding first guess to the innovtion Optiml weight is bckground error vrince multiplied by inverse of totl vrince Precision of nlysis is sum of precisions of bckground nd observtion Error vrince of nlysis is error vrince of bckground reduced by - optiml weight
36 Simple Assimiltion Cycle Observtion used once nd then discrded Forecst phse to updte Anlysis phse to updte Obtin bckground s b t i+ M [ t i ] nd nd Obtin vrince of bckground s b b b i b i+ σ b σ σ t σ t lterntively σ t i+ σ t i
37 Simple Klmn Filter,,,,,,, M covrince i s : e r r o r bckground Forecst where M M hen bised! not model, ] [ before step s Anlysis Q M M M Q t M t i i b i b m i m i t i i t b i b m m i t i t + > < + + > < σ ε σ ε ε ε ε ε ε
38 Multivrite Dt Assimiltion
39 Multivrite Cse x n x x t x stte vector y m y y t n vector observtio y
40 Stte Vectors x stte vector column mtrix x t true stte x b bckground stte x nlysis, estimte of x t
41 Ingredients of Good Estimte of the Stte Vector nlysis Strt from good first guess forecst from previous good nlysis Allow for errors in observtions nd first guess give most weight to dt you trust Anlysis should be smooth Anlysis should respect known physicl lws
42 Some Useful Mtrix Properties rnspose of product : A B B A Inverse of product : A B B A Inverse of trnspose : A A Positive definitene s s for symmetrix mtrix A : x, the sclr x A x > 0, unless x 0 this property is conserved through inversion
43 Observtions Observtions re gthered into n y observtion vector, clled the observtion vector Usully fewer observtions thn vribles in the model; they re irregulrly spced; nd my be of different kind to those in the model Introduce n observtion opertor to mp from model stte spce to observtion spce x H x
44 Errors
45 Vrince becomes Covrince Mtrix Errors in x i re often correlted sptil structure in flow dynmicl or chemicl reltionships Vrince for sclr cse becomes Covrince Mtrix for vector cse COV Digonl elements re the vrinces of x i Off-digonl elements re covrinces between x i nd x j Observtion of x i ffects estimte of x j
46 he Error Covrince Mtrix > < > < > < > < > < > < > < > < > < > < n n n n n n n n e e e e e e e e e e e e e e e e e e e e e e e e εε ε ε P i i e i e σ > <
47 Bckground Errors hey re the estimtion errors of the bckground stte: ε b x b x t B verge bis covrince < < ε b > ε < ε > ε < ε > b b >
48 Observtion Errors hey contin errors in the observtion process instrumentl error, errors in the design of H, nd representtiveness errors, ie discretizton errors tht prevent x t from being perfect representtion of the true stte ε y H x < ε > o t o R < ε < ε > ε < ε > o o o o >
49 Control Vribles We my not be ble to solve the nlysis problem for ll components of the model stte eg cloud-relted vribles, or need to reduce resolution he work spce is then not the model spce but the sub-spce in which we correct x b, clled control-vrible spce x x b + δx
50 Innovtions nd Residuls Key to dt ssimiltion is the use of differences between observtions nd the stte vector of the system We cll y H x b the innovtion We cll y H x the nlysis residul Give importnt informtion
51 Anlysis Errors hey re the estimtion errors of the nlysis stte tht we wnt to minimize ε x x t Covrince mtrix A
52 Using the Error Covrince Mtrix Recll tht n error covrince mtrix x for the error in hs the form: C C < εε > y H x H If where is mtrix, then the error covrince for y C HCH y is given by:
53 BLUE Estimtor he BLUE estimtor is given by: he nlysis error covrince mtrix is: Note tht: b b + + R H B H B H K x y K x x H K H B I A + + R H H R H B R H B H B H
54 Sttisticl Interpoltion with Lest Squres Estimtion Clled Best Liner Unbised Estimtor BLUE Simplified versions of this lgorithm yield the most common lgorithms used tody in meteorology nd ocenogrphy
55 Assumptions Used in BLUE Linerized observtion opertor: H x H x b H x x b B R nd re positive definite Errors re unbised: < x x >< y H x > b t t Errors re uncorrelted: < x x y H x > b t t Liner nlysis: corrections to bckground depend linerly on bckground obs Optiml nlysis: minimum vrince estimte 0 0
56 Optiml Interpoltion observtion observtion opertor x x b + K y H x b nlysis linerity H H bckground forecst mtrix inverse K B H H B H + R limited re
57 y H x b t obs point dt void x b
58 END
Predict Global Earth Temperature using Linier Regression
Predict Globl Erth Temperture using Linier Regression Edwin Swndi Sijbt (23516012) Progrm Studi Mgister Informtik Sekolh Teknik Elektro dn Informtik ITB Jl. Gnesh 10 Bndung 40132, Indonesi 23516012@std.stei.itb.c.id
More informationMultivariate problems and matrix algebra
University of Ferrr Stefno Bonnini Multivrite problems nd mtrix lgebr Multivrite problems Multivrite sttisticl nlysis dels with dt contining observtions on two or more chrcteristics (vribles) ech mesured
More informationCMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
More informationA Matrix Algebra Primer
A Mtrix Algebr Primer Mtrices, Vectors nd Sclr Multipliction he mtrix, D, represents dt orgnized into rows nd columns where the rows represent one vrible, e.g. time, nd the columns represent second vrible,
More informationLecture 21: Order statistics
Lecture : Order sttistics Suppose we hve N mesurements of sclr, x i =, N Tke ll mesurements nd sort them into scending order x x x 3 x N Define the mesured running integrl S N (x) = 0 for x < x = i/n for
More information1B40 Practical Skills
B40 Prcticl Skills Comining uncertinties from severl quntities error propgtion We usully encounter situtions where the result of n experiment is given in terms of two (or more) quntities. We then need
More informationSatellite Retrieval Data Assimilation
tellite etrievl Dt Assimiltion odgers C. D. Inverse Methods for Atmospheric ounding: Theor nd Prctice World cientific Pu. Co. Hckensck N.J. 2000 Chpter 3 nd Chpter 8 Dve uhl Artist depiction of NAA terr
More informationStudent Activity 3: Single Factor ANOVA
MATH 40 Student Activity 3: Single Fctor ANOVA Some Bsic Concepts In designed experiment, two or more tretments, or combintions of tretments, is pplied to experimentl units The number of tretments, whether
More informationTests for the Ratio of Two Poisson Rates
Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson
More informationNon-Linear & Logistic Regression
Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find
More informationSolution for Assignment 1 : Intro to Probability and Statistics, PAC learning
Solution for Assignment 1 : Intro to Probbility nd Sttistics, PAC lerning 10-701/15-781: Mchine Lerning (Fll 004) Due: Sept. 30th 004, Thursdy, Strt of clss Question 1. Bsic Probbility ( 18 pts) 1.1 (
More information1 Linear Least Squares
Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving
More informationIncorporating Ensemble Covariance in the Gridpoint Statistical Interpolation Variational Minimization: A Mathematical Framework
2990 M O N T H L Y W E A T H E R R E V I E W VOLUME 38 Incorporting Ensemble Covrince in the Gridpoint Sttisticl Interpoltion Vritionl Minimiztion: A Mthemticl Frmework XUGUANG WANG School of Meteorology,
More informationContinuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More informationDiscrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17
EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More informationDiscrete Mathematics and Probability Theory Summer 2014 James Cook Note 17
CS 70 Discrete Mthemtics nd Proility Theory Summer 2014 Jmes Cook Note 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion, y tking
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationReinforcement learning II
CS 1675 Introduction to Mchine Lerning Lecture 26 Reinforcement lerning II Milos Huskrecht milos@cs.pitt.edu 5329 Sennott Squre Reinforcement lerning Bsics: Input x Lerner Output Reinforcement r Critic
More informationEstimation of Binomial Distribution in the Light of Future Data
British Journl of Mthemtics & Computer Science 102: 1-7, 2015, Article no.bjmcs.19191 ISSN: 2231-0851 SCIENCEDOMAIN interntionl www.sciencedomin.org Estimtion of Binomil Distribution in the Light of Future
More informationLECTURE NOTE #12 PROF. ALAN YUILLE
LECTURE NOTE #12 PROF. ALAN YUILLE 1. Clustering, K-mens, nd EM Tsk: set of unlbeled dt D = {x 1,..., x n } Decompose into clsses w 1,..., w M where M is unknown. Lern clss models p(x w)) Discovery of
More informationThe steps of the hypothesis test
ttisticl Methods I (EXT 7005) Pge 78 Mosquito species Time of dy A B C Mid morning 0.0088 5.4900 5.5000 Mid Afternoon.3400 0.0300 0.8700 Dusk 0.600 5.400 3.000 The Chi squre test sttistic is the sum of
More informationCBE 291b - Computation And Optimization For Engineers
The University of Western Ontrio Fculty of Engineering Science Deprtment of Chemicl nd Biochemicl Engineering CBE 9b - Computtion And Optimiztion For Engineers Mtlb Project Introduction Prof. A. Jutn Jn
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationObservability of flow dependent structure functions and their use in data assimilation
Oservility of flow dependent structure functions nd their use in dt ssimiltion Pierre Guthier Bsed on work done in collortion with Cristin Lupu (MSc 2006, PhD thesis 2010, UQAM) nd Stéphne Lroche (Env.
More informationGNSS Data Assimilation and GNSS Tomography for Global and Regional Models
GNSS Dt Assimiltion nd GNSS Tomogrphy for Globl nd Regionl Models M. Bender, K. Stephn, A. Rhodin, R. Potthst Germn Wether Service (DWD) Interntionl Symposium on Dt Assimiltion, Munich 214 27. Februry
More informationReview of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
More informationMatrix Algebra. Matrix Addition, Scalar Multiplication and Transposition. Linear Algebra I 24
Mtrix lger Mtrix ddition, Sclr Multipliction nd rnsposition Mtrix lger Section.. Mtrix ddition, Sclr Multipliction nd rnsposition rectngulr rry of numers is clled mtrix ( the plurl is mtrices ) nd the
More informationDriving Cycle Construction of City Road for Hybrid Bus Based on Markov Process Deng Pan1, a, Fengchun Sun1,b*, Hongwen He1, c, Jiankun Peng1, d
Interntionl Industril Informtics nd Computer Engineering Conference (IIICEC 15) Driving Cycle Construction of City Rod for Hybrid Bus Bsed on Mrkov Process Deng Pn1,, Fengchun Sun1,b*, Hongwen He1, c,
More informationCS 188 Introduction to Artificial Intelligence Fall 2018 Note 7
CS 188 Introduction to Artificil Intelligence Fll 2018 Note 7 These lecture notes re hevily bsed on notes originlly written by Nikhil Shrm. Decision Networks In the third note, we lerned bout gme trees
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Statistical Physics I Spring Term Solutions to Problem Set #1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Deprtment 8.044 Sttisticl Physics I Spring Term 03 Problem : Doping Semiconductor Solutions to Problem Set # ) Mentlly integrte the function p(x) given in
More informationBases for Vector Spaces
Bses for Vector Spces 2-26-25 A set is independent if, roughly speking, there is no redundncy in the set: You cn t uild ny vector in the set s liner comintion of the others A set spns if you cn uild everything
More informationTutorial 4. b a. h(f) = a b a ln 1. b a dx = ln(b a) nats = log(b a) bits. = ln λ + 1 nats. = log e λ bits. = ln 1 2 ln λ + 1. nats. = ln 2e. bits.
Tutoril 4 Exercises on Differentil Entropy. Evlute the differentil entropy h(x) f ln f for the following: () The uniform distribution, f(x) b. (b) The exponentil density, f(x) λe λx, x 0. (c) The Lplce
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationMaximum Likelihood Estimation for Allele Frequencies. Biostatistics 666
Mximum Likelihood Estimtion for Allele Frequencies Biosttistics 666 Previous Series of Lectures: Introduction to Colescent Models Comuttionlly efficient frmework Alterntive to forwrd simultions Amenble
More informationChapter 2 Fundamental Concepts
Chpter 2 Fundmentl Concepts This chpter describes the fundmentl concepts in the theory of time series models In prticulr we introduce the concepts of stochstic process, men nd covrince function, sttionry
More informationRobust Predictions in Games with Incomplete Information
Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn nd Stephen Morris April 2011 Introduction in gmes of incomplete informtion, privte informtion represents informtion bout: pyo environment
More informationMIXED MODELS (Sections ) I) In the unrestricted model, interactions are treated as in the random effects model:
1 2 MIXED MODELS (Sections 17.7 17.8) Exmple: Suppose tht in the fiber breking strength exmple, the four mchines used were the only ones of interest, but the interest ws over wide rnge of opertors, nd
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More informationMath 270A: Numerical Linear Algebra
Mth 70A: Numericl Liner Algebr Instructor: Michel Holst Fll Qurter 014 Homework Assignment #3 Due Give to TA t lest few dys before finl if you wnt feedbck. Exercise 3.1. (The Bsic Liner Method for Liner
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationRobust Predictions in Games with Incomplete Information
Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn nd Stephen Morris Collegio Crlo Alberto, Turin 16 Mrch 2011 Introduction Gme Theoretic Predictions re very sensitive to "higher order
More informationChapter 3 Polynomials
Dr M DRAIEF As described in the introduction of Chpter 1, pplictions of solving liner equtions rise in number of different settings In prticulr, we will in this chpter focus on the problem of modelling
More information19 Optimal behavior: Game theory
Intro. to Artificil Intelligence: Dle Schuurmns, Relu Ptrscu 1 19 Optiml behvior: Gme theory Adversril stte dynmics hve to ccount for worst cse Compute policy π : S A tht mximizes minimum rewrd Let S (,
More informationOrdinary differential equations
Ordinry differentil equtions Introduction to Synthetic Biology E Nvrro A Montgud P Fernndez de Cordob JF Urchueguí Overview Introduction-Modelling Bsic concepts to understnd n ODE. Description nd properties
More informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationTHE UNIVERSITY OF CHICAGO Booth School of Business Business 41914, Spring Quarter 2015, Mr. Ruey S. Tsay
THE UNIVERSITY OF CHICAGO Booth School of Business Business 494, Spring Qurter 05, Mr. Ruey S. Tsy Reference: Chpter of the textbook. Introduction Lecture 3: Vector Autoregressive (VAR) Models Vector utoregressive
More informationA NOTE ON ESTIMATION OF THE GLOBAL INTENSITY OF A CYCLIC POISSON PROCESS IN THE PRESENCE OF LINEAR TREND
A NOTE ON ESTIMATION OF THE GLOBAL INTENSITY OF A CYCLIC POISSON PROCESS IN THE PRESENCE OF LINEAR TREND I WAYAN MANGKU Deprtment of Mthemtics, Fculty of Mthemtics nd Nturl Sciences, Bogor Agriculturl
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics
SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationDetection and Estimation Theory
ESE 54 Detection nd Estimtion heory Joseph A. O Sullivn Smuel C. Schs Professor Electronic Systems nd Signls Reserch Lbortory Electricl nd Systems Engineering Wshington University Urbuer Hll 34-935-473
More informationSUMMER KNOWHOW STUDY AND LEARNING CENTRE
SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18
More informationRobust Predictions in Games with Incomplete Information
Robust Predictions in Gmes with Incomplete Informtion Dirk Bergemnn nd Stephen Morris Northwestern University Mrch, 30th, 2011 Introduction gme theoretic predictions re very sensitive to "higher order
More informationNOTES AND CORRESPONDENCE. Ensemble Square Root Filters*
JULY 2003 NOES AND CORRESPONDENCE 1485 NOES AND CORRESPONDENCE Ensemble Squre Root Filters* MICHAEL K. IPPE Interntionl Reserch Institute or Climte Prediction, Plisdes, New Yor JEFFREY L. ANDERSON GFDL,
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationDecision Networks. CS 188: Artificial Intelligence Fall Example: Decision Networks. Decision Networks. Decisions as Outcome Trees
CS 188: Artificil Intelligence Fll 2011 Decision Networks ME: choose the ction which mximizes the expected utility given the evidence mbrell Lecture 17: Decision Digrms 10/27/2011 Cn directly opertionlize
More informationProblem. Statement. variable Y. Method: Step 1: Step 2: y d dy. Find F ( Step 3: Find f = Y. Solution: Assume
Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom r vrible. Define new rndom vrible Y = g. Find the pdf of Y. Method: Step : Step : Step 3: Plot Y = g( ). Find F ( y) by mpping
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More informationCHM Physical Chemistry I Chapter 1 - Supplementary Material
CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion
More informationLecture 3 Gaussian Probability Distribution
Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationMethod: Step 1: Step 2: Find f. Step 3: = Y dy. Solution: 0, ( ) 0, y. Assume
Functions of Rndom Vrible Problem Sttement We know the pdf ( or cdf ) of rndom vrible. Define new rndom vrible Y g( ) ). Find the pdf of Y. Method: Step : Step : Step 3: Plot Y g( ). Find F ( ) b mpping
More informationNumerical Methods I Orthogonal Polynomials
Numericl Methods I Orthogonl Polynomils Aleksndr Donev Cournt Institute, NYU 1 donev@cournt.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fll 2014 Nov 6th, 2014 A. Donev (Cournt Institute) Lecture IX
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationPre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs
Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationX Z Y Table 1: Possibles values for Y = XZ. 1, p
ECE 534: Elements of Informtion Theory, Fll 00 Homework 7 Solutions ll by Kenneth Plcio Bus October 4, 00. Problem 7.3. Binry multiplier chnnel () Consider the chnnel Y = XZ, where X nd Z re independent
More informationLecture 4: Piecewise Cubic Interpolation
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 4: Piecewise Cubic Interpoltion Compiled 5 September In this lecture we consider piecewise cubic interpoltion
More informationHere we study square linear systems and properties of their coefficient matrices as they relate to the solution set of the linear system.
Section 24 Nonsingulr Liner Systems Here we study squre liner systems nd properties of their coefficient mtrices s they relte to the solution set of the liner system Let A be n n Then we know from previous
More informationUNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction
Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from
More information13: Diffusion in 2 Energy Groups
3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationElementary Linear Algebra
Elementry Liner Algebr Anton & Rorres, 1 th Edition Lecture Set 5 Chpter 4: Prt II Generl Vector Spces 163 คณ ตศาสตร ว ศวกรรม 3 สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา 1/2555 163 คณตศาสตรวศวกรรม 3 สาขาวชาวศวกรรมคอมพวเตอร
More informationSome multivariate methods
/7/ Outline Some multivrite methods VERIE CRDENS, PH.D. SSOCIE DJUNC PROFESSOR DEPREN OF RDIOOGY ND BIOEDIC IGING Useful liner lgebr Principl Components nlysis (PC) Independent Components nlysis (IC) Joint
More informationNumerical Integration
Chpter 1 Numericl Integrtion Numericl differentition methods compute pproximtions to the derivtive of function from known vlues of the function. Numericl integrtion uses the sme informtion to compute numericl
More informationStein-Rule Estimation and Generalized Shrinkage Methods for Forecasting Using Many Predictors
Stein-Rule Estimtion nd Generlized Shrinkge Methods for Forecsting Using Mny Predictors Eric Hillebrnd CREATES Arhus University, Denmrk ehillebrnd@cretes.u.dk Te-Hwy Lee University of Cliforni, Riverside
More informationBest Approximation in the 2-norm
Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
More informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationJack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah
1. Born-Oppenheimer pprox.- energy surfces 2. Men-field (Hrtree-Fock) theory- orbitls 3. Pros nd cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usully does HF-how? 6. Bsis sets nd nottions 7. MPn,
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationDATA ASSIMILATION IN A FLOOD MODELLING SYSTEM USING THE ENSEMBLE KALMAN FILTER
CMWRXVI DATA ASSIMILATION IN A FLOOD MODELLING SYSTEM USING THE ENSEMBLE KALMAN FILTER HENRIK MADSEN, JOHAN HARTNACK, JACOB V.T. SØRENSEN DHI Wter & Environment, Agern Allé 5, DK-2970 Hørsholm, Denmr ABSTRACT
More informationINTRODUCTION TO LINEAR ALGEBRA
ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR
More informationSufficient condition on noise correlations for scalable quantum computing
Sufficient condition on noise correltions for sclble quntum computing John Presill, 2 Februry 202 Is quntum computing sclble? The ccurcy threshold theorem for quntum computtion estblishes tht sclbility
More informationMeasuring Electron Work Function in Metal
n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd
More informationClassical Mechanics. From Molecular to Con/nuum Physics I WS 11/12 Emiliano Ippoli/ October, 2011
Clssicl Mechnics From Moleculr to Con/nuum Physics I WS 11/12 Emilino Ippoli/ October, 2011 Wednesdy, October 12, 2011 Review Mthemtics... Physics Bsic thermodynmics Temperture, idel gs, kinetic gs theory,
More information1 Online Learning and Regret Minimization
2.997 Decision-Mking in Lrge-Scle Systems My 10 MIT, Spring 2004 Hndout #29 Lecture Note 24 1 Online Lerning nd Regret Minimiztion In this lecture, we consider the problem of sequentil decision mking in
More informationProbability Distributions for Gradient Directions in Uncertain 3D Scalar Fields
Technicl Report 7.8. Technische Universität München Probbility Distributions for Grdient Directions in Uncertin 3D Sclr Fields Tobis Pfffelmoser, Mihel Mihi, nd Rüdiger Westermnn Computer Grphics nd Visuliztion
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be rel-vlues nd smooth The pproximtion of n integrl by numericl
More informationECON 331 Lecture Notes: Ch 4 and Ch 5
Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve
More informationIntegral equations, eigenvalue, function interpolation
Integrl equtions, eigenvlue, function interpoltion Mrcin Chrząszcz mchrzsz@cernch Monte Crlo methods, 26 My, 2016 1 / Mrcin Chrząszcz (Universität Zürich) Integrl equtions, eigenvlue, function interpoltion
More informationLecture 20: Numerical Integration III
cs4: introduction to numericl nlysis /8/0 Lecture 0: Numericl Integrtion III Instructor: Professor Amos Ron Scribes: Mrk Cowlishw, Yunpeng Li, Nthnel Fillmore For the lst few lectures we hve discussed
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationTesting categorized bivariate normality with two-stage. polychoric correlation estimates
Testing ctegorized bivrite normlity with two-stge polychoric correltion estimtes Albert Mydeu-Olivres Dept. of Psychology University of Brcelon Address correspondence to: Albert Mydeu-Olivres. Fculty of
More information