Dirichlet forms methods for error calculus and sensitivity analysis


 Darleen Lloyd
 2 years ago
 Views:
Transcription
1 Dirichlet forms methods for error calculus and sensitivity analysis Nicolas BOULEAU, Osaka university, november 2004 These lectures propose tools for studying sensitivity of models to scalar or functional parameters. A Dirichlet forms based language is developed in order to manage the propagation of the variances and the biases of errors through mathematical models. Examples will be given in physics, numerical analysis and finance. In the two first lectures, the intuitive calculus of Gauss for the propagation of errors will be connected with the rigorous mathematical framework of error structures. Then the main features of error structures will be studied until infinite products in order to construct error structures on functional spaces and on the Monte Carlo space. The third lecture will be devoted to error calculus on the Wiener space with the classical OrnsteinUhlenbeck structure or generalized Mehlertype structures with applications to SDE and finance. In the fourth lecture will be tackled the question of identifying an error structure. The main role of the Fisher information will be exposed and also other methods based on asymptotic Hopftype theorems or based on Donsker invariance theorem. Reference : N. Bouleau, Error Calculus for Finance and Physics, De Gruyter 2003.
2 First lecture Propagation of errors : from Gauss to Dirichlet forms A) Propagation of errors  1 Historical outlook (or landscape)  the propagation calculus of Gauss  its coherence property, non coherence of other formulae  2 The propagation of errors  by non linear maps : variances and bias  bias by quadratic map  intuitive error calculus  3 Examples of propagation calculations  Gaussian variables  triangle  oscillographe  4 Examples of dynamical systems sensitivity analysis  Feigenbaum transition  piecewise linear system  Lorenz attractor B) Error structures  Languages with or without extension tool : the example of probability theory  Adding an extension tool to the language of Gauss : the idea  Definition of error structures  Examples  Comparison of approaches Second lecture Error structures and sensitivity analysis C) Properties of error structures  recalling the definition  Lipschitzian calculus  images  densities and Dloc  finite and infinite products  the gradient and the sharp  Integration by part formulae D) Application in simulation : error calculus on the Monte Carlo space  structures with or without border terms  Sensitivity analysis of a Markov chain E) Application in numerical analysis  sensitivity of an ODE to a functional coefficient  comments on finite elements methods
3 Third lecture New tools for finance F) Error structures on the Wiener space G) error structures on the Poisson space H) sensitivity analysis of an SDE, application to finance I) A non classical approach to finance Fourth lecture Links with statistics and empirical data J) Identifying an error structure  The link with the Fisher information and its stability  Natural error structures for some dynamical systems, extension of the "arbitrary functions method" K) Convergence in Dirichlet law and transition from finite to infinite dimension  Central limit theorem  Convergence of an erroneous random walk to the erroneous Brownian motion : extension of Donsker theorem and applications.
4 First lecture Propagation of errors : from Gauss to Dirichlet forms A) Propagation of errors  1/ Historical outlook (or landscape)  the propagation calculus of Gauss  its coherence property, non coherence of other formulae  2/ The propagation of errors  by non linear maps : variances and bias  bias by quadratic map  intuitive error calculus  3/ Examples of propagation calculations  simulation of Gaussian variables  triangle  4/ Examples of dynamical systems sensitivity analysis  Feigenbaum transition  piecewise linear system  Lorenz attractor B) Error structures  Languages with or without extension tool : the example of probability theory  Adding an extension tool to the language of Gauss : the idea  Definition of error structures  Examples  Comparison of approaches
5 Historical insights, error theory and least squares Legendre Nouvelles méthodes pour la détermination des orbites des planètes (1805) Gauss Theoria motus corporum coelestium (1809) Laplace Théorie analytique des probabilités (1811) Least squares method for chosing the best value Famous argument leading to the normal law for the errors Least squares method for solving linear systems
6
7
8
9 . Now, it is clear that if the errors are correlated, Gauss formula becomes σ 2 U = ij F F σ ij V i V j and in general σ ij depend on the values of the V i s, so that we obtain the general formula σ 2 U = ij F V i F V j σ ij (V 1, V 2,...)
10 2. The propagation of errors
11 Intuitive notion of error structure The preceding example shows that the quadratic error operator Ɣ naturally polarizes into a bilinear operator (as the covariance operator in probability theory), which is a firstorder differential operator. 1. We thus adopt the following temporary definition of an error structure: An error structure is a space equipped with an operator Ɣ acting upon real functions (,Ɣ) and satisfying the following properties: a) Symmetry Ɣ[F, G] = Ɣ[G, F]; b) Bilinearity c) Positivity [ Ɣ λ i F i, ] µ j G j = λ i µ j Ɣ [ F i, G j ]; i j ij Ɣ [ F ] = Ɣ [ F, F ] 0 d) Functional calculus on regular functions Ɣ[ (F 1,...,F p ), (G 1,...,G q )] = i (F 1,...,F p ) j (G 1,...,G q )Ɣ[F i, G j ]. i, j 2. In order to take in account the biases, we also have to introduce a bias operato a linear operator acting on regular functions through a second order functional calculus involving Ɣ : A[ (F 1,...,F p )] = i (F 1,...,F p )A[F i ] i + 1 ij 2 (F 1,...,F p )Ɣ[F i, F j ] ij
12 Example Let us consider the usual way of simulating two normal random variables { X = R cos 2πV Y = R sin 2πV with R = 2 log U in other words U = e R2 2. Then (X, Y ) is a reduced normal pair when U and V are independent uniformly distributed on [0, 1]. Let us suppose that the variances of the errors on U and V are such that i) Γ[V,V]=1 ii) Γ[R,R]=4π 2 R 2 iii) G[V,R]=0. hypothesis i) means that the error on V does nt depend on V, hypothesis ii) means that the proportional error on R is constant, it is equivalent, by the functional calculus, to the hypothesis G[U,U]=16π 2 U 2 log 2 U, and hypothesis iii) means that the errors on Vand Rare uncorrelated. Then supposing G satisfies the functional calculus, we get G[X,X] =cos 2 2πV.G[R,R]+R 2 4π 2 sin 2 2πV =4π 2 R 2 G[Y,Y] =4π 2 R 2 G[X,Y] =cos2πv sin2πv.g[r,r]+4π 2 R 2 ( sin2πv cos2πv)=0 With these hypotheses on U and V and their errors, we can conclude that the variances of the errors on X and Y depend only on the radius R and the coerror on X and Y vanishes. If we think (U, V ) or (R, V ) as the data of the model and (X, Y ) as the output, we see that this error calculus may be seen as a sensitivity analysis but dealing with more information than simple derivation since we obtain cosensitivity as well.
13 error calculation for a triangle. Suppose we are drawing atriangle with agraduated rule and protractor: we take the polar angle of O A, say θ 1, and set O A = l 1 ; next we take the angle (O A, AB), say θ 2, and set AB = l 2. y B θ 2 A O θ 1 x 1) Select hypotheses on errors l 1, l 2 and θ 1, θ 2 and their errors can be modeled as follows: ( (0, L) 2 (0, π) 2, B ( (0, L) 2 (0, π) 2), dl ) 1 dl 2 dθ 1 dθ 2 L L π π, ID, Ɣ where { ID = and Ɣ[ f ] = l 2 1 f L 2 ( dl1 L ( f l 1 dl 2 L dθ 1 π ) dθ 2 : π f, f, f, f ( L 2 dl1 dl 2 dθ 1 l 1 l 2 θ 1 θ 2 L L π ) 2 ( ) f f f 2 ( ) f 2 +l 1 l 2 +l f ( ) f f 2 +, l 1 l 2 l 2 θ 1 θ 1 θ 2 θ 2 This quadratic error operator indicates that the errors on lengths l 1, l 2 are uncorrelated with those on angles θ 1, θ 2 (i.e. no term in f f l i θ j ). Such a hypothesis proves natural when measurements are conducted using different instruments. The bilinear operator associated with Ɣ is Ɣ[ f, g] = l 2 f g ( f l 1 l 1 2 l g 1l 2 + f ) g + l 2 f g 2 l 1 l 2 l 2 l 1 l 2 l 2 + f g + 1 ( f g + f ) g + f g. θ 1 θ 1 2 θ 1 θ 2 θ 2 θ 1 θ 2 θ 2 )} dθ 2 π
14 2) Compute the errors on significant quantities using functional calculus on Ɣ Take point B for instance: X B = l 1 cos θ 1 + l 2 cos(θ 1 + θ 2 ), Y B = l 1 sin θ 1 + l 2 sin(θ 1 + θ 2 ) Ɣ[X B ] = l l 1l 2 (cos θ sin θ 1 sin(θ 1 + θ 2 )) + l 2 2 (1 + 2 sin2 (θ 1 + θ 2 )) Ɣ[Y B ] = l l 1l 2 (cos θ cos θ 1 cos(θ 1 + θ 2 )) + l 2 2 (1 + 2 cos2 (θ 1 + θ 2 )) Ɣ[X B, Y B ] = l 1 l 2 sin(2θ 1 + θ 2 ) l 2 2 sin(2θ 1 + 2θ 2 ). For the area of the triangle, the formula area(o AB) = 1 2 l 1l 2 sin θ 2 yields: Ɣ[area(O AB)] = 1 4 l2 1 l2 2 (1 + 2 sin2 θ 2 ). The proportional error on the triangle area (Ɣ[area(O AB)]) 1/2 ( ) 1 1/2 = area(o AB) sin θ 2 reaches a minimum at θ 2 = π when the triangle is rectangular. From O B 2 = l l 1l 2 cos θ 2 + l 2 2, we obtain Ɣ[O B 2 ] = 4 [ (l l2 2 )2 + 3(l l2 2 )l ] 1l 2 cos θ 2 + 2l 2 1 l2 2 cos2 θ 2 = 4O B 2 (O B 2 l 1 l 2 cos θ 2 ) and by Ɣ[O B] = Ɣ[O B2 ] 4O B 2, we have: Ɣ[O B] = 1 l 1l 2 cos θ 2 O B 2 O B 2 thereby providing the result that the proportional error on O B is (Ɣ[O B])1/2 minimal when l 1 = l 2 and θ 2 = 0 and in this case = 3 O B 2.
15 Cathodic tube An oscillograph is modeled in the following way. After acceleration by an electric field, electrons arrive at point O 1 at a speed v 0 > 0 orthogonal to plane P 1. Between parallel planes P 1 and P 2, a magnetic field B orthogonal to O 1 O 2 is acting; its components on O 2 x and O 2 y are ( B 1, B 2 ). P 1 y P 2 P 3 v 0 O 1 O 2 O 3 x M a α Equation of the model. The physics of the problem is classical. The gravity force is negligible, the Lorenz force q v B is orthogonal to v such that the modulus v remains constant and equal to v 0, and the electrons describe a circle of radius R = mv 0 e B. If θ is the angle of the trajectory with O 1 O 2 as it passes through P 2, we then have: θ = arcsin a R O 2 A = R(1 cos θ) and A = ( O 2 A B 2 B, O 2 A B ) 1. B θ R P 2 M P 1 θ P 3 A O 1 O 2 O 3 a Figure in the plane of the trajectory α
16 The position of M is thus given by: M = (X, Y ) ( ) mv0 B2 X = (1 cos θ) + d tan θ e ( B ) B mv0 B1 Y = (1 cos θ) + d tan θ e B B θ = arcsin ae B mv 0 To study the sensitivity of point M to the magnetic field B, we assume that B varies in [ λ, λ] 2, equipped with the error structure ( [ λ, λ] 2, B ( [ λ, λ] 2), IP, ID, u u ) u 2 2. We can now compute the quadratic error on M, i.e. the matrix [ Ɣ ] ( ) ( M, M t Ɣ[X] Ɣ[X, Y ] ( X B = = 1 ) 2 + ( X B 2 ) 2 = Ɣ[X, Y ] Ɣ[Y ] X B 1 Y B 1 + X B 2 Y B 2 (1) X Y B 1 B 1 + X Y B 2 B 2 ( Y B 1 ) 2 + ( Y B 2 ) 2 ). Using some approximations to simplfy the calculations, we obtain [ Ɣ ] M, M t = (2) = ( 4β 2 B1 2 = B2 2 + ( α + β B β ) 2 ( ( ) B2 2 2β B 1 B 2 α + 3β B B2)) 2 ( ( )) 2β B 1 B 2 α + 3β B B2 2 4β 2 B1 2B2 2 + ( α + β B β ) 2. B2 1 It follows in particular, by computing the determinant, that the law of M is absolutely continuous.
17 If we now suppose that the inaccuracy on the magnetic field stems from a noise in the electric circuit responsible for generating B and that this noise is centered: A [ B 1 ] = A [ B2 ] = 0, we can compute the bias of the errors on M = (X, Y ) which yields A[X] = 1 2 X 2 B1 2 Ɣ [ ] 1 2 X B B2 2 Ɣ [ ] B 2 A[Y ] = 1 2 Y 2 B1 2 Ɣ [ ] 1 2 Y B B2 2 Ɣ [ ] B 2 A[X] = 4β B 2 A[Y ] = 4β B 1. By comparison with (2), we can observe that: A [ O3 M ] = y 4β α + β B 2 O3 M. bias O 3 x The appearance of biases in the absence of bias on the hypotheses is specifically due to the fact that the method considers the errors, although infinitesimal, to be random quantities. This feature will be highlighted in the following table.
Coefficient of Potential and Capacitance
Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationElectromagnetism  Lecture 2. Electric Fields
Electromagnetism  Lecture 2 Electric Fields Review of Vector Calculus Differential form of Gauss s Law Poisson s and Laplace s Equations Solutions of Poisson s Equation Methods of Calculating Electric
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationFEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL
FEGYVERNEKI SÁNDOR, PROBABILITY THEORY AND MATHEmATICAL STATIsTICs 4 IV. RANDOm VECTORs 1. JOINTLY DIsTRIBUTED RANDOm VARIABLEs If are two rom variables defined on the same sample space we define the joint
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSAMAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More information3. INNER PRODUCT SPACES
. INNER PRODUCT SPACES.. Definition So far we have studied abstract vector spaces. These are a generalisation of the geometric spaces R and R. But these have more structure than just that of a vector space.
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More information2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
More informationMATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
More informationPCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
More informationMaster s Theory Exam Spring 2006
Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem
More informationON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE
i93 c J SYSTEMS OF CURVES 695 ON CERTAIN DOUBLY INFINITE SYSTEMS OF CURVES ON A SURFACE BY C H. ROWE. Introduction. A system of co 2 curves having been given on a surface, let us consider a variable curvilinear
More informationInstitute of Actuaries of India Subject CT3 Probability and Mathematical Statistics
Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in
More informationGraduate Programs in Statistics
Graduate Programs in Statistics Course Titles STAT 100 CALCULUS AND MATR IX ALGEBRA FOR STATISTICS. Differential and integral calculus; infinite series; matrix algebra STAT 195 INTRODUCTION TO MATHEMATICAL
More information6 J  vector electric current density (A/m2 )
Determination of Antenna Radiation Fields Using Potential Functions Sources of Antenna Radiation Fields 6 J  vector electric current density (A/m2 ) M  vector magnetic current density (V/m 2 ) Some problems
More informationMATHEMATICS (CLASSES XI XII)
MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More information* Magnetic Scalar Potential * Magnetic Vector Potential. PPT No. 19
* Magnetic Scalar Potential * Magnetic Vector Potential PPT No. 19 Magnetic Potentials The Magnetic Potential is a method of representing the Magnetic field by using a quantity called Potential instead
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationLINE INTEGRALS OF VECTOR FUNCTIONS: GREEN S THEOREM. Contents. 2. Green s Theorem 3
LINE INTEGRALS OF VETOR FUNTIONS: GREEN S THEOREM ontents 1. A differential criterion for conservative vector fields 1 2. Green s Theorem 3 1. A differential criterion for conservative vector fields We
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationProbability and Statistics
Probability and Statistics Syllabus for the TEMPUS SEE PhD Course (Podgorica, April 4 29, 2011) Franz Kappel 1 Institute for Mathematics and Scientific Computing University of Graz Žaneta Popeska 2 Faculty
More informationFURTHER VECTORS (MEI)
Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level  MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: 97 Mathematics
More informationA QUICK GUIDE TO THE FORMULAS OF MULTIVARIABLE CALCULUS
A QUIK GUIDE TO THE FOMULAS OF MULTIVAIABLE ALULUS ontents 1. Analytic Geometry 2 1.1. Definition of a Vector 2 1.2. Scalar Product 2 1.3. Properties of the Scalar Product 2 1.4. Length and Unit Vectors
More informationDEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x
Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of
More informationChapter 19. General Matrices. An n m matrix is an array. a 11 a 12 a 1m a 21 a 22 a 2m A = a n1 a n2 a nm. The matrix A has n row vectors
Chapter 9. General Matrices An n m matrix is an array a a a m a a a m... = [a ij]. a n a n a nm The matrix A has n row vectors and m column vectors row i (A) = [a i, a i,..., a im ] R m a j a j a nj col
More informationMathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 PreAlgebra 4 Hours
MAT 051 PreAlgebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT
More information13.4 THE CROSS PRODUCT
710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product
More informationLecture L6  Intrinsic Coordinates
S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6  Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationPROBLEM SET. Practice Problems for Exam #1. Math 2350, Fall 2004. Sept. 30, 2004 ANSWERS
PROBLEM SET Practice Problems for Exam #1 Math 350, Fall 004 Sept. 30, 004 ANSWERS i Problem 1. The position vector of a particle is given by Rt) = t, t, t 3 ). Find the velocity and acceleration vectors
More informationThe Fourth International DERIVETI92/89 Conference Liverpool, U.K., 1215 July 2000. Derive 5: The Easiest... Just Got Better!
The Fourth International DERIVETI9/89 Conference Liverpool, U.K., 5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue NotreDame Ouest Montréal
More informationSCHWEITZER ENGINEERING LABORATORIES, COMERCIAL LTDA.
Pocket book of Electrical Engineering Formulas Content 1. Elementary Algebra and Geometry 1. Fundamental Properties (real numbers) 1 2. Exponents 2 3. Fractional Exponents 2 4. Irrational Exponents 2 5.
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationReview Sheet for Test 1
Review Sheet for Test 1 Math 26100 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And
More informationStatistics Graduate Courses
Statistics Graduate Courses STAT 7002Topics in StatisticsBiological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.
More informationCommunication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 2, FEBRUARY 2002 359 Communication on the Grassmann Manifold: A Geometric Approach to the Noncoherent MultipleAntenna Channel Lizhong Zheng, Student
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationDynamic Process Modeling. Process Dynamics and Control
Dynamic Process Modeling Process Dynamics and Control 1 Description of process dynamics Classes of models What do we need for control? Modeling for control Mechanical Systems Modeling Electrical circuits
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special DistributionsVI Today, I am going to introduce
More informationApplications to Data Smoothing and Image Processing I
Applications to Data Smoothing and Image Processing I MA 348 Kurt Bryan Signals and Images Let t denote time and consider a signal a(t) on some time interval, say t. We ll assume that the signal a(t) is
More informationL 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has
The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:
More information3. Reaction Diffusion Equations Consider the following ODE model for population growth
3. Reaction Diffusion Equations Consider the following ODE model for population growth u t a u t u t, u 0 u 0 where u t denotes the population size at time t, and a u plays the role of the population dependent
More informationG.A. Pavliotis. Department of Mathematics. Imperial College London
EE1 MATHEMATICS NUMERICAL METHODS G.A. Pavliotis Department of Mathematics Imperial College London 1. Numerical solution of nonlinear equations (iterative processes). 2. Numerical evaluation of integrals.
More informationProbability and Statistics
CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b  0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute  Systems and Modeling GIGA  Bioinformatics ULg kristel.vansteen@ulg.ac.be
More informationSection 8.8. 1. The given line has equations. x = 3 + t(13 3) = 3 + 10t, y = 2 + t(3 + 2) = 2 + 5t, z = 7 + t( 8 7) = 7 15t.
. The given line has equations Section 8.8 x + t( ) + 0t, y + t( + ) + t, z 7 + t( 8 7) 7 t. The line meets the plane y 0 in the point (x, 0, z), where 0 + t, or t /. The corresponding values for x and
More informationTwo vectors are equal if they have the same length and direction. They do not
Vectors define vectors Some physical quantities, such as temperature, length, and mass, can be specified by a single number called a scalar. Other physical quantities, such as force and velocity, must
More informationSection 1.4. Lines, Planes, and Hyperplanes. The Calculus of Functions of Several Variables
The Calculus of Functions of Several Variables Section 1.4 Lines, Planes, Hyperplanes In this section we will add to our basic geometric understing of R n by studying lines planes. If we do this carefully,
More information1 Determinants and the Solvability of Linear Systems
1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely sidestepped
More informationMean Value Coordinates
Mean Value Coordinates Michael S. Floater Abstract: We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More information11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space
11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationProbability and Random Variables. Generation of random variables (r.v.)
Probability and Random Variables Method for generating random variables with a specified probability distribution function. Gaussian And Markov Processes Characterization of Stationary Random Process Linearly
More informationEXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL
EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist
More informationARBITRAGEFREE OPTION PRICING MODELS. Denis Bell. University of North Florida
ARBITRAGEFREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic
More informationVariance Reduction. Pricing American Options. Monte Carlo Option Pricing. Delta and Common Random Numbers
Variance Reduction The statistical efficiency of Monte Carlo simulation can be measured by the variance of its output If this variance can be lowered without changing the expected value, fewer replications
More informationThe Delta Method and Applications
Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationChapter 1  Matrices & Determinants
Chapter 1  Matrices & Determinants Arthur Cayley (August 16, 1821  January 26, 1895) was a British Mathematician and Founder of the Modern British School of Pure Mathematics. As a child, Cayley enjoyed
More informationAlgebra 1 Course Information
Course Information Course Description: Students will study patterns, relations, and functions, and focus on the use of mathematical models to understand and analyze quantitative relationships. Through
More informationMA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem
MA 323 Geometric Modelling Course Notes: Day 02 Model Construction Problem David L. Finn November 30th, 2004 In the next few days, we will introduce some of the basic problems in geometric modelling, and
More informationPreCalculus Semester 1 Course Syllabus
PreCalculus Semester 1 Course Syllabus The Plano ISD eschool Mission is to create a borderless classroom based on a positive studentteacher relationship that fosters independent, innovative critical
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationRecall that two vectors in are perpendicular or orthogonal provided that their dot
Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal
More informationEMS. National Presentation GrandDuchy of L U X E M B 0 U R G
Niveaux de référence pour l'enseignement des mathématiques en Europe Reference levels in School Mathematics Education in Europe EMS European Mathematical Society http://www.emis.de/ Committee on Mathematics
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More informationInner Product Spaces
Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and
More informationHIGH SCHOOL: GEOMETRY (Page 1 of 4)
HIGH SCHOOL: GEOMETRY (Page 1 of 4) Geometry is a complete college preparatory course of plane and solid geometry. It is recommended that there be a strand of algebra review woven throughout the course
More information12.5 Equations of Lines and Planes
Instructor: Longfei Li Math 43 Lecture Notes.5 Equations of Lines and Planes What do we need to determine a line? D: a point on the line: P 0 (x 0, y 0 ) direction (slope): k 3D: a point on the line: P
More informationLecture 8: Signal Detection and Noise Assumption
ECE 83 Fall Statistical Signal Processing instructor: R. Nowak, scribe: Feng Ju Lecture 8: Signal Detection and Noise Assumption Signal Detection : X = W H : X = S + W where W N(, σ I n n and S = [s, s,...,
More informationCITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION
No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August
More informationReview Jeopardy. Blue vs. Orange. Review Jeopardy
Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 03 Jeopardy Round $200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?
More informationDeterminants, Areas and Volumes
Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a twodimensional object such as a region of the plane and the volume of a threedimensional object such as a solid
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationDiablo Valley College Catalog 20142015
Mathematics MATH Michael Norris, Interim Dean Math and Computer Science Division Math Building, Room 267 Possible career opportunities Mathematicians work in a variety of fields, among them statistics,
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationIntroduction to MonteCarlo Methods
Introduction to MonteCarlo Methods Bernard Lapeyre Halmstad January 2007 MonteCarlo methods are extensively used in financial institutions to compute European options prices to evaluate sensitivities
More informationSome Comments on the Derivative of a Vector with applications to angular momentum and curvature. E. L. Lady (October 18, 2000)
Some Comments on the Derivative of a Vector with applications to angular momentum and curvature E. L. Lady (October 18, 2000) Finding the formula in polar coordinates for the angular momentum of a moving
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationState of Stress at Point
State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,
More informationUnderstanding and Applying Kalman Filtering
Understanding and Applying Kalman Filtering Lindsay Kleeman Department of Electrical and Computer Systems Engineering Monash University, Clayton 1 Introduction Objectives: 1. Provide a basic understanding
More informationHøgskolen i Narvik Sivilingeniørutdanningen STE6237 ELEMENTMETODER. Oppgaver
Høgskolen i Narvik Sivilingeniørutdanningen STE637 ELEMENTMETODER Oppgaver Klasse: 4.ID, 4.IT Ekstern Professor: Gregory A. Chechkin email: chechkin@mech.math.msu.su Narvik 6 PART I Task. Consider twopoint
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationMath 526: Brownian Motion Notes
Math 526: Brownian Motion Notes Definition. Mike Ludkovski, 27, all rights reserved. A stochastic process (X t ) is called Brownian motion if:. The map t X t (ω) is continuous for every ω. 2. (X t X t
More informationMATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!
MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Prealgebra Algebra Precalculus Calculus Statistics
More information