Covariance and Correlation
|
|
- Lindsay Thomas
- 5 years ago
- Views:
Transcription
1 and Statistics 3513 Fall 008 Mike Anderson Abstract and correlation are measures of association; how strongly one random variable is related to another. Page 1 of 8
2 1. is a measure of association, how much one random variable changes with respect to another random variable. It is defined as Cov(X, Y ) E [(X µ x )(Y µ y )] We often see covariance in the sums and differences of random variables Similarly and in general V [X + Y ] E [ (X + Y µ x µ y ) ] E [ (X µ x ) ] + E [ (Y µ y ) ] + E [(X µ x )(Y µ y )] V [X] + V [Y ] + Cov(X, Y ) V [X Y ] V [X] + V [Y ] Cov(X, Y ) [ ] V X i V [X i ] + Cov(X i, X j ) i i i<j Page of 8
3 SAT Scores The College Boards publish a lot of data on SAT scores every year, but some obvious statistics are missing. For example, on their website are the means and standard deviations for the various subject scores (math, critical reading, and writing) as well as the composite scores (CR+M, CR+M+W), but nowhere are there measure of association, say between math and critical reading scores. Fortunately, there is enough data to calculate a covariance: subject µ σ M CR CR+M Refresher: A Normal Probability σcr+m σcr + σm + Cov(CR, M) Cov(CR, M) 1 ( σ CR+M σcr σm ) Current admission standards at UTSA are such that a student with combined SAT score, CR+M, of 100 or better, is eligible for admission, regardless of high school class standing. What proportion of students who took the 008 SAT exam are eligible for admission to UTSA? Page 3 of 8
4 3. Variance- Matrix When dealing with multiple random variables, it s convenient to represent variance and covariance as a single mathematical object, the variance-covariance matrix. If we look at the SAT data and let X m represent the math score and X r represent the reading score, then we have E [X] ( E [Xm ] E [X r ] ) ( ) ( V [X] 3.1. Using the Matrix: Linear Combinations V [X m ] Cov(X m, X r ) Cov(X m, X r ) V [X r ] ) ( ) The variance-covariance matrix simplifies variance calculations for linear combinations of random variables. If A is a linear transformation on the random vector X then E [AX] AE [X] V [AX] AV [X] A T The matrix A can be a single linear combination (a row vector), or it can be a set of linear combinations (a matrix). When A is a matrix, the result above will include the covariances between each pair of transformed variables. Consider the sum and difference of two random variables [ ] [ ] ( ) 1 1 X1 X1 + X Y AX 1 1 X X 1 X The variance-covariance matrix for Y is Σ Y AΣ X A T [ ] [ σ 1 σ 1 σ 1 σ ] [ ] Page 4 of 8
5 3.. The Sum and Difference in SAT Scores Consider the sum and difference of the Math and Critical Reading SAT scores. From our previous result we see that [ σ Σ Y 1 + σ 1 + σ σ1 σ σ1 σ σ1 σ 1 + σ X m + X r N(1017, 11 ) X m X r N(13, σ ) ] Questions: What is σ? What is the probability that a person s math and reading scores differ by more than 00 points? Page 5 of 8
6 Definition is just covariance rescaled to the interval (-1, +1): 4.. Examples ρ XY Cov(X, Y ) V [X] V [Y ] From the previous example about SAT scores, we found V [CR] 11 V [M] 116 Cov(CR, M) so the correlation is ρ CR,M Page 6 of 8
7 5. Principal Components (OPTIONAL) Take another look at the variance-covariance matrix for the sum and difference of the math and reading scores: [ ] [ ] σ Σ ± 1 + σ 1 + σ σ1 σ 44, σ1 σ σ1 σ 1 + σ 91 7, 479 The covariance is quite small compared to either of the two variances. Might it be possible to find two linear combinations a weighted sum and difference that have zero covariance? 5.1. Rotation Matrices Yes we can. The key is to use an orthonormal transformation, or rotation matrix: [ ] cos θ sin θ R θ sin θ cos θ This is a length-preserving transformation in -D. To get a better idea of what R θ does, answer these Questions: What is R θ? On graph paper, plot the points (column vectors) X and R π/4 X: X [ ] Page 7 of 8
8 5.. Rotating to Zero Now apply the rotation to our known variance-covariance matrix: [ ] [ ] [ R θ ΣR T cos θ sin θ σ θ 1 σ 1 cos θ sin θ sin θ cos θ σ 1 σ sin θ cos θ ] Then find θ such that the covariance term is zero. These identities might be useful sin θ sin θ cos θ cos θ cos θ sin θ Page 8 of 8
Dimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas
Dimensionality Reduction: PCA Nicholas Ruozzi University of Texas at Dallas Eigenvalues λ is an eigenvalue of a matrix A R n n if the linear system Ax = λx has at least one non-zero solution If Ax = λx
More informationPrinciple Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA
Principle Components Analysis (PCA) Relationship Between a Linear Combination of Variables and Axes Rotation for PCA Principle Components Analysis: Uses one group of variables (we will call this X) In
More informationProbability Distribution for a normal random variable x:
Chapter5 Continuous Random Variables 5.3 The Normal Distribution Probability Distribution for a normal random variable x: 1. It is and about its mean µ. 2. (the that x falls in the interval a < x < b is
More informationPsychology 310 Exam1 FormA Student Name:
Psychology 310 Exam1 FormA Student Name: 1 Compute the sample mean X forthefollowing5numbers: 1,4,2,3,4 (a) 2. 8 (b) 3.00 (c) 2. 24 (d) 1. 4 (e) None of the above are correct 2 Compute the sample variance
More informationMethods for sparse analysis of high-dimensional data, II
Methods for sparse analysis of high-dimensional data, II Rachel Ward May 23, 2011 High dimensional data with low-dimensional structure 300 by 300 pixel images = 90, 000 dimensions 2 / 47 High dimensional
More informationEXAM # 3 PLEASE SHOW ALL WORK!
Stat 311, Summer 2018 Name EXAM # 3 PLEASE SHOW ALL WORK! Problem Points Grade 1 30 2 20 3 20 4 30 Total 100 1. A socioeconomic study analyzes two discrete random variables in a certain population of households
More informationAlgebra 2 Notes Systems of Equations and Inequalities Unit 03c. System of Equations in Three Variables
System of Equations in Three Variables Big Idea A system of equations in three variables consists of multiple planes graphed on the same coordinate plane. The solutions to these systems consists of a single
More informationFINM 331: MULTIVARIATE DATA ANALYSIS FALL 2017 PROBLEM SET 3
FINM 331: MULTIVARIATE DATA ANALYSIS FALL 2017 PROBLEM SET 3 The required files for all problems can be found in: http://www.stat.uchicago.edu/~lekheng/courses/331/hw3/ The file name indicates which problem
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationConcept Category 4. Quadratic Equations
Concept Category 4 Quadratic Equations 1 Solving Quadratic Equations by the Square Root Property Square Root Property We previously have used factoring to solve quadratic equations. This chapter will introduce
More informationUnit 2. Describing Data: Numerical
Unit 2 Describing Data: Numerical Describing Data Numerically Describing Data Numerically Central Tendency Arithmetic Mean Median Mode Variation Range Interquartile Range Variance Standard Deviation Coefficient
More informationMethods for sparse analysis of high-dimensional data, II
Methods for sparse analysis of high-dimensional data, II Rachel Ward May 26, 2011 High dimensional data with low-dimensional structure 300 by 300 pixel images = 90, 000 dimensions 2 / 55 High dimensional
More informationThere are two things that are particularly nice about the first basis
Orthogonality and the Gram-Schmidt Process In Chapter 4, we spent a great deal of time studying the problem of finding a basis for a vector space We know that a basis for a vector space can potentially
More informationVectors and Matrices Statistics with Vectors and Matrices
Vectors and Matrices Statistics with Vectors and Matrices Lecture 3 September 7, 005 Analysis Lecture #3-9/7/005 Slide 1 of 55 Today s Lecture Vectors and Matrices (Supplement A - augmented with SAS proc
More informationAbsolute Value Inequalities 2.5, Functions 3.6
Absolute Value Inequalities 2.5, Functions 3.6 Fall 2013 - Math 1010 (Math 1010) M 1010 3.6 1 / 17 Roadmap 2.5 - Review of absolute value inequalities. 3.6 - Functions: Relations, Functions 3.6 - Evaluating
More informationMATH Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited).
MATH 311-504 Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited). Definition. A mapping f : V 1 V 2 is one-to-one if it maps different elements from V 1 to different
More informationCORELATION - Pearson-r - Spearman-rho
CORELATION - Pearson-r - Spearman-rho Scatter Diagram A scatter diagram is a graph that shows that the relationship between two variables measured on the same individual. Each individual in the set is
More informationMidterm 1 practice UCLA: Math 32B, Winter 2017
Midterm 1 practice UCLA: Math 32B, Winter 2017 Instructor: Noah White Date: Version: practice This exam has 4 questions, for a total of 40 points. Please print your working and answers neatly. Write your
More informationMath 1314 Week #14 Notes
Math 3 Week # Notes Section 5.: A system of equations consists of two or more equations. A solution to a system of equations is a point that satisfies all the equations in the system. In this chapter,
More informationAP CALCULUS BC 2006 SCORING GUIDELINES (Form B) Question 2
AP CALCULUS BC 2006 SCORING GUIDELINES (Form B) Question 2 An object moving along a curve in the xy-plane is at position ( x() t, y() t ) at time t, where dx t tan( e ) for t 0. At time t =, the object
More informationFinal exam (practice) UCLA: Math 31B, Spring 2017
Instructor: Noah White Date: Final exam (practice) UCLA: Math 3B, Spring 207 This exam has 8 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions in the
More informationProblem Score Problem Score 1 /12 6 /12 2 /12 7 /12 3 /12 8 /12 4 /12 9 /12 5 /12 10 /12 Total /120
EE103/CME103: Introduction to Matrix Methods October 27 2016 S. Boyd Midterm Exam This is an in-class 80 minute midterm. You may not use any books, notes, or computer programs (e.g., Julia). Throughout
More information11. Regression and Least Squares
11. Regression and Least Squares Prof. Tesler Math 186 Winter 2016 Prof. Tesler Ch. 11: Linear Regression Math 186 / Winter 2016 1 / 23 Regression Given n points ( 1, 1 ), ( 2, 2 ),..., we want to determine
More informationPrincipal Components Theory Notes
Principal Components Theory Notes Charles J. Geyer August 29, 2007 1 Introduction These are class notes for Stat 5601 (nonparametrics) taught at the University of Minnesota, Spring 2006. This not a theory
More information7. The Multivariate Normal Distribution
of 5 7/6/2009 5:56 AM Virtual Laboratories > 5. Special Distributions > 2 3 4 5 6 7 8 9 0 2 3 4 5 7. The Multivariate Normal Distribution The Bivariate Normal Distribution Definition Suppose that U and
More informationAP CALCULUS BC 2009 SCORING GUIDELINES
AP CALCULUS BC 009 SCORING GUIDELINES Question 6 The Maclaurin series for by f( x) = x e is 3 n x x x x e = 1 + x + + + + +. The continuous function f is defined 6 n! ( x 1) e 1 for x 1 and f () 1 = 1.
More informationPrincipal Components Analysis (PCA) and Singular Value Decomposition (SVD) with applications to Microarrays
Principal Components Analysis (PCA) and Singular Value Decomposition (SVD) with applications to Microarrays Prof. Tesler Math 283 Fall 2015 Prof. Tesler Principal Components Analysis Math 283 / Fall 2015
More informationPRINCIPAL COMPONENTS ANALYSIS
121 CHAPTER 11 PRINCIPAL COMPONENTS ANALYSIS We now have the tools necessary to discuss one of the most important concepts in mathematical statistics: Principal Components Analysis (PCA). PCA involves
More informationCORRELATION. suppose you get r 0. Does that mean there is no correlation between the data sets? many aspects of the data may a ect the value of r
Introduction to Statistics in Psychology PS 1 Professor Greg Francis Lecture 11 correlation Is there a relationship between IQ and problem solving ability? CORRELATION suppose you get r 0. Does that mean
More informationChapter 5. The multivariate normal distribution. Probability Theory. Linear transformations. The mean vector and the covariance matrix
Probability Theory Linear transformations A transformation is said to be linear if every single function in the transformation is a linear combination. Chapter 5 The multivariate normal distribution When
More informationLinear Algebra & Geometry why is linear algebra useful in computer vision?
Linear Algebra & Geometry why is linear algebra useful in computer vision? References: -Any book on linear algebra! -[HZ] chapters 2, 4 Some of the slides in this lecture are courtesy to Prof. Octavia
More informationMATH 1553 PRACTICE MIDTERM 3 (VERSION B)
MATH 1553 PRACTICE MIDTERM 3 (VERSION B) Name Section 1 2 3 4 5 Total Please read all instructions carefully before beginning. Each problem is worth 10 points. The maximum score on this exam is 50 points.
More informationLecture # 3 Orthogonal Matrices and Matrix Norms. We repeat the definition an orthogonal set and orthornormal set.
Lecture # 3 Orthogonal Matrices and Matrix Norms We repeat the definition an orthogonal set and orthornormal set. Definition A set of k vectors {u, u 2,..., u k }, where each u i R n, is said to be an
More informationj=1 u 1jv 1j. 1/ 2 Lemma 1. An orthogonal set of vectors must be linearly independent.
Lecture Notes: Orthogonal and Symmetric Matrices Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Orthogonal Matrix Definition. Let u = [u
More informationSSEA Math 51 Track Final Exam August 30, Problem Total Points Score
Name: This is the final exam for the Math 5 track at SSEA. Answer as many problems as possible to the best of your ability; do not worry if you are not able to answer all of the problems. Partial credit
More informationJennie F. Snapp Math Lesson Plans 6 th Grade Date: December 31-4, Notes Topic & Standard Objective Homework Monday
Date: December 31-4, 2019 No School No School Exponents 6.EE.1 Write and evaluate numerical expressions involving whole number exponents. Exponents Orders of Operations Students will write expressions
More informationMAS223 Statistical Inference and Modelling Exercises
MAS223 Statistical Inference and Modelling Exercises The exercises are grouped into sections, corresponding to chapters of the lecture notes Within each section exercises are divided into warm-up questions,
More informationProperties of Linear Transformations from R n to R m
Properties of Linear Transformations from R n to R m MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Topic Overview Relationship between the properties of a matrix transformation
More informationL3: Review of linear algebra and MATLAB
L3: Review of linear algebra and MATLAB Vector and matrix notation Vectors Matrices Vector spaces Linear transformations Eigenvalues and eigenvectors MATLAB primer CSCE 666 Pattern Analysis Ricardo Gutierrez-Osuna
More informationMATH 33A LECTURE 3 PRACTICE MIDTERM I
MATH A LECTURE PRACTICE MIDTERM I Please note: Show your work Correct answers not accompanied by sufficent explanations will receive little or no credit (except on multiple-choice problems) Please call
More informationHomework 2. Solutions T =
Homework. s Let {e x, e y, e z } be an orthonormal basis in E. Consider the following ordered triples: a) {e x, e x + e y, 5e z }, b) {e y, e x, 5e z }, c) {e y, e x, e z }, d) {e y, e x, 5e z }, e) {
More information18.S096 Problem Set 7 Fall 2013 Factor Models Due Date: 11/14/2013. [ ] variance: E[X] =, and Cov[X] = Σ = =
18.S096 Problem Set 7 Fall 2013 Factor Models Due Date: 11/14/2013 1. Consider a bivariate random variable: [ ] X X = 1 X 2 with mean and co [ ] variance: [ ] [ α1 Σ 1,1 Σ 1,2 σ 2 ρσ 1 σ E[X] =, and Cov[X]
More informationA Probability Review
A Probability Review Outline: A probability review Shorthand notation: RV stands for random variable EE 527, Detection and Estimation Theory, # 0b 1 A Probability Review Reading: Go over handouts 2 5 in
More informationEXAM. Exam 1. Math 5316, Fall December 2, 2012
EXAM Exam Math 536, Fall 22 December 2, 22 Write all of your answers on separate sheets of paper. You can keep the exam questions. This is a takehome exam, to be worked individually. You can use your notes.
More informationSTA 2101/442 Assignment 3 1
STA 2101/442 Assignment 3 1 These questions are practice for the midterm and final exam, and are not to be handed in. 1. Suppose X 1,..., X n are a random sample from a distribution with mean µ and variance
More informationAssessing the relation between language comprehension and performance in general chemistry. Appendices
Assessing the relation between language comprehension and performance in general chemistry Daniel T. Pyburn a, Samuel Pazicni* a, Victor A. Benassi b, and Elizabeth E. Tappin c a Department of Chemistry,
More informationLecture 11. Multivariate Normal theory
10. Lecture 11. Multivariate Normal theory Lecture 11. Multivariate Normal theory 1 (1 1) 11. Multivariate Normal theory 11.1. Properties of means and covariances of vectors Properties of means and covariances
More informationBasic Concepts in Matrix Algebra
Basic Concepts in Matrix Algebra An column array of p elements is called a vector of dimension p and is written as x p 1 = x 1 x 2. x p. The transpose of the column vector x p 1 is row vector x = [x 1
More informationFinite Mathematics Chapter 2. where a, b, c, d, h, and k are real numbers and neither a and b nor c and d are both zero.
Finite Mathematics Chapter 2 Section 2.1 Systems of Linear Equations: An Introduction Systems of Equations Recall that a system of two linear equations in two variables may be written in the general form
More informationAP CALCULUS AB 2007 SCORING GUIDELINES (Form B)
AP CALCULUS AB 27 SCORING GUIDELINES (Form B) Question 2 A particle moves along the x-axis so that its velocity v at time 2 t is given by vt () = sin ( t ). The graph of v is shown above for t 5 π. The
More information11 Correlation and Regression
Chapter 11 Correlation and Regression August 21, 2017 1 11 Correlation and Regression When comparing two variables, sometimes one variable (the explanatory variable) can be used to help predict the value
More informationEXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. 1. Determinants
EXERCISES ON DETERMINANTS, EIGENVALUES AND EIGENVECTORS. Determinants Ex... Let A = 0 4 4 2 0 and B = 0 3 0. (a) Compute 0 0 0 0 A. (b) Compute det(2a 2 B), det(4a + B), det(2(a 3 B 2 )). 0 t Ex..2. For
More informationIntroduction to Computational Finance and Financial Econometrics Matrix Algebra Review
You can t see this text! Introduction to Computational Finance and Financial Econometrics Matrix Algebra Review Eric Zivot Spring 2015 Eric Zivot (Copyright 2015) Matrix Algebra Review 1 / 54 Outline 1
More informationMeasuring Associations : Pearson s correlation
Measuring Associations : Pearson s correlation Scatter Diagram A scatter diagram is a graph that shows that the relationship between two variables measured on the same individual. Each individual in the
More informationSTA 431s17 Assignment Eight 1
STA 43s7 Assignment Eight The first three questions of this assignment are about how instrumental variables can help with measurement error and omitted variables at the same time; see Lecture slide set
More informationPart A. BASIC INFORMATION: (Please print) 1. Name: 2. Please mark all courses you have taken in high school or college :
THE PLACEMENT TEST FOR PRECALCULUS/CALCULUS I - Fall, 1 Please read the following carefully: Your score on this test (in Part B), your math SAT or ACT score, and the information you are asked to provide
More informationAP Physics B Math Competancy Test
AP Physics B Math Competancy Test The following test is designed to allow you, the student, to determine if your math skills are adequate for the AP Physics B course offered by PHC Prep Academy. Be aware
More informationMath Exam 2, October 14, 2008
Math 96 - Exam 2, October 4, 28 Name: Problem (5 points Find all solutions to the following system of linear equations, check your work: x + x 2 x 3 2x 2 2x 3 2 x x 2 + x 3 2 Solution Let s perform Gaussian
More informationEngage Education Foundation
E Free Exam for 2006-15 VCE study design Engage Education Foundation Units 3 and 4 Further Maths: Exam 2 Practice Exam Solutions Stop! Don t look at these solutions until you have attempted the exam. Any
More informationElementary maths for GMT
Elementary maths for GMT Linear Algebra Part 1: Vectors, Representations Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear Algebra: tuples, triples... of numbers
More informationPractice Exam. 2x 1 + 4x 2 + 2x 3 = 4 x 1 + 2x 2 + 3x 3 = 1 2x 1 + 3x 2 + 4x 3 = 5
Practice Exam. Solve the linear system using an augmented matrix. State whether the solution is unique, there are no solutions or whether there are infinitely many solutions. If the solution is unique,
More informationREVIEW 8/2/2017 陈芳华东师大英语系
REVIEW Hypothesis testing starts with a null hypothesis and a null distribution. We compare what we have to the null distribution, if the result is too extreme to belong to the null distribution (p
More information1) [3pts] 2. Simplify the expression. Give your answer as a reduced fraction. No credit for decimal answers. ( ) 2) [4pts] ( 3 2 ) 1
Math 097 Winter 2018 Final Exam (Form A) Name: Instructor s Name: Score: /100 (+ 3 bonus) 1. Evaluate 2(x h) 3 when x = 2 and h = 5. 1) 2. Simplify the expression. Give your answer as a reduced fraction.
More informationCorrelation: Relationships between Variables
Correlation Correlation: Relationships between Variables So far, nearly all of our discussion of inferential statistics has focused on testing for differences between group means However, researchers are
More information384 PU M.Sc Five year Integrated M.Sc Programmed (Mathematics, Computer Science,Statistics)
384 PU M.Sc Five year Integrated M.Sc Programmed (Mathematics, Computer Science,Statistics) 1 of 1 146 PU_216_384_E 2 cos 1 π 4 cos 1 cos1 2 of 1 15 PU_216_384_E 1! 1 1 3 of 1 15 PU_216_384_E 1 4 of 1
More informationECE 5615/4615 Computer Project
Set #1p Due Friday March 17, 017 ECE 5615/4615 Computer Project The details of this first computer project are described below. This being a form of take-home exam means that each person is to do his/her
More informationMATH10212 Linear Algebra B Homework Week 4
MATH22 Linear Algebra B Homework Week 4 Students are strongly advised to acquire a copy of the Textbook: D. C. Lay Linear Algebra and its Applications. Pearson, 26. ISBN -52-2873-4. Normally, homework
More informationAP CALCULUS AB 2009 SCORING GUIDELINES (Form B) Question 2. or meters 2 :
AP CALCULUS AB 2009 SCORING GUIDELINES (Form B) Question 2 A storm washed away sand from a beach, causing the edge of the water to get closer to a nearby road. The rate at which the distance between the
More informationHistograms. Mark Scheme. Save My Exams! The Home of Revision For more awesome GCSE and A level resources, visit us at
Save My Exams! The Home of Revision For more awesome GCSE and A level resources, visit us at www.savemyexams.co.uk/ Histograms Mark Scheme Level Subject Exam Board Topic Sub Topic Booklet IGCSE Maths Edexcel
More informationAnnouncements September 19
Announcements September 19 Please complete the mid-semester CIOS survey this week The first midterm will take place during recitation a week from Friday, September 3 It covers Chapter 1, sections 1 5 and
More informationAnnouncements: You can turn in homework until 6pm, slot on wall across from 2202 Bren. Make sure you use the correct slot! (Stats 8, closest to wall)
Announcements: You can turn in homework until 6pm, slot on wall across from 2202 Bren. Make sure you use the correct slot! (Stats 8, closest to wall) We will cover Chs. 5 and 6 first, then 3 and 4. Mon,
More informationMath 415 Exam I. Name: Student ID: Calculators, books and notes are not allowed!
Math 415 Exam I Calculators, books and notes are not allowed! Name: Student ID: Score: Math 415 Exam I (20pts) 1. Let A be a square matrix satisfying A 2 = 2A. Find the determinant of A. Sol. From A 2
More informationSection 1.8/1.9. Linear Transformations
Section 1.8/1.9 Linear Transformations Motivation Let A be a matrix, and consider the matrix equation b = Ax. If we vary x, we can think of this as a function of x. Many functions in real life the linear
More informationPre-College Workbook
Pre-College Workbook Bexhill College Maths Department The questions in this workbook represent a non-exhaustive selection of skills required to succeed on the A Level mathematics course - it is your responsibility
More informationREFRESHER. William Stallings
BASIC MATH REFRESHER William Stallings Trigonometric Identities...2 Logarithms and Exponentials...4 Log Scales...5 Vectors, Matrices, and Determinants...7 Arithmetic...7 Determinants...8 Inverse of a Matrix...9
More informationMathematics 206 Solutions for HWK 23 Section 6.3 p358
Mathematics 6 Solutions for HWK Section Problem 9. Given T(x, y, z) = (x 9y + z,6x + 5y z) and v = (,,), use the standard matrix for the linear transformation T to find the image of the vector v. Note
More information2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course.
1. Solving Linear Equations 2. Solving Linear Systems of Equations 3. Multiplying Polynomials and Solving Quadratics 4. Writing the Equation of a Line 5. Laws of Exponents and Scientific Notation 6. Solving
More informationStatistics 202: Data Mining. c Jonathan Taylor. Week 2 Based in part on slides from textbook, slides of Susan Holmes. October 3, / 1
Week 2 Based in part on slides from textbook, slides of Susan Holmes October 3, 2012 1 / 1 Part I Other datatypes, preprocessing 2 / 1 Other datatypes Document data You might start with a collection of
More informationChapter 6 The Standard Deviation as a Ruler and the Normal Model
Chapter 6 The Standard Deviation as a Ruler and the Normal Model Overview Key Concepts Understand how adding (subtracting) a constant or multiplying (dividing) by a constant changes the center and/or spread
More informationExam 1 - Math Solutions
Exam 1 - Math 3200 - Solutions Spring 2013 1. Without actually expanding, find the coefficient of x y 2 z 3 in the expansion of (2x y z) 6. (A) 120 (B) 60 (C) 30 (D) 20 (E) 10 (F) 10 (G) 20 (H) 30 (I)
More informationPart I. Other datatypes, preprocessing. Other datatypes. Other datatypes. Week 2 Based in part on slides from textbook, slides of Susan Holmes
Week 2 Based in part on slides from textbook, slides of Susan Holmes Part I Other datatypes, preprocessing October 3, 2012 1 / 1 2 / 1 Other datatypes Other datatypes Document data You might start with
More informationMathematical foundations - linear algebra
Mathematical foundations - linear algebra Andrea Passerini passerini@disi.unitn.it Machine Learning Vector space Definition (over reals) A set X is called a vector space over IR if addition and scalar
More informationSTUDENT NAME: STUDENT SIGNATURE: STUDENT ID NUMBER: SECTION NUMBER RECITATION INSTRUCTOR:
MA262 FINAL EXAM SPRING 2016 MAY 2, 2016 TEST NUMBER 01 INSTRUCTIONS: 1. Do not open the exam booklet until you are instructed to do so. 2. Before you open the booklet fill in the information below and
More informationExponential Family and Maximum Likelihood, Gaussian Mixture Models and the EM Algorithm. by Korbinian Schwinger
Exponential Family and Maximum Likelihood, Gaussian Mixture Models and the EM Algorithm by Korbinian Schwinger Overview Exponential Family Maximum Likelihood The EM Algorithm Gaussian Mixture Models Exponential
More informationMath Matrix Algebra
Math 44 - Matrix Algebra Review notes - (Alberto Bressan, Spring 7) sec: Orthogonal diagonalization of symmetric matrices When we seek to diagonalize a general n n matrix A, two difficulties may arise:
More informationAP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 2. the
AP CALCULUS AB 2006 SCORING GUIDELINES (Form B) Question 2 Let f be the function defined for x 0 with f ( 0) = 5 and f, the ( x 4) 2 first derivative of f, given by f ( x) = e sin ( x ). The graph of y
More informationMultivariate Statistical Analysis
Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 4 for Applied Multivariate Analysis Outline 1 Eigen values and eigen vectors Characteristic equation Some properties of eigendecompositions
More informationSystem of Linear Equations
Math 20F Linear Algebra Lecture 2 1 System of Linear Equations Slide 1 Definition 1 Fix a set of numbers a ij, b i, where i = 1,, m and j = 1,, n A system of m linear equations in n variables x j, is given
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationEcon 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms. 1 Diagonalization and Change of Basis
Econ 204 Supplement to Section 3.6 Diagonalization and Quadratic Forms De La Fuente notes that, if an n n matrix has n distinct eigenvalues, it can be diagonalized. In this supplement, we will provide
More informationThe Matrix Algebra of Sample Statistics
The Matrix Algebra of Sample Statistics James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) The Matrix Algebra of Sample Statistics
More informationx. Figure 1: Examples of univariate Gaussian pdfs N (x; µ, σ 2 ).
.8.6 µ =, σ = 1 µ = 1, σ = 1 / µ =, σ =.. 3 1 1 3 x Figure 1: Examples of univariate Gaussian pdfs N (x; µ, σ ). The Gaussian distribution Probably the most-important distribution in all of statistics
More informationb) (1) Using the results of part (a), let Q be the matrix with column vectors b j and A be the matrix with column vectors v j :
Exercise assignment 2 Each exercise assignment has two parts. The first part consists of 3 5 elementary problems for a maximum of 10 points from each assignment. For the second part consisting of problems
More informationFinal exam (practice 1) UCLA: Math 32B, Spring 2018
Instructor: Noah White Date: Final exam (practice 1) UCLA: Math 32B, Spring 2018 This exam has 7 questions, for a total of 80 points. Please print your working and answers neatly. Write your solutions
More informationWavelet Transform And Principal Component Analysis Based Feature Extraction
Wavelet Transform And Principal Component Analysis Based Feature Extraction Keyun Tong June 3, 2010 As the amount of information grows rapidly and widely, feature extraction become an indispensable technique
More information1. Select the unique answer (choice) for each problem. Write only the answer.
MATH 5 Practice Problem Set Spring 7. Select the unique answer (choice) for each problem. Write only the answer. () Determine all the values of a for which the system has infinitely many solutions: x +
More informationMidterm 1 revision source for MATH 227, Introduction to Linear Algebra
Midterm revision source for MATH 227, Introduction to Linear Algebra 5 March 29, LJB page 2: Some notes on the Pearson correlation coefficient page 3: Practice Midterm Exam page 4: Spring 27 Midterm page
More informationAssignment 1 Math 5341 Linear Algebra Review. Give complete answers to each of the following questions. Show all of your work.
Assignment 1 Math 5341 Linear Algebra Review Give complete answers to each of the following questions Show all of your work Note: You might struggle with some of these questions, either because it has
More informationMath 308 Practice Final Exam Page and vector y =
Math 308 Practice Final Exam Page Problem : Solving a linear equation 2 0 2 5 Given matrix A = 3 7 0 0 and vector y = 8. 4 0 0 9 (a) Solve Ax = y (if the equation is consistent) and write the general solution
More informationRaquel Prado. Name: Department of Applied Mathematics and Statistics AMS-131. Spring 2010
Raquel Prado Name: Department of Applied Mathematics and Statistics AMS-131. Spring 2010 Final Exam (Type B) The midterm is closed-book, you are only allowed to use one page of notes and a calculator.
More information