MATH590: Approximation in R d

Size: px
Start display at page:

Download "MATH590: Approximation in R d"

Transcription

1 MATH9: Approximation in R d Abstract The methods of linear algebra are used to distinguish between dierent gaits. Table of contents Karhunen-Loève theorem ? 2 Singular-value decomposition ? 3 Covariance matrices ? 4 Data-driven bases ? Application to gait analysis ? Karhunen-Loève theorem A probability space is a triplet (; F ; P ) with: a sample space of all possible outcomes; F a set of events that is a set of subsets of ; P : F! R a probability function for each event. Rather improperly named, a random variable X:! E, is a function dened on a sample space with values in a measurable space (e.g., R d ). For some measurable subset S E, the probability of X 2 S is Pr(X 2 S) = P (f! 2 jx(!) 2 S g) A stochastic process X t (!) is indexed collection of random variables. Often the index is time, and X t : R! E. A centered stochastic process has mean value zero with E the expectation operator. E[X t (!)] = ; The Karhunen-Loève theorem arms the existence of a canonical description of a stochastic process as a linear combination of random variables Z k with time-dependent coecients e k (t), or, conversely, as a linear combination of time-varying functions e k (t) with random coecients Z k X t (!) = X(t;!) = X k= Z k e k (t):

2 2 Singular-value decomposition As will be explained in detail in the STC module, the singular value decomposition is a discrete form of the Karhunen-Loève theorem. Let the matrix X denote multiple samples of some real-valued, centered stochastic process X(t ;! ) X(t ;! 2 ) ::: X(t ;! N ) X = B X(t 2 ;! ) X(t 2 ;! 2 ) ::: X(t 2 ;! N ) A 2 RmN : X(t m ;! ) X(t m ;! 2 ) ::: X(t m ;! N ) There exist orthogonal matrices U 2 R mm, V 2 R N N, and the quasi-diagonal positive matrix = diag( ; :::; r ; ; :::; ) 2 R + mn such that X = UV T ; known as the singular value decomposition (SVD). Introducing the column vectors of U; V U = ( u u 2 ::: u m ); V = ( v v 2 ::: v N ); the SVD can be rewritten in two important forms: Sum of rank- updates. X = P r T k= k u k v k Bases for linear operator X. X ( v ::: v N ) = ( Xv ::: Xv r ::: Xv N ) = ( u ::: r u r ::: ): 3 Covariance matrices The covariance of two centered random variables X(t i ;!) = X i ; X(t j ;!) = X j is cov [X i ; X j ] = E[X i X j ]; typically approximated through a statistic from N observations E[X i X j ] = XN X N i (! k )X j (! k ) k= The covariance matrix C of m centered random variables X(t ;!); :::; X(t m ;!) has entries C ij = cov [X i ; X j ]; 2

3 and is expressed as the matrix product C = N XXT 2 R mm : By construction the covariance matrix is symmetric positive denite (spd) and therefore admits an orthogonal eigendecomposition m C = UU T = X k= k u k u k T : Assume (by re-labeling if necessary) that > 2 > ::: > m. It is often the case that p eigenvalues dominate over all others (the strongly correlated modes) > 2 > ::: > p p+ > > m ; and the correlation matrix is approximated by the rst p rank- updates p C = X k= k u k u k T : 4 Data-driven bases The dominant correlated modes form a natural basis set for analysis of data. Rather than solving the covariance matrix eigenproblem the SVD of X is used since X = UV T ; NC = XX T = UV T V T U T = U T U T : Application to gait analysis Consider the data obtained from many individual gait measurements (either from dierent individuals or at dierent times for the same individual). The goal is to identify dierences w.r.t. a mean gait and use those dierences to () identify either an individual or (2) a particular type of walking (climbing stairs versus level walking). The procedure is demonstrated here for the second problem. dir='/home/student/courses/math9/numdata'; chdir(dir); LastName='Mitran'; d=textread(strcat(lastname,'.data')); mu=max(size(d))/7; disp(mu) 49 3

4 /home/student/courses/math9/numdata data=reshape(d,[7,mu])'; data(:6,:7) Interpolate to obtain even-spaced data, and plot the data. t=data(,); t=data(mu,); ni = 2^ceil(log2(mu)); dt=(t-t)/ni; ti=(:ni-)*dt; ti=ti'; ai=interp(data(:,),data(:,3),ti); fid=fopen('periods.data','w'); fprintf(fid,'%f %f\n',[ti ai]'); cd '/home/student/courses/math9/numdata' set style line lt 2 lc rgb "x64" lw 3 plot 'periods.data' u :2 w l ls C A periods.data u : Find the Fourier spectrum of the vertical acceleration data. 4

5 Ai=fft(ai); PAi=log(Ai.*conj(Ai)); fid=fopen('spectrum.data','w'); fprintf(fid,'%f\n',pai(:ni/4)); cd '/home/student/courses/math9/numdata' set style line lt 2 lc rgb "x64" lw 3 plot 'spectrum.data' ls 8 spectrum.data There are several peaks within the Fourier spectrum. Seek the peak corresponding to a natural step period of approximately T s = (t t )/n Steps =.s. This turns out to be close to the global peak as shown in the following calculations ks=floor((t-t)/(.)) 32 [PAimx imx]=max(pai); [PAimx imx] ( ) N=imx-; Ts=(t-t)/N.4679

6 Isolate the steps, nd an average step waveform g, and compute the centered data matrix X m = floor(ni/n); t=(:m-)'*dt; a = reshape(ai(2:m*n+),[m,n]); g = mean(a')'; X = a - repmat(g,,n); fid=fopen('steps.data','w'); i=; while(i<n) fprintf(fid,'%f %f\n',[t a(:,i)]'); fprintf(fid,'\n'); i=i+; endwhile; fid=fopen('gstep.data','w'); fprintf(fid,'%f %f\n',[t g]'); cd '/home/student/courses/math9/numdata' set style line lt 2 lc rgb "x64" lw 3 set style line 2 lt 2 lc rgb "xff" lw 6 plot 'steps.data' u :2 w l ls, 'gstep.data' u :2 w l ls 2 steps.data u :2 gstep.data u : Find the natural basis for the problem by computing the SVD of X, and investigate the dominant singular values [U,S,V]=svd(X,); (diag(s)(:6))' 6

7 ( ) fid=fopen('gaits.data','w'); fprintf(fid,'%f %f %f %f\n',[t U(:,:2) g]'); From the above each step can be characterized by the rst two components u ; u 2. Plot these modes and compare to the average gait g. cd '/home/student/courses/math9/numdata' set style line lt 2 lc rgb "x64" lw 3 set style line 2 lt 2 lc rgb "x6464" lw 3 set style line 3 lt 2 lc rgb "xff" lw 3 plot 'gaits.data' u :2 w l ls, '' u :3 w l ls 2, '' u :4 w l ls 3.6. gaits.data u :2 u :3 u : Find the coordinates c of each step along these directions. c=x'*u(:,:2); fid=fopen('coefs.data','w'); fprintf(fid,'%f %f\n',c'); size(c) ( ) 7

8 cd '/home/student/courses/math9/numdata' set style line lt 2 lc rgb "x64" lw 3 plot 'coefs.data' u :2 w p ls coefs.data u : At this point, an economical representation of each of the N steps has been obtained, and the next question is to classify the observed results. This forms the objective of clustering analysis that relies on the mathematical theory of sets, as presented in the next module. 8

Lecture 3: Review of Linear Algebra

Lecture 3: Review of Linear Algebra ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters, transforms,

More information

Lecture 3: Review of Linear Algebra

Lecture 3: Review of Linear Algebra ECE 83 Fall 2 Statistical Signal Processing instructor: R Nowak, scribe: R Nowak Lecture 3: Review of Linear Algebra Very often in this course we will represent signals as vectors and operators (eg, filters,

More information

Principal Component Analysis CS498

Principal Component Analysis CS498 Principal Component Analysis CS498 Today s lecture Adaptive Feature Extraction Principal Component Analysis How, why, when, which A dual goal Find a good representation The features part Reduce redundancy

More information

linearly indepedent eigenvectors as the multiplicity of the root, but in general there may be no more than one. For further discussion, assume matrice

linearly indepedent eigenvectors as the multiplicity of the root, but in general there may be no more than one. For further discussion, assume matrice 3. Eigenvalues and Eigenvectors, Spectral Representation 3.. Eigenvalues and Eigenvectors A vector ' is eigenvector of a matrix K, if K' is parallel to ' and ' 6, i.e., K' k' k is the eigenvalue. If is

More information

LECTURE 16: PCA AND SVD

LECTURE 16: PCA AND SVD Instructor: Sael Lee CS549 Computational Biology LECTURE 16: PCA AND SVD Resource: PCA Slide by Iyad Batal Chapter 12 of PRML Shlens, J. (2003). A tutorial on principal component analysis. CONTENT Principal

More information

1. The Polar Decomposition

1. The Polar Decomposition A PERSONAL INTERVIEW WITH THE SINGULAR VALUE DECOMPOSITION MATAN GAVISH Part. Theory. The Polar Decomposition In what follows, F denotes either R or C. The vector space F n is an inner product space with

More information

Dimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas

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 information

Singular Value Decomposition and Principal Component Analysis (PCA) I

Singular Value Decomposition and Principal Component Analysis (PCA) I Singular Value Decomposition and Principal Component Analysis (PCA) I Prof Ned Wingreen MOL 40/50 Microarray review Data per array: 0000 genes, I (green) i,i (red) i 000 000+ data points! The expression

More information

Linear Regression and Its Applications

Linear Regression and Its Applications Linear Regression and Its Applications Predrag Radivojac October 13, 2014 Given a data set D = {(x i, y i )} n the objective is to learn the relationship between features and the target. We usually start

More information

Second-Order Inference for Gaussian Random Curves

Second-Order Inference for Gaussian Random Curves Second-Order Inference for Gaussian Random Curves With Application to DNA Minicircles Victor Panaretos David Kraus John Maddocks Ecole Polytechnique Fédérale de Lausanne Panaretos, Kraus, Maddocks (EPFL)

More information

Polynomial Chaos and Karhunen-Loeve Expansion

Polynomial Chaos and Karhunen-Loeve Expansion Polynomial Chaos and Karhunen-Loeve Expansion 1) Random Variables Consider a system that is modeled by R = M(x, t, X) where X is a random variable. We are interested in determining the probability of the

More information

Economics 620, Lecture 5: exp

Economics 620, Lecture 5: exp 1 Economics 620, Lecture 5: The K-Variable Linear Model II Third assumption (Normality): y; q(x; 2 I N ) 1 ) p(y) = (2 2 ) exp (N=2) 1 2 2(y X)0 (y X) where N is the sample size. The log likelihood function

More information

EIGENVALUES AND SINGULAR VALUE DECOMPOSITION

EIGENVALUES AND SINGULAR VALUE DECOMPOSITION APPENDIX B EIGENVALUES AND SINGULAR VALUE DECOMPOSITION B.1 LINEAR EQUATIONS AND INVERSES Problems of linear estimation can be written in terms of a linear matrix equation whose solution provides the required

More information

Multivariate Statistical Analysis

Multivariate 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 information

A Least Squares Formulation for Canonical Correlation Analysis

A Least Squares Formulation for Canonical Correlation Analysis A Least Squares Formulation for Canonical Correlation Analysis Liang Sun, Shuiwang Ji, and Jieping Ye Department of Computer Science and Engineering Arizona State University Motivation Canonical Correlation

More information

be a Householder matrix. Then prove the followings H = I 2 uut Hu = (I 2 uu u T u )u = u 2 uut u

be a Householder matrix. Then prove the followings H = I 2 uut Hu = (I 2 uu u T u )u = u 2 uut u MATH 434/534 Theoretical Assignment 7 Solution Chapter 7 (71) Let H = I 2uuT Hu = u (ii) Hv = v if = 0 be a Householder matrix Then prove the followings H = I 2 uut Hu = (I 2 uu )u = u 2 uut u = u 2u =

More information

Linear Algebra Review. Vectors

Linear Algebra Review. Vectors Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors

More information

PCA and admixture models

PCA and admixture models PCA and admixture models CM226: Machine Learning for Bioinformatics. Fall 2016 Sriram Sankararaman Acknowledgments: Fei Sha, Ameet Talwalkar, Alkes Price PCA and admixture models 1 / 57 Announcements HW1

More information

Large Scale Data Analysis Using Deep Learning

Large Scale Data Analysis Using Deep Learning Large Scale Data Analysis Using Deep Learning Linear Algebra U Kang Seoul National University U Kang 1 In This Lecture Overview of linear algebra (but, not a comprehensive survey) Focused on the subset

More information

1 Vectors. Notes for Bindel, Spring 2017 Numerical Analysis (CS 4220)

1 Vectors. Notes for Bindel, Spring 2017 Numerical Analysis (CS 4220) Notes for 2017-01-30 Most of mathematics is best learned by doing. Linear algebra is no exception. You have had a previous class in which you learned the basics of linear algebra, and you will have plenty

More information

EE731 Lecture Notes: Matrix Computations for Signal Processing

EE731 Lecture Notes: Matrix Computations for Signal Processing EE731 Lecture Notes: Matrix Computations for Signal Processing James P. Reilly c Department of Electrical and Computer Engineering McMaster University October 17, 005 Lecture 3 3 he Singular Value Decomposition

More information

Bare minimum on matrix algebra. Psychology 588: Covariance structure and factor models

Bare minimum on matrix algebra. Psychology 588: Covariance structure and factor models Bare minimum on matrix algebra Psychology 588: Covariance structure and factor models Matrix multiplication 2 Consider three notations for linear combinations y11 y1 m x11 x 1p b11 b 1m y y x x b b n1

More information

Linear Algebra (Review) Volker Tresp 2017

Linear Algebra (Review) Volker Tresp 2017 Linear Algebra (Review) Volker Tresp 2017 1 Vectors k is a scalar (a number) c is a column vector. Thus in two dimensions, c = ( c1 c 2 ) (Advanced: More precisely, a vector is defined in a vector space.

More information

. a m1 a mn. a 1 a 2 a = a n

. a m1 a mn. a 1 a 2 a = a n Biostat 140655, 2008: Matrix Algebra Review 1 Definition: An m n matrix, A m n, is a rectangular array of real numbers with m rows and n columns Element in the i th row and the j th column is denoted by

More information

E = UV W (9.1) = I Q > V W

E = UV W (9.1) = I Q > V W 91 9. EOFs, SVD A common statistical tool in oceanography, meteorology and climate research are the so-called empirical orthogonal functions (EOFs). Anyone, in any scientific field, working with large

More information

1 Linear Algebra Problems

1 Linear Algebra Problems Linear Algebra Problems. Let A be the conjugate transpose of the complex matrix A; i.e., A = A t : A is said to be Hermitian if A = A; real symmetric if A is real and A t = A; skew-hermitian if A = A and

More information

Linear Algebra (Review) Volker Tresp 2018

Linear Algebra (Review) Volker Tresp 2018 Linear Algebra (Review) Volker Tresp 2018 1 Vectors k, M, N are scalars A one-dimensional array c is a column vector. Thus in two dimensions, ( ) c1 c = c 2 c i is the i-th component of c c T = (c 1, c

More information

https://goo.gl/kfxweg KYOTO UNIVERSITY Statistical Machine Learning Theory Sparsity Hisashi Kashima kashima@i.kyoto-u.ac.jp DEPARTMENT OF INTELLIGENCE SCIENCE AND TECHNOLOGY 1 KYOTO UNIVERSITY Topics:

More information

Channel capacity estimation using free probability theory

Channel capacity estimation using free probability theory Channel capacity estimation using free probability theory January 008 Problem at hand The capacity per receiving antenna of a channel with n m channel matrix H and signal to noise ratio ρ = 1 σ is given

More information

Structure in Data. A major objective in data analysis is to identify interesting features or structure in the data.

Structure in Data. A major objective in data analysis is to identify interesting features or structure in the data. Structure in Data A major objective in data analysis is to identify interesting features or structure in the data. The graphical methods are very useful in discovering structure. There are basically two

More information

2. Review of Linear Algebra

2. Review of Linear Algebra 2. Review of Linear Algebra ECE 83, Spring 217 In this course we will represent signals as vectors and operators (e.g., filters, transforms, etc) as matrices. This lecture reviews basic concepts from linear

More information

MLCC 2015 Dimensionality Reduction and PCA

MLCC 2015 Dimensionality Reduction and PCA MLCC 2015 Dimensionality Reduction and PCA Lorenzo Rosasco UNIGE-MIT-IIT June 25, 2015 Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component

More information

Review of Some Concepts from Linear Algebra: Part 2

Review of Some Concepts from Linear Algebra: Part 2 Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set

More information

Concentration Ellipsoids

Concentration Ellipsoids Concentration Ellipsoids ECE275A Lecture Supplement Fall 2008 Kenneth Kreutz Delgado Electrical and Computer Engineering Jacobs School of Engineering University of California, San Diego VERSION LSECE275CE

More information

STAT 100C: Linear models

STAT 100C: Linear models STAT 100C: Linear models Arash A. Amini April 27, 2018 1 / 1 Table of Contents 2 / 1 Linear Algebra Review Read 3.1 and 3.2 from text. 1. Fundamental subspace (rank-nullity, etc.) Im(X ) = ker(x T ) R

More information

Independent Component Analysis and Its Application on Accelerator Physics

Independent Component Analysis and Its Application on Accelerator Physics Independent Component Analysis and Its Application on Accelerator Physics Xiaoying Pang LA-UR-12-20069 ICA and PCA Similarities: Blind source separation method (BSS) no model Observed signals are linear

More information

Reduced rank regression in cointegrated models

Reduced rank regression in cointegrated models Journal of Econometrics 06 (2002) 203 26 www.elsevier.com/locate/econbase Reduced rank regression in cointegrated models.w. Anderson Department of Statistics, Stanford University, Stanford, CA 94305-4065,

More information

Throughout these notes we assume V, W are finite dimensional inner product spaces over C.

Throughout these notes we assume V, W are finite dimensional inner product spaces over C. Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal

More information

Least-Squares Rigid Motion Using SVD

Least-Squares Rigid Motion Using SVD Least-Squares igid Motion Using SVD Olga Sorkine Abstract his note summarizes the steps to computing the rigid transformation that aligns two sets of points Key words: Shape matching, rigid alignment,

More information

5. Random Vectors. probabilities. characteristic function. cross correlation, cross covariance. Gaussian random vectors. functions of random vectors

5. Random Vectors. probabilities. characteristic function. cross correlation, cross covariance. Gaussian random vectors. functions of random vectors EE401 (Semester 1) 5. Random Vectors Jitkomut Songsiri probabilities characteristic function cross correlation, cross covariance Gaussian random vectors functions of random vectors 5-1 Random vectors we

More information

Numerical Methods I: Eigenvalues and eigenvectors

Numerical Methods I: Eigenvalues and eigenvectors 1/25 Numerical Methods I: Eigenvalues and eigenvectors Georg Stadler Courant Institute, NYU stadler@cims.nyu.edu November 2, 2017 Overview 2/25 Conditioning Eigenvalues and eigenvectors How hard are they

More information

q-alg/ Mar 96

q-alg/ Mar 96 Integrality of Two Variable Kostka Functions Friedrich Knop* Department of Mathematics, Rutgers University, New Brunswick NJ 08903, USA knop@math.rutgers.edu 1. Introduction q-alg/9603027 29 Mar 96 Macdonald

More information

2 Tikhonov Regularization and ERM

2 Tikhonov Regularization and ERM Introduction Here we discusses how a class of regularization methods originally designed to solve ill-posed inverse problems give rise to regularized learning algorithms. These algorithms are kernel methods

More information

CS 340 Lec. 6: Linear Dimensionality Reduction

CS 340 Lec. 6: Linear Dimensionality Reduction CS 340 Lec. 6: Linear Dimensionality Reduction AD January 2011 AD () January 2011 1 / 46 Linear Dimensionality Reduction Introduction & Motivation Brief Review of Linear Algebra Principal Component Analysis

More information

Linear Least Squares. Using SVD Decomposition.

Linear Least Squares. Using SVD Decomposition. Linear Least Squares. Using SVD Decomposition. Dmitriy Leykekhman Spring 2011 Goals SVD-decomposition. Solving LLS with SVD-decomposition. D. Leykekhman Linear Least Squares 1 SVD Decomposition. For any

More information

Karhunen-Loève decomposition of Gaussian measures on Banach spaces

Karhunen-Loève decomposition of Gaussian measures on Banach spaces Karhunen-Loève decomposition of Gaussian measures on Banach spaces Jean-Charles Croix GT APSSE - April 2017, the 13th joint work with Xavier Bay. 1 / 29 Sommaire 1 Preliminaries on Gaussian processes 2

More information

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................

More information

Probabilistic Latent Semantic Analysis

Probabilistic Latent Semantic Analysis Probabilistic Latent Semantic Analysis Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr

More information

Principal Component Analysis

Principal Component Analysis Principal Component Analysis November 24, 2015 From data to operators Given is data set X consisting of N vectors x n R D. Without loss of generality, assume x n = 0 (subtract mean). Let P be D N matrix

More information

Lecture 6: Methods for high-dimensional problems

Lecture 6: Methods for high-dimensional problems Lecture 6: Methods for high-dimensional problems Hector Corrada Bravo and Rafael A. Irizarry March, 2010 In this Section we will discuss methods where data lies on high-dimensional spaces. In particular,

More information

Positive Definite Matrix

Positive Definite Matrix 1/29 Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Positive Definite, Negative Definite, Indefinite 2/29 Pure Quadratic Function

More information

Theorem A.1. If A is any nonzero m x n matrix, then A is equivalent to a partitioned matrix of the form. k k n-k. m-k k m-k n-k

Theorem A.1. If A is any nonzero m x n matrix, then A is equivalent to a partitioned matrix of the form. k k n-k. m-k k m-k n-k I. REVIEW OF LINEAR ALGEBRA A. Equivalence Definition A1. If A and B are two m x n matrices, then A is equivalent to B if we can obtain B from A by a finite sequence of elementary row or elementary column

More information

Linear Algebra - Part II

Linear Algebra - Part II Linear Algebra - Part II Projection, Eigendecomposition, SVD (Adapted from Sargur Srihari s slides) Brief Review from Part 1 Symmetric Matrix: A = A T Orthogonal Matrix: A T A = AA T = I and A 1 = A T

More information

Eigenvalue problems and optimization

Eigenvalue problems and optimization Notes for 2016-04-27 Seeking structure For the past three weeks, we have discussed rather general-purpose optimization methods for nonlinear equation solving and optimization. In practice, of course, we

More information

Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques

Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques Institut für Numerische Mathematik und Optimierung Karhunen-Loève Approximation of Random Fields Using Hierarchical Matrix Techniques Oliver Ernst Computational Methods with Applications Harrachov, CR,

More information

Data Mining Techniques

Data Mining Techniques Data Mining Techniques CS 6220 - Section 3 - Fall 2016 Lecture 12 Jan-Willem van de Meent (credit: Yijun Zhao, Percy Liang) DIMENSIONALITY REDUCTION Borrowing from: Percy Liang (Stanford) Linear Dimensionality

More information

Quick Tour of Linear Algebra and Graph Theory

Quick Tour of Linear Algebra and Graph Theory Quick Tour of Linear Algebra and Graph Theory CS224W: Social and Information Network Analysis Fall 2014 David Hallac Based on Peter Lofgren, Yu Wayne Wu, and Borja Pelato s previous versions Matrices and

More information

Lecture 13. Principal Component Analysis. Brett Bernstein. April 25, CDS at NYU. Brett Bernstein (CDS at NYU) Lecture 13 April 25, / 26

Lecture 13. Principal Component Analysis. Brett Bernstein. April 25, CDS at NYU. Brett Bernstein (CDS at NYU) Lecture 13 April 25, / 26 Principal Component Analysis Brett Bernstein CDS at NYU April 25, 2017 Brett Bernstein (CDS at NYU) Lecture 13 April 25, 2017 1 / 26 Initial Question Intro Question Question Let S R n n be symmetric. 1

More information

Lecture 1: Review of linear algebra

Lecture 1: Review of linear algebra Lecture 1: Review of linear algebra Linear functions and linearization Inverse matrix, least-squares and least-norm solutions Subspaces, basis, and dimension Change of basis and similarity transformations

More information

LinGloss. A glossary of linear algebra

LinGloss. A glossary of linear algebra LinGloss A glossary of linear algebra Contents: Decompositions Types of Matrices Theorems Other objects? Quasi-triangular A matrix A is quasi-triangular iff it is a triangular matrix except its diagonal

More information

Eigenvalues and diagonalization

Eigenvalues and diagonalization Eigenvalues and diagonalization Patrick Breheny November 15 Patrick Breheny BST 764: Applied Statistical Modeling 1/20 Introduction The next topic in our course, principal components analysis, revolves

More information

. Consider the linear system dx= =! = " a b # x y! : (a) For what values of a and b do solutions oscillate (i.e., do both x(t) and y(t) pass through z

. Consider the linear system dx= =! =  a b # x y! : (a) For what values of a and b do solutions oscillate (i.e., do both x(t) and y(t) pass through z Preliminary Exam { 1999 Morning Part Instructions: No calculators or crib sheets are allowed. Do as many problems as you can. Justify your answers as much as you can but very briey. 1. For positive real

More information

Chapter 7: Symmetric Matrices and Quadratic Forms

Chapter 7: Symmetric Matrices and Quadratic Forms Chapter 7: Symmetric Matrices and Quadratic Forms (Last Updated: December, 06) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). A few theorems have been moved

More information

Contents. Acknowledgments

Contents. Acknowledgments Table of Preface Acknowledgments Notation page xii xx xxi 1 Signals and systems 1 1.1 Continuous and discrete signals 1 1.2 Unit step and nascent delta functions 4 1.3 Relationship between complex exponentials

More information

Preliminary Examination, Numerical Analysis, August 2016

Preliminary Examination, Numerical Analysis, August 2016 Preliminary Examination, Numerical Analysis, August 2016 Instructions: This exam is closed books and notes. The time allowed is three hours and you need to work on any three out of questions 1-4 and any

More information

Math Camp Notes: Linear Algebra I

Math Camp Notes: Linear Algebra I Math Camp Notes: Linear Algebra I Basic Matrix Operations and Properties Consider two n m matrices: a a m A = a n a nm Then the basic matrix operations are as follows: a + b a m + b m A + B = a n + b n

More information

Knowledge Discovery and Data Mining 1 (VO) ( )

Knowledge Discovery and Data Mining 1 (VO) ( ) Knowledge Discovery and Data Mining 1 (VO) (707.003) Review of Linear Algebra Denis Helic KTI, TU Graz Oct 9, 2014 Denis Helic (KTI, TU Graz) KDDM1 Oct 9, 2014 1 / 74 Big picture: KDDM Probability Theory

More information

Unsupervised Machine Learning and Data Mining. DS 5230 / DS Fall Lecture 7. Jan-Willem van de Meent

Unsupervised Machine Learning and Data Mining. DS 5230 / DS Fall Lecture 7. Jan-Willem van de Meent Unsupervised Machine Learning and Data Mining DS 5230 / DS 4420 - Fall 2018 Lecture 7 Jan-Willem van de Meent DIMENSIONALITY REDUCTION Borrowing from: Percy Liang (Stanford) Dimensionality Reduction Goal:

More information

= a. a = Let us now study what is a? c ( a A a )

= a. a = Let us now study what is a? c ( a A a ) 7636S ADVANCED QUANTUM MECHANICS Solutions 1 Spring 010 1 Warm up a Show that the eigenvalues of a Hermitian operator A are real and that the eigenkets of A corresponding to dierent eigenvalues are orthogonal

More information

Calculus and linear algebra for biomedical engineering Week 3: Matrices, linear systems of equations, and the Gauss algorithm

Calculus and linear algebra for biomedical engineering Week 3: Matrices, linear systems of equations, and the Gauss algorithm Calculus and linear algebra for biomedical engineering Week 3: Matrices, linear systems of equations, and the Gauss algorithm Hartmut Führ fuehr@matha.rwth-aachen.de Lehrstuhl A für Mathematik, RWTH Aachen

More information

Frank C Porter and Ilya Narsky: Statistical Analysis Techniques in Particle Physics Chap. c /9/9 page 147 le-tex

Frank C Porter and Ilya Narsky: Statistical Analysis Techniques in Particle Physics Chap. c /9/9 page 147 le-tex Frank C Porter and Ilya Narsky: Statistical Analysis Techniques in Particle Physics Chap. c08 2013/9/9 page 147 le-tex 8.3 Principal Component Analysis (PCA) 147 Figure 8.1 Principal and independent components

More information

Principal Component Analysis and Singular Value Decomposition. Volker Tresp, Clemens Otte Summer 2014

Principal Component Analysis and Singular Value Decomposition. Volker Tresp, Clemens Otte Summer 2014 Principal Component Analysis and Singular Value Decomposition Volker Tresp, Clemens Otte Summer 2014 1 Motivation So far we always argued for a high-dimensional feature space Still, in some cases it makes

More information

Signal Analysis. Principal Component Analysis

Signal Analysis. Principal Component Analysis Multi dimensional Signal Analysis Lecture 2E Principal Component Analysis Subspace representation Note! Given avector space V of dimension N a scalar product defined by G 0 a subspace U of dimension M

More information

PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS

PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS PROOF OF TWO MATRIX THEOREMS VIA TRIANGULAR FACTORIZATIONS ROY MATHIAS Abstract. We present elementary proofs of the Cauchy-Binet Theorem on determinants and of the fact that the eigenvalues of a matrix

More information

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T.

Notes on singular value decomposition for Math 54. Recall that if A is a symmetric n n matrix, then A has real eigenvalues A = P DP 1 A = P DP T. Notes on singular value decomposition for Math 54 Recall that if A is a symmetric n n matrix, then A has real eigenvalues λ 1,, λ n (possibly repeated), and R n has an orthonormal basis v 1,, v n, where

More information

Singular Value Decompsition

Singular Value Decompsition Singular Value Decompsition Massoud Malek One of the most useful results from linear algebra, is a matrix decomposition known as the singular value decomposition It has many useful applications in almost

More information

Image Registration Lecture 2: Vectors and Matrices

Image Registration Lecture 2: Vectors and Matrices Image Registration Lecture 2: Vectors and Matrices Prof. Charlene Tsai Lecture Overview Vectors Matrices Basics Orthogonal matrices Singular Value Decomposition (SVD) 2 1 Preliminary Comments Some of this

More information

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition

Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing

More information

Economics 472. Lecture 10. where we will refer to y t as a m-vector of endogenous variables, x t as a q-vector of exogenous variables,

Economics 472. Lecture 10. where we will refer to y t as a m-vector of endogenous variables, x t as a q-vector of exogenous variables, University of Illinois Fall 998 Department of Economics Roger Koenker Economics 472 Lecture Introduction to Dynamic Simultaneous Equation Models In this lecture we will introduce some simple dynamic simultaneous

More information

FINM 331: MULTIVARIATE DATA ANALYSIS FALL 2017 PROBLEM SET 3

FINM 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 information

Assignment #10: Diagonalization of Symmetric Matrices, Quadratic Forms, Optimization, Singular Value Decomposition. Name:

Assignment #10: Diagonalization of Symmetric Matrices, Quadratic Forms, Optimization, Singular Value Decomposition. Name: Assignment #10: Diagonalization of Symmetric Matrices, Quadratic Forms, Optimization, Singular Value Decomposition Due date: Friday, May 4, 2018 (1:35pm) Name: Section Number Assignment #10: Diagonalization

More information

Outline Introduction: Problem Description Diculties Algebraic Structure: Algebraic Varieties Rank Decient Toeplitz Matrices Constructing Lower Rank St

Outline Introduction: Problem Description Diculties Algebraic Structure: Algebraic Varieties Rank Decient Toeplitz Matrices Constructing Lower Rank St Structured Lower Rank Approximation by Moody T. Chu (NCSU) joint with Robert E. Funderlic (NCSU) and Robert J. Plemmons (Wake Forest) March 5, 1998 Outline Introduction: Problem Description Diculties Algebraic

More information

The purpose of this section is to derive the asymptotic distribution of the Pearson chi-square statistic. k (n j np j ) 2. np j.

The purpose of this section is to derive the asymptotic distribution of the Pearson chi-square statistic. k (n j np j ) 2. np j. Chapter 9 Pearson s chi-square test 9. Null hypothesis asymptotics Let X, X 2, be independent from a multinomial(, p) distribution, where p is a k-vector with nonnegative entries that sum to one. That

More information

(Linear equations) Applied Linear Algebra in Geoscience Using MATLAB

(Linear equations) Applied Linear Algebra in Geoscience Using MATLAB Applied Linear Algebra in Geoscience Using MATLAB (Linear equations) Contents Getting Started Creating Arrays Mathematical Operations with Arrays Using Script Files and Managing Data Two-Dimensional Plots

More information

MATH 3795 Lecture 10. Regularized Linear Least Squares.

MATH 3795 Lecture 10. Regularized Linear Least Squares. MATH 3795 Lecture 10. Regularized Linear Least Squares. Dmitriy Leykekhman Fall 2008 Goals Understanding the regularization. D. Leykekhman - MATH 3795 Introduction to Computational Mathematics Linear Least

More information

CS 143 Linear Algebra Review

CS 143 Linear Algebra Review CS 143 Linear Algebra Review Stefan Roth September 29, 2003 Introductory Remarks This review does not aim at mathematical rigor very much, but instead at ease of understanding and conciseness. Please see

More information

G METHOD IN ACTION: FROM EXACT SAMPLING TO APPROXIMATE ONE

G METHOD IN ACTION: FROM EXACT SAMPLING TO APPROXIMATE ONE G METHOD IN ACTION: FROM EXACT SAMPLING TO APPROXIMATE ONE UDREA PÄUN Communicated by Marius Iosifescu The main contribution of this work is the unication, by G method using Markov chains, therefore, a

More information

Solving Large Nonlinear Sparse Systems

Solving Large Nonlinear Sparse Systems Solving Large Nonlinear Sparse Systems Fred W. Wubs and Jonas Thies Computational Mechanics & Numerical Mathematics University of Groningen, the Netherlands f.w.wubs@rug.nl Centre for Interdisciplinary

More information

A TOUR OF LINEAR ALGEBRA FOR JDEP 384H

A TOUR OF LINEAR ALGEBRA FOR JDEP 384H A TOUR OF LINEAR ALGEBRA FOR JDEP 384H Contents Solving Systems 1 Matrix Arithmetic 3 The Basic Rules of Matrix Arithmetic 4 Norms and Dot Products 5 Norms 5 Dot Products 6 Linear Programming 7 Eigenvectors

More information

Math 314 Lecture Notes Section 006 Fall 2006

Math 314 Lecture Notes Section 006 Fall 2006 Math 314 Lecture Notes Section 006 Fall 2006 CHAPTER 1 Linear Systems of Equations First Day: (1) Welcome (2) Pass out information sheets (3) Take roll (4) Open up home page and have students do same

More information

Lecture 02 Linear Algebra Basics

Lecture 02 Linear Algebra Basics Introduction to Computational Data Analysis CX4240, 2019 Spring Lecture 02 Linear Algebra Basics Chao Zhang College of Computing Georgia Tech These slides are based on slides from Le Song and Andres Mendez-Vazquez.

More information

A Statistical Look at Spectral Graph Analysis. Deep Mukhopadhyay

A Statistical Look at Spectral Graph Analysis. Deep Mukhopadhyay A Statistical Look at Spectral Graph Analysis Deep Mukhopadhyay Department of Statistics, Temple University Office: Speakman 335 deep@temple.edu http://sites.temple.edu/deepstat/ Graph Signal Processing

More information

Def. The euclidian distance between two points x = (x 1,...,x p ) t and y = (y 1,...,y p ) t in the p-dimensional space R p is defined as

Def. The euclidian distance between two points x = (x 1,...,x p ) t and y = (y 1,...,y p ) t in the p-dimensional space R p is defined as MAHALANOBIS DISTANCE Def. The euclidian distance between two points x = (x 1,...,x p ) t and y = (y 1,...,y p ) t in the p-dimensional space R p is defined as d E (x, y) = (x 1 y 1 ) 2 + +(x p y p ) 2

More information

Linear Algebra Methods for Data Mining

Linear Algebra Methods for Data Mining Linear Algebra Methods for Data Mining Saara Hyvönen, Saara.Hyvonen@cs.helsinki.fi Spring 2007 The Singular Value Decomposition (SVD) continued Linear Algebra Methods for Data Mining, Spring 2007, University

More information

Linear Algebra Methods for Data Mining

Linear Algebra Methods for Data Mining Linear Algebra Methods for Data Mining Saara Hyvönen, Saara.Hyvonen@cs.helsinki.fi Spring 2007 Linear Discriminant Analysis Linear Algebra Methods for Data Mining, Spring 2007, University of Helsinki Principal

More information

hapter 8 Simulation/Realization 8 Introduction Given an nth-order state-space description of the form x_ (t) = f (x(t) u(t) t) (state evolution equati

hapter 8 Simulation/Realization 8 Introduction Given an nth-order state-space description of the form x_ (t) = f (x(t) u(t) t) (state evolution equati Lectures on Dynamic Systems and ontrol Mohammed Dahleh Munther Dahleh George Verghese Department of Electrical Engineering and omputer Science Massachuasetts Institute of Technology c hapter 8 Simulation/Realization

More information

Lecture 8. Principal Component Analysis. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. December 13, 2016

Lecture 8. Principal Component Analysis. Luigi Freda. ALCOR Lab DIAG University of Rome La Sapienza. December 13, 2016 Lecture 8 Principal Component Analysis Luigi Freda ALCOR Lab DIAG University of Rome La Sapienza December 13, 2016 Luigi Freda ( La Sapienza University) Lecture 8 December 13, 2016 1 / 31 Outline 1 Eigen

More information

Data Mining and Analysis: Fundamental Concepts and Algorithms

Data Mining and Analysis: Fundamental Concepts and Algorithms : Fundamental Concepts and Algorithms dataminingbook.info Mohammed J. Zaki 1 Wagner Meira Jr. 2 1 Department of Computer Science Rensselaer Polytechnic Institute, Troy, NY, USA 2 Department of Computer

More information

Statistical Learning & Applications. f w (x) =< f w, K x > H = w T x. α i α j < x i x T i, x j x T j. = < α i x i x T i, α j x j x T j > F

Statistical Learning & Applications. f w (x) =< f w, K x > H = w T x. α i α j < x i x T i, x j x T j. = < α i x i x T i, α j x j x T j > F CR2: Statistical Learning & Applications Examples of Kernels and Unsupervised Learning Lecturer: Julien Mairal Scribes: Rémi De Joannis de Verclos & Karthik Srikanta Kernel Inventory Linear Kernel The

More information

Principal Component Analysis

Principal Component Analysis I.T. Jolliffe Principal Component Analysis Second Edition With 28 Illustrations Springer Contents Preface to the Second Edition Preface to the First Edition Acknowledgments List of Figures List of Tables

More information