Matrix Decomposition and its Application in Statistics Nishith

Matrix Decomposition and its Application in Statistics Nishith Kumar Lecturer Department of Statistics Begum Rokeya University, Rangpur. Email: nk. bru 09@gmail. com 1

Overview • • Introduction LU decomposition QR decomposition Cholesky decomposition Jordan Decomposition Spectral decomposition Singular value decomposition Applications 2

Introduction This Lecture covers relevant matrix decompositions, basic numerical methods, its computation and some of its applications. Decompositions provide a numerically stable way to solve a system of linear equations, as shown already in [Wampler, 1970], and to invert a matrix. Additionally, they provide an important tool for analyzing the numerical stability of a system. Some of most frequently used decompositions are the LU, QR, Cholesky, Jordan, Spectral decomposition and Singular value decompositions. 3

Easy to solve system (Cont. ) Some linear system that can be easily solved The solution: 4

Easy to solve system (Cont. ) Lower triangular matrix: Solution: This system is solved using forward substitution 5

Easy to solve system (Cont. ) Upper Triangular Matrix: Solution: This system is solved using Backward substitution 6

LU Decomposition LU decomposition was originally derived as a decomposition of quadratic and bilinear forms. Lagrange, in the very first paper in his collected works( 1759) derives the algorithm we call Gaussian elimination. Later Turing introduced the LU decomposition of a matrix in 1948 that is used to solve the system of linear equation. Let A be a m × m with nonsingular square matrix. Then there exists two matrices L and U such that, where L is a lower triangular matrix and U is an upper triangular matrix. and Where, J-L Lagrange A. M. Turing (1736 – 1813) (1912 -1954) 7

How to decompose A=LU? A … U (upper triangular) U = Ek E 1 A A = (E 1)-1 (Ek)-1 U U E 2 E 1 A If each such elementary matrix Ei is a lower triangular matrices, it can be proved that (E 1)-1, , (Ek)-1 are lower triangular, and (E 1)-1 (Ek)-1 is a lower triangular matrix. Let L=(E 1)-1 (Ek)-1 then A=LU. 8

Calculation of L and U (cont. ) = Now reducing the first column we have 9

Calculation of L and U (cont. ) Now Therefore, = =LU If A is a Non singular matrix then for each L (lower triangular matrix) the upper triangular matrix is unique but an LU decomposition is not unique. There can be more than one such LU decomposition for a matrix. Such as = =LU 10

Calculationof of. LLand U U (cont. ) Thus LU decomposition is not unique. Since we compute LU decomposition by elementary transformation so if we change L then U will be changed such that A=LU To find out the unique LU decomposition, it is necessary to put some restriction on L and U matrices. For example, we can require the lower triangular matrix L to be a unit one (i. e. set all the entries of its main diagonal to ones). LU Decomposition in R: • library(Matrix) • x<-matrix(c(3, 2, 1, 9, 3, 4, 4, 2, 5 ), ncol=3, nrow=3) • expand(lu(x)) 11

Calculation of L and U (cont. ) • Note: there also generalizations of LU to non-square and singular matrices, such as rank revealing LU factorization. • [Pan, C. T. (2000). On the existence and computation of rank revealing LU factorizations. Linear Algebra and its Applications, 316: 199 -222. • Miranian, L. and Gu, M. (2003). Strong rank revealing LU factorizations. Linear Algebra and its Applications, 367: 1 -16. ] • Uses: The LU decomposition is most commonly used in the solution of systems of simultaneous linear equations. We can also find determinant easily by using LU decomposition (Product of the diagonal element of upper and lower triangular matrix). 12

Solving system of linear equation using LU decomposition Suppose we would like to solve a m×m system AX = b. Then we can find a LU-decomposition for A, then to solve AX =b, it is enough to solve the systems Thus the system LY = b can be solved by the method of forward substitution and the system UX = Y can be solved by the method of backward substitution. To illustrate, we give some examples Consider the given system AX = b, where and 13

Solving system of linear equation using LU decomposition We have seen A = LU, where Thus, to solve AX = b, we first solve LY = b by forward substitution Then 14

Solving system of linear equation using LU decomposition Now, we solve UX =Y by backward substitution then 15

QR Decomposition Jørgen Pedersen Gram (1850 – 1916) Firstly QR decomposition originated with Gram(1883). Later Erhard Schmidt (1907) proved the QR Decomposition Theorem Erhard Schmidt (1876 -1959) If A is a m×n matrix with linearly independent columns, then A can be decomposed as , where Q is a m×n matrix whose columns form an orthonormal basis for the column space of A and R is an nonsingular upper triangular matrix. 16

QR-Decomposition (Cont. ) Theorem : If A is a m×n matrix with linearly independent columns, then A can be decomposed as , where Q is a m×n matrix whose columns form an orthonormal basis for the column space of A and R is an nonsingular upper triangular matrix. Proof: Suppose A=[u 1 | u 2|. . . | un] and rank (A) = n. Apply the Gram-Schmidt process to {u 1, u 2 , . . . , un} and the orthogonal vectors v 1, v 2 , . . . , vn are Let for i=1, 2, . . . , n. Thus q 1, q 2 , . . . , qn form a orthonormal basis for the column space of A. 17

QR-Decomposition (Cont. ) Now, i. e. , Thus ui is orthogonal to qj for j>i; 18

QR-Decomposition (Cont. ) Let Q= [q 1 q 2. . . qn] , so Q is a m×n matrix whose columns form an orthonormal basis for the column space of A. Now, i. e. , A=QR. Where, Thus A can be decomposed as A=QR , where R is an upper triangular and nonsingular matrix. 19

QR Decomposition Example: Find the QR decomposition of 20

Calculation of QR Decomposition Applying Gram-Schmidt process of computing QR decomposition 1 st Step: 2 nd Step: 3 rd Step: 21

Calculation of QR Decomposition 4 th Step: 5 th Step: 6 th Step: 22

Calculation of QR Decomposition Therefore, A=QR R code for QR Decomposition: x<-matrix(c(1, 2, 3, 2, 5, 4, 3, 4, 9), ncol=3, nrow=3) qrstr <- qr(x) Q<-qr. Q(qrstr) R<-qr. R(qrstr) Uses: QR decomposition is widely used in computer codes to find the eigenvalues of a matrix, to solve linear systems, and to find least squares approximations. 23

Least square solution using QR Decomposition The least square solution of b is Let X=QR. Then Therefore, 24

Cholesky Decomposition Cholesky died from wounds received on the battle field on 31 August 1918 at 5 o'clock in the morning in the North of France. After his death one of his fellow officers, Commandant Benoit, published Cholesky's method of computing solutions to the normal equations for some least squares data fitting problems published in the Bulletin géodesique in 1924. Which is known as Cholesky Decomposition Cholesky Decomposition: If A is a real, symmetric and positive definite matrix then there exists a unique lower triangular matrix L with positive diagonal element such that . Andre-Louis Cholesky 1875 -1918 25

Cholesky Decomposition Theorem: If A is a n×n real, symmetric and positive definite matrix then there exists a unique lower triangular matrix G with positive diagonal element such that . Proof: Since A is a n×n real and positive definite so it has a LU decomposition, A=LU. Also let the lower triangular matrix L to be a unit one (i. e. set all the entries of its main diagonal to ones). So in that case LU decomposition is unique. Let us suppose observe that . is a unit upper triangular matrix. Thus, A=LDMT. Since A is Symmetric so, A=AT. i. e. , LDMT =MDLT. From the uniqueness we have L=M. So, A=LDLT. Since A is positive definite so all diagonal elements of D are positive. Let then we can write A=GGT. 26

Cholesky Decomposition (Cont. ) Procedure To find out the cholesky decomposition Suppose We need to solve the equation 27

Example of Cholesky Decomposition Suppose For k from 1 to n Then Cholesky Decomposition For j from k+1 to n Now, 28

R code for Cholesky Decomposition • x<-matrix(c(4, 2, -2, 2, 10, 2, -2, 2, 5), ncol=3, nrow=3) • cl<-chol(x) • If we Decompose A as LDLT then and 29

Application of Cholesky Decomposition is used to solve the system of linear equation Ax=b, where A is real symmetric and positive definite. In regression analysis it could be used to estimate the parameter if XTX is positive definite. In Kernel principal component analysis, Cholesky decomposition is also used (Weiya Shi; Yue-Fei Guo; 2010) 30

Characteristic Roots and Characteristics Vectors Any nonzero vector x is said to be a characteristic vector of a matrix A, If there exist a number λ such that Ax= λx; Where A is a square matrix, also then λ is said to be a characteristic root of the matrix A corresponding to the characteristic vector x. Characteristic root is unique but characteristic vector is not unique. We calculate characteristics root λ from the characteristic equation |A- λI|=0 For λ= λi the characteristics vector is the solution of x from the following homogeneous system of linear equation (A- λi. I)x=0 Theorem: If A is a real symmetric matrix and λi and λj are two distinct latent root of A then the corresponding latent vector xi and xj are orthogonal. 31

Multiplicity Algebraic Multiplicity: The number of repetitions of a certain eigenvalue. If, for a certain matrix, λ={3, 3, 4}, then the algebraic multiplicity of 3 would be 2 (as it appears twice) and the algebraic multiplicity of 4 would be 1 (as it appears once). This type of multiplicity is normally represented by the Greek letter α, where α(λi) represents the algebraic multiplicity of λi. Geometric Multiplicity: the geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors associated with it. 32

Jordan Decomposition Camille Jordan (1870) • Let A be any n×n matrix then there exists a nonsingular matrix P and JK(λ) a k×k matrix form Such that Camille Jordan (1838 -1921) where k 1+k 2+ … + kr =n. Also λi , i=1, 2, . . . , r are the characteristic roots And ki are the algebraic multiplicity of λi , Jordan Decomposition is used in Differential equation and time series analysis. 33

Spectral Decomposition A. L. Cauchy established the Spectral Decomposition in 1829. CAUCHY, A. L. (1789 -1857) Let A be a m × m real symmetric matrix. Then there exists an orthogonal matrix P such that or , where Λ is a diagonal matrix. 34

Spectral Decomposition and Principal component Analysis (Cont. ) By using spectral decomposition we can write In multivariate analysis our data is a matrix. Suppose our data is X matrix. Suppose X is mean centered i. e. , and the variance covariance matrix is ∑. The variance covariance matrix ∑ is real and symmetric. Using spectral decomposition we can write ∑=PΛPT. Where Λ is a diagonal matrix. Also tr(∑) = Total variation of Data =tr(Λ) 35

Spectral Decomposition and Principal component Analysis (Cont. ) The Principal component transformation is the transformation Y=(X-µ)P Where, v E(Yi)=0 v V(Yi)=λi v Cov(Yi , Yj)=0 if i ≠ j v V(Y 1) ≥ V(Y 2) ≥. . . ≥ V(Yn) v v 36

R code for Spectral Decomposition x<-matrix(c(1, 2, 3, 2, 5, 4, 3, 4, 9), ncol=3, nrow=3) eigen(x) Application: Ø For Data Reduction. Ø Image Processing and Compression. Ø K-Selection for K-means clustering Ø Multivariate Outliers Detection Ø Noise Filtering Ø Trend detection in the observations. 37

Historical background of SVD There are five mathematicians who were responsible for establishing the existence of the singular value decomposition and developing its theory. Eugenio Beltrami (1835 -1899) Camille Jordan (1838 -1921) James Joseph Sylvester (1814 -1897) Erhard Schmidt (1876 -1959) Hermann Weyl (1885 -1955) The Singular Value Decomposition was originally developed by two mathematician in the mid to late 1800’s 1. Eugenio Beltrami , 2. Camille Jordan Several other mathematicians took part in the final developments of the SVD including James Joseph Sylvester, Erhard Schmidt and Hermann Weyl who studied the SVD into the mid-1900’s. C. Eckart and G. Young prove low rank approximation of SVD (1936). 38 C. Eckart

What is SVD? Any real (m×n) matrix X, where (n≤ m), can be decomposed, X = UΛVT v U is a (m×n) column orthonormal matrix (UTU=I), containing the eigenvectors of the symmetric matrix XXT. v Λ is a (n×n ) diagonal matrix, containing the singular values of matrix X. The number of non zero diagonal elements of Λ corresponds to the rank of X. v VT is a (n×n ) row orthonormal matrix (VTV=I), containing the eigenvectors of the symmetric matrix XTX. 39

Singular Value Decomposition (Cont. ) Theorem (Singular Value Decomposition) : Let X be m×n of rank r, r ≤ n ≤ m. Then there exist matrices U , V and a diagonal matrix Λ , with positive diagonal elements such that, Proof: Since X is m × n of rank r, r ≤ n ≤ m. So XXT and XTX both of rank r ( by using the concept of Grammian matrix ) and of dimension m × m and n × n respectively. Since XXT is real symmetric matrix so we can write by spectral decomposition, Where Q and D are respectively, the matrices of characteristic vectors and corresponding characteristic roots of XXT. Again since XTX is real symmetric matrix so we can write by spectral decomposition, 40

Singular Value Decomposition (Cont. ) Where R is the (orthogonal) matrix of characteristic vectors and M is diagonal matrix of the corresponding characteristic roots. Since XXT and XTX are both of rank r, only r of their characteristic roots are positive, the remaining being zero. Hence we can write, Also we can write, 41

Singular Value Decomposition (Cont. ) We know that the nonzero characteristic roots of XXT and XTX are equal so Partition Q, R conformably with D and M, respectively i. e. , ; such that Qr is m × r , Rr is n × r and correspond respectively to the nonzero characteristic roots of XXT and XTX. Now take Where are the positive characteristic roots of XXT and hence those of XTX as well (by using the concept of 42 grammian matrix. )

Singular Value Decomposition (Cont. ) Now define, Now we shall show that S=X thus completing the proof. Similarly, From the first relation above we conclude that for an arbitrary orthogonal matrix, say P 1 , While from the second we conclude that for an arbitrary orthogonal matrix, say P 2 43 We must have

Singular Value Decomposition (Cont. ) The preceding, however, implies that for arbitrary orthogonal matrices P 1 , P 2 the matrix X satisfies Which in turn implies that, Thus 44

R Code for Singular Value Decomposition x<-matrix(c(1, 2, 3, 2, 5, 4, 3, 4, 9), ncol=3, nrow=3) sv<-svd(x) D<-sv$d U<-sv$u V<-sv$v 45

Decomposition in Diagram Matrix A Full column rank Lu decomposition Not always unique QR Decomposition Rectangular Square Symmetric PD Cholesky Decomposition Asymmetric AM>GM Spectral Decomposition AM=GM Jordan Decomposition SVD Similar Diagonalization P-1 AP=Λ 46

Properties Of SVD Rewriting the SVD where r = rank of A λi = the i-th diagonal element of Λ. ui and vi are the i-th columns of U and V respectively. 47

Proprieties of SVD Low rank Approximation Theorem: If A=UΛVT is the SVD of A and the singular values are sorted as , then for any l <r, the best rank-l approximation to A is ; Low rank approximation technique is very much important for data compression. 48

Low-rank Approximation • SVD can be used to compute optimal low-rank approximations. • Approximation of A is à of rank k such that Frobenius norm If are the characteristics roots of ATA then à and X are both m n matrices. 49

Low-rank Approximation • Solution via SVD set smallest r-k singular values to zero K=2 column notation: sum of rank 1 matrices 50

Approximation error • How good (bad) is this approximation? • It’s the best possible, measured by the Frobenius norm of the error: • where the λi are ordered such that λi λi+1. Now 51

Row approximation and column approximation Suppose Ri and cj represent the i-th row and j-th column of A. The SVD of A and is The SVD equation for Ri is We can approximate Ri by ; l<r where i = 1, …, m. Also the SVD equation for Cj is, where j = 1, 2, …, n We can also approximate Cj by ; l<r 52

Least square solution in inconsistent system By using SVD we can solve the inconsistent system. This gives the least square solution. The least square solution where Ag be the MP inverse of A. 53

The SVD of Ag is This can be written as Where 54

Basic Results of SVD 55

SVD based PCA If we reduced variable by using SVD then it performs like PCA. Suppose X is a mean centered data matrix, Then X using SVD, X=UΛVT we can write- XV = UΛ Suppose Y = XV = UΛ Then the first columns of Y represents the first principal component score and so on. o SVD Based PC is more Numerically Stable. o If no. of variables is greater than no. of observations then SVD based PCA will give efficient result(Antti Niemistö, Statistical Analysis of Gene Expression Microarray Data, 2005) 56

Application of SVD Ø Ø Ø Ø Data Reduction both variables and observations. Solving linear least square Problems Image Processing and Compression. K-Selection for K-means clustering Multivariate Outliers Detection Noise Filtering Trend detection in the observations and the variables. 57

Origin of biplot § § § Gabriel (1971) One of the most important advances in data analysis in recent decades Currently… § > 50, 000 web pages § Numerous academic publications § Included in most statistical analysis packages § Still a very new technique to most scientists Prof. Ruben Gabriel, “The founder of biplot” Courtesy of Prof. Purificación Galindo University of Salamanca, Spain 58

What is a biplot? • “Biplot” = “bi” + “plot” – “plot” • scatter plot of two rows OR of two columns, or • scatter plot summarizing the rows OR the columns – “bi” • BOTH rows AND columns • 1 biplot >> 2 plots 59

Practical definition of a biplot “Any two-way table can be analyzed using a 2 D-biplot as soon as it can be sufficiently approximated by a rank-2 matrix. ” (Gabriel, 1971) (Now 3 D-biplots are also possible…) Matrix decomposition P(4, 3) G(3, 2) E(2, 3) E 1 G 2 G 1 E 2 G 4 E 3 G-by-E table G 3 60

Singular Value Decomposition (SVD) & Singular Value Partitioning (SVP) The ‘rank’ of Y, i. e. , the minimum number of PC required to fully represent Y Matrix characterising the rows “Singular values” SVD: Matrix characterising the columns Common uses value of f f=1 SVP: f=0 f=1/2 Rows scores Plot Column scores Biplot Plot 61

Biplot v The simplest biplot is to show the first two PCs together with the projections of the axes of the original variables v x-axis represents the scores for the first principal component v Y-axis the scores for the second principal component. v The original variables are represented by arrows which graphically indicate the proportion of the original variance explained by the first two principal components. v The direction of the arrows indicates the relative loadings on the first and second principal components. v Biplot analysis can help to understand the multivariate data i) Graphically ii) Effectively iii) Conveniently. 62

Biplot of Iris Data 1= Setosa 2= Versicolor 3= Virginica 63

Image Compression Example Pansy Flower image, collected from http: //www. ats. ucla. edu/stat/r/code/pansy. jpg This image is 600× 465 pixels 64

Singular values of flowers image Plot of the singular values 65

Low rank Approximation to flowers image Rank-1 approximation Rank- 5 approximation 66

Low rank Approximation to flowers image Rank-20 approximation Rank-30 approximation 67

Low rank Approximation to flowers image Rank-50 approximation Rank-80 approximation 68

Low rank Approximation to flowers image Rank-100 approximation Rank-120 approximation 69

Low rank Approximation to flowers image Rank-150 approximation True Image 70

Outlier Detection Using SVD Nishith and Nasser (2007, MSc. Thesis) propose a graphical method of outliers detection using SVD. It is suitable for both general multivariate data and regression data. For this we construct the scatter plots of first two PC’s, and first PC and third PC. We also make a box in the scatter plot whose range lies median(1 st. PC) ± 3 × mad(1 st. PC) in the X-axis and median(2 nd. PC/3 rd. PC) ± 3 × mad(2 nd. PC/3 rd. PC) in the Yaxis. Where mad = median absolute deviation. The points that are outside the box can be considered as extreme outliers. The points outside one side of the box is termed as outliers. Along with the box we may construct another smaller box bounded by 2. 5/2 MAD line 71

Outlier Detection Using SVD (Cont. ) HAWKINS-BRADU-KASS (1984) DATA Data set containing 75 observations with 14 influential observations. Among them there are ten high leverage outliers (cases 1 -10) and for high leverage points (cases 11 -14) -Imon (2005). Scatter plot of Hawkins, Bradu and kass data (a) scatter plot of first two PC’s and (b) scatter plot of first and third PC. 72

Outlier Detection Using SVD (Cont. ) MODIFIED BROWN DATA Data set given by Brown (1980). Ryan (1997) pointed out that the original data on the 53 patients which contains 1 outlier (observation number 24). Scatter plot of modified Brown data (a) scatter plot of first two PC’s and (b) scatter plot of first and third PC. Imon and Hadi(2005) modified this data set by putting two more outliers as cases 54 and 55. Also they showed that observations 24, 54 and 55 are outliers by using generalized standardized Pearson residual (GSPR) 73

Cluster Detection Using SVD Singular Value Decomposition is also used for cluster detection (Nishith, Nasser and Suboron, 2011). The methods for clustering data using first three PC’s are given below, median (1 st PC) ± k × mad (1 st PC) in the X-axis and median (2 nd PC/3 rd PC) ± k × mad (2 nd PC/3 rd PC) in the Y-axis. Where mad = median absolute deviation. The value of k = 1, 2, 3. 74

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Principals stations in climate data 76

Climatic Variables The climatic variables are, 1. Rainfall (RF) mm 2. Daily mean temperature (T-MEAN)0 C 3. Maximum temperature (T-MAX)0 C 4. Minimum temperature (T-MIN)0 C 5. Day-time temperature (T-DAY)0 C 6. Night-time temperature (T-NIGHT)0 C 7. Daily mean water vapor pressure (VP) MBAR 8. Daily mean wind speed (WS) m/sec 9. Hours of bright sunshine as percentage of maximum possible sunshine hours (MPS)% 10. Solar radiation (SR) cal/cm 2/day 77

Consequences of SVD Generally many missing values may present in the data. It may also contain unusual observations. Both types of problem can not handle Classical singular value decomposition. Robust singular value decomposition can solve both types of problems. Robust singular value decomposition can be obtained by alternating L 1 regression approach (Douglas M. Hawkins, Li Liu, and S. Stanley Young, (2001)). 78

The Alternating L 1 Regression Algorithm for Robust Singular Value Decomposition. There is no obvious choice of the initial values of Initialize the leading left singular vector Fit the L 1 regression coefficient cj by minimizing ; j=1, 2, …, p Calculate the resulting estimate of the left eigenvector ui=d/ ║d║ Calculate right singular vector v 1=c/║c║ , where ║. ║ refers to Euclidean norm. Again fit the L 1 regression coefficient di by minimizing ; i=1, 2, …. , n Iterate this process untill it converge. For the second and subsequent of the SVD, we replaced X by a deflated matrix 79 obtained by subtracting the most recently found them in the SVD X X-λkukvk. T

Clustering weather stations on Map Using RSVD 80

References • • • Brown B. W. , Jr. (1980). Prediction analysis for binary data. in Biostatistics Casebook, R. G. Miller, Jr. , B. Efron, B. W. Brown, Jr. , L. E. Moses (Eds. ), New York: Wiley. Dhrymes, Phoebus J. (1984), Mathematics for Econometrics, 2 nd ed. Springer Verlag, New York. Hawkins D. M. , Bradu D. and Kass G. V. (1984), Location of several outliers in multiple regression data using elemental sets. Technometrics, 20, 197 -208. Imon A. H. M. R. (2005). Identifying multiple influential observations in linear Regression. Journal of Applied Statistics 32, 73 – 90. Kumar, N. , Nasser, M. , and Sarker, S. C. , 2011. “A New Singular Value Decomposition Based Robust Graphical Clustering Technique and Its Application in Climatic Data” Journal of Geography and Geology, Canadian Center of Science and Education , Vol-3, No. 1, 227 -238. Ryan T. P. (1997). Modern Regression Methods, Wiley, New York. • Stewart, G. W. (1998). Matrix Algorithms, Vol 1. Basic Decompositions, Siam, Philadelphia. • Matrix Decomposition. http: //fedc. wiwi. huberlin. de/xplore/ebooks/html/csa/node 36. html 81

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