Matrix Decomposition and its Application in Statistics

1
Matrix Decomposition and its Application in
Statistics

• Nishith Kumar
• Lecturer
• Department of Statistics
• Begum Rokeya University, Rangpur.
• Email nk.bru09_at_gmail.com

2
Overview

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

3
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.

4
Easy to solve system (Cont.)
Some linear system that can be easily solved
The solution
5
Easy to solve system (Cont.)
Lower triangular matrix
Solution This system is solved using forward
substitution
6
Easy to solve system (Cont.)
Upper Triangular Matrix
Solution This system is solved using Backward
substitution
7
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 (1736 1813)
A. M. Turing (1912-1954)
8
How to decompose ALU?
A ? ?U (upper triangular) ? U Ek ??? E1 A
? A (E1)-1 ??? (Ek)-1 U If each such
elementary matrix Ei is a lower triangular
matrices,it can be proved that (E1)-1, ???,
(Ek)-1 are lower triangular, and(E1)-1 ???
(Ek)-1 is a lower triangular matrix.Let L(E1)-1
??? (Ek)-1 then ALU.
U E2 E1 A
9
Calculation of L and U (cont.)

• Now reducing the first column we have

10
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

11
Calculation of L and U (cont.)
Calculation of L and 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 ALU
• 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)
• xlt-matrix(c(3,2,1, 9,3,4,4,2,5 ),ncol3,nrow3)
• expand(lu(x))

12
Calculation of L and U (cont.)

• Note there are 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).

13
Solving system of linear equation using LU
decomposition

• Suppose we would like to solve a  mm 
  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

14
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

15
Solving system of linear equation using LU
decomposition

• Now, we solve UX Y by backward substitution
• then

16
QR Decomposition
Firstly QR decomposition originated with
Gram(1883). Later Erhard Schmidt (1907) proved
the QR Decomposition Theorem
Jørgen Pedersen Gram (1850 1916)
Erhard Schmidt (1876-1959)

• If A is a mn matrix with linearly
  independent columns, then A can be decomposed as
  , where Q is a mn matrix
  whose columns form an orthonormal basis for the
  column space of A and R is an nonsingular upper
  triangular matrix.

17
QR-Decomposition (Cont.)

• Theorem If A is a mn matrix with
  linearly independent columns, then A can be
  decomposed as , where Q is a
  mn matrix whose columns form an orthonormal
  basis for the column space of A and R is an
  nonsingular upper triangular matrix.
• Proof Suppose Au1 u2 . . . un and
  rank (A) n.
• Apply the Gram-Schmidt process to
  u1, u2 , . . . ,un and the
• orthogonal vectors v1, v2 , . . .
  ,vn are
• Let for i1,2,. .
  ., n. Thus q1, q2 , . . . ,qn form a orthonormal
• basis for
  the column space of A.

18
QR-Decomposition (Cont.)

• Now,
• i.e.,
• Thus ui is orthogonal to qj for jgti

19
QR-Decomposition (Cont.)

• Let Q q1 q2 . . . qn , so Q is a mn
  matrix whose columns form an
• orthonormal basis for the column space of A
  .
• Now,
• i.e., AQR.
• Where,
• Thus A can be decomposed as AQR , where R is an
  upper triangular and nonsingular matrix.

20
QR Decomposition

• Example Find the QR decomposition of

21
Calculation of QR Decomposition

• Applying Gram-Schmidt process of computing QR
  decomposition
• 1st Step
• 2nd Step
• 3rd Step

22
Calculation of QR Decomposition

• 4th Step
• 5th Step
• 6th Step

23
Calculation of QR Decomposition

• Therefore, AQR
• R code for QR Decomposition
• xlt-matrix(c(1,2,3, 2,5,4, 3,4,9),ncol3,nrow3)
• qrstr lt- qr(x)
• Qlt-qr.Q(qrstr)
• Rlt-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.

24
Least square solution using QR Decomposition

• The least square solution of b is
• Let XQR. Then
• Therefore,

25
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
26
Cholesky Decomposition

• Theorem If A is a nn 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 nn real and positive
  definite so it has a LU decomposition, ALU. 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, ALDMT .Since A is Symmetric so, AAT
  . i.e., LDMT MDLT. From the uniqueness we have
  LM. So, ALDLT . Since A is positive definite so
  all diagonal elements of D are positive. Let
• then we can write AGGT.

27
Cholesky Decomposition (Cont.)

• Procedure To find out the cholesky decomposition
  
• Suppose
• We need to solve
• the equation

28
Example of Cholesky Decomposition
For k from 1 to n
For j from k1 to n

• Suppose
• Then Cholesky Decomposition
• Now,

29
R code for Cholesky Decomposition

• xlt-matrix(c(4,2,-2, 2,10,2, -2,2,5),ncol3,nrow3)
  
• cllt-chol(x)
• If we Decompose A as LDLT then
• and

30
Application of Cholesky Decomposition

• Cholesky Decomposition is used to solve the
  system of linear equation Axb, 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)

31
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 ? such that Ax ?x
• Where A is a square matrix, also then ? 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 ? from the
  characteristic equation A- ?I0
• For ? ?i the characteristics vector is the
  solution of x from the following homogeneous
  system of linear equation (A- ?iI)x0

Theorem If A is a real symmetric matrix and ?i
and ?j are two distinct latent root of A then the
corresponding latent vector xi and xj are
orthogonal.
32
Multiplicity

• Algebraic Multiplicity The number of
  repetitions of a certain eigenvalue. If, for a
  certain matrix, ?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 a, where a(?i) represents the algebraic
  multiplicity of ?i.
• Geometric Multiplicity the geometric
  multiplicity of an eigenvalue is the number of
  linearly independent eigenvectors associated with
  it.

33
Jordan Decomposition Camille Jordan (1870)

• Let A be any nn matrix then there exists a
  nonsingular matrix P and JK(?) a kk matrix form
• Such that

Camille Jordan (1838-1921)
where k1k2 kr n. Also ?i , i1,2,. . ., r
are the characteristic roots And ki are the
algebraic multiplicity of ?i ,
Jordan Decomposition is used in Differential
equation and time series analysis.
34
Spectral Decomposition
A. L. Cauchy established the Spectral
Decomposition in 1829.

• Let A be a m m real symmetric matrix. Then
  there exists an orthogonal matrix P such that
• or , where
  ? is a diagonal matrix.

CAUCHY, A.L.(1789-1857)
35
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 ?P?PT
  . Where ? is a diagonal matrix.
• Also
• tr(?) Total variation of Data tr(?)

36
Spectral Decomposition and Principal component
Analysis (Cont.)

• The Principal component transformation is the
  transformation
• Y(X-µ)P
• Where,
• E(Yi)0
• V(Yi)?i
• Cov(Yi ,Yj)0 if i ? j
• V(Y1) V(Y2) . . . V(Yn)

37
R code for Spectral Decomposition

• xlt-matrix(c(1,2,3, 2,5,4, 3,4,9),ncol3,nrow3)
• eigen(x)
• Application
• For Data Reduction.
• Image Processing and Compression.
• K-Selection for K-means clustering
• Multivariate Outliers Detection
• Noise Filtering
• Trend detection in the observations.

38
Historical background of SVD

• There are five mathematicians who were
  responsible for establishing the existence of the
  
• singular value decomposition and developing its
  theory.

Camille Jordan (1838-1921)
James Joseph Sylvester (1814-1897)
Erhard Schmidt (1876-1959)
Hermann Weyl (1885-1955)
Eugenio Beltrami (1835-1899)
The Singular Value Decomposition was originally
developed by two mathematician in the mid to
late 1800s 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-1900s. C.Eckart and G. Young prove low rank
approximation of SVD (1936).
C.Eckart
39
What is SVD?

• Any real (mn) matrix X, where (n m), can be
• decomposed,
• X U?VT
• U is a (mn) column orthonormal matrix (UTUI),
  containing the eigenvectors of the symmetric
  matrix XXT.
• ? is a (nn ) diagonal matrix, containing the
  singular values of matrix X. The number of non
  zero diagonal elements of ? corresponds to the
  rank of X.
• VT is a (nn ) row orthonormal matrix (VTVI),
  containing the eigenvectors of the symmetric
  matrix XTX.

40
Singular Value Decomposition (Cont.)

• Theorem (Singular Value Decomposition) Let X be
  mn of rank r, r n m. Then there exist
  matrices U , V and a diagonal matrix ? , 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,

41
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,

42
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 grammian
  matrix.)

43
Singular Value Decomposition (Cont.)

• Now define,
• Now we shall show that SX thus completing the
  proof.
• Similarly,
• From the first relation above we conclude that
  for an arbitrary orthogonal matrix, say P1 ,
• While from the second we conclude that for an
  arbitrary orthogonal matrix, say P2
• We must have

44
Singular Value Decomposition (Cont.)

• The preceding, however, implies that for
  arbitrary orthogonal matrices P1 , P2 the matrix
  X satisfies
• Which in turn implies that,
• Thus

45
R Code for Singular Value Decomposition

• xlt-matrix(c(1,2,3, 2,5,4, 3,4,9),ncol3,nrow3)
• svlt-svd(x)
• Dlt-svd
• Ult-svu
• Vlt-svv

46
Decomposition in Diagram
Matrix A
Full column rank
Lu decomposition Not always unique
QR Decomposition
Rectangular
Square
Asymmetric
SVD
Symmetric
AMGM
AMgtGM
PD
Similar Diagonalization P-1AP?
Jordan Decomposition
Cholesky Decomposition
Spectral Decomposition
47
Properties Of SVD

• Rewriting the SVD
• where
• r rank of A
• ?i the i-th diagonal element of
  ?.
• ui and vi are the i-th columns of U
  and V respectively.

48
Proprieties of SVDLow rank Approximation

• Theorem If AU?VT is the SVD of A and the
• singular values are sorted as
  ,
• then for any l ltr, the best rank-l approximation
• to A is
• Low rank approximation technique is very much
• important for data compression.

49
Low-rank Approximation

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

Frobenius norm
50
Low-rank Approximation

• Solution via SVD

set smallest r-k singular values to zero

K2
51
Approximation error

• How good (bad) is this approximation?
• Its the best possible, measured by the Frobenius
  norm of the error
• where the ?i are ordered such that ?i ? ?i1.

Now
52
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
  lltr
• where i 1,,m.

Also the SVD equation for Cj is, where j 1, 2,
, n
We can also approximate Cj by
lltr
53
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.

54

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

55
Basic Results of SVD
56
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, XU?VT
• we can write- XV U?
• Suppose Y XV U?
• Then the first columns of Y represents the first
• principal component score and so on.
• SVD Based PC is more Numerically Stable.
• 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)

57
Application of SVD

• 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.

58
Origin of biplot

• Gabriel (1971)
• One of the most important advances in data
  analysis in recent decades
• Currently
• gt 50,000 web pages
• Numerous academic publications
• Included in most statistical analysis packages
• Still a very new technique to most scientists

59
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 gtgt 2 plots

60
Practical definition of a biplotAny two-way
table can be analyzed using a 2D-biplot as soon
as it can be sufficiently approximated by a
rank-2 matrix. (Gabriel, 1971)
(Now 3D-biplots are also possible)
Matrix decomposition
E1
G1
G2
P(4, 3) G(3, 2) E(2, 3)
E2
G4
E3
G3
G-by-E table
61
Singular Value Decomposition (SVD) Singular
Value Partitioning (SVP)
SVD
Common uses value of f
f1
SVP
f0
f1/2
Rows scores
Column scores
62
Biplot

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

63
Biplot of Iris Data
1 Setosa 2 Versicolor 3 Virginica
64
Image Compression Example

• Pansy Flower image, collected from
• http//www.ats.ucla.edu/stat/r/code/pansy.jpg
• This image is 600465 pixels

65
Singular values of flowers image

• Plot of the singular values

66
Low rank Approximation to flowers image
Rank- 5 approximation

• Rank-1 approximation

67
Low rank Approximation to flowers image

• Rank-20 approximation

Rank-30 approximation
68
Low rank Approximation to flowers image

• Rank-50 approximation

Rank-80 approximation
69
Low rank Approximation to flowers image

• Rank-100 approximation

Rank-120 approximation
70
Low rank Approximation to flowers image
True Image

• Rank-150 approximation

71
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 PCs, and first PC
  and third PC. We also make a box in the scatter
  plot whose range lies
• median(1stPC) 3 mad(1stPC) in the X-axis
  and median(2ndPC/3rdPC) 3 mad(2ndPC/3rdPC) in
  the Y-axis.
• 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

72
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 PCs and (b) scatter
plot of first and third PC.
73
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).
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)
Scatter plot of modified Brown data (a) scatter
plot of first two PCs and (b) scatter plot of
first and third PC.
74
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 PCs are given below,
• median (1st PC) k mad (1st PC) in the
  X-axis and median (2nd PC/3rd PC) k mad (2nd
  PC/3rd PC) in the Y-axis.
• Where mad median absolute deviation. The
  value of k 1, 2, 3.

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76
Principals stations in climate data
77
Climatic Variables

• The climatic variables are,
• Rainfall (RF) mm
• Daily mean temperature (T-MEAN)0C
• Maximum temperature (T-MAX)0C
• Minimum temperature (T-MIN)0C
• Day-time temperature (T-DAY)0C
• Night-time temperature (T-NIGHT)0C
• Daily mean water vapor pressure (VP) MBAR
• Daily mean wind speed (WS) m/sec
• Hours of bright sunshine as percentage of maximum
  possible sunshine hours (MPS)
• Solar radiation (SR) cal/cm2/day

78
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 L1 regression approach
  (Douglas M. Hawkins, Li Liu, and S. Stanley
  Young, (2001)).

79
The Alternating L1 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 L1 regression coefficient cj by
minimizing
j1,2,,p
Calculate right singular vector v1c/c , where
. refers to Euclidean norm.
Again fit the L1 regression coefficient di by
minimizing i1,2,.,n
Calculate the resulting estimate of the left
eigenvector uid/ d
Iterate this process untill it converge.
For the second and subsequent of the SVD, we
replaced X by a deflated matrix obtained by
subtracting the most recently found them in the
SVD X X-?kukvkT
80
Clustering weather stations on MapUsing RSVD
81
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, 2nd 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.hu-berlin.d
  e/xplore/ebooks/html/csa/node36.html

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