introduction to sas vectors and matrices · pdf fileintroduction to sas. vectors and matrices...
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![Page 1: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/1.jpg)
Lecture #2 - 8/31/2005 Slide 1 of 55
Introduction to SASVectors and Matrices
Lecture 2
August 31, 2005Multivariate Analysis
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Overview
● Today’s Lecture
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 2 of 55
Today’s Lecture
■ Introduction to SAS.
■ Vectors and Matrices (Supplement 2A - augmented withSAS proc iml).
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Lecture #2 - 8/31/2005 Slide 3 of 55
SAS■ To start SAS, on a Windows PC, go to Start...All Programs...The SAS
System...The SAS System for Windows V8.
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Lecture #2 - 8/31/2005 Slide 4 of 55
Main Program Window■ The SAS program looks like this (some helpful commands are shown with
the arrows):
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Overview
Introduction to SAS
● Main Program Window
● SAS Editor
● SAS Code
● First SAS Program
● Run
● SAS Data Libraries
● SAS Procedures
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 5 of 55
SAS Editor■ To run SAS you must create a file of SAS code, which the
SAS processor reads and uses to run the program.
■ Simply type your SAS code into the Program Editor window.
■ For our example today, we will create (and save to) a newSAS code file, so to do that be sure to have your curserinside of the editor window and go to File...Save...
■ SAS code files usually end with the extension *.sas.
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Overview
Introduction to SAS
● Main Program Window
● SAS Editor
● SAS Code
● First SAS Program
● Run
● SAS Data Libraries
● SAS Procedures
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 6 of 55
SAS Code Basics■ For use in day-to-day statistical applications, SAS code
consists of two components:
◆ Data steps (where data input usually happens.
◆ Proc steps (where statistical analyses usually happen.
■ General exceptions to these rules exist.
■ All statements terminate with a semicolon (this is usuallywhere errors can occur).
■ Commented code can begin with:
◆ An asterisk (*) for single lines - terminated with asemicolon.
◆ /* for multiple lines, terminated with an ending */.
■ Enough talk...how about an example? Type the following intothe SAS Editor:
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Lecture #2 - 8/31/2005 Slide 7 of 55
First SAS Programtitle ’First SAS Program’;data example1;input id gender $ age;cards;1 F 232 M 203 F 184 F .5 M 196 M 217 M 258 F 249 F 2010 M 22;
proc print;var id gender age;run;
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Overview
Introduction to SAS
● Main Program Window
● SAS Editor
● SAS Code
● First SAS Program
● Run
● SAS Data Libraries
● SAS Procedures
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 8 of 55
First SAS ProgramNotice the color scheme of the SAS Enhanced Editor (note: ifyou do not see color, do not panic, you may not have theEnhanced Editor installed).
■ Items in light blue are command words (like “input” or “var”.
■ Items in dark blue are procedural words (like “proc,” “data,” or“run”).
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Overview
Introduction to SAS
● Main Program Window
● SAS Editor
● SAS Code
● First SAS Program
● Run
● SAS Data Libraries
● SAS Procedures
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 9 of 55
Run the SAS Program■ To run the program you just entered, press the running man
icon at the center of the top of the SAS main program.
■ Once you press the “run” button, the log window will becomeactive, giving you information about the program as itexecutes.
■ In this window you will see errors (in red), or warnings (ingreen I think).
■ As multivariate progresses, you will become aware ofinstances where warnings will be present because ofproblems in your analysis.
■ Also now appearing is a new window called output...this iswhere the output of the procedure that was just run isdisplayed.
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Overview
Introduction to SAS
● Main Program Window
● SAS Editor
● SAS Code
● First SAS Program
● Run
● SAS Data Libraries
● SAS Procedures
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 10 of 55
SAS Data Libraries■ The data set you just entered is now part of a SAS data
library that can be referenced at any point during theremainder of the program.
■ The default SAS data library is called “work,” and can beaccessed by clicking through the explorer window on the lefthand side of the program.
■ Double click on Libraries...Work...Example1 and you will seethe data displayed in a grid.
■ To familiarize you with SAS, here are some handouts (alsoavailable on the BlackBoard Site)...
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Lecture #2 - 8/31/2005 Slide 11 of 55
SAS Procedures■ The bulk of the statistical work done in SAS is through procedure statements.
■ Proc statements follow a flexible syntax that typically has the following:
proc -statement_name- data=-data_name- [options];var [included variables];[options];run;
■ The names and options are all found in the SAS manual, which is freely(shhh) available online at:http://www.id.unizh.ch/software/unix/statmath/sas/sasdoc/main.htm
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Lecture #2 - 8/31/2005 Slide 12 of 55
SAS Procedures Example■ Using the Example1 data set, type the following into the editor:
proc sort data=example1;by gender;run;
proc univariate data=example1 plots;by gender;var age;run;
■ The univariate procedure produces univariate statistics, the manual entry canbe at found:
http://www.id.unizh.ch/software/unix/statmath/sas/sasdoc/proc/z0146802.htm
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 13 of 55
Matrix Introduction
■ Imagine you are interested in studying the culturaldifferences in student achievement in high school.
■ Your study gets IRB approval and you work hard to getparental approval.
■ You go into school and collect data using many differentinstruments.
■ You then input the data into your favorite stat package, likeSAS (or MS Excel).
■ How do you think the data is stored?
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 14 of 55
Why Learn Matrix Algebra?
■ Nearly all multivariate techniques are founded in matrixalgebra.
■ Often times when statistics break down, the cause of thefailure can be traced through matrix procedures.
■ If you are trying to apply a new method, most of the technicalstatistical literature uses matrix algebra assuming a basic toadvanced knowledge of matrices - you may need to readthese articles.
■ Have you seen:◆ (X′X)−1X′y?◆ ΛΦΛ
′ + Ψ?◆ F1F2F3LF−1
3 F−12 P(F−1
2 )′(F−13 )′L′F′
3F′
2F′
1 + U2?
■ Warning: using matrix algebra lingo is a great way to endconversations or break up parties.
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 15 of 55
Definitions
■ We begin this class with some general definitions (fromdictionary.com):
◆ Matrix:
1. A rectangular array of numeric or algebraic quantitiessubject to mathematical operations.
2. The substrate on or within which a fungus grows.
◆ Algebra:
1. A branch of mathematics in which symbols, usuallyletters of the alphabet, represent numbers or membersof a specified set and are used to represent quantitiesand to express general relationships that hold for allmembers of the set.
2. A set together with a pair of binary operations definedon the set. Usually, the set and the operations includean identity element, and the operations arecommutative or associative.
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 16 of 55
Proc IML
■ To help demonstrate the topics we will discuss today, I will beshowing examples in SAS proc iml.
■ The Interactive Matrix Language (IML) is a scientificcomputing package in SAS that typically used forcomplicated statistical routines.
■ Of course, other matrix programs exist - for many statisticalapplications MATLAB is very useful.
■ SPSS and SAS both have matrix computing capabilities, but(in my opinion) neither are as efficient, as user friendly, or asflexible as MATLAB.
◆ It is better to leave most of the statistical computing to thecomputer scientists.
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 17 of 55
Proc IML Basics
■ Proc IML is a proc step in SAS that runs without needing touse a preliminary data step.
■ To use IML, make sure the following are placed in a SAScode file.
proc iml;
reset print;
quit;
■ The “reset print;” line makes every result get printed in theoutput window.
■ The IML code will go between the “reset print;” and the “quit;”
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 18 of 55
Matrices
■ Away from the definition, a matrix is simply a rectangular wayof storing data.
■ Matrices can have unlimited dimensions, however for ourpurposes, all matrices will be in two dimensions:
◆ Rows
◆ Columns
■ Matrices are symbolized by boldface font in text, usuallywith capital letters.
A =
[
4 7 5
6 6 3
]
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 19 of 55
Vectors
■ A vector is a matrix where one dimension is equal to sizeone.
◆ Column vector: A column vector is a matrix of size r × 1.
◆ Row vector: A row vector is a matrix of size 1 × c.
■ Vectors allow for geometric representations of matrices.
■ The Pearson correlation coefficient is a function of the anglebetween vectors.
■ Much of the statistical theory underlying linear models(ANOVA-type) can be conceptualized by projections ofvectors (think of the dependent variable Y as a columnvector).
■ Vectors are typically symbolized by boldface font in text,usually with lowercase letters.
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 20 of 55
Scalars
■ A scalar is a matrix of size 1 × 1.
■ Scalars can be thought of as any single value.
■ The difficult concept to get used to is seeing a number as amatrix:
A =[
2.759]
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 21 of 55
Matrix Elements
■ A matrix is composed of a set of elements, each denoted it’srow and column position within the matrix.
■ For a matrix A of size r × c, each element is denoted by:
aij
◆ The first subscript is the index for the rows: i = 1, . . . , r.◆ The second subscript is the index for the columns:
j = 1, . . . , c.
A =
a11 a12 . . . a1c
a21 a22 . . . a1c
......
......
ar1 ar2 . . . arc
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 22 of 55
Transpose
■ The transpose of a matrix is simply the switching of theindices for rows and columns.
■ An element aij in the original matrix (in the ith row and jth
column) would be aji in the transposed matrix (in the jth rowand the ith column).
■ If the original matrix was of size i × j the transposed matrixwould be of size j × i.
A =
[
4 7 5
6 6 3
]
A′ =
4 6
7 6
5 3
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Overview
Introduction to SAS
Matrix Algebra
● Introduction
● Why?
● Definitions
● Matrix Computing
● Proc IML Basics
● Matrices
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 23 of 55
Types of Matrices
■ Square Matrix: A matrix that as the same number of rowsand columns.
◆ Correlation and covariance matrices are examples ofsquare matrices.
■ Diagonal Matrix: A diagonal matrix is a square matrix withnon-zero elements down the diagonal and zero values forthe off-diagonal elements.
A =
2.759 0 0
0 1.643 0
0 0 0.879
■ Symmetric Matrix: A symmetric matrix is a square matrixwhere aij = aji for all elements in i and j.
◆ Correlation/covariance and distance matrices areexamples of symmetric matrices.
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Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 24 of 55
Algebraic Operations
■ As mentioned in the definition at the beginning of class,algebra is simply a set of math that defines basic operations.
◆ Identity
◆ Zero
◆ Addition
◆ Subtraction
◆ Multiplication
◆ Division
■ Matrix algebra is simply the use of these operations withmatrices.
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Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 25 of 55
Matrix Addition
■ Matrix addition is very much like scalar addition, the onlyconstraint is that the two matrices must be of the same size(same number of rows and columns).
■ The resulting matrix contains elements that are simply theresult of adding two scalars.
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Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 26 of 55
Matrix Addition
A =
a11 a12
a21 a22
a31 a32
a41 a42
B =
b11 b12
b21 b22
b31 b32
b41 b42
A + B =
a11 + b11 a12 + b12
a21 + b21 a22 + b22
a31 + b31 a32 + b32
a41 + b41 a42 + b42
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Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 27 of 55
IML Addition
proc iml;reset print;
A={10 15, 11 9, 1 -6};B={5 2, 1 0, 10 7};C=A+B;
quit;
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Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 28 of 55
Matrix Subtraction
■ Matrix subtraction is identical to matrix addition, with theexception that all elements of the new matrix are thesubtracted elements of the previous matrices.
■ Again, the only constraint is that the two matrices must be ofthe same size (same number of rows and columns).
■ The resulting matrix contains elements that are simply theresult of subtracting two scalars.
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Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 29 of 55
Matrix Subtraction
A =
a11 a12
a21 a22
a31 a32
a41 a42
B =
b11 b12
b21 b22
b31 b32
b41 b42
A − B =
a11 − b11 a12 − b12
a21 − b21 a22 − b22
a31 − b31 a32 − b32
a41 − b41 a42 − b42
![Page 30: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/30.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 30 of 55
IML Subtraction
proc iml;reset print;
A={10 15, 11 9, 1 -6};B={5 2, 1 0, 10 7};C=A-B;
quit;
![Page 31: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/31.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 31 of 55
Matrix Multiplication
■ Unlike matrix addition and subtraction, matrix multiplicationis much more complicated.
■ Matrix multiplication results in a new matrix that can be ofdiffering size from either of the two original matrices.
■ Matrix multiplication is defined only for matrices where thenumber of columns of the first matrix is equal to the numberof rows of the second matrix.
■ The resulting matrix as the same number of rows as the firstmatrix, and the same number of columns as the secondmatrix.
A B = C(r × c) (c × k) (r × k)
![Page 32: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/32.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 32 of 55
Matrix Multiplication
A =
a11 a12
a21 a22
a31 a32
a41 a42
B =
[
b11 b12 b13
b21 b22 b23
]
AB =
a11b11 + a12b21 a11b12 + a12b22 a11b13 + a12b23
a21b11 + a22b21 a21b12 + a22b22 a21b13 + a22b23
a31b11 + a32b21 a31b12 + a32b22 a31b13 + a32b23
a41b11 + a42b21 a41b12 + a42b22 a41b13 + a42b23
![Page 33: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/33.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 32 of 55
Matrix Multiplication
A =
a11 a12
a21 a22
a31 a32
a41 a42
B =
[
b11 b12 b13
b21 b22 b23
]
AB =
a11b11 + a12b21 a11b12 + a12b22 a11b13 + a12b23
a21b11 + a22b21 a21b12 + a22b22 a21b13 + a22b23
a31b11 + a32b21 a31b12 + a32b22 a31b13 + a32b23
a41b11 + a42b21 a41b12 + a42b22 a41b13 + a42b23
![Page 34: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/34.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 32 of 55
Matrix Multiplication
A =
a11 a12
a21 a22
a31 a32
a41 a42
B =
[
b11 b12 b13
b21 b22 b23
]
AB =
a11b11 + a12b21 a11b12 + a12b22 a11b13 + a12b23
a21b11 + a22b21 a21b12 + a22b22 a21b13 + a22b23
a31b11 + a32b21 a31b12 + a32b22 a31b13 + a32b23
a41b11 + a42b21 a41b12 + a42b22 a41b13 + a42b23
![Page 35: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/35.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 33 of 55
IML Multiplication
proc iml;reset print;
A={10 15, 11 9, 1 -6};B={5 2, 1 0, 10 7};C=A*T(B);D=T(B)*(A);
quit;
![Page 36: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/36.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 34 of 55
Multiplication and Summation
■ Because of the additive nature induced by matrixmultiplication, many statistical formulas that use:
∑
can be expressed by matrix notation.
■ For instance, consider a single variable Xi, with i = 1, . . . , Nobservations.
■ Putting the set of observations into the column vector X, ofsize N × 1, we can show that:
N∑
i=1
X2 = X′ X
![Page 37: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/37.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 35 of 55
Matrix Multiplication by Scalar
■ Recall that a scalar is simply a matrix of size (1 × 1).■ Matrix multiplication by a scalar causes all elements of the
matrix to be multiplied by the scalar.■ The resulting matrix has all elements multiplied by the scalar.
A =
a11 a12
a21 a22
a31 a32
a41 a42
s × A =
s × a11 s × a12
s × a21 s × a22
s × a31 s × a32
s × a41 s × a42
![Page 38: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/38.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 36 of 55
Identity Matrix
■ The identity matrix is defined as a matrix that whenmultiplied with another matrix produces that original matrix:
A I = A
I A = A
■ The identity matrix is simply a square matrix that has alloff-diagonal elements equal to zero, and all diagonalelements equal to one.
I(3×3) =
1 0 0
0 1 0
0 0 1
![Page 39: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/39.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 37 of 55
Zero Matrix
■ The zero matrix is defined as a matrix that when multipliedwith another matrix produces the matrix:
A 0 = 0
0 A = 0
■ The zero matrix is simply a square matrix that has allelements equal to zero.
0(3×3) =
0 0 0
0 0 0
0 0 0
![Page 40: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/40.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 38 of 55
Matrix “Division”: The Inverse
■ Recall from basic math that:
a
b=
1
ba = b−1a
■ And that:a
a= 1
■ Matrix inverses are just like division in basic math.
![Page 41: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/41.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 39 of 55
The Inverse
■ For a square matrix, an inverse matrix is simply the matrixthat when pre-multiplied with another matrix produces theidentity matrix:
A−1A = I
■ Matrix inverse calculation is complicated and unnecessarysince computers are much more efficient at finding inversesof matrices.
■ One point of emphasis: just like in regular division, divisionby zero is undefined.
■ By analogy - not all matrices can be inverted.
![Page 42: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/42.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
● Algebra
● Addition
● Subtraction
● Multiplication
● Identity
● Zero
● “Division”
● Singular Matrices
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 40 of 55
Singular Matrices
■ A matrix that cannot be inverted is called a singular matrix.
■ In statistics, common causes of singular matrices are foundby linear dependence among the rows or columns of asquare matrix.
■ Linear dependence can be cause by combinations ofvariables, or by variables with extreme correlations (eithernear 1.00 or -1.00).
![Page 43: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/43.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
● Linear Combinations
● Linear Dependencies
● Vector Length
● Inner Product
● Angle Between Vectors
● Vector Projections
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 41 of 55
Linear Combinations
■ Vectors can be combined by adding multiples:
y = a1x1 + a2x2 + . . . + akxk
■ The resulting vector, y, is called a linear combination.
■ All for k vectors, the set of all possible linear combinations iscalled their span.
![Page 44: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/44.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
● Linear Combinations
● Linear Dependencies
● Vector Length
● Inner Product
● Angle Between Vectors
● Vector Projections
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 42 of 55
Linear Dependencies
■ A set of vectors are said to be linearly dependent ifa1, a2, . . . , ak exist, and:
◆ a1x1 + a2x2 + . . . + akxk = 0.
◆ a1, a2, . . . , ak are not all zero.
■ Such linear dependencies occur when a linear combinationis added to the vector set.
■ Matrices comprised of a set of linearly dependent vectorsare singular.
■ A set of linearly independent vectors forms what is called abasis for the vector space.
■ Any vector in the vector space can then be expressed as alinear combination of the basis vectors.
![Page 45: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/45.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
● Linear Combinations
● Linear Dependencies
● Vector Length
● Inner Product
● Angle Between Vectors
● Vector Projections
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 43 of 55
Vector Length
■ The length of a vector emanating from the origin is given bythe Pythagorean formula:
Lx =√
x21 + x2
2 + . . . + x2k
![Page 46: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/46.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
● Linear Combinations
● Linear Dependencies
● Vector Length
● Inner Product
● Angle Between Vectors
● Vector Projections
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 44 of 55
Inner Product
■ The inner (or dot) product of two vectors x and y is the sumof element-by-element multiplication:
x′y = x1y1 + x2y2 + . . . + xkyk
![Page 47: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/47.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
● Linear Combinations
● Linear Dependencies
● Vector Length
● Inner Product
● Angle Between Vectors
● Vector Projections
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 45 of 55
Vector Angle
■ The angle formed between two vectors x and y is
cos(θ) =x′y√
x′x√
y′y
■ If x′y = 0, vectors x and y are perpendicular, as noted byx⊥y
■ All basis vectors are perpendicular.
![Page 48: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/48.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
● Linear Combinations
● Linear Dependencies
● Vector Length
● Inner Product
● Angle Between Vectors
● Vector Projections
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 46 of 55
Vector Projections
■ The projection of a vector x onto a vector y is given by:
x′yL2
yy
■ Through such projections, a set of linear independentvectors can be created from any set of vectors.
■ One process used to create such vectors is through theGram-Schmidt Process.
![Page 49: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/49.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 47 of 55
Matrix Properties
The following are some algebraic properties of matrices:
■ (A + B) + C = A + (B + C) - Associative
■ A + B = B + A - Commutative
■ c(A + B) = cA + cB - Distributive
■ (c + d)A = cA + dA
■ (A + B)′ = A′ + B′
■ (cd)A = c(dA)
■ (cA)′ = cA′
![Page 50: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/50.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
Lecture #2 - 8/31/2005 Slide 48 of 55
Matrix Properties
The following are more algebraic properties of matrices:
■ c(AB) = (cA)B
■ A(BC) = (AB)C
■ A(B + C) = AB + AC
■ (B + C)A = BA + CA
■ (AB)′ = B′A′
■ For xj such that Ax j is defined:
◆
n∑
j=1
Ax j = An
∑
j=1
xj
◆
n∑
j=1
(Ax j)(Ax j)′ = A
n∑
j=1
xjx′
j
A′
![Page 51: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/51.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
● Advanced Topics
● Determinants
● Orthogonality
● Eigenspaces
● Decompositions
Wrapping Up
Lecture #2 - 8/31/2005 Slide 49 of 55
Advanced Matrix Functions/Operations
■ We end our matrix discussion with some advanced topics.
■ All of these topics are related to multivariate analyses.
■ None of these will seem too straight forward today, but nextweek we will use some of these results to demonstrateproperties of sample statistics.
![Page 52: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/52.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
● Advanced Topics
● Determinants
● Orthogonality
● Eigenspaces
● Decompositions
Wrapping Up
Lecture #2 - 8/31/2005 Slide 50 of 55
Matrix Determinants
■ A square matrix can be characterized by a scalar valuecalled a determinant.
det A = |A|■ Much like the matrix inverse, calculation of the determinant is
very complicated and tedious, and is best left to computers.
■ What can be learned from determinants is if a matrix issingular.
■ Matrices with determinants that are greater than zero aresaid to be “positive definite,” a byproduct of which is that apositive matrix is non-singular.
![Page 53: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/53.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
● Advanced Topics
● Determinants
● Orthogonality
● Eigenspaces
● Decompositions
Wrapping Up
Lecture #2 - 8/31/2005 Slide 51 of 55
Matrix Orthogonality
■ A square matrix (A) is said to be orthogonal if:
AA ′ = A′A = I
■ Orthogonal matrices are characterized by two properties:
1. The product of all row vector multiples is the zero matrix(perpendicular vectors).
2. For each row vector, the sum of all elements is one.
![Page 54: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/54.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
● Advanced Topics
● Determinants
● Orthogonality
● Eigenspaces
● Decompositions
Wrapping Up
Lecture #2 - 8/31/2005 Slide 52 of 55
Eigenvalues and Eigenvectors
■ A square matrix can be decomposed into a set ofeigenvalues and eigenvectors.
Ax = λx
■ From a statistical standpoint:◆ Principal components are comprised of linear combination
of a set of variables weighed by the eigenvectors.
◆ The eigenvalues represent the proportion of varianceaccounted for by specific principal components.
◆ Each principal component is orthogonal to the next,producing a set of uncorrelated variables that may beused for regression purposes.
![Page 55: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/55.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
● Advanced Topics
● Determinants
● Orthogonality
● Eigenspaces
● Decompositions
Wrapping Up
Lecture #2 - 8/31/2005 Slide 53 of 55
Spectral Decompositions
■ Imagine that a matrix A is of size k × k.
■ A then has:
◆ k eigenvalues: λi, i = 1, . . . , k.
◆ k eigenvectors: ei, i = 1, . . . , k (each of size k × 1.
■ A can be expressed by:
A =
k∑
i=1
λieie′
i
■ This expression is called the Spectral Decomposition, whereA is decomposed into k parts.
■ One can find A−1 by taking 1λ
in the spectral decomposition.
![Page 56: Introduction to SAS Vectors and Matrices · PDF fileIntroduction to SAS. Vectors and Matrices (Supplement 2A - augmented with SAS proc iml). Lecture #2 - 8/31/2005 Slide 3 of 55 SAS](https://reader033.vdocuments.net/reader033/viewer/2022050902/5a78c4157f8b9a70648c8a61/html5/thumbnails/56.jpg)
Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
● Final Thought
● Next Class
Lecture #2 - 8/31/2005 Slide 54 of 55
Final Thought
■ The SAS learning curve issmall, but once you havethe basics, all you need isthe manual and you can doalmost anything.
■ Proc IML is useful formatrices, but probably notso useful to you.
■ Matrix algebra makes the technical things in life easier.
■ The applications of matrices will be demonstrated throughoutthe rest of this course.
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Overview
Introduction to SAS
Matrix Algebra
Algebra
Vector Definitions
Matrix Properties
Advanced Topics
Wrapping Up
● Final Thought
● Next Class
Lecture #2 - 8/31/2005 Slide 55 of 55
Next Time
■ Applications of matrix algebra in statistics (Chapter 2 sectionsix and on).
■ Geometric implications of multivariate descriptive statistics(more applications).