Matrix Approach to Simple Linear Regression

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Matrix Approach to Simple
Linear Regression
KNNL – Chapter 5
Matrices
• Definition: A matrix is a rectangular array of numbers
or symbolic elements
• In many applications, the rows of a matrix will
represent individuals cases (people, items, plants,
animals,...) and columns will represent attributes or
characteristics
• The dimension of a matrix is it number of rows and
columns, often denoted as r x c (r rows by c columns)
• Can be represented in full form or abbreviated form:
 a11
a
 21

A
 ai1


 ar1
a12
a22
a1 j
a2 j
ai 2
aij
ar 2
arj
a1c 
a2 c 

   aij  i  1,..., r ; j  1,..., c
aic 


arc 
Special Types of Matrices
Square Matrix: Number of rows = # of Columns
r  c
 20 32 50 
b 
b
A  12 28 42 
B   11 12 
b21 b22 
 28 46 60 
Vector: Matrix with one column (column vector) or one row (row vector)
 d1 
d 
D   2
E '  17 31
F '   f1 f 2 f 3 
 d3 
 
d4 
Transpose: Matrix formed by interchanging rows and columns of a matrix (use "prime" to denote transpose)
57 
C   24 
18 
6 15 22 
G

23
8 13 25 
6 8 
G '  15 13 
32
 22 25
h1c 
hr1 
 h11
 h11
   h  i  1,..., r ; j  1,..., c  H '  
   h  j  1,..., c; i  1,..., r
H  
ij




  ji 
r c
c r
 hr1
 h1c
hrc 
hrc 
Matrix Equality: Matrices of the same dimension, and corresponding elements in same cells are all equal:
 b11 b12 
4 6
A
=
B

b
  b11  4, b12  6, b21  12, b22  10

b
12
10


 21 22 
Regression Examples - Toluca Data
X
 Y1 
Y 
Response Vector: Y   2 
n1
 
 
Yn 
Y'  Y1 Y2
Yn 
1 n
1 X 1 
1 X 
2
Design Matrix: X  
n 2




1 X n 
1
1 
1
X'  

2 n
X
X
X
2
n
 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
80
30
50
90
70
60
120
80
100
50
40
70
90
20
110
100
30
50
90
110
30
90
40
80
70
Y
399
121
221
376
361
224
546
352
353
157
160
252
389
113
435
420
212
268
377
421
273
468
244
342
323
Matrix Addition and Subtraction
Addition and Subtraction of 2 Matrices of Common Dimension:
70   6 7 
7  0   2 7
4 7
 2 0
 42
 42
C 
D

C

D


C

D


14 6 
10  14 12  6   24 18
10  14 12  6    4 6 
22
22
22
22
10
12





 


 

a1c 
b1c 
 a11
b11
   a  i  1,..., r ; j  1,..., c B  
  b  i  1,..., r ; j  1,..., c
A  
ij 



  ij 
r c
r c
 ar1
br1
arc 
brc 
a1c  b1c 
 a11  b11

   a  b  i  1,..., r ; j  1,..., c
AB  
  ij ij 
r c
 ar1  br1
arc  brc 
a1c  b1c 
 a11  b11
   a  b  i  1,..., r ; j  1,..., c
A  B  
  ij ij 
r c
 ar1  br1
arc  brc 
Regression Example:
 Y1 
Y 
Y   2
n1
 
 
Yn 
 E Y1 


E
Y


2

E Y  


n1


 E Yn 
 1 
 
ε   2
n1
 
 
 n 
Y  E Y  ε
n1
n1
n1
 Y1   E Y1    1   E Y1  1 
Y   E Y     E Y   
 2   2    2  2 
since  2   


   

  
   

  
Yn   E Yn   n   E Yn    n 
Matrix Multiplication
Multiplication of a Matrix by a Scalar (single number):
 2 1
 3(2) 3(1)   6 3 
k 3 A

k
A






 2 7 
3(2) 3(7)   6 21
Multiplication of a Matrix by a Matrix (#cols(A) = #rows(B)):
If c A  rB : A B  AB =  abij  i  1,..., rA ; j  1,..., cB
rA c A rB cB
rA cB
abij  sum of the products of the c A  rB elements of i th row of A and jth column of B :
2 5 
A   3 1
32
 0 7 
 3 1
B 

22
2 4 
2(1)  5(4)  16 18 
 2(3)  5(2)
A B  AB  3(3)  (1)(2) 3(1)  (1)(4)    7 7 
32 2 2
32
 0(3)  7(2)
0(1)  7(4)  14 28 
 c

If c A  rB  c : A B  AB =  abij  =   aik bkj  i  1,..., rA ; j  1,..., cB
rA c A rB cB
rA cB
 k 1

Matrix Multiplication Examples - I
Simultaneous Equations: a11 x1  a12 x2  y1 a21 x1  a22 x2  y2
 a11 a12   x1   y1 
(2 equations: x1 , x2 unknown): 
 



 a21 a22   x2   y2 
 a11 x1  a12 x2   y1 
 
    AX = Y

 a21 x1  a22 x2   y2 
4
2
Sum of Squares: 42   2   32   4 2 3  2    29
 3 
1 X 1 
 0  1 X 1 
1 X  
   X 


2
1 2
0
 0
Regression Equation (Expected Values): 


  1  





1
X



X
n
1 n

 0
Matrix Multiplication Examples - II
Matrices used in simple linear regression (that generalize to multiple regression):
Y ' Y  Y1 Y2
1
X'X  
 X1
1
X'Y  
 X1
1
X2
1
X2
 Y1 
Y  n
Yn   2    Yi 2
  i 1
 
Yn 
1 X 1  
n
1  1 X 2  
 n

 
Xn  

  X i
1 X n   i 1
 Y1   n

Y
i 
1  Y2   
i 1




n
Xn    

X
Y
   i i 

Yn   i 1

Xi 

i 1

n
2
Xi 

i 1

n
1 X 1 
  0  1 X 1 
1 X  
   X 


2
1 2
0
 0
X  


  1  





1 X n 
  0  1 X n 
Special Matrix Types
Symmetric Matrix: Square matrix with a transpose equal to itself: A = A':
 6 19 8
 6 19 8
A  19 14 3 
A'  19 14 3   A
 8 3 1 
 8 3 1 
Diagonal Matrix: Square matrix with all off-diagonal elements equal to 0:
4 0 0
b1 0 0 
A   0 1 0 
B   0 b2 0 
Note:Diagonal matrices are symmetric (not vice versa)
 0 0 2 
 0 0 b3 
Identity Matrix: Diagonal matrix with all diagonal elements equal to 1 (acts like multiplying a scalar by 1):
1 0 0 
 a11 a12 a13 
 a11 a12 a13 




I  0 1 0 
A   a21 a22 a23   IA  AI  A   a21 a22 a23 
33
33
0 0 1 
 a31 a32 a33 
 a31 a32 a33 
Scalar Matrix: Diagonal matrix with all diagonal elements equal to a single number"
k
0

0

0
0 0 0
1 0 0 0 

0 1 0 0 
k 0 0
k I
k
44
0 0 1 0 
0 k 0



0 0 k
0 0 0 1 
1-Vector and matrix and zero-vector:
1
1
1  
r 1


1
1
J  
r r
1
1


1
0
0
0  
r 1
 
 
0
Note: 1 ' 1  1 1
1r r 1
1
1
1    r


1
1
1
1 1 '    1 1
r 11r


1
1
1  
1
1
 J
 r r
1
Linear Dependence and Rank of a Matrix
• Linear Dependence: When a linear function of the
columns (rows) of a matrix produces a zero vector
(one or more columns (rows) can be written as linear
function of the other columns (rows))
• Rank of a matrix: Number of linearly independent
columns (rows) of the matrix. Rank cannot exceed
the minimum of the number of rows or columns of
the matrix. rank(A) ≤ min(rA,ca)
• A matrix if full rank if rank(A) = min(rA,ca)
1
A 
22
 4
4
B 
22
4
3 
 A1
12   21
A 2 
21 
3 
 B1 B 2 

12   21 21 
3A1  A 2  0
0B1  0B 2  0
Columns of A are linearly dependent rank(A ) = 1
Columns of B are linearly independent rank(B) = 2
Matrix Inverse
• Note: For scalars (except 0), when we multiply a
number, by its reciprocal, we get 1:
2(1/2)=1
x(1/x)=x(x-1)=1
• In matrix form if A is a square matrix and full rank (all
rows and columns are linearly independent), then A
has an inverse: A-1 such that: A-1 A = A A-1 = I
2 8 
A

 4 2 
2
 36
A -1  
4
 36
8
36 

2 
36 
2
 36
A -1 A  
4
 36
8
 4 32


36  2 8   36 36


2   4 2   8 8

36 
 36 36
16 16 

36 36  1 0 
I

32 4  0 1 

36 36 
0
0 
000
0  0  0  1 0 0 
4 1 / 4   0  0
4 0 0
1/ 4






-1
-1
B   0 2 0 B   0 1/ 2 0  BB   0  0  0
0   2  1/ 2   0
0  0  0   0 1 0   I
 0  0  0
 0 0 6 
 0
0
1/ 6 
000
0  0  6 1/ 6   0 0 1 
Computing an Inverse of 2x2 Matrix
a12 
a
A   11
  full rank (columns/rows are linearly independent)
22
 a21 a22 
Determinant of A  A  a11a22  a12 a21
Note: If A is not full rank (for some value k ): a11  ka12
a21  ka22
 A  a11a22  a12 a21  ka12 a22  a12 ka22  0
 a22 a12 
-1
 a
 Thus A does not exist if A is not full rank
22
a
 21
11 
While there are rules for general r  r matrices, we will use computers to solve them
A 1 
1
A
Regression Example:
1 X 1 

1 X 
 n
2
  X'X  
X


 n


 X i
 i 1
1 X n 

 X'X 
1

1
n

n X i  X
i 1

2
 n 2
  Xi
 i 1
 n
  X i
 i 1
2
1
X
  n
2
n
Xi  X


1
i 1
  X'X   
X

  n
2
  Xi  X
 i 1




2

 n
 
 n
2
  Xi  
n
n
 n

i 1


2
2


X'X  n X i    X i   n  X i 
 n X i  X
 i 1

n
i 1
i 1
 i 1 







Xi 

i 1
 
n

X i2 

i 1

n

n

 X i 
i 1


n 



n
2
Xi  X 


i 1

1

n
2 
Xi  X 

i 1

X





n
Note:  X i  n X
i 1

n
i 1
Xi  X

2
n

2
n
n
i 1
i 1

  X i2  n X   X i2   X i  X
i 1
2

2
 nX
2
Use of Inverse Matrix – Solving Simultaneous Equations
AY = C where A and C are matrices of of constants, Y is matrix of unknowns
 A -1 AY  A -1C  Y = A -1C
Equation 1: 12 y1  6 y2  48
(assuming A is square and full rank)
Equation 2: 10y1  2 y2  12
 y1 
 48
Y 
C 
Y = A -1C
12 
 y2 
6 
 2 6  1  2
1
-1
 A 
 


12(2)  6(10)  10 12  84 10 12 
12 6 
A

10

2


6   48 1  96  72  1 168   2 
1 2
Y=A C 
 
 
 





84 10 12  12  84  480  144 84 336  4
-1
Note the wisdom of waiting to divide by | A | at end of calculation!
Useful Matrix Results
All rules assume that the matrices are conformable to operations:
Addition Rules:
AB BA
( A  B )  C  A  ( B  C)
Multiplication Rules:
(AB)C  A(BC)
C( A  B)  CA + CB
k ( A  B)  kA  kB k  scalar
Transpose Rules:
( A ') '  A
( A  B) '  A ' B '
( AB) '  B'A'
(ABC)' = C'B'A'
Inverse Rules (Full Rank, Square Matrices):
(AB)-1 = B -1 A -1
(ABC)-1 = C-1B -1 A -1
(A -1 )-1 = A
(A')-1 = (A -1 )'
Random Vectors and Matrices
Shown for case of n=3, generalizes to any n:
Random variables: Y1 , Y2 , Y3
 Y1 
 Y  Y2 
Y3 
 E Y1 


Expectation: E Y   E Y2 
 E Y3 
In general: E Y   E Yij  i  1,..., n; j  1,..., p
n p
Variance-Covariance Matrix for a Random Vector:
  Y1  E Y1 




 2 Y  E  Y  E Y  Y  E Y '  E  Y2  E Y2  Y1  E Y1 Y2  E Y2  Y3  E Y3   
 Y  E Y  

3 
 3


2

Y1  E Y1



 E  Y2  E Y2  Y1  E Y1


  Y3  E Y3  Y1  E Y1


Y  E Y  Y  E Y  Y  E Y  Y  E Y     
Y  E Y 
Y  E Y  Y  E Y    
  
Y

E
Y
Y

E
Y
Y

E
Y
  
 
 


 
1
1
2
2
1
1
3
3
2
1
2
2
3
3
21
2
2
2
2
3
3
2
2
3
3
31
 12  13 

 22  23   
 32  32 
Linear Regression Example (n=3)
Error terms are assumed to be independent, with mean 0, constant variance  2 :
 E  i   0
 1 
ε   2 
 3 
Y = Xβ + ε
 2  i    2
0
E ε   0 
 0 
  i ,  j   0  i  j
 2 0
0


2
2
σ ε   0 
0    2I
2
0
0



E Y  E Xβ + ε  Xβ  E ε  Xβ
 2 0
0


σ 2 Y  σ 2 Xβ + ε  σ 2 ε   0  2 0    2 I
2
0
0



Mean and Variance of Linear Functions of Y
A  matrix of fixed constants Y  random vector
k n
n1
 a11
W  AY  
k 1
 ak 1
a1n   Y1 
W1   a11Y1  ...  a1nYn 
    random vector: W     

 
  

Wk   ak 1Y1  ...  aknYn 
akn  Yn 
 E W1   E a11Y1  ...  a1nYn    a11 E Y1  ...  a1n E Yn  

 
 

E W  


 
 

 E Wk   E ak 1Y1  ...  aknYn   ak1 E Y1  ...  akn E Yn 
a1n   E Y1 
 a11


 
  AE Y

 ak 1
akn   E Yn 
 
 E  A  Y - E Y    Y - E Y  'A' = AE  Y - E Y   Y - E Y  ' A' = Aσ


σ 2 W = E  AY - AE Y  AY - AE Y ' = E  A  Y - E Y    A  Y - E Y   ' =
2
Y A'
Multivariate Normal Distribution
  12  12
 Y1 
 1 

Y 
 
2


2
Y   2
μ = E Y   2 
Σ = σ 2 Y   21

 
 

 
 
 1n  2 n
Yn 
 n 
Multivariate Normal Density function:
f  Y    2 
 n /2
Σ
1/2
 1

exp    Y - μ  'Σ -1  Y - μ  
 2

 Yi ~ N  i ,  i2  i  1,..., n
 Yi , Y j    ij i  j
Note, if A is a (full rank) matrix of fixed constants:
W = AY ~ N  Aμ, AΣA'
 1n 

 2n 

2 
 n 
Y ~ N  μ, Σ 
Simple Linear Regression in Matrix Form
Simple Linear Regression Model: Yi   0  1 X i   i
i  1,..., n
 Y1   0  1 X 1  1 
Y     X   
0
1 2
2
 2




Y




X


n
0
1
n
n


Defining:
 Y1 
Y 
Y=  2 
 
 
Yn 
1 X 1 
1 X 
2
X




1 X n 
 
β   0
 1 
 1 
 
ε   2   Y = Xβ + ε
 
 
 n 
Assuming constant variance, and independence of error terms  i :
 2 0

0 2
2
2

σ Y = σ ε 


0
0
0

0
  2I
 n n

2
Further, assuming normal distributio n for error terms  i : Y ~ N  Xβ,  2 I 
  0  1 X 1 
   X 
1 2
since: Xβ   0
 E Y




  0  1 X n 
Estimating Parameters by Least Squares
Normal equations obtained from:
n
Q Q
,
and setting each equal to 0:
 0 1
n
nb0  b1  X i   Yi
i 1
n
i 1
n
n
b0  X i  b1  X   X iYi
i 1
i 1
2
i
i 1

 n
Note: In matrix form: X ' X   n

 X i
 i 1

Xi 

i 1

n

X i2 

i 1

n
 n

Y

i


b 
i 1
 Defining b   0 
X'Y   n


 b1 
  X iYi 
 i 1

 X'Xb = X'Y  b =  X'X  X'Y
-1
Based on matrix form:
Q   Y - Xβ  '  Y - Xβ  = Y'Y - Y'Xβ - β'X'Y + β'X'Xβ 
n
n
n
n


2
2
 Y ' Y  2   0  Yi  1  X iYi   n 0  2  0 1  X i  1  X i2
i 1
i 1
i 1
 i 1

n
n
 Q  


2
Y

2
n


2

X


i
0
1
i
   


i 1
i 1
0



  2 X ' Y  2 X ' X 
(Q) 

n
n
 Q   n
β
2

2
X
Y

2

X

2

X



i
i
0
i
1
i
   

i 1
i 1

 1   i 1
Setting this equal to zero, and replacing  with b  X ' Xb  X ' Y
Fitted Values and Residuals
^
^
Y i  b0  b1 X i
ei  Yi  Y i
In Matrix form:
^ 
 Y 1   b0  b1 X 1 
 ^  b  b X 
^
-1
-1
Y  Y 2    0 1 2   Xb = X  X'X  X'Y = HY
H = X  X'X  X'

  

 ^  b b X 
Y n   0 1 n 
 
H is called the "hat" or "projection" matrix, note that H is idempotent (HH = H ) and symmetric(H = H'):
HH = X  X'X  X'X  X'X  X'  X'I  X'X  X'  X  X'X  X'  H
-1
-1
-1
-1
^


^ 
Y

Y
1
 1
  Y1   Y 1 
^ 
Y   ^ 

^
e  Y2  Y 2    2   Y 2   Y - Y = Y - Xb = Y - HY = (I - H)Y

    

   ^ 
^
Y  Y n  Yn  Y n 
 
 n


^


H' = X  X'X  X' ' = X  X'X  X' = H
-1
-1

^
Note: E Y = E HY = HE Y = HXβ = X  X'X  X'Xβ = Xβ
σ 2 Y = Hσ 2IH'   2 H
E e = E  I  H  Y =  I  H  E Y =  I  H  Xβ  Xβ - Xβ = 0
σ 2 e =  I  H  σ 2I  I  H  '   2  I  H 

^
s Y  MSE H
2
-1
s 2 e = MSE  I  H 
Analysis of Variance
n

Total (Corrected) Sum of Squares: SSTO   Yi  Y
i 1

2
 n 
  Yi 
n
  Yi 2   i 1 
n
i 1
2
2
n
Note: Y ' Y   Yi 2
i 1
 n 
  Yi 
 i 1    1  Y'JY
 
n
n
1
J  
n n
1
1


1
 1 
1
 SSTO  Y ' Y    Y'JY  Y ' I    J  Y
n
 n 
SSE  e'e =  Y - Xb  '  Y - Xb  = Y'Y - Y'Xb - b'X'Y + b'X'Xb = Y'Y - b'X'Y = Y'  I - H  Y
since b'X'Y = Y'X'  X'X  X'Y = Y'HY
-1
2
 n 
  Yi 

1
1 
SSR  SSTO  SSE  b'X'Y   i 1   Y'HY    Y'JY  Y '  H    J  Y
n
n
n 

Note that SSTO, SSR, and SSE are all QUADRATIC FORMS: Y'AY for symmetric matrices A
Inferences in Linear Regression
b =  X'X  X'Y Þ E b =  X'X  X'E Y =  X'X  X'Xβ = β
-1
-1
-1
σ 2 b =  X'X  X'σ 2 Y X  X'X  = σ 2  X'X  X'IX  X'X  = σ 2  X'X 
-1
Recall:
 X'X 
-1
-1
2
1
X
  n
2
n
Xi  X


i 1

X


n

2
  Xi  X
 i 1




-1

 n

2
Xi  X 


i 1

1

n
2 
Xi  X 

i 1



^
1 
Xh   
Xh 

^
s 2 b  MSE  X'X 





^
s 2 Y h  Xh's 2 b Xh  MSE Xh'  X'X  Xh
Predicted New Response at X  X h :
Y h  b0  b1 X h  Xh'b
-1
2
 MSE
X MSE
 n

 n
Xi  X


2
i 1
 s b  
X MSE


n

2
Xi  X


i 1

X
Estimated Mean Response at X  X h :
Y h  b0  b1 X h  Xh'b
-1

s 2 pred  MSE 1  Xh'  X'X  Xh
-1

-1


-1

 n

2
Xi  X 


i 1

MSE

n
2 
Xi  X 

i 1

X MSE

2




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