Data, Covariance, and Correlation Matrix
Nathaniel E. Helwig
Assistant Professor of Psychology and Statistics
University of Minnesota (Twin Cities)
Updated 16-Jan-2017
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 1
Copyright
Copyright c 2017 by Nathaniel E. Helwig
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 2
Outline of Notes
1) The Data Matrix
3) The Correlation Matrix
Definition
Definition
Properties
Properties
R code
R code
2) The Covariance Matrix
4) Miscellaneous Topics
Definition
Crossproduct calculations
Properties
Vec and Kronecker
R code
Visualizing data
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 3
The Data Matrix
The Data Matrix
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 4
The Data Matrix
Definition
The Organization of Data
The data matrix refers to the array of numbers


x11 x12 · · · x1p
x21 x22 · · · x2p 


x31 x32 · · · x3p 
X=

 ..
..
.. 
..
 .
.
.
. 
xn1 xn2 · · ·
xnp
where xij is the j-th variable collected from the i-th item (e.g., subject).
items/subjects are rows
variables are columns
X is a data matrix of order n × p (# items by # variables).
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 5
The Data Matrix
Definition
Collection of Column Vectors
We can view a data matrix as a collection of column vectors:



X = x1 x2 · · ·

xp 
where xj is the j-th column of X for j ∈ {1, . . . , p}.
The n × 1 vector xj gives the j-th variable’s scores for the n items.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 6
The Data Matrix
Definition
Collection of Row Vectors
We can view a data matrix as a collection of row vectors:


x01


x02


X=

..


.
x0n
where x0i is the i-th row of X for i ∈ {1, . . . , n}.
The 1 × p vector x0i gives the i-th item’s scores for the p variables.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 7
The Data Matrix
Properties
Calculating Variable (Column) Means
The sample mean of the j-th variable is given by
n
x̄j =
1X
xij
n
i=1
= n−1 10n xj
where
1n denotes an n × 1 vector of ones
xj denotes the j-th column of X
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 8
The Data Matrix
Properties
Calculating Item (Row) Means
The sample mean of the i-th item is given by
p
x̄i =
1X
xij
p
=
p−1 x0i 1p
j=1
where
1p denotes an p × 1 vector of ones
x0i denotes the i-th row of X
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 9
The Data Matrix
R Code
Data Frame and Matrix Classes in R
> data(mtcars)
> class(mtcars)
[1] "data.frame"
> dim(mtcars)
[1] 32 11
> head(mtcars)
mpg cyl disp hp drat
wt qsec vs am gear carb
Mazda RX4
21.0
6 160 110 3.90 2.620 16.46 0 1
4
4
Mazda RX4 Wag
21.0
6 160 110 3.90 2.875 17.02 0 1
4
4
Datsun 710
22.8
4 108 93 3.85 2.320 18.61 1 1
4
1
Hornet 4 Drive
21.4
6 258 110 3.08 3.215 19.44 1 0
3
1
Hornet Sportabout 18.7
8 360 175 3.15 3.440 17.02 0 0
3
2
Valiant
18.1
6 225 105 2.76 3.460 20.22 1 0
3
1
> X <- as.matrix(mtcars)
> class(X)
[1] "matrix"
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 10
The Data Matrix
R Code
Row and Column Means
> # get row means (3 ways)
> rowMeans(X)[1:3]
Mazda RX4 Mazda RX4 Wag
Datsun 710
29.90727
29.98136
23.59818
> c(mean(X[1,]), mean(X[2,]), mean(X[3,]))
[1] 29.90727 29.98136 23.59818
> apply(X,1,mean)[1:3]
Mazda RX4 Mazda RX4 Wag
Datsun 710
29.90727
29.98136
23.59818
> # get column means (3 ways)
> colMeans(X)[1:3]
mpg
cyl
disp
20.09062
6.18750 230.72188
> c(mean(X[,1]), mean(X[,2]), mean(X[,3]))
[1] 20.09062
6.18750 230.72188
> apply(X,2,mean)[1:3]
mpg
cyl
disp
20.09062
6.18750 230.72188
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 11
The Data Matrix
R Code
Other Row and Column Functions
> # get column medians
> apply(X,2,median)[1:3]
mpg
cyl disp
19.2
6.0 196.3
> c(median(X[,1]), median(X[,2]), median(X[,3]))
[1] 19.2
6.0 196.3
> # get column ranges
> apply(X,2,range)[,1:3]
mpg cyl disp
[1,] 10.4
4 71.1
[2,] 33.9
8 472.0
> cbind(range(X[,1]), range(X[,2]), range(X[,3]))
[,1] [,2] [,3]
[1,] 10.4
4 71.1
[2,] 33.9
8 472.0
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 12
The Covariance Matrix
The Covariance Matrix
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 13
The Covariance Matrix
Definition
The Covariation of Data
The covariance matrix refers to the symmetric array of numbers
 2

s1 s12 s13 · · · s1p
s21 s2 s23 · · · s2p 
2


2


S = s31 s32 s3 · · · s3p 
 ..
..
..
.
.
..
.. 
 .

.
.
2
sp1 sp2 sp3 · · · sp
where
sj2 = (1/n)
Pn
i=1 (xij
− x̄j )2 is the variance of the j-th variable
Pn
sjk = (1/n) i=1 (xij − x̄j )(xik − x̄k ) is the covariance between the
j-th and k -th variables
P
x̄j = (1/n) ni=1 xij is the mean of the j-th variable
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 14
The Covariance Matrix
Definition
Covariance Matrix from Data Matrix
We can calculate the covariance matrix such as
S=
1 0
X Xc
n c
where Xc = X − 1n x̄0 = CX with
x̄0 = (x̄1 , . . . , x̄p ) denoting the vector of variable means
C = In − n−1 1n 10n denoting a centering matrix
Note that the centered matrix Xc has the form


x11 − x̄1 x12 − x̄2 · · · x1p − x̄p
x21 − x̄1 x22 − x̄2 · · · x2p − x̄p 


x31 − x̄1 x32 − x̄2 · · · x3p − x̄p 
Xc = 



..
..
..
..


.
.
.
.
xn1 − x̄1 xn2 − x̄2 · · ·
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
xnp − x̄p
Updated 16-Jan-2017 : Slide 15
The Covariance Matrix
Properties
Variances are Nonnegative
Variances are sums-of-squares, which implies that sj2 ≥ 0 ∀j.
sj2 > 0 as long as there does not exist an α such that xj = α1n
This implies that. . .
tr(S) ≥ 0 where tr(·) denotes the matrix trace function
Pp
j=1 λj ≥ 0 where (λ1 , . . . , λp ) are the eigenvalues of S
If n < p, then λj = 0 for at least one j ∈ {1, . . . , p}. If n ≥ p and the p
columns of X are linearly independent, then λj > 0 for all j ∈ {1, . . . , p}.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 16
The Covariance Matrix
Properties
The Cauchy-Schwarz Inequality
From the Cauchy-Schwarz inequality we have that
sjk2 ≤ sj2 sk2
with the equality holding if and only if xj and xk are linearly dependent.
We could also write the Cauchy-Schwarz inequality as
|sjk | ≤ sj sk
where sj and sk denote the standard deviations of the variables.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 17
The Covariance Matrix
R Code
Covariance Matrix by Hand (hard way)
>
>
>
>
>
n <- nrow(X)
C <- diag(n) - matrix(1/n, n, n)
Xc <- C %*% X
S <- t(Xc) %*% Xc / (n-1)
S[1:3,1:6]
mpg
cyl
disp
hp
drat
wt
mpg
36.324103 -9.172379 -633.0972 -320.7321
2.1950635 -5.116685
cyl
-9.172379
3.189516
199.6603 101.9315 -0.6683669
1.367371
disp -633.097208 199.660282 15360.7998 6721.1587 -47.0640192 107.684204
# or #
> Xc <- scale(X, center=TRUE, scale=FALSE)
> S <- t(Xc) %*% Xc / (n-1)
> S[1:3,1:6]
mpg
cyl
disp
hp
drat
wt
mpg
36.324103 -9.172379 -633.0972 -320.7321
2.1950635 -5.116685
cyl
-9.172379
3.189516
199.6603 101.9315 -0.6683669
1.367371
disp -633.097208 199.660282 15360.7998 6721.1587 -47.0640192 107.684204
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 18
The Covariance Matrix
R Code
Covariance Matrix using cov Function (easy way)
# calculate covariance matrix
> S <- cov(X)
> dim(S)
[1] 11 11
# check variance
> S[1,1]
[1] 36.3241
> var(X[,1])
[1] 36.3241
> sum((X[,1]-mean(X[,1]))^2) / (n-1)
[1] 36.3241
# check covariance
> S[1:3,1:6]
mpg
cyl
disp
hp
drat
wt
mpg
36.324103 -9.172379 -633.0972 -320.7321
2.1950635 -5.116685
cyl
-9.172379
3.189516
199.6603 101.9315 -0.6683669
1.367371
disp -633.097208 199.660282 15360.7998 6721.1587 -47.0640192 107.684204
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 19
The Correlation Matrix
The Correlation Matrix
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 20
The Correlation Matrix
Definition
The Correlation of Data
The correlation matrix refers to the symmetric array of numbers


1 r12 r13 · · · r1p
r21 1 r23 · · · r2p 


r31 r32 1 · · · r3p 
R=

 ..
..
..
.. 
..
 .
.
.
.
. 
rp1 rp2 rp3 · · ·
where
1
Pn
sjk
i=1 (xij − x̄j )(xik − x̄k )
qP
rjk =
= qP
sj sk
n
n
2
2
i=1 (xij − x̄j )
i=1 (xik − x̄k )
is the Pearson correlation coefficient between variables xj and xk .
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 21
The Correlation Matrix
Definition
Correlation Matrix from Data Matrix
We can calculate the correlation matrix such as
1
R = X0s Xs
n
where Xs = CXD−1 with
C = In − n−1 1n 10n denoting a centering matrix
D = diag(s1 , . . . , sp ) denoting a diagonal scaling matrix
Note that the standardized matrix Xs has the form


(x11 − x̄1 )/s1 (x12 − x̄2 )/s2 · · · (x1p − x̄p )/sp
(x21 − x̄1 )/s1 (x22 − x̄2 )/s2 · · · (x2p − x̄p )/sp 




Xs = (x31 − x̄1 )/s1 (x32 − x̄2 )/s2 · · · (x3p − x̄p )/sp 


..
..
..
..


.
.
.
.
(xn1 − x̄1 )/s1 (xn2 − x̄2 )/s2 · · ·
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
(xnp − x̄p )/sp
Updated 16-Jan-2017 : Slide 22
The Correlation Matrix
Properties
Correlation of a Variable with Itself is One
Assuming that sj2 > 0 for all j ∈ {1, . . . , p}, we have that
Pn
− x̄j )(xik − x̄k )
qP
=
n
2
2
i=1 (xij − x̄j )
i=1 (xik − x̄k )
Cor(xj , xk ) = qP
n
i=1 (xij
1
rjk
if j = k
if j 6= k
Because rjk = 1 whenever j = k , we know that
tr(R) = p where tr(·) denotes the matrix trace function
Pp
j=1 λj = p where (λ1 , . . . , λp ) are the eigenvalues of R
We also know that the eigenvalues satisfy
λj = 0 for at least one j ∈ {1, . . . , p} if n < p
λj > 0 ∀j if columns of X are linearly independent
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 23
The Correlation Matrix
Properties
The Cauchy-Schwarz Inequality (revisited)
Reminder: the Cauchy-Schwarz inequality implies that
sjk2 ≤ sj2 sk2
with the equality holding if and only if xj and xk are linearly dependent.
Rearranging the terms, we have that
sjk2
sj2 sk2
≤1
←→
rjk2 ≤ 1
which implies that |rjk | ≤ 1 with equality holding if and only if
xj = α1n + βxk for some scalars α ∈ R and β ∈ R.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 24
The Correlation Matrix
R Code
Correlation Matrix by Hand (hard way)
>
>
>
>
>
>
n <- nrow(X)
C <- diag(n) - matrix(1/n, n, n)
D <- diag(apply(X, 2, sd))
Xs <- C %*% X %*% solve(D)
R <- t(Xs) %*% Xs / (n-1)
R[1:3,1:6]
[,1]
[,2]
[,3]
[,4]
[,5]
[,6]
[1,] 1.0000000 -0.8521620 -0.8475514 -0.7761684 0.6811719 -0.8676594
[2,] -0.8521620 1.0000000 0.9020329 0.8324475 -0.6999381 0.7824958
[3,] -0.8475514 0.9020329 1.0000000 0.7909486 -0.7102139 0.8879799
# or #
> Xs <- scale(X, center=TRUE, scale=TRUE)
> R <- t(Xs) %*% Xs / (n-1)
> R[1:3,1:6]
mpg
cyl
disp
hp
drat
wt
mpg
1.0000000 -0.8521620 -0.8475514 -0.7761684 0.6811719 -0.8676594
cyl -0.8521620 1.0000000 0.9020329 0.8324475 -0.6999381 0.7824958
disp -0.8475514 0.9020329 1.0000000 0.7909486 -0.7102139 0.8879799
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 25
The Correlation Matrix
R Code
Correlation Matrix using cor Function (easy way)
# calculate correlation matrix
> R <- cor(X)
> dim(R)
[1] 11 11
# check correlation of mpg and cyl
> R[1,2]
[1] -0.852162
> cor(X[,1],X[,2])
[1] -0.852162
> cov(X[,1],X[,2]) / (sd(X[,1]) * sd(X[,2]))
[1] -0.852162
# check correlations
> R[1:3,1:6]
mpg
cyl
disp
hp
drat
wt
mpg
1.0000000 -0.8521620 -0.8475514 -0.7761684 0.6811719 -0.8676594
cyl -0.8521620 1.0000000 0.9020329 0.8324475 -0.6999381 0.7824958
disp -0.8475514 0.9020329 1.0000000 0.7909486 -0.7102139 0.8879799
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 26
Miscellaneous Topics
Miscellaneous Topics
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 27
Miscellaneous Topics
Crossproduct Calculations
Two Types of Matrix Crossproducts
We often need to calculate one of two different types of crossproducts:
X0 Y = “regular” crossproduct of X and Y
XY0 = “transpose” crossproduct of X and Y
Regular crossproduct is X0 being post-multipled by Y.
Transpose crossproduct is X being post-multipled by Y0 .
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 28
Miscellaneous Topics
Crossproduct Calculations
Simple and Efficient Crossproducts in R
> X <- matrix(rnorm(2*3),2,3)
> Y <- matrix(rnorm(2*3),2,3)
> t(X) %*% Y
[,1]
[,2]
[,3]
[1,] 0.1342302 -1.8181837 -1.107821
[2,] 1.1014703 -0.6619466 -1.356606
[3,] 0.8760823 -1.0077151 -1.340044
> crossprod(X, Y)
[,1]
[,2]
[,3]
[1,] 0.1342302 -1.8181837 -1.107821
[2,] 1.1014703 -0.6619466 -1.356606
[3,] 0.8760823 -1.0077151 -1.340044
> X %*% t(Y)
[,1]
[,2]
[1,] 0.8364239 3.227566
[2,] -1.3899946 -2.704184
> tcrossprod(X, Y)
[,1]
[,2]
[1,] 0.8364239 3.227566
[2,] -1.3899946 -2.704184
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 29
Miscellaneous Topics
Vec Operator and Kronecker Product
Turning a Matrix into a Vector
The vectorization (vec) operator turns a matrix into a vector:
vec(X) = x11 , x21 , . . . , xn1 , x12 , . . . , xn2 , . . . , x1p , . . . , xnp
0
where the vectorization is done column-by-column.
In R, we just use the combine function c to vectorize a matrix
> X <- matrix(1:6,2,3)
> X
[,1] [,2] [,3]
[1,]
1
3
5
[2,]
2
4
6
> c(X)
[1] 1 2 3 4 5 6
> c(t(X))
[1] 1 3 5 2 4 6
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 30
Miscellaneous Topics
Vec Operator and Kronecker Product
vec Operator Properties
Some useful properties of the vec(·) operator include:
vec(a0 ) = vec(a) = a for any vector a ∈ Rm
vec(ab0 ) = b ⊗ a for any vectors a ∈ Rm and b ∈ Rn
vec(A)0 vec(B) = tr(A0 B) for any matrices A, B ∈ Rm×n
vec(ABC) = (C0 ⊗ A)vec(B) if the product ABC exists
Note: ⊗ is the Kronecker product, which is defined on the next slide.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 31
Miscellaneous Topics
Vec Operator and Kronecker Product
Kronecker Product of Two Matrices
Given X = {xij }n×p and Y = {yij }m×q , the Kronecker product is


x11 Y x12 Y · · · x1p Y
x21 Y x22 Y · · · x2p Y


X⊗Y= .
..
.. 
..
 ..
.
.
. 
xn1 Y xn2 Y · · ·
xnp Y
which is a matrix of order mn × pq.
In R, the kronecker function calculates Kronecker products
> X <- matrix(1:4,2,2)
> Y <- matrix(5:10,2,3)
> kronecker(X, Y)
[,1] [,2] [,3] [,4] [,5] [,6]
[1,]
5
7
9
15
21
27
[2,]
6
8
10
18
24
30
[3,]
10
14
18
20
28
36
[4,]
12
16
20
24
32
40
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 32
Miscellaneous Topics
Vec Operator and Kronecker Product
Kronecker Product Properties
Some useful properties of the Kronecker product include:
1
A ⊗ a = Aa = aA = a ⊗ A for any a ∈ R and A ∈ Rm×n
2
(A ⊗ B)0 = A0 ⊗ B0 for any matrices A ∈ Rm×n and B ∈ Rp×q
3
a0 ⊗ b = ba0 = b ⊗ a0 for any vectors a ∈ Rm and b ∈ Rn
4
tr(A ⊗ B) = tr(A)tr(B) for any matrices A ∈ Rm×m and B ∈ Rp×p
5
(A ⊗ B)−1 = A−1 ⊗ B−1 for any invertible matrices A and B
6
(A ⊗ B)† = A† ⊗ B† where (·)† is Moore-Penrose pseudoinverse
7
|A ⊗ B| = |A|p |B|m for any matrices A ∈ Rm×m and B ∈ Rp×p
8
rank(A ⊗ B) = rank(A)rank(B) for any matrices A and B
9
A ⊗ (B + C) = A ⊗ B + A ⊗ C for any matrices A, B, and C
10
(A + B) ⊗ C = A ⊗ C + B ⊗ C for any matrices A, B, and C
11
(A ⊗ B) ⊗ C = A ⊗ (B ⊗ C) for any matrices A, B, and C
12
(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD) for any matrices A, B, C, and D
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 33
Miscellaneous Topics
Vec Operator and Kronecker Product
Common Application of Vec and Kronecker
Suppose the rows of X are iid samples from some multivariate
distribution with mean µ = (µ1 , . . . , µp )0 and covariance matrix Σ.
iid
xi ∼ (µ, Σ) where xi is the i-th row of X
If we let y = vec(X0 ), then the expectation and covariance are
E(y) = 1n ⊗ µ is the mean vector
V (y) = In ⊗ Σ is the covariance matrix
Note that the covariance matrix is block diagonal


Σ 0 ··· 0
0 Σ · · · 0


In ⊗ Σ =  . . .
. . ... 
 .. ..

0
0
···
Σ
given that data from different subjects are assumed to be independent.
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 34
Miscellaneous Topics
Visualizing Multivariate Data
Two Versions of a Scatterplot in R
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plot(mtcars$hp, mtcars$mpg, xlab="HP", ylab="MPG")
library(car)
scatterplot(mtcars$hp, mtcars$mpg, xlab="HP", ylab="MPG")
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 35
Miscellaneous Topics
Visualizing Multivariate Data
Two Versions of a Scatterplot Matrix in R
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cylint <- as.integer(factor(mtcars$cyl))
pairs(~mpg+disp+hp+wt, data=mtcars, col=cylint, pch=cylint)
library(car)
scatterplotMatrix(~mpg+disp+hp+wt|cyl, data=mtcars)
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 36
Miscellaneous Topics
Visualizing Multivariate Data
Three-Dimensional Scatterplot in R
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library(scatterplot3d)
sp3d <- scatterplot3d(mtcars$hp, mtcars$wt, mtcars$mpg,
type="h", color=cylint, pch=cylint,
xlab="HP", ylab="WT", zlab="MPG")
fitmod <- lm(mpg ~ hp + wt, data=mtcars)
sp3d$plane3d(fitmod)
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 37
Miscellaneous Topics
Visualizing Multivariate Data
2
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Color Image (Heat Map) Plots in R
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fitmod <- lm(mpg ~ hp + wt, data=mtcars)
hpseq <- seq(50, 330, by=20)
wtseq <- seq(1.5, 5.4, length=15)
newdata <- expand.grid(hp=hpseq, wt=wtseq)
fit <- predict(fitmod, newdata)
fitmat <- matrix(fit, 15, 15)
image(hpseq, wtseq, fitmat, xlab="HP", ylab="WT")
library(bigsplines)
imagebar(hpseq, wtseq, fitmat, xlab="HP", ylab="WT", zlab="MPG", col=heat.colors(12), ncolor=12)
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 38
Miscellaneous Topics
Visualizing Multivariate Data
cyl
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0.42 −0.59 −0.43 −0.71 0.09 −0.17 qsec
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Correlation Matrix Plot in R
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0.6 −0.52 −0.59 −0.24 0.71 −0.69 −0.23 0.17
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am
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0.48 −0.49 −0.56 −0.13 0.7 −0.58 −0.21 0.21 0.79 gear
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−0.55 0.53 0.39 0.75 −0.09 0.43 −0.66 −0.57 0.06 0.27 carb
−1
−1
cmat <- cor(mtcars)
library(corrplot)
corrplot(cmat, method="circle")
corrplot.mixed(cmat, lower="number", upper="ellipse")
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 39
Miscellaneous Topics
Visualizing Multivariate Data
1.0
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mpg
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Correlation Matrix Color Image (Heat Map) in R
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disp
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disp
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drat
0.68 −0.7 −0.71−0.45
−0.87 0.78 0.89 0.66 −0.71
qsec
qsec
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z
drat
−0.78 0.83 0.79
0.42 −0.59−0.43−0.71 0.09 −0.17
−0.5
am
am
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0.66 −0.81−0.71−0.72 0.44 −0.55 0.74
0.6 −0.52−0.59−0.24 0.71 −0.69−0.23 0.17
mpg
disp
drat
qsec
am
carb
−0.55 0.53 0.39 0.75 −0.09 0.43 −0.66−0.57 0.06 0.27
mpg
disp
drat
qsec
am
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carb
−1.0
carb
0.48 −0.49−0.56−0.13 0.7 −0.58−0.21 0.21 0.79
carb
cmat <- cor(mtcars)
p <- nrow(cmat)
library(RColorBrewer)
imagebar(1:p, 1:p, cmat[,p:1], axes=F, zlim=c(-1,1), xlab="", ylab="", col=brewer.pal(7, "RdBu"))
axis(1, 1:p, labels=rownames(cmat))
axis(2, p:1, labels=colnames(cmat))
for(k in 1:p) { for(j in 1:k) { if(j < k) text(j, p+1-k, labels=round(cmat[j,k],2), cex=0.75) } }
Nathaniel E. Helwig (U of Minnesota)
Data, Covariance, and Correlation Matrix
Updated 16-Jan-2017 : Slide 40