HW1 Solution
3.0
#1.6)
(a) Plot the marginal dot diagrams for all the variables.
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NO2
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solar radiation
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HS
Figure 1:
Dot plots of the variables in the air pollution dataset.
(b) Construct the x̄, Sn , and R arrays, and interpret the entries in R.
1
Wind
7.50
x̄
Solar
73.86
Table 1:
Wind
2.50
-2.78
-0.38
-0.46
-0.59
-2.23
0.17
Wind
Solar
CO
NO
NO2
O3
HC
Table 2:
Wind
Solar
CO
NO
NO2
O3
HC
CO
4.55
NO
2.19
NO2
10.05
O3
9.40
HC
3.10
The sample means of variables.
Solar
-2.78
300.52
3.91
-1.39
6.76
30.79
0.62
CO
-0.38
3.91
1.52
0.67
2.31
2.82
0.14
NO
-0.46
-1.39
0.67
1.18
1.09
-0.81
0.18
NO2
-0.59
6.76
2.31
1.09
11.36
3.13
1.04
O3
-2.23
30.79
2.82
-0.81
3.13
30.98
0.59
HC
0.17
0.62
0.14
0.18
1.04
0.59
0.48
The sample variance-covariance matrix of the variables.
Wind
1.00
-0.10
-0.19
-0.27
-0.11
-0.25
0.16
Table 3:
Solar
-0.10
1.00
0.18
-0.07
0.12
0.32
0.05
CO
-0.19
0.18
1.00
0.50
0.56
0.41
0.17
NO
-0.27
-0.07
0.50
1.00
0.30
-0.13
0.23
NO2
-0.11
0.12
0.56
0.30
1.00
0.17
0.45
O3
-0.25
0.32
0.41
-0.13
0.17
1.00
0.15
HC
0.16
0.05
0.17
0.23
0.45
0.15
1.00
The sample correlation matrix of the variables.
The majority of the variables have only weak linear associations, with correlations close to zero. The pollutants are mostly positively correlated with
each other. Wind is negative correlated with pollutants, while solar radiation is
positively correlated with pollutants.
2
Windy and Sunny
Windy and Not Sunny
1
4
6
10
14
15
7
16
22
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19
23
26
28
20
21
30
31
35
42
36
Not Windy and Sunny
2
3
5
9
12
25
27
37
38
39
40
Figure 2:
Not Windy and Not Sunny
8
11
13
24
29
32
33
34
41
Star plots of the air pollution variables.
To investigate if there is an effect on air pollution with wind and the sun, we
can divide the wind and solar radiation variable in half by the median and then
make the star plots in Figure 2. From the stars, we can see that solar radiation
have some effects on the air pollution. When it is sunny, most pollutants are
on a relative low level. And within each group, the patterns are quite different.
So there are a fair amount of variation in each of the four groups, as there were
days with very little pollution in each group, as well as days with quite a bit of
air pollution.
3
#1.17)
In this dataset the first three are measured in seconds, while the last four
are measured in minutes.
x̄
100m
11.62
200m
23.64
Table 4:
100m
200m
400m
800m
1500m
3000m
Marathon
Table 5:
100m
200m
400m
800m
1500m
3000m
Marathon
100m
0.20
0.48
1.01
0.04
0.11
0.28
9.44
400m
53.41
800m
2.08
1500m
4.33
3000m
9.45
Marathon
173.25
The sample means for the track record.
200m
0.48
1.23
2.55
0.09
0.26
0.65
23.18
400m
1.01
2.55
7.17
0.26
0.70
1.72
57.49
800m
0.04
0.09
0.26
0.01
0.03
0.08
2.57
1500m
0.11
0.26
0.70
0.03
0.11
0.27
8.88
3000m
0.28
0.65
1.72
0.08
0.27
0.68
22.57
Marathon
9.44
23.18
57.49
2.57
8.88
22.57
925.96
The sample variance covariance matrix for the track record variables.
100m
1.00
0.95
0.83
0.73
0.73
0.74
0.69
Table 6:
200m
0.95
1.00
0.86
0.72
0.70
0.71
0.69
400m
0.83
0.86
1.00
0.90
0.79
0.78
0.71
800m
0.73
0.72
0.90
1.00
0.90
0.86
0.78
1500m
0.73
0.70
0.79
0.90
1.00
0.97
0.88
3000m
0.74
0.71
0.78
0.86
0.97
1.00
0.90
Marathon
0.69
0.69
0.71
0.78
0.88
0.90
1.00
The sample correlation matrix for the track record variables.
All the seven variables are strongly positively correlated. And the correlations tend to be larger when distances are close to each other.For example he
correlation between 100m and 200m is 0.95, while the correlation between 100m
and marathon is 0.69. This makes sense, since runners are good at races of
similar length.
4
#1.18)
1
100m
8.62
Table 7:
100m
200m
400m
800m
1500m
3000m
Marathon
200m
8.48
400m
7.51
800m
6.44
1500m
5.81
3000m
5.33
Marathon
4.15
The sample mean track records measured in meters/second.
100m
0.11
0.12
0.10
0.08
0.10
0.10
0.13
200m
0.12
0.15
0.13
0.09
0.11
0.12
0.16
400m
0.10
0.13
0.14
0.11
0.12
0.12
0.15
800m
0.08
0.09
0.11
0.11
0.12
0.12
0.15
1500m
0.10
0.11
0.12
0.12
0.16
0.16
0.20
3000m
0.10
0.12
0.12
0.12
0.16
0.17
0.21
Marathon
0.13
0.16
0.15
0.15
0.20
0.21
0.32
Table 8:
The sample variance covariance matrix of track records measured
in meters/second.
100m
200m
400m
800m
1500m
3000m
Marathon
100m
1.00
0.95
0.84
0.73
0.74
0.75
0.72
200m
0.95
1.00
0.86
0.73
0.72
0.72
0.71
400m
0.84
0.86
1.00
0.90
0.80
0.78
0.71
800m
0.73
0.73
0.90
1.00
0.92
0.87
0.79
1500m
0.74
0.72
0.80
0.92
1.00
0.96
0.86
3000m
0.75
0.72
0.78
0.87
0.96
1.00
0.89
Marathon
0.72
0.71
0.71
0.79
0.86
0.89
1.00
Table 9:
The sample correlation matrix of track records measured in meters/second.
The results are similar with those I obtained in Exercise 1.17. All have
strong, positive linear relationships with each other, and races that are closer
together in distance have a stronger relationship. The differences in correlation
are slightly less maybe because all the races were measured in the same units
now.
Compute the sample variance-covariance matrix (call it S). Obtain the spectral decomposition (also called the eigenvalue decomposition: use eigen if you
are using R) of the variance covariance matrix. Next post-multiply the observation matrix (call it X) with P. Plot the pairwise scatter plots of the first three
columns.
5
−7.4
−7.0
−6.6
−6.2
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var 1
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Figure 3:
−16
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−15
−14
−2.2
−1.8
−1.4
pairwise scatter plots of the first three columns of Y=X*P.
From the pairwise scatter plots, we could not find any obvious relationships
among these three variables. The first two columns may have some negative
relationship while the last two columns have a slightly positive relationship.
#1.26 (a)
x̄
Breed
4.38
SaleP
1742.43
Table 10:
YearlingHT
50.52
FFBody
995.95
PctFFBody
70.88
Frame
6.32
The sample means of the variables in the bulls dataset.
6
Back.fat
0.20
SaleHT
54.13
Salewt
1555.29
Breed
SaleP
YearlingHT
FFBody
PctFF
Frame
Back.fat
SaleHT
Salewt
Breed
9.68
-434.74
2.83
117.83
4.80
1.25
-0.17
3.04
46.94
Table 11:
SaleP
-434.74
388133.66
456.47
5890.60
-229.47
276.42
15.44
486.97
25645.89
YearlingHT
2.83
456.47
3.00
100.13
2.96
1.51
-0.05
2.98
82.81
FFBody
117.83
5890.60
100.13
8594.34
209.50
51.95
-1.40
129.94
6680.31
PctFF
4.80
-229.47
2.96
209.50
10.69
1.46
-0.14
3.41
83.93
Frame
1.25
276.42
1.51
51.95
1.46
0.86
-0.02
1.49
44.32
Back.fat
-0.17
15.44
-0.05
-1.40
-0.14
-0.02
0.01
-0.05
2.41
SaleHT
3.04
486.97
2.98
129.94
3.41
1.49
-0.05
4.02
147.29
Salewt
46.94
25645.89
82.81
6680.31
83.93
44.32
2.41
147.29
16850.66
The sample variance covariance matrix of the variables in the bulls
dataset.
Breed
SaleP
YearlingHT
FFBody
PctFFBody
Frame
Back.fat
SaleHT
Salewt
Breed
1.00
-0.22
0.52
0.41
0.47
0.43
-0.62
0.49
0.12
Table 12:
SaleP
-0.22
1.00
0.42
0.10
-0.11
0.48
0.28
0.39
0.32
YearlingHT
0.52
0.42
1.00
0.62
0.52
0.94
-0.34
0.86
0.37
FFBody
0.41
0.10
0.62
1.00
0.69
0.60
-0.17
0.70
0.56
PctFFBody
0.47
-0.11
0.52
0.69
1.00
0.48
-0.49
0.52
0.20
Frame
0.43
0.48
0.94
0.60
0.48
1.00
-0.26
0.80
0.37
Back.fat
-0.62
0.28
-0.34
-0.17
-0.49
-0.26
1.00
-0.28
0.21
The sample correlation matrix of the variables in the bulls dataset.
Only a few variables(Frame, Yearling height, and Sale Height) have strong
relationships with each other. I do not think the breeds are well separated in
this system since all the correlations between breed and other variables are not
strong. The best potential variable to distinguish between breeds is back fat,
which has the strongest linear relationship with breed.
(b)
I did not find any obvious outliers from Figure 4. From the three dimensional
plot, we can observe that most bulls with breed 8 (Simental) have less back fat
and larger frame. And the values of back fat and frame in breed 1 (Angus) are
more spread out.
7
SaleHT
0.49
0.39
0.86
0.70
0.52
0.80
-0.28
1.00
0.57
Salewt
0.12
0.32
0.37
0.56
0.20
0.37
0.21
0.57
1.00
Figure 4:
A three dimensional plot.
(c)
This time the points are more closely clustered, so it is more clearly to
separate these three breeds. Bulls with breed 8 (Simental) have higher fat free
body weight and higher sale height. And the values of fat free body weight and
sale height in breed 1 (Angus) are more spread out.
8
Figure 5:
A three dimensional plot.
9
#2.20)
0.526
A
= PΛ P =
0.851
0.526
A−1/2 = P Λ−1/2 P 0 =
0.851
1.376 0.325
0.761
A1/2 A−1/2 =
0.325 1.701 −0.145
1/2
1/2
0
2 1
A=
1 3
−0.851 1.902
0
0.526 0.851
1.376 0.325
=
0.526
0
1.176 −0.851 0.526
0.325 1.701
−0.851 0.526
0
0.526 0.851
0.761 −0.145
=
0.526
0
0.851 −0.851 0.526
−0.145 0.616
−0.145
0.761 −0.145 1.376 0.325
1 0
=
=
=I
0.616
−0.145 0.616
0.325 1.701
0 1
# 2.23)
√
V
1/2
ρV
1/2


= 

σ11
√

σ22
..
.
√
1
 ρ12

  ..
 .
ρ12
1
..
.
...
...
..
.
 √
σ11
ρ1p

ρ2p 

..  
. 
σpp
ρ1p ρ2p . . .
1

√ √ 
√ √
σ11
ρ12 σ11 σ22 . . . ρ1p σ11 σpp
√ √
ρ12 √σ11 √σ22
σ22
. . . ρ2p σ22 σpp 


= 

..
..
..
..


.
.
.
.
√ √
√ √
ρ1p σ11 σpp ρ2p σ22 σpp . . .
σpp


σ11 σ12 . . . σ1p
σ12 σ22 . . . σ2p 


=  .
..
..  = Σ
..
 ..
.
.
. 
σ1p σ2p . . . σpp
10
√

σ22
..
.
√




σpp