# [R] problem with PCA

David L Carlson dcarlson at tamu.edu
Mon Mar 13 16:18:33 CET 2017

```The manual is talking about the angle between the variables in p dimensional space were p is the number of variables. The angle can appear differently in 2-dimensions depending on your viewing angle (which dimensions you are ignoring, think of how the 2-dimensional shadow of a 3-dimensional object changes as the sun moves across the sky). Imagine two vectors originating at the origin on a plane, one pointing northeast and one pointing southeast. Now rotate those vectors in a 3rd dimension by bringing the northeast vector toward you. As you do that the southeast vector will move away and the angle between the 2 will appear to decrease. When you have rotated 90 degrees toward you, the vectors will be on top of one another, their angle appears to have changed from 90 degrees to 0 degrees.

As you indicated, in your first example, your variables are only moderately correlated. The first 2 components capture only 78 percent of the variation among the original 4 variables:

> summary(pca.mx_fus)
Importance of components:
PC1      PC2      PC3     PC4
Standard deviation     1.5264   0.8950   0.7234 0.58793
Proportion of Variance 0.5825   0.2003   0.1308 0.08642
Cumulative Proportion  0.5825 <<0.7828>> 0.9136 1.00000

In your second example, the variables are more highly correlated and the first 2 components capture almost 98 percent of the variation among the original 5 variables. As a result, plotting the first two variables gives you a better perspective on variation in the original 5 dimensions:

> summary(pca.tb)
Importance of components:
PC1      PC2       PC3     PC4       PC5
Standard deviation     1.9694   1.0063   0.32647 0.04681 3.868e-17
Proportion of Variance 0.7757   0.2025   0.02132 0.00044 0.000e+00
Cumulative Proportion  0.7757 <<0.9782>> 0.99956 1.00000 1.000e+00

-------------------------------------
David L Carlson
Department of Anthropology
Texas A&M University
College Station, TX 77840-4352

From: Denis Francisci [mailto:denis.francisci at gmail.com]
Sent: Saturday, March 11, 2017 3:21 AM
To: David L Carlson <dcarlson at tamu.edu>
Cc: R-help Mailing List <r-help at r-project.org>
Subject: Re: [R] problem with PCA

If I understood the relative positions of variable arrows don't reflect the coefficient of correlation of the original variables. In fact these positions change if I use different PC axes.
But in some manual about PCA in R I read: "Pairs of variables that form acute angles at the origin, close to 0°, should be highly and positively correlated; variables close to right angles tend to have low correlation; variables at obtuse angles, close to 180°, tend to have high negative correlation".

And If I do a fictional test, it seems true:

tb<-data.frame(
c(1,2,3,4,5,6,7,8,9), #orig data
c(2,4,5,8,10,12,14,16,18),#strong positive correlation
c(25,29,52,63,110,111,148,161,300),#weakly correlation
c(-1,-2,-3,-4,-5,-6,-7,-8,-9),#strong negative correlation
c(3,8,4,6,1,3,2,5,7)#not correlation
)
names(tb)<-c("orig","corr+","corr+2","corr-","random")

pca<-prcomp(as.matrix(tb),scale=T)
biplot(pca,choices = c(1,2))

On the first 2 PC the positions of arrows reflect perfectly the original correlations.

My data behaviour differently, maybe because my original variables are not strong correlated?

2017-03-10 15:49 GMT+01:00 David L Carlson <dcarlson at tamu.edu>:
This is more a question about principal components analysis than about R. You have 4 variables and they are moderately correlated with one another (weight and hole are only .2). When the data consist of measurements, this usually suggests that the overall size of the object is being partly measured by each variable. In your case object size is measured by the first principle component (PC1) with larger objects having more negative scores so larger objects are on the left and smaller ones are on the right of the biplot.

The biplot can only display 2 of the 4 dimensions of your data at one time. In the first 2 dimensions, diam and height are close together, but in the 3rd dimension (PC3), they are on opposite sides of the component. If you plot different pairs of dimensions (e.g. 1 with 3 or 2 with 3, see below), the arrows will look different because you are looking from different directions.

> pca
Standard deviations:
[1] 1.5264292 0.8950379 0.7233671 0.5879295

Rotation:
PC1         PC2         PC3        PC4
height -0.5210224 -0.06545193  0.80018012 -0.2897646
diam   -0.5473677  0.06309163 -0.57146893 -0.6081376
hole   -0.4598646 -0.70952862 -0.17476677  0.5045297
weight -0.4663141  0.69878797 -0.05090785  0.5400508

> biplot(pca, choices=c(1, 3))
> biplot(pca, choices=c(2, 3))

-------------------------------------
David L Carlson
Department of Anthropology
Texas A&M University
College Station, TX 77840-4352

-----Original Message-----
From: R-help [mailto:r-help-bounces at r-project.org] On Behalf Of Denis Francisci
Sent: Friday, March 10, 2017 4:45 AM
To: R-help Mailing List <r-help at r-project.org>
Subject: [R] problem with PCA

Hi all.
I'm newbie in PCA by I don't understand a behaviour of R.
I have this data matrix:

>mx_fus
height diam  hole  weight
1    2.3  3.5  1.1   18
2    2.0  3.5  0.9   17
3    3.8  4.3  0.7   34
4    2.1  3.4  0.9   15
5    2.3  3.8  1.0   19
6    2.2  3.8  1.0   19
7    3.2  4.4  0.9   34
8    3.0  4.3  1.0   30
9    2.8  3.9  0.9   21
10   3.3  4.2  1.1   33
11   2.3  3.9  0.9   25
12   2.3  3.3  0.5   17
13   0.9  2.4  0.4   10
14   1.4  2.4  0.5   10
15   2.2  3.6  0.7   22
16   2.9  3.8  0.8   30
17   2.9  3.5  0.6   27
18   2.3  3.5  0.5   24
19   1.8  2.3  0.5   29
20   1.4  2.5  0.6   34
21   0.8  2.3  0.6   21
22   1.8  2.4  0.6   23
23   1.5  2.2  0.6    7
24   0.9  1.7  0.4   14
25   2.1  2.2  0.5   25
26   1.3  2.4  0.6   33
27   1.3  2.7  0.4   39
28   0.5  2.2  0.5   13
29   1.4  4.2  0.8   23
30   1.6  2.0  0.4   30
31   1.4  2.2  0.6   25
32   1.8  2.5  0.6   28
33   1.4  2.6  0.6   41
34   1.6  2.3  0.3   32
35   1.6  2.5  0.5   41
36   2.8  2.9  0.8   47
37   0.6  2.5  0.8   21
38   1.6  2.8  0.7   13
39   1.7  3.3  0.8   17
40   1.6  3.9  1.9   20
41   1.4  4.7  0.9   26
42   1.2  4.2  0.7   21
43   3.5  4.2  0.9   47
44   2.3  3.6  0.7   24
45   2.3  3.4  0.4   21
46   1.9  2.6  0.7   14
47   1.9  3.0  0.7   15
48   2.7  3.7  0.9   26
49   3.0  3.8  0.7   35
50   1.2  2.0  0.7    5
51   1.6  2.5  0.5   15
52   1.3  2.6  0.5   16
53   2.5  3.9  0.9   32
54   0.9  3.3  0.6    9
55   1.8  2.4  0.5   17
56   2.4  3.7  1.1   30
57   2.1  3.5  1.1   22
58   2.6  3.9  1.0   38
59   2.6  3.6  1.0   27
60   2.6  4.1  1.0   34
61   2.9  3.6  0.8   32
62   2.6  3.3  0.7   22
63   1.8  2.5  0.7   26
64   3.0  2.8  1.3    2
65   0.5  2.2  0.4    3
66   1.9  3.4  0.7   14
67   1.4  3.8  0.9   18
68   2.0  4.0  1.0   30
69   3.1  4.0  1.3   21
70   2.5  4.0  0.8   19
71   2.5  4.5  1.0   20
72   1.8  3.5  1.4   18
73   2.1  3.5  1.4   25
74   1.5  2.6  0.5    9
75   2.8  3.2  1.2   16
76   1.0  5.0  0.3   32
77   0.3  5.8  0.5   56
78   0.5  1.5  0.2    1
79   0.7  1.4  0.2    1
80   0.5  1.3  0.2    1
81   0.7  3.3  0.4    7
82   1.9  4.7  1.0   24
83   3.1  4.2  0.9   49
84   2.8  3.6  0.7   28
85   2.7  3.2  0.7   29
86   3.0  4.0  0.9   36
87   1.7  2.7  0.7   14
88   1.5  2.9  0.7   18
89   2.9  3.5  0.7   30
90   3.0  3.4  0.8   30
91   2.0  2.8  0.5   14
92   2.4  3.5  0.7   24
93   0.8  4.1  0.6   12
94   1.7  2.5  0.5   23
95   1.4  2.4  0.8   31
96   1.5  2.7  0.4   20
97   2.6  3.7  0.6   31
98   2.6  3.0  0.6   18
99   2.5  5.0  0.7   40
100  2.5  3.7  0.5   30
101  2.4  2.9  0.7   17
102  2.3  3.0  0.5   15
103  2.2  3.3  0.6   19
104  1.5  2.1  0.5    5
105  2.0  2.2  0.5   10
106  2.6  3.5  0.6   26
107  2.3  3.0  0.6   15
108  2.5  4.5  0.7   40
109  2.1  3.1  0.5   15
110  1.3  2.1  0.8   14
111  0.8  2.5  0.2    5
112  0.6  3.1  0.7    8

I perform a PCA in R

>pca<-prcomp(mx_fus,scale=TRUE)
>biplot(pca, choices = c(1,2), cex=0.7)

The biplot put the arrows of diam and height very near on the first
component axis.
So I understand that these 2 variables are well represented in the PC1 and
they are correlated each other.
But if I test the correlation, the value o correlation coefficient is low

>cor(mx_fus[,1],mx_fus[,2])
0.4828185

Why the plot says a thing and correlation function says the opposite?
Two near arrows don't represent a strong correlation between the 2
variables (as I read in some manuals), but only with the component axis?