problem with PCA

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?

Than's in advance


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