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.franci...@gmail.com]
Sent: Saturday, March 11, 2017 3:21 AM
To: David L Carlson <dcarl...@tamu.edu>
Cc: R-help Mailing List <r-help@r-project.org>
Subject: Re: [R] problem with PCA
Thank you David for your answer.
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 <dcarl...@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-boun...@r-project.org] On Behalf Of Denis Francisci
Sent: Friday, March 10, 2017 4:45 AM
To: R-help Mailing List <r-help@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
[[alternative HTML version deleted]]
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
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