[R] prcomp - principal components in R

[zubin]

Hello, not understanding the output of prcomp, I reduce the number of
components and the output continues to show cumulative 100% of the
variance explained, which can't be the case dropping from 8 components
to 3.

How do i get the output in terms of the cumulative % of the total
variance, so when i go from total solution of 8 (8 variables in the data
set), to a reduced number of components, i can evaluate % of variance
explained, or am I missing something??

8 variables in the data set

> princ = prcomp(df[,-1],rotate="varimax",scale=TRUE)
> summary(princ)
Importance of components:
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8
Standard deviation 1.381 1.247 1.211 0.994 0.927 0.764 0.6708 0.4366
Proportion of Variance 0.238 0.194 0.183 0.124 0.107 0.073 0.0562 0.0238
Cumulative Proportion 0.238 0.433 0.616 0.740 0.847 0.920 0.9762 *1.0000*

> princ = prcomp(df[,-1],rotate="varimax",scale=TRUE,tol=.75)
> summary(princ)

Importance of components:
PC1 PC2 PC3
Standard deviation 1.381 1.247 1.211
Proportion of Variance 0.387 0.316 0.297
Cumulative Proportion 0.387 0.703 *1.000*

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[stephen sefick]

principal components is a data reduction technique. It looks like
you have three axes that account for 100%. Make this reporducible.

--
Stephen Sefick

Let's not spend our time and resources thinking about things that are
so little or so large that all they really do for us is puff us up and
make us feel like gods. We are mammals, and have not exhausted the
annoying little problems of being mammals.

-K. Mullis

[zubin]

okay, an extreme case, only 1 component, explains 100%, something weird
going on..

> princ = prcomp(df[,-1],rotate="varimax",scale=TRUE,tol=.95)

> summary(princ)
Importance of components:
PC1

Standard deviation 1.38
Proportion of Variance 1.00
Cumulative Proportion 1.00

[stephen sefick]

Look at it linearly?

--
Stephen Sefick

Let's not spend our time and resources thinking about things that are
so little or so large that all they really do for us is puff us up and
make us feel like gods. We are mammals, and have not exhausted the
annoying little problems of being mammals.

-K. Mullis

______________________________________________

[Daniel Malter]

In the first PCA you ask how much variance of the EIGHT (!) variables is
captured by the first, second,..., eigth principal component.

In the second PCA you ask how much variance of the THREE (!) variables is
captured by the first, second, and third principal component.

Of course you need only as many PCs as there are variables to capture 100 %
of the variance. Your "problem" thus comes from the fact that you have eight
variables in the first PCA, which requires eight PCs to capture 100%, and
that you have only three variables in the second PCA, which naturally only
requires three PCs to capture 100% of the variance.

So it's more, yes, you are missing something in this case, rather than that
something is wrong with the analyses.

HTH,
Daniel

-------------------------
cuncta stricte discussurus
-------------------------

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Von: r-help-...@r-project.org [mailto:r-help-...@r-project.org] Im
Auftrag von zubin
Gesendet: Monday, November 09, 2009 12:37 PM
An: r-h...@r-project.org
Betreff: [R] prcomp - principal components in R

[zubin]

All 8 variables are still in the analysis, i am just reducing the number
of components being estimated i thought..

Example 1 component 8 variables, there is no way 1 component explains
100% of the variance of the 8 variable data set.

> princ = prcomp(df[,-1],rotate="varimax",scale=TRUE,tol=.95)

> summary(princ)
Importance of components:
PC1

Standard deviation 1.38
Proportion of Variance 1.00
Cumulative Proportion 1.00

> summary(princ)

Rotation:
PC1
VIX0 -0.08217686
UUP0 -0.18881983
USO0 0.26647346
GLD0 0.26983923
HYG0 0.60674758
term0 0.18220237
spread0 0.61614047
TNX0 0.18111684

[mark...@verizon.net]

Hi: I'm not familar with prcomp but with the principal components function
in bill revelle's psych package , one can specify the number of components
one wants to use to build the "closest" covariance matrix I don't know
what tol is doing in your example but it's not doing that.
                                     Â
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             mark

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[Tony Plate]

The output of summary prcomp displays the cumulative amount of variance explained relative to the total variance explained by the principal components PRESENT in the object. So, it is always guaranteed to be at 100% for the last principal component present. You can see this from the code in summary.prcomp() (see this code with getAnywhere("summary.prcomp")).

Here's how to get the output you want (the last line in the transcript below):

> set.seed(1)
> summary(pc1 <- prcomp(x))

Importance of components:
PC1 PC2 PC3 PC4 PC5

Standard deviation 1.175 1.058 0.976 0.916 0.850
Proportion of Variance 0.275 0.223 0.190 0.167 0.144
Cumulative Proportion 0.275 0.498 0.688 0.856 1.000
> summary(pc2 <- prcomp(x, tol=0.8))

Importance of components:
PC1 PC2 PC3

Standard deviation 1.17 1.058 0.976
Proportion of Variance 0.40 0.324 0.276
Cumulative Proportion 0.40 0.724 1.000
> pc2$sdev
[1] 1.1749061 1.0581362 0.9759016
> pc1$sdev
[1] 1.1749061 1.0581362 0.9759016 0.9164905 0.8503122
> svd(scale(x, center=T, scale=F))$d / sqrt(nrow(x)-1)
[1] 1.1749061 1.0581362 0.9759016 0.9164905 0.8503122
> cumsum(pc1$sdev^2) / sum((svd(scale(x, center=T, scale=F))$d / sqrt(nrow(x)-1))^2)
[1] 0.2752317 0.4984734 0.6883643 0.8558386 1.0000000
>
> # output in terms of the cumulative % of the total variance
> cumsum(pc2$sdev^2) / sum((svd(scale(x, center=T, scale=F))$d / sqrt(nrow(x)-1))^2)
[1] 0.2752317 0.4984734 0.6883643
>

It's probably better to get prcomp to compute all the components in the first place, because the SVD is the bulk of the computation anyway (so doing it again will be slower for large matrices.) Then just look at the most important principal components. However, there may be a shortcut for computing the values of D in the SVD of a matrix -- you could look for that if you have demanding computations (e.g., the sqrts of the eigen values of the covariance matrix of scaled x: sqrt(eigen(var(scale(x, center=T, scale=F)), only.values=T)$values)).

-- Tony Plate