R help - Apr 2012 - pca biplot.princomp has a bug?

x=rmvnorm(2000, rep(0, 6), diag(c(5, rep(1,5))))
    x=scale(x, center=T, scale=F)
    pc <- princomp(x)
    biplot(pc)

There are a bunch of red arrows plotted, what do they mean? I knew that the
first arrow labelled with "Var1" should be pointing the most varying
direction of the data-set (if we think them as 2000 data points, each being
a vector of size 6). I also read from somewhere, the most varying direction
should be the direction of the 1st eigen vector.

However, reading into the code of biplot in R. The line about the arrows is:

 if(var.axes)
  arrows(0, 0, y[,1L] * 0.8, y[,2L] * 0.8, col = col[2L],

Where `y` is the actually the loadings matrix, which is the eigenvector
matrix. So it looks like the 1st arrow is actually pointing from `(0, 0)`
to `(y[1, 1], y[1, 2])`. I understand that we are trying to plot a high
dimensional arrow onto a 2D plane. That's why we are taking the 1st and 2nd
element of the `y[1, ]` vector. However what I don't understand is:

Shouldn't the 1st eigenvector direction be the vector denoted by `y[, 1]`,
instead of `y[1, ]`?  (Again, here `y` is the eigenvector matrix, obtained
by PCA or by eigendecomposition of `t(x) %*% x`.) i.e. the eigenvectors
should be column vectors, not those horizontal vectors.

Even though we are plotting them on 2D plane, we should draw the 1st
direction to be from `(0, 0)` pointing to `(y[1, 1], y[2, 1])`?



I think the R code biplot.princomp has a bug: the loading matrix
(eigenvector matrix) should be transposed before being sent into
biplot.princomp... Any thoughts?

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The arrows are not pointing in the most-varying direction of the data.  The
principal components are pointing in the most-varying direction of the
data.  But you are not plotting the data on the original scale, you are
plotting the data on the rotated scale, and thus the horizontal axis is the
most-varying direction of the data.

The arrows are pointing in the direction of the variables, as projected
into the 2-d plane of the biplot.

There is no bug.

Kevin Wright



On Wed, Apr 25, 2012 at 11:29 AM, Michael <comtech.usa@gmail.com> wrote:
>    x=rmvnorm(2000, rep(0, 6), diag(c(5, rep(1,5))))
>    x=scale(x, center=T, scale=F)
>    pc <- princomp(x)
>    biplot(pc)
>
> There are a bunch of red arrows plotted, what do they mean? I knew that the
> first arrow labelled with "Var1" should be pointing the most
varying
> direction of the data-set (if we think them as 2000 data points, each being
> a vector of size 6). I also read from somewhere, the most varying direction
> should be the direction of the 1st eigen vector.
>
> However, reading into the code of biplot in R. The line about the arrows
> is:
>
>  if(var.axes)
>  arrows(0, 0, y[,1L] * 0.8, y[,2L] * 0.8, col = col[2L],
>
> Where `y` is the actually the loadings matrix, which is the eigenvector
> matrix. So it looks like the 1st arrow is actually pointing from `(0, 0)`
> to `(y[1, 1], y[1, 2])`. I understand that we are trying to plot a high
> dimensional arrow onto a 2D plane. That's why we are taking the 1st and
2nd
> element of the `y[1, ]` vector. However what I don't understand is:
>
> Shouldn't the 1st eigenvector direction be the vector denoted by `y[,
1]`,
> instead of `y[1, ]`?  (Again, here `y` is the eigenvector matrix, obtained
> by PCA or by eigendecomposition of `t(x) %*% x`.) i.e. the eigenvectors
> should be column vectors, not those horizontal vectors.
>
> Even though we are plotting them on 2D plane, we should draw the 1st
> direction to be from `(0, 0)` pointing to `(y[1, 1], y[2, 1])`?
>
>
>
> I think the R code biplot.princomp has a bug: the loading matrix
> (eigenvector matrix) should be transposed before being sent into
> biplot.princomp... Any thoughts?
>
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>
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>


-- 
Kevin Wright

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