principal components to original units

[Meritas]

Hi,

I have derived the principal component score series (corelation matrix, other options default). I realize the series are standardized.
How should one transform the principal components into the units of original data?

Would the PCs from covariance matrix be in original data units?


Thank you in advance!

[EViews Glenn]

Yes. Make sure when you compute the components that you normalize the loadings.

[Meritas]

Thanks!

And what about correlation - matrix derived PC scores ? How to transform them to original units ?

[EViews Glenn]

The scores for the components are intimately related to whichever matrix you decompose. Since the correlation matrix is unit free, the information to make the scores in the original data units isn't available. In principle, we could have an option that computes scores from the decomposition of the correlation matrix using scaling information from the original data, but since you can choose to compute the decomposition of the covariance matrix using a single option, there's really no reason to add the extra complexity.

[Bigbrotherjx]

Yes. Make sure when you compute the components that you normalize the loadings.

When you say normalise, do you mean take the element in the eigenvector and divide by the sum of the the elements?

I used Eviews to generate a series from principal components directly...is this in original units?

[EViews Glenn]

There are options in the scores calculation for what normalization you want. Normalizing the loadings means that the scores will have the scale that you want. Normalizing the scores means that they will have covariances proportional to unity. The documentation describes this setting in greater detail.

The default should be for the scores to have the scale you want.

[Bigbrotherjx]

I'm using Eviews 5 and I don't see any options when I use principal components, only the choice between decomposing the correlation matrix and covariance matrix.

What I mean is that when the series is generated automatically by Eviews, it is a different scale to if I generated it myself using the eigenvector that I am given.

[EViews Glenn]

I just did a quick test with EViews 5. If you compute the PC of the covariances and make the scores, the diagonal of the covariance of the scores is indeed equal to the eigenvalues.

[Bigbrotherjx]

I just did a quick test with EViews 5. If you compute the PC of the covariances and make the scores, the diagonal of the covariance of the scores is indeed equal to the eigenvalues.

I used the correlation matrix, and basically the elements of the first eigenvector were all around 0.3 or so, which 10 elements. If I generate a series using these elements, then obviously it will be much larger than the original series in terms of magnitude. I am correct in generating the series of the common factor by weighting each series by the corresponding element of the 1st eigenvector right?

Also, I'm not sure what you mean by scores.

[EViews Glenn]

As I noted earlier, you can't use the correlation matrix. You have to use the covariance matrix.

[Bigbrotherjx]

If I am trying to capture common variation between equivalent variables in different countries, am I right in thinking that standardizing the data i.e. using the correlation rather than covariance matrix is desirable if the variances of variables differ much?

[EViews Glenn]

That's really a question for another forum.

I will say, without doing the math, that my guess is that weighting the scores from the correlation pcomp with the eigenvalues from the covariance pcomp is probably mixing apples and oranges.

[Bigbrotherjx]

I'm not sure what you mean by mixing apples and oranges...my question is basically asking whether I should tick "covariance" or "correlation" in the principal components menu. In what sense am I mixing up the scores and eigenvalues?

[Bigbrotherjx]

Also, slightly different question. Does principal components require the underlying series to be stationary?

Cheers.

[EViews Glenn]

To be honest, I think I'm repeating the same thing over and over. If you want the scores to be in the original scale, you want to use the covariances. The variances of those scores will match the eigenvalues from the covariance calculation.