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I am currently reviewing principal components from my recent output. For each principal component I know you get an eigenvalue which represents how much of the variance is explained. If I was to take the cumulative sum of the eigenvalues such that I wanted say 75% of variance explained would it make sense to sum the respective principal components?

E.g. if cumulative sum of pc1, pc2, pc3 accounted for 75% of variance (from summing eigenvalues) could I sum pc1 + pc2 + pc3?

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  • $\begingroup$ For what purpose? $\endgroup$
    – chl
    Commented Feb 25, 2021 at 19:43
  • $\begingroup$ The first principal component is the direction along which the data has maximum variance. In your example, if your hypothesis is correct, the data would have greater variance in the direction pc1 + pc2 + pc3. $\endgroup$
    – Jonny Lomond
    Commented Feb 25, 2021 at 20:10

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Not only you could, but we also do. A typical example in finance risk is adding the shift (pc1) and tilt (pc2) components to create shift and tilt scenarios on the yield curves. Here's a random page from interweb. The bullet point 4 in "Practical applications" section is what I meant. Another example is a bottom table on p.49 here.

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  • $\begingroup$ Are you sure you are writing about the sum and not a linear combination? Maybe the distinction comes down to whether one considers the PCs to be normalized to unit length or to the square root of the eigenvalues. $\endgroup$
    – whuber ♦
    Commented Feb 25, 2021 at 19:58
  • $\begingroup$ @whuber sum is a linear combination $\endgroup$
    – Aksakal
    Commented Feb 25, 2021 at 20:00
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    $\begingroup$ Of course--but it's a particular linear combination that, for unit-normalized PCs, rarely has any meaning. $\endgroup$
    – whuber ♦
    Commented Feb 25, 2021 at 21:07

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