# (PCA) How do I find out if principal components (number of significant coefficients) change over time?

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lil brain on 13 Apr 2022
Edited: William Rose on 3 Jun 2022
Hi,
I have an experiement where I am expecting a dimensionality reduction over time. In this experiment there are five phases.
I am interested in finding out if the prinicpal components change throughout these phases (over time).
I would like to test this by running the PCA on each of these phases to see if they differ.
One idea I had was to determine the number of significant coefficients in a principal component. Could someone tell me if and how that is possible?
If anyone has a different idea I would be happy to hear that as well!
Thanks!

William Rose on 3 Jun 2022
If your hypothesis is true, then the percent of variance explained by the first M principal components (where you can choose M=1, 2, 3 or whatever) should increase with time. You can check this easily.
The vector explained has the percent of variance explained by each principal component (PC). Of course these will always add up to 100%, when you include all the components, but the first few PCs will contain a disproportionately large share. If your hypothesis is true, the sum of the first few elements in explained will grow with time.
See the help for pca.
William Rose on 3 Jun 2022
Edited: William Rose on 3 Jun 2022
[edit: correct typos]
Deciding how many compnents are significant, in PCA, is a non-trivial assignment. There is no straightforward significance test, or p-value, for a principal component. Which is unlike the coefficients in a typical multiple regression, for example, where there are t-statistics and p-values associated with each component in the regression.
In the case of PCA, various tests have been proposed for deciding how many components to keep (i.e. how many components are significant). The "broken stick" method has found favor and is simpler to implement than other good methods. Therefore it is what I recommend.
1. Donald Jackson (1993) reviewed and carefully tested 10 different approaches. Article here or here (may be paywalled). Jackson writes (pp. 2211-2212) "The broken-stick method correctly assessed the dimensionality of the data matrices. .... This method provided a good combination of simplicity of calculation and accurate evaluation of dimensionality relative to the other statistical approaches." and he writes (p.2212) "The most promising approaches to component evaluation are the broken-stick model and the bootstrapped eigenvalue-eigenvector method. The broken-stick approach has the advantage of being simple to calculate."
2. "Dimension reduction: Guidelines for retaining principal components".

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