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I have a data set, I calculate the correlation matrix and get eigen values for PCA. I want to intuitively understand following features

  1. Number of significant eigenvalues. In my dataset, some matrices have only 3 or 4 significant eigenvalues(the value drops from like 140-200 to 1). However some have relatively large number of significant eigenvalues. After 20-50 eigenvalues, the magnitude drops to 1. What is this telling me ?

  2. Magnitude of first few eigenvalues. If I take 3-4 largest values and some data subset have large eigen values and some have less, what is this telling me ?

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  • $\begingroup$ Reading you question it sounds like you are referring to the scores rather than the eigenvalues. There is one eigenvalue per eigenvector, not multiple. The eigenvalue is the scalar that scales the unit vector eigenvectors to the mean magnitude that they occur in the dataset (see en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors). The scores are the specific value for that eigenvector per sample. If I have interpreted correctly then the current answer is inappropriate. If the sww has interpreted it correctly then you should edit your question to align with the answer. $\endgroup$
    – ReneBt
    Commented May 16, 2018 at 8:31

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  1. If the number of eigenvalues are small, this means your dimension is effectively that number, or those are the only latent features which control your data.
  2. If the magnitude of the first few is large it means you will be able to better approximate your data with those as compared to if they were small and many. This shows that there were indeed a few dimensions which were responsible for most of your data generation.
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  • $\begingroup$ If I have two signals and I take correlation and then take its eigenvalues. If I have few effective eigen values, does that mean that the two signals are strongly correlated ? $\endgroup$
    – jav321
    Commented May 16, 2018 at 5:07
  • $\begingroup$ @jav321 few eigenvalues only means the correlation structure is simple. The strength of the correlation is based on % variance explained by those few eigenvalues. $\endgroup$
    – ReneBt
    Commented May 21, 2018 at 15:46

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