Timeline for Are explanatory variables considered random in PCA?
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 11, 2014 at 19:48 | comment | added | ttnphns | @Alexis, maths operations imply something beyond what they show. (Just like words. If it were not so, no philosophy could be ever build on both.) Computing correlations imply standardizing variables, even though you don't calculate those. | |
| Jun 11, 2014 at 19:39 | comment | added | Alexis | @ttnphns I thought I was picking on the math? The results of PCA on the correlation matrix are the same whether or not one standardizes ones variables (see my previous comment for why). If PCA of the correlation matrix required standardization, then that mathematical fact would not hold. Ergo, the OP's first sentence is mathematically false. | |
| Jun 11, 2014 at 19:27 | comment | added | ttnphns |
@Alexis, don't pick on words, please. Interpretation of all variables contributing equally to the PCA concept of "total variance." is ornate and somewhat obscured wording of "variables are taken as having equal variance" (1 or another magnitude - no difference).
|
|
| Jun 11, 2014 at 19:12 | comment | added | Alexis | @ttnphns The problem is that $\text{cor}(\mathbf{X}) = \text{cor}(a\mathbf{X}+b)$ for any real $0 < a < \infty$ and any $-\infty < b < \infty$. So, no: PCA on correlation matrices does not require standardized variables, rather it requires an interpretation of all variables contributing equally to the PCA concept of "total variance." I get where you are coming from. I just don't agree with you. XO, Senhora Diehard :D | |
| Jun 11, 2014 at 19:07 | comment | added | ttnphns |
@Alexis, please don't be diehard. I'm sooner with gung, not with you. PCA on correlations is PCA of standardized variables. The problem with your stand is that actual magnitude for variance. The thing is that when you do PCA on correlations, the "actual magnitude" for you is 1. Don't like it? Then don't do PCA on corrrelations.
|
|
| Jun 11, 2014 at 18:05 | comment | added | Alexis | @gung, I am afraid I disagree: PCA of the covariance matrix explicitly links proportion of variance to the actual magnitude of each variable's variance. PCA of the correlation matrix does not actually transform (standardize) any of the actual variables, which actually may have incomparable variances. | |
| Jun 11, 2014 at 17:20 | history | edited | gung - Reinstate Monica | CC BY-SA 3.0 |
light editing
|
| Jun 11, 2014 at 17:20 | comment | added | gung - Reinstate Monica | @Alexis, saying that it is false is too strong. When the correlation matrix is being analyzed, in essence the variables have all been standardized. Thus the variables all have variances equal to 1 & "the sum of the variances of the principal components is [still] equal to the sum of the variances of the explanatory variables". The question is still sensible even in the case of the correlation matrix. | |
| Jun 11, 2014 at 5:38 | comment | added | ttnphns | PCA is simply an applied mathematical method. It itself doesn't assume that the variables being analyzed are random. PCA is indifferent to words "sample", "population" etc. | |
| Jun 10, 2014 at 22:46 | comment | added | Alexis | Well it is misleading in that it is often false. | |
| Jun 10, 2014 at 22:38 | comment | added | Pasato | I hope it wasn't too misleading. PCA was just an example to display the problem I had trouble with :) | |
| Jun 10, 2014 at 22:05 | comment | added | Alexis | The claim in your first sentence—"One of properties of PCA states that sum of the variances of the principal components is equal to the sum of the variances of the explanatory variables."—is only true when performing PCA on the covariance matrix $\mathbf{\Sigma}$. However, when performing PCA on the correlation matrix $\mathbf{R}$, the assumption is that each variable contributes precisely 1 unit of variance to total variance. | |
| Jun 10, 2014 at 21:47 | vote | accept | Pasato | ||
| Jun 10, 2014 at 21:45 | answer | added | gung - Reinstate Monica | timeline score: 3 | |
| Jun 10, 2014 at 21:40 | history | edited | gung - Reinstate Monica | CC BY-SA 3.0 |
edited tags; edited & formatted
|
| Jun 10, 2014 at 21:31 | history | asked | Pasato | CC BY-SA 3.0 |