Hi, thanks for your post. In the equation following this sentence "This means that the variance of the dataset projected onto the first Principal", could you explain why the expression on the left is equal to the right? Thanks!
Could you develop mathematically the afirmation: "By dividing out the singular avlue (and multplying it by (n-1)^(1/2) to deal with the fact that we started with the covariance) we obtain a transformed dataset which is in some sense spherical"? How do you get Y = (n-1)^(1/2)*UE^-1?
A way to understand the paragraph it is to write out U in terms of its column vectors and multiply that use the definitions of \Sigma and u_i. You can also take the covariance of the transformed data set, and it should come out to be the identity matrix!
Remarkbox
Hi, thanks for your post. In the equation following this sentence "This means that the variance of the dataset projected onto the first Principal", could you explain why the expression on the left is equal to the right? Thanks!
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You can use the equation for S from the box "What Is the Covariance Matrix?" to expand the right hand side, then use
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Helpful
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Hi, thanks for your post, very insightful.
Could it be that the U matrix represents the PCA taking variables as vectors instead of observations?
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Hi and thanks. I'm not sure what you mean about U, but I'll think about it if you clarify that for me.
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Hi, thanks again for your post.
Could you develop mathematically the afirmation: "By dividing out the singular avlue (and multplying it by (n-1)^(1/2) to deal with the fact that we started with the covariance) we obtain a transformed dataset which is in some sense spherical"? How do you get Y = (n-1)^(1/2)*UE^-1?
Thanks
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A way to understand the paragraph it is to write out U in terms of its column vectors and multiply that use the definitions of \Sigma and u_i. You can also take the covariance of the transformed data set, and it should come out to be the identity matrix!
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Thanks for the post. There is a mistake in Fig. 3, the dimension of $Sigma$ must be $d\times d$
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Thanks. It's actually Fig. 2, and the dimension of U is incorrect in the figure: it should be $n \times n$.
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Hi,
I could not figure out how did you go from
to this
(i.e. last two equations of the post) I would appreciate if you could explain.
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