This post is going to extend on the topic scaling introduced to you in the last post.
I had given you a statement that scaling using error variances actually is the best scaling technique possible , although I hadn't given any proofs , the examples that has been demonstrated in the assignments stands for itself .
After all the proof of the pudding is in the eating . One more advantage this scaling gives is that we can precisely find the rank of the data matrix , using this type of a scaling . An intuition regarding this can be given as follows , since the last few eigenvalues of the data matrix are are supposed to be zero , it is only the errors that make it non zero therefore scaling the data matrix using error variances should make the last set of eigenvalues = 1 . This is simply a hand waving or may be a even worse kind of an explanation , but it can be proved ,that the above statement is true .
Now that I had made some statements , I present to you the algorithm for the MLPCA briefly introduced to you in the last post.Although how we find the error variances , is yet to be discussed , I had given you one particular method for finding the error variances but such a method simply puts more load on the experimenter side which most people wouldn't actually like , so is there any method for actually finding the error variances from the data itself? It turns out that yes there is a method to do it , it is called the IPCA.
Now using $L$ , we get
$Z_{s} = L^{-1} Z$, Now applying SVD to $Z_{s}$ , we get
$Z_{s} = U_{s} S_{s} V_{s} ^{'}$
Now taking the last few vectors of $C_{s} = U_{s}^{( : ,(k+1) : end)}$ corresponding to singular values = 1 , we actually get the constraint set of equations with respect to the scaled matrix $Z_{s}$ . Where k is the row rank of the data matrix.
Now to converting the constraint set to original matrix we get :
$C = L^{-1}C_{s}$ . Here C is the constraint set corresponding to the original matrix .
Now , still we hadn't proposed the IPCA , which will actually help you in finding the error covariance matrix with a catch , which we shall discuss in the following blog posts.
I had given you a statement that scaling using error variances actually is the best scaling technique possible , although I hadn't given any proofs , the examples that has been demonstrated in the assignments stands for itself .
After all the proof of the pudding is in the eating . One more advantage this scaling gives is that we can precisely find the rank of the data matrix , using this type of a scaling . An intuition regarding this can be given as follows , since the last few eigenvalues of the data matrix are are supposed to be zero , it is only the errors that make it non zero therefore scaling the data matrix using error variances should make the last set of eigenvalues = 1 . This is simply a hand waving or may be a even worse kind of an explanation , but it can be proved ,that the above statement is true .
Now that I had made some statements , I present to you the algorithm for the MLPCA briefly introduced to you in the last post.Although how we find the error variances , is yet to be discussed , I had given you one particular method for finding the error variances but such a method simply puts more load on the experimenter side which most people wouldn't actually like , so is there any method for actually finding the error variances from the data itself? It turns out that yes there is a method to do it , it is called the IPCA.
MLPCA
Algorithm : Let \( \Sigma _{e}\) be the error covariance matrix of the measurements , $ L^{'}L = \Sigma _{e} $ . If the error covariances turn out to be zero then $\Sigma_{e}$ is nothing but a diagonal matrix , the following algorithm will do nothing but scaling .Now using $L$ , we get
$Z_{s} = L^{-1} Z$, Now applying SVD to $Z_{s}$ , we get
$Z_{s} = U_{s} S_{s} V_{s} ^{'}$
Now taking the last few vectors of $C_{s} = U_{s}^{( : ,(k+1) : end)}$ corresponding to singular values = 1 , we actually get the constraint set of equations with respect to the scaled matrix $Z_{s}$ . Where k is the row rank of the data matrix.
Now to converting the constraint set to original matrix we get :
$C = L^{-1}C_{s}$ . Here C is the constraint set corresponding to the original matrix .
Now , still we hadn't proposed the IPCA , which will actually help you in finding the error covariance matrix with a catch , which we shall discuss in the following blog posts.
The Auto Scaling algorithm
It is similar to the MLPCA algorithm but instead of $\Sigma _{e}$ being the error covariance matrix , it is simply the covariance matrix of the data matrix .
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