# Comparing original variables with characteristic values of diagonalized variance-covariance matrix

If I have a reference data set comprising repeated measurements of 3 variables of a system in state $A$. Given new observations of these variables for a different system I would like to classify individual observations as being in state $A$ or not.

My initial inclination would be to compare the new value of each variable to the distribution in the reference state. However, a manuscript I am reading for a similar type of analysis suggests to instead construct a $3 \times 3$ covariance matrix based on the reference state, and diagonalize this covariance matrix. I am advised that I can compare each new set of measurements of the 3 variables against this diagonalized matrix. My understanding is that to follow this method, I need to examine the relative deviation of the new variables ($x_i$) from the eigenvalues ($\lambda_i$) of the new matrix and determine if this deviation is within a certain tolerance.

$\sum_{i=1}^3 \left(x_i - \sqrt{\lambda_i}\right)^2$

It seems that in this case I am comparing a set of new measurements $x_i$ against the loadings of a set of transformed variables (according to my interpretation of PCA).

Is this a common method of classification, and is there a term for this type of analysis? I was not aware that the singular values (square root of eigenvalues) would be directly comparable to the original variables.

Thank you in advance.

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There must be something missing from this recipe you are following, because it appears to make no statistical sense and in many cases will be a terrible procedure. Is this "manuscript" accessible to the rest of the world to consult? – whuber♦ Nov 27 '11 at 23:33
@whuber, thank you for this response. I am not sure if the author would like his manuscript scrutinized on a public forum, but I will try to contact him directly. But thank you for your feedback, I thought something was amiss and I may have misinterpreted this portion of the manuscript. – crippledlambda Nov 28 '11 at 3:43

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Perhaps you could quote the relevant part anonymously. If there is a possibility of multiple interpretations, we can suggest other ways to make sense of it, and if there is an error in the manuscript, you can share that with its author, who in either case should be grateful for the opportunity to clarify or correct their work. – whuber♦ Nov 28 '11 at 15:43
Thanks so much - I spoke with the author and it turns out that the new observations should also be projected onto the new basis set for comparison. – crippledlambda Dec 2 '11 at 5:22
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