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If I have a data set with 6 variables (A, B, C, D) all measured in different units (eg. m, kg, s, K) and I perform principal component analysis to transform this into two Principal Axes, how do I express the units of those axes?

For example: PC1= aA[m] + bB[s] + cC[K] + dD[kg] (a,b,c,d are constants from the eigenvectors)

What are the units of PC1? Does it even make sense to give this a unit? Can I relate it to some physical aspect of the system or is it too abstract?

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Usually, we standardize variables before doing principal-components analysis. This ensures that each variable as treated as equally important regardless of scale; e.g., a variable won't become 1000 times as important if you convert it from kilograms to grams. A side-effect is that the original units are lost.

Since each principal component can be written as a linear combination of the original variables, you could still do something like equate 1 unit of the component to so-and-so many kilograms of A plus such-and-such many liters of B and so on. This is unlikely to be enlightening, though.

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