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I am learning statistics mainly from a book : Elements de statistique, from three author of the Brussels University. 5th Edition

On the PCA chapter, however, I need to read from other sources, especially to understand how to describe what the axis represents from individual and variable analysis. I then read the whole courses that can be found everywhere over the Internet. And I have a trouble.

My book wrotes :

La droite qui ajuste le mieux le nuage N* des individus est celle autour de laquelle l'inertie de N* est minimale.
(The line that adjusts the best the cloud N* of individuals is the one around which the inertia of N* is minimal).

And the problem is : almost all other sources I can read are searching for the maximal inertia to find the best axes.
I can't figure my book could be wrong (on it's 5th edition such a bug would had been found), so what is it trying to explain to me ? And why do I find the contrary elsewhere ?

gunes
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Marc Le Bihan
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  • There are many questions in SO dealing with PCA, its derivations and interpretations. For example, see https://stats.stackexchange.com/questions/2691/making-sense-of-principal-component-analysis-eigenvectors-eigenvalues/6271#6271 – jpmuc Dec 22 '19 at 19:34
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    Although it is quite similar, I don't think this specific question has an answer in the linked post, where there is no mention of *inertia*. – gunes Dec 23 '19 at 19:49

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In PCA, we choose the projection direction(s) that captures the most variance; this is well explained in the linked posts. In some sources, inertia is described as variance. So, the directions with most variance are the directions with most inertia. I can't review the definitions in the book to see if they're talking about the same concept or not since it's in French. However, from a physics perspective, if we think the data points as a cloud of mass and seek for an axis around which rotating the object is easiest, it's going to be the direction of the first PC. For a reference, check the moment of inertia of a rectangular prism/cuboid. The axis with least inertia will be the longest dimension.

gunes
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