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I've conducted principal component analysis (PCA) with FactoMineR R package on my data set. I have a general question:

Given that the first and the second dimensions of PCA are orthogonal, is it possible to say that these are opposite patterns?

For example, can I interpret the results as: "the behavior that is characterized in the first dimension is the opposite behavior to the one that is characterized in the second dimension"? I know there are several questions about orthogonal components, but none of them answers this question explicitly.

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  • $\begingroup$ On the contrary. Orthogonal components may be seen as totally "independent" of each other, like apples and oranges. Items measuring "opposite", by definitiuon, behaviours will tend to be tied with the same component, with opposite polars of it. $\endgroup$
    – ttnphns
    Commented Jun 25, 2015 at 12:43
  • $\begingroup$ I would concur with @ttnphns, with the proviso that "independent" be replaced by "uncorrelated." What this question might come down to is what you actually mean by "opposite behavior." Could you give a description or example of what that might be? $\endgroup$
    – whuber ♦
    Commented Jun 25, 2015 at 14:53
  • $\begingroup$ my data set contains information about academic prestige mesurements and public involvement measurements (with some supplementary variables) of academic faculties. the PCA shows that there are two major patterns: the first characterised as the academic measurements and the second as the public involevement. $\endgroup$
    – tzipy
    Commented Jun 25, 2015 at 18:19
  • $\begingroup$ how do I interpret the results (beside that there are two patterns in the academy)? should I say that academic presige and public envolevement are un correlated or they are opposite behavior, which by that I mean that people who publish and been recognized in the academy has no (or little) appearance in bublic discourse, or there is no connection between the two patterns $\endgroup$
    – tzipy
    Commented Jun 25, 2015 at 18:23

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I would try to reply using a simple example. Consider we have data where each record corresponds to a height and weight of a person. PCA might discover direction $(1,1)$ as the first component. This can be interpreted as overall size of a person. If you go in this direction, the person is taller and heavier. A complementary dimension would be $(1,-1)$ which means: height grows, but weight decreases. This direction can be interpreted as correction of the previous one: what cannot be distinguished by $(1,1)$ will be distinguished by $(1,-1)$.

We cannot speak opposites, rather about complements. The further dimensions add new information about the location of your data. This happens for original coordinates, too: could we say that X-axis is opposite to Y-axis?

The trick of PCA consists in transformation of axes so the first directions provides most information about the data location.

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