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I'm wondering which of the namings is right: Principal components analysis or principal component analysis.

When I googled "principal component analysis" I got 526,000,000 related results, whereas when I googled "principal components analysis" I got 482,000,000. So the former outnumbers the latter on Google, and indeed when I typed "principal components analysis" google only showed the websites that contain "principal component analysis" in its title, including Wikipedia.

However, PCA is written as "principal components analysis" in the famous "Deep Learning" book by Ian Goodfellow, and as far as I know "principal components analysis" is more widely used in biological literatures.

Although I always assume no algorithmic differences whichever people use, I want to make it clear which one is more preferably used and why.

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    $\begingroup$ I put the s in bold as it took me a while to notice the difference between the two expressions. $\endgroup$ Dec 21, 2020 at 6:45
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    $\begingroup$ The downvotes here aren't explained but can be imputed as implying (1) this is trivial (2) this is off-topic as a matter of (English) usage. Against those imputed arguments I would reply (1) it's a small point, but deserves its moment (2) I wouldn't want to delegate or relegate this discussion to another SE site where people knowing almost nothing about statistics predominate. There are many people here well-informed about language as well as statistics. $\endgroup$
    – Nick Cox
    Dec 21, 2020 at 11:26
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    $\begingroup$ Note that the number of results reported by Google is close to meaningless, since it depends heavily on your personal search engine bubble. For instance, I get 413 million results for "principal component analysis" and 472 million results for "principal components analysis" (both without the quotes), so for me, the pattern is reversed! $\endgroup$ Dec 22, 2020 at 16:10
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    $\begingroup$ @StephanKolassa: the easy way to avoid the effect of personalized search engine bubble is to repeat the search when logged out, and ideally from other geos/IPs, and in general with language preferences set to a wide set of languages. $\endgroup$
    – smci
    Dec 23, 2020 at 10:16

5 Answers 5

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Ian Jolliffe discusses this on p.viii of the 2002 second edition of his Principal Component Analysis (New York: Springer) -- which, as you can see immediately, jumps one way. He expresses a definite preference for that form principal component analysis as similar to say factor analysis or cluster analysis and cites evidence that it is more common any way. Fortuitously, but fortunately for this question, this material is visible on my local Amazon site, and perhaps on yours too.

I add that the form independent component analysis seems overwhelmingly preponderant for that approach, although whether this is, as it were, independent evidence might be in doubt.

It's not evident from the title but J.E. Jackson's A User's Guide to Principal Components (New York: John Wiley, 1991) has the same choice.

A grab sample of multivariate books from my shelves suggests a majority for the singular form.

An argument I would respect might be that in most cases the point is to calculate several principal components, but a similar point could be made for several factors or several clusters. I suggest that the variants factors analysis and clusters analysis, which I can't recall ever seeing in print, would typically be regarded as non-standard or typos by reviewers, copy-editors or editors.

I can't see that principal components analysis is wrong in any sense, statistically or linguistically, and it is certainly often seen, but I would suggest following leading authorities and using principal component analysis unless you have arguments to the contrary or consider your own taste paramount.

I write as a native (British) English speaker and have no idea on whether there are arguments the other way in any other language -- perhaps through grammatical rules, as the mathematics and statistics of PCA are universal. I hope for comments in that direction.

If in doubt, define PCA once and refer to that thereafter, and hope that anyone passionate for the form you don't use doesn't notice. Or write about empirical orthogonal functions.

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    $\begingroup$ All the cool kids are doing EOF these days, 'PCA' is so 2019... $\endgroup$ Dec 21, 2020 at 15:01
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NGrams from Google Books suggests they were similar in frequency of use from 1960 to about 1982, after which the without-s form started to be more popular.

This suggests to me that neither form is wrong but the without-s form may be slightly more comfortable to say. enter image description here

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  • When Hotelling first used the words 'analysis', 'component(s)' and 'principal' together. It was in the plural form.

    But it wasn't exactly as PCA yet and it was 'method of principal components' or in the title 'Analysis of a complex of statistical variables into principal components.'

  • In most other languages the plural form is used. For instance, in French it is 'Analyse en composantes principales' and in German it is 'Hauptkomponentenanalyse'.

However, in the English language the singular form is more common.

The reason for the singular form is because this is the common way in English language. See for the canonical question on the English forum:

When are attributive nouns plural?.

And there are many other related question asking for instance for 'number(s) analysis'.

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    $\begingroup$ (+1) Helpful details. FWIW, Hotelling gave the name but Karl Pearson is often given credit for the basic idea. I think I've seen ACP in French language literature. Is there an abbreviation e.g. HKA in German language literature? $\endgroup$
    – Nick Cox
    Dec 22, 2020 at 13:27
  • $\begingroup$ @NickCox I am not sure about German literature, but in Dutch it is often just called PCA. $\endgroup$ Dec 22, 2020 at 13:34
  • $\begingroup$ Actually I am not sure about my last comment. English is the language used at Dutch universities nowadays, and maybe I am not old enough that I had to learn about the Dutch word on Wikipedia. $\endgroup$ Dec 22, 2020 at 17:44
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I learned it as "principal components analysis" and I find some others insisting that the singular is better.

You buy books at a "book store", not at a "books store", and you won't touch something with a ten-foot pole, not a ten-feet pole, etc. That is a trait of the English language. In German it's exactly the other way around: the plural is used for things like this. So "principal component analysis" follows the pattern usually followed in English.

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    $\begingroup$ "cattle market" is an exception to the rule, although,perhaps because "bull market" means something different and "cow market" would not be a good word for anywhere where bulls or bullocks were also sold. Still, my guess is that exceptions are rare. The bigger question is why principal components analysis is so common if it's malformed. $\endgroup$
    – Nick Cox
    Dec 22, 2020 at 17:54
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    $\begingroup$ @NickCox : The singular of "cattle" seems to be "head of cattle". $\endgroup$ Dec 22, 2020 at 18:04
  • $\begingroup$ Indeed, but whoever says or writes "a head of cattle" or "one head of cattle". It's a singular only by virtue of multiplication by a number making it a plural. (Just extending your comment, not disagreeing with it.) $\endgroup$
    – Nick Cox
    Dec 22, 2020 at 18:11
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    $\begingroup$ Historically "cattle" meant any domesticated animal farmed for its meat. In other words it was a collective noun, not the plural of "cow or bull". (Similarly "corn" in British English is still a collective noun for any type of cereal grain, not just a synonym for "maize" as it is in American English). $\endgroup$
    – alephzero
    Dec 22, 2020 at 22:16
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    $\begingroup$ @Scortchi and friends: Interestingly digressive digressions, and indeed I started it. I'd assert that in contemporary (British) English, cattle only connotes bovine species, so that e.g sheep, goats, llamas no longer qualify. It's quite true that much wider usages can be found historically, which I didn't know. "Livestock" seems to be the wider referent. Whether livestock includes say poultry, I wouldn't know. A paltry matter. $\endgroup$
    – Nick Cox
    Dec 24, 2020 at 11:19
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If you want to rely on google search as a "majority vote" criterion, I think you should put quotes around the three words. This what I get:

  • "Principal component analysis": 12,400,000 results

  • "Principal componentS analysis": 2,880,000 results

Wikipedia's article is also without S.

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