Talk:Principal component analysis

X changed to be the standard way round for a data matrix
I have changed the introduction of X, so it is now defined to be the standard way round for data matrices -- i.e. with each row representing measurements from a single sample, and each column representing a variable that was measured, rather than the other way round which was unnecessarily confusing.

I've then gone through the text and made the appropriate changes diff -- I hope I got them all!

The article is still pretty horrible, but this may make it a little easier to take forward. Jheald (talk) 19:09, 17 June 2013 (UTC)

Could someone please have another go through the article in that relation? There are still occurrences of X^T as the data matrix. Also, if possible, it would be nice to have a reference (textbook) with notation consistent to one used in the article. 138.51.113.194 (talk) 01:33, 13 November 2019 (UTC)

The z-score bit is messed up
Since 2006, the bit on converting input data to z-scores has been confused and messed up. Here, for example, the retired editor who did most of the detailed article writing around that time moved that bit from one place it didn't fit to another. It certainly doesn't fit where it is now, since the eigenvectors are based on the transformed data that they're used to transform.

The easies fix is to just take out that optional section that is wrong and doesn't fit. A better mix might be to put that option up front in defining the B matrix (as B = Z instead of what it is now), after which the rest would make sense; that is, the option is between the present $$\mathbf{B} = \mathbf{X} - \mathbf{h}\mathbf{u}^{T} $$ and a version that divides each row of that by its standard deviation.

Any preferences, or anyone willing to work it over? Dicklyon (talk) 06:58, 12 February 2018 (UTC)

For now, I took it out, not seeing a good way to fix it or a good reason to keep it. Dicklyon (talk) 16:28, 13 February 2018 (UTC)


 * It's still messed up, and I need to fix it, since the last step with the KLT refers to using the Z that is now not defined. It's not clear what's the best way to use B or Z depending on your application, but I did add a few words about what normalizing the input data variance can be good for. Dicklyon (talk) 18:10, 23 August 2020 (UTC)

Dividing each attribute by its standard deviation changes principle components
This is/was suggested as a pre-processing step in the third paragraph, but it may change the directions of the principle components. Consider vectors [1,2] and [-1,-2]. The standard deviation across the values in the first coordinate is 1, and across the second is 2. Dividing, we get [1,1],[-1,-1], which will have a different set of principle components. Does this not defeat the purpose? AP295 (talk) 22:00, 26 March 2020 (UTC)


 * Centering the data is an essential step in PCA, so that could be mentioned in the intro if someone feels it must. On the other hand, dividing each variable by its standard deviation may give you different principle components, the first of which may not preserve as much variance of the original data. I've searched around a bit, and it looks like the practice of using a correlation matrix isn't entirely uncommon, but it will not give you the same results with respect to your original data. If anyone has a good source on this, please let me know. The preceding sentence already describes the algorithm as an eigendecomposition or SVD of the covariance or correlation matrix, which are not affected by the variable means anyway, so the sentence was at least redundant. I propose re-writing that sentence as well to make it clear that using the correlation matrix may not yield the same results.AP295 (talk) 12:32, 27 March 2020 (UTC)
 * This is original research. It is "principal" not "principle". And of course it can be done with normalized vectors in which case it has the covariance. If you revert I will bring it to ANI.Limit-theorem (talk) 19:29, 27 March 2020 (UTC)
 * It is not original research. Do you agree that the first principal component should maximize the variance of the projected data points? PCA is defined as an optimization problem, and if instead of using the covariance matrix, you use the correlation matrix, the transformation you get may not be an optimal solution to this objective. See "Pattern Recognition and Machine Learning" by Bishop. AP295 (talk) 21:15, 27 March 2020 (UTC)
 * I think what is talking about is that the sources that you are adding may be primary sources.  We have a policy against that.  You need to find an appropriate secondary reliable source (such as a journal article or a news article) that supports your addition, unfortunately, that is not what you have been doing.  Instead, your edit seems to be backed by "that is the truth".  Read this essay for an understanding of what that means.  I have reverted your edit again, reinstating your edit constitutes edit warring.  I do not want an admin to block you, but it is important that you comply with Wikipedia's content policies when you are editing.  Thank you.  Aasim 21:40, 27 March 2020 (UTC)
 * @ we clearly use both techniques and the page mentions it. Reported very disuptive editing, 3RR, with original research on WP:AN.Limit-theorem (talk) 21:43, 27 March 2020 (UTC)
 * @ Who's "we"? Please re-consider my point. Are there any software packages that compute PCA using the correlation matrix instead of the covariance matrix? AP295 (talk)
 * @ I'm not sure I understand what you mean. I didn't add any content or cite any new sources in the article. I only removed something that was inconsistent with the usual definition of PCA.AP295 (talk) 21:56, 27 March 2020 (UTC)
 * Wolfram's Mathematica does both covariance and correlation, and any book on Z scores used in finance and risk management. Limit-theorem (talk) 22:33, 27 March 2020 (UTC)
 * @ Whether you apply PCA to z-scores or any other sort of data has nothing to do with PCA itself. At least make clear the distinction between necessary steps of the algorithm, like mean subtraction, and the sort of data you're using it on. Again, using the correlation matrix may not yield a solution to the maximum-variance/minimum-projection-error objective with respect to the original data. — Preceding unsigned comment added by AP295 (talk • contribs) 22:49, 27 March 2020 (UTC)
 * I give up. It did not hit you that some methods use normalized variance "– and, possibly, normalizing each variable's variance to make it equal to 1; see Z-scores.[4]". The text does not deny that other nonnormalized methods exist, in ML or other. You are bordering on vandalism and will end up being blocked if you continue violating encyclopedic standards.Limit-theorem (talk) 22:55, 27 March 2020 (UTC)
 * @ I understand what you mean, but using the correlation matrix changes the algorithm in a fundamental way, and I think this distinction should at least be made clear to the reader who might not understand the difference otherwise. AP295 (talk) 23:12, 27 March 2020 (UTC)
 * It does not change anything methodologically since you are dealing with normalized vectors and derive their attributes. People can retranslate from the normalized vector to the raw one. Limit-theorem (talk) 11:49, 28 March 2020 (UTC)
 * @ Please see my summary below. It seems we got off on the wrong foot. I'm not claiming that using PCA on standardized data is unsound or methodologically incorrect. AP295 (talk) 12:28, 28 March 2020 (UTC)

I'd like to summarize my point and make a few things clear, please correct me if I'm mistaken about any of these points. Computing PCA using a correlation matrix instead of a covariance matrix will give you the same set of principal components that you'd get by standardizing your data and then applying the usual form of PCA. Rather than a "different method" of PCA, this could be viewed as applying PCA to a specific sort of data or as a pre-processing step unrelated to PCA itself (as opposed to mean-subtraction, which is a necessary step if you want to compute the covariance matrix using a matrix product). My concern is that the intro does not make this distinction clear. PCA is formulated as an optimization problem and using the correlation matrix may not yield the same solution to the maximum-variance/minimum-projection-error objective with respect to an arbitrary collection of data. Conflating the mathematical definition of PCA with a specific use makes the intro less clear. AP295 (talk) 12:20, 28 March 2020 (UTC)


 * You are now doing WP:OR. Wikipedia is not about this, but about reflecting the professional literature. If you disagree with the literature, please publish and it will make its way to WP. Limit-theorem (talk) 12:49, 28 March 2020 (UTC)
 * @ This is not original research. Two sections above this one, another user remarked that the intro should be re-written and the problem stated mathematically, and I agree with them. Without a precise definition of PCA to work from, we're not getting anywhere. AP295 (talk) 13:01, 28 March 2020 (UTC)
 * I made sure it says that there are two methods, unambiguously. Limit-theorem (talk) 21:33, 28 March 2020 (UTC)

That's a bit better, but I still think the intro needs work. I'll leave it alone for now, but you may find this code interesting:

The variance in the third coordinate is very high compared to the others, so for the first principal component, you'd want a vector that lies close to or on that axis. However, because the correlation between different variables will be very small, the correlation matrix will be close to the identity matrix, where the "first" eigenvector is very sensitive to small differences. So the results will likely be different, and using the correlation matrix, they might also differ substantially between each run even though you have large samples of the same distribution, whereas the first PC you get from eigendecomposition of the covariance matrix will be fairly consistent. Again, I'm not at all claiming that PCA can't or shouldn't be applied to z-scores. Just that in this case, using a correlation matrix might not give someone the results they expect.

This condition was described by Karl Pearson himself: "This sphericity of distribution of points in space involves the vanishing of all the correlations between the variables and the equality of all their standard-deviations", in the original 1901 paper. AP295 (talk) 23:32, 29 March 2020 (UTC)

One more comment: Compare the first sentence of this article with how Pearson introduced PCA:

Wikipedia: Principal component analysis (PCA) is a statistical procedure that uses an orthogonal transformation to convert a set of observations of possibly correlated variables into a set of values of linearly uncorrelated variables called principal components.

Pearson: In many physical, statistical, and biological investigations it is desirable to represent a system of points in plane, three, or higher dimensioned space by the "best-fitting" straight line or plane.

In my opinion, the latter is a much clearer way to motivate the problem and convey its geometric intuition. AP295 (talk) 15:03, 30 March 2020 (UTC)

Maybe something like this: "Given a set of points in two, three, or higher dimensional space, a "best fitting" line can be defined as one that minimizes the average squared distance from a point to the line. The next best-fitting line can be similarly chosen from directions perpendicular to the first. Repeating this process yields an orthogonal basis in which individual dimensions of the data are uncorrelated." AP295 (talk) 14:34, 3 April 2020 (UTC)

I'd like to replace the first paragraph with: ''Given a collection of points in two, three, or higher dimensional space, a "best fitting" line can be defined as one that minimizes the average squared distance from a point to the line. The next best-fitting line can be similarly chosen from directions perpendicular to the first. Repeating this process yields an orthogonal basis in which different individual dimensions of the data are uncorrelated. These basis vectors are called Principal Components, and several related procedures Principal component analysis (PCA).'' I think it reads much more clearly than the current paragraph and describes PCA as the solution to a specific objective rather than a "statistical procedure", without being any less precise.

Any suggestions/objections? AP295 (talk) 14:29, 4 April 2020 (UTC)


 * You are right! The Pearson's definition of Principal Components as the best approximation of the data cloud with minimisation of MSE (Mean squared error) is much more seminal that the definition of this article. Of course, minimisation of MSE is equivalent to maximisation of the variance of projection (a simple consequence of the Pythagorean theorem), but the approximation approach is much clearer and allows multiple nonlinear generalisations. BTW, the history of the "variance" reformulation of the original PCA could be interesting: who and what for destroyed the Pearson's approach and created this viral "variance" definition? Concerning your definition: It is correct, but the iterative algorithm of selection of Principal Components should be, perhaps, a consequence of the basic definition: the best approximation of data by k-dimensional linear manifolds (if we follow the original approach). For the centred data set, these k-dimensional linear manifolds form a complete flag of subspaces. Its orthogonal basis is called the principal components basis. ...Agor153 (talk) 12:10, 5 April 2020 (UTC)
 * Thanks for your comment! My reference on the topic (Bishop) shows how the solution can be derived in each way, from maximum-variance and from minimum MSE projection-distance. Deriving a solution from the maximum variance objective seems a little bit simpler, but I think stating the objective geometrically, as a line or plane that minimizes squared distances, grounds the problem in terms anyone can understand. It's true that PCA is usually not computed in an explicitly iterative fashion like that (as far as I'm aware), but it's the most precise and accessible explanation I can come up with using so many words. The first k principal components form a basis for the best-fitting k-dimensional subspace, but an orthogonal basis for the best-fitting k-dimensional subspace doesn't necessarily comprise the first k principle components (correct me if I'm mistaken on this). And iteratively constructing/defining a basis should be familiar ground for anyone who has taken a course on linear algebra. Thanks again for your comments. I'm not a mathematician so second opinions are welcome. I'll wait a bit longer before editing the article. AP295 (talk) 13:48, 5 April 2020 (UTC)
 * Fine. The only comment I have is: the orthonormal basis of a complete flag of subspaces is practically unique: the first vector belongs to the 1D subspace from the flag, the second vector is in its orthogonal complement in the 2D subspace from the flag, etc. The only uncertainty is in the signs - each vector can be multiplied by -1. Therefore, if we have the k-dimensional planes of the best approximation then the basis we speak about is uniquely defined (up to multiplications of some vectors by -1).Agor153 (talk) 13:58, 5 April 2020 (UTC)
 * Thanks again. I replaced the opening paragraph but the article is still in a rough state. It reads more like a rough literature survey than an encyclopedic entry. Many things are repeated in different sections, and the article is littered with stray observations. AP295 (talk) 12:24, 7 April 2020 (UTC)

@ @ FYI, the only reason I can think of to use a correlation matrix instead of computing the z-scores beforehand is to save a few division ops. If you have m data points with dimension n, then computing the z-scores would require m*n divisions, but after computing the covariance matrix, dividing out their standard deviations to get the correlation matrix would take O(n*n) ops. Since it's often the case that n<<m, this could be more efficient. As far as I know, Mathematica is closed source so we don't actually know what its PCA function does with that argument. It might just compute the z-scores and then use SVD. I'm curious why they do this, so if anyone knows, feel free to comment. Anyway, someone re-formatted the second paragraph but I still think it's misleading. Using the correlation matrix is not an equivalent formulation and that should be made clear. Again, I think that specific applications of PCA (e.g. to z-scores) and their specific pre-processing steps should not be conflated with the algorithm itself. Something to the effect of "The principal components are the eigenvectors of the covariance matrix obtained from the data, or equivalently the singular vectors from SVD of the data matrix." might be clearer and more appropriate for the intro. AP295 (talk) 15:49, 24 April 2020 (UTC)

See the section above where I note this topic has long been horribly messed up. It should be fixed, not removed, since normalizing by variance is indeed a very common part of PCA. Yes, it affects the components – it makes them independent of the measurement units, for example – and yes if you think of it as pre-process before PCA than you can keep your concept of PCA more pure. But omitting it or denying it is not helpful. I'm not familiar with covariance vs correlation as the way to explain this option, but if that's what sources do, it's fine for us to do so. I'd rather see the difference integrated as a small optional difference though. Dicklyon (talk) 19:15, 23 August 2020 (UTC)

Entry paragraph
Hello, the first sentence of the article was more about "how" than "what", so I created another one concerning "what" PCA is. We should further consider postponing the "how" discussion in my opinion; one does not need to know how to do something until s/he knows what it is and why s/he should use it. "Why" could follow "what", e.g. the sentence beginning with "PCA is mostly used as a tool..." could be the second sentence. What do you think? --Gufosowa (talk) 08:28, 24 July 2020 (UTC)


 * @ PCA is most easily understood as successively fitting lines to a set of points in space. Why start it off with "In statistics,"? It's not exclusively a statistical technique, and you don't always use the principal components for dimensionality reduction either. The article is in serious need of repair. Its unnecessarily long and obscures what is otherwise a very simple and elegant concept with jargon and a lack of continuity. If you want to repair it, the intro (and every other section) could use some work, but I think the opening paragraph was just fine the way it was. It was similar to how Pearson introduced PCA and I think he put it best. AP295 (talk) 15:24, 3 August 2020 (UTC)
 * you have not answered my main discussion point. If you have any issues with the definition sentence you can modify it. If you think that it doesn't only belong to statistics, then you can add anything else that it belongs to. There was not a mention of dimensionality reduction anyway.
 * "...fitting lines to a set of points in space" is "linear regression". The current entry paragraph talks about all other things which have their own pages in Wikipedia, until defining PCA at the very end. This does not fit the Wikipedia standard at all. Here are some examples of good entry paragraphs: 1 2 3
 * Wikipedia is an encyclopedia, it is not a textbook. So the discussion of "what" should precede the discussion of "how". Do you have any objections to that? --Gufosowa (talk) 17:12, 3 August 2020 (UTC)
 * It is fallacious to think that the reason linear regression is called linear regression is that it involves fitting lines. Polynomial regression, for example, is an instance of linear regression, so called for good reasons. Michael Hardy (talk) 04:16, 1 September 2020 (UTC)
 * @ The definition I gavein the intro is equivalent to the maximum variance formulation, and not equivalent to linear regression. You cannot fit a vertical line with linear regression, for example. I believe it will be clearest to other English speakers the way I've written it, so I do object to any changes . Defining "principal component analysis" requires defining what a "principal component" is, and I've condensed that into as short and accessible an explanation as I could. AP295 (talk) 17:22, 3 August 2020 (UTC)
 * @ Again, the rest of the article does need a lot of work, so I don't mind at all if you want to improve other sections. Even the rest of the intro is a bit misleading and will need to be rewritten at some point. AP295 (talk) 17:27, 3 August 2020 (UTC)
 * If it is equal to maximum variance, then you can simply say "maximum variance" in the definition. We don't need to explain every fundamental concept. If someone does not know what variance is, they can follow the link to the variance page. Otherwise we would need to recite everything from Pythagoras. This article is about PCA, and PCA only.
 * I do not like the current entry paragraph because of these two reasons: 1) It discusses "how" before "what" 2) It delays "PCA" to discuss other things that have their own pages.
 * Hence, I created a sentence that talks about "what" and gives "PCA" right away (Please compare to link 1 link 2 link 3):
 * "In statistics, principal component analysis (PCA) is a method to project data in a higher dimensional space into a lower dimensional space by maximizing the variance of each dimension."
 * It is open for modifications. If you have any oppositions to 1) and 2) please share. --Gufosowa (talk) 17:46, 3 August 2020 (UTC)
 * @ It doesn't start with the "how", it starts with a definition of "best fitting line", which is the objective that PCA optimizes. It is immediately clear to most readers what that means. Yes, PCA maximizes the variance of the projected data, but why lead with that? I don't think it's unreasonable to choose the simpler of two equivalent definitions to state in the intro. Nothing is "delayed" by this introduction, I think the reader can bear with it for the whole four sentences before at long last they come to the definition of "Principal component". If you're concerned about brevity, the rest of the article could use a lot of work in that regard. AP295 (talk) 17:59, 3 August 2020 (UTC)
 * @ I'll add that I think readability is far more important than rigid stylistic convention. Most people use Wikipedia as a learning tool. You don't have to shoehorn the entire definition into the first sentence. AP295 (talk) 18:21, 3 August 2020 (UTC)
 * Starting with the definition of "best fitting line" is what I mean by delaying.
 * Starting with "what" is not only for the stylistic convention, but also for a more healthy approach to a topic. As I mentioned above "one does not need to know how to do something until s/he knows what it is and why s/he should use it". Anyway, our positions are clear and I started to repeat myself. Let's hear other Wikipedists' opinions, too. --Gufosowa (talk) 19:49, 3 August 2020 (UTC)
 * @ It is absolutely necessary to define "best fitting line". Linear regression minimizes squared vertical distances between the points and the line i.e. (f(x)-y)^2. The first principle component minimizes squared (orthogonal) distances. Perhaps a graphic would help: https://services.math.duke.edu/education/modules2/materials/test/test/errors.gif Maybe I'll make something similar for this article. AP295 (talk) 20:14, 3 August 2020 (UTC)

Dear, , , I pinged you as you are active users who wrote to this talk page previously. Could you please give an input to resolve this discussion? It is stuck for more than a week. Thank you. --Gufosowa (talk) 08:50, 12 August 2020 (UTC)
 * Yes, I agree with you that the current description in the lead is more a process than a definition. I'd saying something like how it was before AP295 changed it in this edit was better, though still not great.  Let's go back and craft a compromise. Dicklyon (talk) 20:46, 12 August 2020 (UTC)
 * The intro in the edit you linked describes PCA as a "statistical procedure" (whatever that is) and is practically a circular definition, so I really can't imagine why you think the current intro is more "procedural" in nature than the old one. That's exactly the sort of description I intended to avoid by first stating the objective function, then defining principal components as a solution to that objective. If you want to change it, I insist you define PCA in terms of the least-squares objective, and define "principal component" in the simplest possible terms as a solution to that objective. Pearson described it geometrically in terms of the objective it optimizes, and I feel that this is much clearer than using jargon like "statistical procedure". I also insist that it not start with the phrase "In statistics," AP295 (talk) 22:16, 12 August 2020 (UTC)
 * I'm not saying it was a great lead, which is why I suggested working on a compromise. It's not clear to me why you object to "In statistics" for starters.  Is there a better way to characterize the field within which this concept is meaningful?  Wasn't Pearson a statistician?  And I always thought of PCA as maximizing the variance captured, as opposed to minimizing the residual, so didn't think of it as a "least squares" thing, though I agree that's one approach.  The basic idea of fitting a line to data seems imperfect, though, as least squares fitting usually starts with an independent variable, which we don't have in this case.  Anyway, there are various ways to go about this.  I suggest each of you find some sources to compare how they go about it. Dicklyon (talk) 00:27, 13 August 2020 (UTC)
 * Pearson was a statistician, and yet he still chose to introduce the concept geometrically. The least-squares approach highlights the difference between PCA and linear regression, and nicely ties it in with linear algebra while remaining (relatively) accessible to the average undergrad. Ideally this article would introduce the problem in terms of the objective (and thus defining "principal component"), solve for the first PC (thus showing it's an eigenvector), and then move on to alternative derivations, efficient calculation of the principal components via SVD/eigendecomposition, applications, etc. I object to the article starting with the phrase "In statistics", because statistics is but one area where PCA is applied (perhaps we can say how it's applied in the "applications" section), and I feel it's a slight psychological deterrent to the average reader. AP295 (talk) 18:22, 14 August 2020 (UTC)
 * How about "In data analysis" then? Dicklyon (talk) 06:44, 15 August 2020 (UTC)
 * Unless there's another field in which "principal component analysis" has an entirely different meaning, I don't see the need to so distinguish PCA in the first sentence of this article. Applications can be addressed in the "applications" section. AP295 (talk) 16:49, 15 August 2020 (UTC)
 * P.S. I would not be opposed to a very clear and well-organized derivation from maximum variance if we can replace the whole intro (and maybe clean up some of the main article) instead of just the first paragraph. The whole article needs work and it's not my intent to be an obstacle here. I'd rather people not make quick and careless edits to the first paragraph though, as I've thought very carefully about how it should read. AP295 (talk) 01:33, 16 August 2020 (UTC)
 * Thank you for invitation to this discussion. I can say the both positions are correct but (1) I prefer Pearson's definition principal components through best fit. Nevertheless, Pearson used "lines and planes of best fit". This means that the plane of k first principal components is defined as a k-dimensional plane of best fit. The iterative process used now in the first paragraph is a simple theorem, a consequence of the definition (the mean point belongs to the line of the first component, the line belongs to the plane, etc.). There is one unpleasant incorrectness now: the iterative procedure is not yet formally defined if the eigenspectrum is degenerated (of course, the correction is obvious but forces us to discuss minor technical issues when we must discuss ideas. The "object-oriented" intro ("what ") seems to be better than the procedure-oriented one. Thus, if my opinion matters> I prefer Pearson's definition through data approximation (especially because it allows immediate non-linear generalisations, while the "variance approach" does not). Nevertheless, this definition is better to introduce correctly, as Pearson did, lines and planes of best fit (this is 'what', not 'how'), and then the iterative procedure can be presented as the simple corollary (with all proper formulations for degenerated spectra).-Agor153 (talk) 20:19, 16 August 2020 (UTC)
 * I'm not sure what you mean. You can always find a "best-fitting" line. There may be more than one direction to choose from but it's the same up to the order they're selected. (edit:  If there aren't enough linearly independent eigenvectors then any orthogonal basis for the remaining subspace can be taken. This doesn't contradict the definition in the first paragraph, although you're correct if you meant that there may not be a unique solution. This seems like a very minor issue that can be addressed elsewhere.) Does WP have any official guidelines on how much technical competence we should expect from the reader? At least, I think the intro should be relatively gentle, and that's why I wrote up the construction that's currently in the first paragraph. The intro is not the place to slam the reader with technical jargon, unnecessary comments, stray observations, etc. We do not want to deter the reader. It should be as simple as reasonably possible. AP295 (talk) 18:50, 17 August 2020 (UTC)
 * I edited the intro to include a definition based on the maximum variance objective, and I think it reads a bit more clearly. This and the first paragraph are really just recursive definitions of the principal components, so I don't consider them to be too "procedural" in nature. However, if anyone objects, please let me know why. I'd really like to delete the rest of the intro after the first couple paragraphs include a simple proof that the first PC is an eigenvector in its place, but I'll wait for comments. AP295 (talk) 17:03, 18 August 2020 (UTC)
 * Some more edits. Hopefully this is a satisfactory compromise. It's starting to look like a good intro in my opinion. The terms "Principal Component" and PCA are clearly defined and both objectives are briefly covered. I don't think we need to be overly pedantic in the intro. Anyway, there's still a lot of repetition in the rest of the article that should be removed at some point. It seems like the article has been continually added to without much attempt at maintaining its continuity/readability. AP295 (talk) 14:30, 19 August 2020 (UTC)
 * @ To be honest, I don't think your edit reads as nicely. The phrase "directions (principal components) that align with most of the variation" is not as precise or succinct as either recursive definition. It is awkward to define PCA without first taking a couple sentences to define "principal components" and I don't think it does the reader any good to force this style. It would actually make a lot more sense to re-title the article "Principal Components" and have "PCA" redirect to the article, since the term "PCA" doesn't seem to have a single concrete definition. Please consider this idea. AP295 (talk) 18:25, 19 August 2020 (UTC)
 * I'm not saying I got it to a great state, but I really think we need to lead with a definition, not a tutorial. And please avoid caplitalization of things that aren't proper names. Dicklyon (talk) 22:48, 19 August 2020 (UTC)
 * @ The article starts with a recursive definition of principal components. If you want to phrase it a bit more formally that's fine by me but it is a (reasonably) precise mathematical definition. AP295 (talk) 22:56, 19 August 2020 (UTC)
 * @ I made a few changes to the intro as they came to mind. It's really the same intro but with slightly better wording, and I hope you'll be convinced that this is a reasonable way to present the topic. Keep in mind that PCA can be defined completely without reference to any concepts from statistics at all. PCA is a change of basis that satisfies the recursive least-squares objective described in the first paragraph, and I feel that's by far the most natural way to introduce it. AP295 (talk) 03:07, 20 August 2020 (UTC)

Arbitrary break
@ @ @ I invite you to read the current intro. I'll take a break from editing the article for now but I feel it's much, much clearer and more precise than the intro linked earlier in the talk page. PCA is not inherently a statistical concept per se, it's essentially just a change of basis that satisfies the objective(s) described in the intro. There's no point in forcing it all into one sentence because it won't be comprehensible by anyone who isn't already familiar with PCA. Please be careful if you edit it. We should aim to communicate, rather than merely document. AP295 (talk) 15:20, 20 August 2020 (UTC)

One final thought for today. On Wikipedia it is conventional that the article's topic should be defined/referenced in the first sentence or shortly thereafter. This is understandable; it would be very inconvenient to "delay" the definition any more than a few short sentences, and I've not done so here. However, readers visit an article to understand the topic of that article. Any material before you reference the topic/title is probably going to be read and carefully considered by the reader. In this sense, it is a good opportunity to communicate important ideas and concepts because it may command a great deal of the reader's attention. AP295 (talk) 16:07, 20 August 2020 (UTC)


 * Thanks for your efforts on this, but I don't think it's important for the reader to study distance from a line before knowing that a principal component is a best-fit direction vector; so I've taken another cut at a normal lead starting with a definition that tries to be clear and neutral with respect to whether one prefers the max variance or min distance approach. Please see what you think, do try to improve it. Dicklyon (talk) 20:52, 21 August 2020 (UTC)
 * I beg to differ. Both you and Gufosowa seem to think that I'm describing regression here. There is no dependent variable in PCA. The terms in the objective function are not residuals., please help. AP295 (talk) 23:00, 21 August 2020 (UTC)
 * That's not what I'm thinking. Residuals are what's left after subtracting off a model, regression or not.  There was no mention of independent or dependent variables in the lead I wrote.  You have reverted attempted improvements too many times now; please stop that. Dicklyon (talk) 04:16, 22 August 2020 (UTC)
 * The terms errors and residuals have a specific meaning. The term "distances" is more appropriate for explaining PCA in a general context where your points may not be a statistical sample, and less likely to confuse someone who isn't a statistician. AP295 (talk) 17:27, 22 August 2020 (UTC)
 * I agree in 's change. We do not have to establish all the background, we assume that the reader knows basic concepts like line, error, dimension, space etc. We already have a best fit line article in Wikipedia, we can simply link to it. If the reader does not understand it and find it important, they may follow the link. Hence, the following part is redundant:
 * "Given a collection of points in two, three, or higher dimensional space, a "best fitting" line can be defined as one that minimizes the average squared distance from a point to the line. For a collection of points in $\mathbb{R}^n$ and $i \leq n$, a direction for the $i^{th}$ best-fitting line can be chosen from directions perpendicular to the first $i-1$ best-fitting lines"
 * It should be removed or postponed to the methodology sections. If is concerned that linear regression works numerically while PCA geometrically, we can add it as a keyword in the sentence. --Gufosowa (talk) 13:22, 22 August 2020 (UTC)
 * Regression is a fundamentally different idea from PCA. I urge you to carefully consider that "redundant" sentence. If it is incorrect, then please explain my mistake. Otherwise I insist that it remain in the article's intro.  AP295 (talk) 16:38, 22 August 2020 (UTC)
 * PS I am trying to help you here. It's disappointing when a simple and elegant concept like PCA is obscured by jargon and careless editing. WP is a great resource and I use it all the time. I don't want anyone to come away from this article confused or with mistaken ideas about what PCA is. AP295 (talk) 16:52, 22 August 2020 (UTC)
 * I don't say that it is incorrect, I say that it is redundant. It does not belong to the top of the PCA article. It has its own article already. If you have concerns, please try to fix it instead of reverting altogether. Thank you. --Gufosowa (talk) 17:46, 22 August 2020 (UTC)
 * But it isn't. It's the base case of PCA's recursive definition. We define the first PC and then we define the ith PC using the first i-1 PCs. This is an extremely common pattern in both computer science and the construction of a basis in linear algebra e.g. the Gram–Schmidt process AP295 (talk) 18:41, 22 August 2020 (UTC)
 * @ @ Since this doesn't seem to be going anywhere, I've made a topic on the dispute resolution noticeboard. — Preceding unsigned comment added by AP295 (talk • contribs)
 * Thanks. And I've notified you of your 3RR limit at your talk page.  It has actually gone somewhere, in that several of us think your approach is wrong and are trying to fix it.  I agree with you that it can perhaps be more clear and precise and jargon free, but within the framework of a normal lead please; don't lead with a tutorial. Dicklyon (talk) 20:13, 22 August 2020 (UTC)

Re errors and residuals, residual is clearly more correct within those definitions, and is in common use, e.g. at Matlab's "PCA Residual" function; or this book that talks about the amount that is unexplained by the pc model—the residual on p.35, and on the next page they examine the sum of squares of the residuals. Dicklyon (talk) 20:26, 22 August 2020 (UTC)
 * The term "residual" occurs in the context of statistics. We don't need to call it that here. Calling the objective "average squared distance" is entirely correct, more general in a mathematical sense, and much more likely to be understood by a casual reader. I don't know what you're thinking. In "Pattern Recognition and Machine Learning", Bishop calls it the "minimum error" formulation, but there's really no harm at all in just calling them distances. AP295 (talk) 20:36, 22 August 2020 (UTC)
 * If you see a way to improve it without throwing out the basic normal lead style, please go ahead. But in discussing differences, it can also be very helpful to link sources that do it more like what you're proposing.  It shouldn't be hard to converge on a good wording.  It's not clear to me what you think the context is here that's broader than statistics. Dicklyon (talk) 21:24, 22 August 2020 (UTC)
 * I've already explained the reasons I think we should use the term "distance" instead of errors or residuals. Let's wait until they open the discussion on the dispute resolution page so we aren't wasting our time. PCA is an uncomplicated concept that anyone can understand and a great example of how linear algebra can be applied to real-world problems. This article is a travesty, so I'm trying to improve its correctness and organization while minimizing the amount of background knowledge it requires from the reader. AP295 (talk) 21:57, 22 August 2020 (UTC)
 * I do appreciate that you're trying to improve. Please explain why you think it's a travesty, after all the years of work by so many other editors.  You might also consider responding to my questions, providing sources, etc., as a way to help move the discussion forward, rather than just insisting on your idiosyncratic approach to the lead.  Re residuals vs distances, you've said The terms errors and residuals have a specific meaning. The term "distances" is more appropriate for explaining PCA in a general context where your points may not be a statistical sample, and less likely to confuse someone who isn't a statistician.  I don't understand why you claim that "distances" is more appropriate, esp. in light of the sources I showed that have "residuals"; and I don't see how not being a statistician is relevant to how confusing any of these terms might be.  I'm not a statistician, and it seems OK to me.  Also not clear why you think points being a "statistical sample" is relevant; you can compute statistics on data no matter where they came from, no?  Dicklyon (talk) 22:47, 22 August 2020 (UTC)
 * The word "distances" is more appropriate because that is what they are. The concept of distance is defined in any metric space (edit: you'd need at least the structure of a normed vector space for PCA to make sense, but my point is that distance is a suitably general and well-understood term). The terms "residual" and "error" are often used in the context of regression, which is different from PCA. They are less general and less likely to be understood by the average reader. AP295 (talk) 00:58, 23 August 2020 (UTC)

AP295 forgot to ping us to the WP:DRN discussion he started about us. Please see at Dispute_resolution_noticeboard and enter your summary of the dispute if you like. I just did mine. Dicklyon (talk) 22:56, 22 August 2020 (UTC)

@ The first PC is a vector with the same direction as a line that best fits the data. This line may not be a vector itself, meaning the line that best fits the data may not pass through the origin. This technicality is not a problem if you say "the direction of a line". Incidentally, this is why the data must be centered if you compute the covariance matrix using an outer product (Gramian matrix). AP295 (talk) 18:02, 23 August 2020 (UTC)
 * Yes, I understand that. Is there a rewording that you suggest then? Dicklyon (talk) 20:16, 23 August 2020 (UTC)
 * If you say so. I already made the change. AP295 (talk) 20:18, 23 August 2020 (UTC)
 * This sentence OK, but they you have "Each subsequent $$i^{th}$$ principal component is a direction of a line that minimizes the sum of squared distances and is orthogonal to the first $$i-1$$ principal components." where I had "Each subsequent $$i^{th}$$ principal component is a direction orthogonal to the first $$i-1$$ that best fits the residual."  It's not clear to me what your distances refer to if you're not operating on residuals from the previous.  Does this work for distances of original data points (perhaps it does, though it's not obvious to me, not how I was thinking of it). Dicklyon (talk) 23:08, 23 August 2020 (UTC)
 * Finding the ith principal component could be viewed as a constrained optimization problem (and in fact you can use Lagrange multipliers to show that it's an eigenvector. Bishop does this in "Pattern Recognition and Machine Learning"). You're trying to find a best-fitting line whose direction is orthogonal to the directions of the first i-1 best-fitting lines, with the base case i=1 being unconstrained*. We have to say "directions of the best fitting line" because we are only concerned with their directions. They don't have to pass through the origin, but it's their directions (usually taken as unit vectors) that form the basis of principal components. AP295 (talk) 03:48, 24 August 2020 (UTC)
 * *Edit- Unconstrained in its direction. Bishop actually uses a normality constraint to enforce ||u||=1. Just didn't want that to be a point of confusion/contention. AP295 (talk) 04:31, 24 August 2020 (UTC)
 * I'll put it another way. For the first principal component, one can take (the direction of) a line (say, defined in terms of a direction (||direction||=1) and translation in R^n as the set of points {c*direction+translation | c in R}) that best fits the data. For the ith, take (the direction of) a line that best fits the data, but only from among those lines whose direction is orthogonal to the direction of the first i-1 lines. AP295 (talk) 04:31, 24 August 2020 (UTC)
 * You mean Bishop's section 12.1.2 on projection error minimization? I don't have access to my copy at work these days, but on Amazon I can see some of the relevant pages.  Unfortunately not page 564 where he might support what you're saying.  I'm not saying you're wrong, just that looking at the residuals makes so much more sense.  Also note that he does the maximum variance formulation first. Dicklyon (talk) 05:03, 24 August 2020 (UTC)
 * @ The "residuals" (which usually refer to scalar values, but I suspect you're talking about the parts that don't lie in the subspace spanned by PCs up to i-1) from projecting onto the i-1 PCs are orthogonal to those i-1 PCs by definition. You can just fit the ith PC under the condition that it be orthogonal to the first i-1 PCs. This is a much more natural way of looking at it. AP295 (talk) 14:08, 24 August 2020 (UTC)
 * I don't see it as more natural. Can you point out a source that does it that way?  Does Bishop?  Sorry, I can't see the relevant page.  And yes of course residuals are vectors; how would they be scalars? Dicklyon (talk) 16:12, 24 August 2020 (UTC)
 * Yes, Bishop does. If $$P_{i-1} = [p_1,p_2,\dots,p_{i-1}]$$ are our first i-1 principal components, then a vector v can be decomposed into $$v = (a_1p_1+a_2p_2 + \dots + a_{i-1}p_{i-1}) + R $$ Where vector R must be orthogonal to the subspace spanned by $$P_{i-1}$$. If you restrict your selection of the ith principal component to only those unit vectors orthogonal to P's columns, then $$ p_i^T v = p_i^T (a_1p_1+a_2p_2 + \dots + a_{i-1}p_{i-1}) + p_i^T R = a_1 p_i^T p_1+ a_2 p_i^T p_2 + \dots + a_{i-1} p_i^T p_{i-1} + p_i^T R = p_i^T R $$ AP295 (talk) 16:31, 24 August 2020 (UTC)
 * Good, that's a convincing demonstration that the ith direction applied to the residual R is the same as the ith direction applied to the original vector. But does he really suggest skipping the computation of R, as a practical matter?  Wouldn't it be easier to compute R than to constrain the ith direction to be orthogonal to all those other directions?  How do you do that exactly?  Or is it just a gedanken thing? Dicklyon (talk) 00:12, 25 August 2020 (UTC)
 * This is a recursive definition, and basically the same one Bishop uses. If you set it all up like Bishop does and set the vector of partial derivatives equal to zero, then you get that it's an eigenvector of the covariance matrix. So, they're computed with eigendecomposition or SVD, as I mentioned in Principal Component Analysis. I did make a small omission in that revision, to the maximum variance explanation. In that case it would be easiest to assume the data are mean-centered and talk only about directions, otherwise the variance has no maximum (a "line" can be translated to make the variance infinite even if we parameterize it as having a unit vector for direction). But it's otherwise correct as far as I know. In any case, the minimum-squared-distances presentation is cleaner. AP295 (talk) 01:07, 25 August 2020 (UTC)
 * Anyway, if you see what I mean, I think we should start from this revision - Principal Component Analysis. We first define "best-fitting line", then define our objective as a series of "best-fitting" lines, then state the solution to that objective. This is the most sensible progression in my opinion, and we remind the reader of several important concepts along the way. AP295 (talk) 17:50, 25 August 2020 (UTC)
 * That might be a good approach for an "Introduction" section; not for the lead. Dicklyon (talk) 21:17, 25 August 2020 (UTC)
 * If the article were renamed "Principal Components" we would not have to hem and haw about what PCA means in the first sentence. We could take the intro in the revision I linked earlier and just put something like this in front of it: "The principal components of a collection of points in R^n are a sequence of n vectors where the ith element is the direction of a line that best fits the data and is orthogonal to the first i-1 elements" AP295 (talk) 22:01, 25 August 2020 (UTC)
 * I didn't rename the article, but I made a few changes along those lines. AP295 (talk) 22:17, 25 August 2020 (UTC)
 * I see my link to the change of basis page is gone from the intro, but think about about how a vector decomposes into a linear combination of basis vectors. This is a fundamental concept and I think everyone could do with a reminder. Please fix the intro if you understand what I'm saying. AP295 (talk) 15:04, 24 August 2020 (UTC)
 * You could do some fixing instead of just complaining and reverting, no? Dicklyon (talk) 16:12, 24 August 2020 (UTC)
 * I've been trying to fix this article since March.

How's this? I think it reads pretty well and it's closer to the usual style. AP295 (talk) 22:27, 25 August 2020 (UTC)
 * Better. I'd avoid the formal mathy "points in $$\mathbb{R}^n$$" in the lead sentence, especially since you then use the word "vector" for the same kind of object.  See if you can simplify that to plain English; it doesn't need to start too formal or precise. Dicklyon (talk) 22:35, 25 August 2020 (UTC)
 * "Points in R^n" is there to let the reader know that our sequence is of equal length as the dimension. — Preceding unsigned comment added by AP295 (talk • contribs) 00:01, 26 August 2020 (UTC)
 * I took a stab at saying it in English. Sadly, the article uses dimension p, not n. Dicklyon (talk) 00:37, 26 August 2020 (UTC)
 * That's fine then. Unless Gufosowa has an issue with it we can probably close the topic on the DRN. I will probably continue to edit the rest of the article when I have time though. AP295 (talk) 01:03, 26 August 2020 (UTC)
 * Good idea. It still needs work on terminological and variable name consistency, and work on the Z thing per discussions above. Dicklyon (talk) 04:27, 26 August 2020 (UTC)
 * It looks better, thank you for improving it. Let's close the DRN topic then. --Gufosowa (talk) 20:08, 26 August 2020 (UTC)

The body of the article is still in need of "de-fragmentation" for lack of a better word. I don't have time to work on it now but as a general game plan, I think editors should try to make the article a bit more compact and coherent. AP295 (talk) 17:27, 28 August 2020 (UTC)

Pronumerals inconsistent between Introduction and Details sections
As at 27 Sept 2021 the introduction uses p for the number of principal components and i to index individual principal components.

The Details section uses $$l$$ for the number of principal components and k to index individual principal components. It uses p for the dimension of the underlying vectors and i to index individual rows in the data matrix.

To avoid creating unnecessary confusion for the first-time reader, I propose changing the Introduction to use pronumerals in the same way as in the Details section, ie replace p by $$l$$ and replace i by k.

I just wanted to check first whether I have misunderstood something. Please let me know if I have. Ajkirk (talk) 22:48, 26 September 2021 (UTC)

Biplots and scree plots
Please add a description in the article Biggerj1 (talk) 23:02, 8 November 2021 (UTC)

Incorrect statement in section 4 Further Considerations
The (now deleted by me) statement "Given a set of points in Euclidean space, the first principal component corresponds to a line that passes through the multidimensional mean and minimizes the sum of squares of the distances of the points from the line" is incorrect. Counterexamples are easy to find, e.g. if mean is 0. Another example is in 2d for a cloud of points with $$\langle y\rangle = 0$$ and $$\langle xy\rangle = 0$$. For this example,

PCs are along $$\hat y$$ and $$\hat x$$, only the $$\hat x$$ one goes through the mean/ center of gravity. But $$\hat y$$ is the largest component if $$\langle x^2\rangle + \langle x\rangle ^2 < \langle y^2 \rangle $$. This example corresponds to the figure.

More generally, for $$X = \sum_{k=1}^P\sigma_k {\hat u}_k {\hat v}^T_k$$ an $$M\times N$$ matrix with $$P$$ singular values, (column) mean is $$\frac1{N}\sum_{k=1}^P\sigma_k {\hat u}_k \sum_j(\hat{v}_k)_j, $$ while first principal component is $${\hat u}_1 $$ — Preceding unsigned comment added by Sunejdk (talk • contribs) 08:45, 5 December 2021 (UTC) Sunejdk (talk) 08:47, 3 December 2021 (UTC)

Much too complex for most users
Sorry (and I'm actually a mathematician myself). I'd have to say all this complex mathematics is all very nice but it actually belongs in a specialist textbook, not in an encyclopaedia. A layman who wants to know what PCA is, what it does and what its limitations are is going to be completely bamboozled by this article.

I've made a start on trying to put some plain language into the article instead of all this far-too-clever mathematics - which I would move to some other strictly technical entry called maybe "Mathematics of Principal Components". Redabyss1 (talk) 07:28, 5 May 2022 (UTC)
 * This logic affects all mathematical pages, which would harm Wikipedia. Wikipedia is an encyclopedia and it is used by students. One must not turn it into tutorial pages. As far as I know, people into PCA know some math (the entire concept is mathematical). But I agree that some clarity in the lede is necessary. Limit-theorem (talk) 10:19, 5 May 2022 (UTC)
 * Not so. Most users pump their PCA straight out of statistical packages and have barely heard of eigenvalues or transposes.
 * I have added four other major uses of PCA from the social sciences and genetics. Having done this, I see the article is too complex and meandering and reflects the input of editors with different interests and priorities. It should be broken up now into two or three articles. I don't think separating out the mathematical component would harm either mathematics or Wikipedia. Redabyss1 (talk) 07:29, 6 May 2022 (UTC)
 * People have no business using PCAs if they do not know what an eigenvalue means. The concept is mathematical.Limit-theorem (talk) 10:09, 6 May 2022 (UTC)
 * Though I do not agree with your proposal, to be more constructive, you should actually provide examples of how you see the topic being split. What are the two or three articles you are proposing, what content should go in each. Give a roadmap for your suggested improvements. Twotontwentyone (talk) 18:41, 23 September 2022 (UTC)
 * I agree with @Limit-theorem that stripping out the math complexity is contrary to the point of Wikipedia (and constitutes a paradigm shift that would necessitate the rewriting of most Wiki pages related to math topics). Wikipedia is collection of human knowledge. Introductions or specialty sections can be used to give reads more of a bird's eye view of the topic, but the richness of information for this topic should not be watered down just because some people are not able to appreciate/use it. Twotontwentyone (talk) 18:39, 23 September 2022 (UTC)
 * There is no need to strip away the math, yet at the same time the article could be much accessible to laypeople and students. The section on Intuition seems the place to provide a non-jargon laden description. That is not achieved by starting off with "PCA can be thought of as fitting a p-dimensional ellipsoid to the data". Would be great if someone could attempt a plain English version. Koedinger (talk) 00:22, 16 December 2022 (UTC)

Applications: Intelligence section doesn't belong.
This paragraph exclusively discusses factor analysis, never PCA. Why is it here? (Article already mentions FA earlier, with appropriate links.) Jmacwiki (talk) 19:26, 11 June 2022 (UTC)

Notation for the SVD uses W instead of V
In the section on how PCA relates to the SVD the SVD is described as X=U Sigma W^T, yet the page for SVD describes it as U Sigma V^T. Is there any reason as to why this page uses W instead of V? If there isn't, then it should be changed to be consistent with the main SVD page. 31.205.215.152 (talk) 00:15, 19 January 2023 (UTC)