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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)

title: Talk:Principal component analysis


text:
Renee Nele

Eva Renée Nele (born 1932) is a German artist who works principally in goldsmithing and metal sculpture.

Life
Nele was born on 15 March 1932 in Berlin, the daughter of Arnold Bode; she grew up in Kassel. She studied under Hans Uhlmann (artist) at the Akademie der Künste in Berlin and under Richard Hamilton at the Central School of Art and Design in London. She also studied at the Atelier Lacourière-Frélaut in Paris. She was a member of Situationist International until 1962. She lives in Frankfurt am Main.

From 1985 to 1990 she taught sculpture at the Internationale Sommerakademie für Bildende Kunst Salzburg in Salzburg, Austria. She also taught at the Goethe-Universität of Frankfurt and – from 1993 – at the University of Giessen in Giessen, in Hesse.

Work
Nele participated in Documenta II, organised by her father in 1959.

Her sculpture The Couple, made in 1961, is in the collection of the Bayerische Staatsgemäldesammlungen in Munich.

Die Rampe, a Holocaust memorial showing faceless cloaked figures descending on a ramp from a railway goods wagon or collapsed at the foot of it, was created in 1982, and was installed on the campus of the University of Kassel in 1985. It was destroyed by arson soon after, and restored in 1987. In 2017 it was moved to the Holländischer Platz campus of the university, much of which is built on land that was formerly the site of factories of the Henschel company, which during the Second World War made extensive use of forced labour. A work titled Windsbraut is in the Dalbergplatz in Frankfurt-Höchst.

In 2008 she was, with the cellist Frank Wolff (cellist), one of two recipients in that year of the Goethe Plaque of the City of Frankfurt.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Renee Nele


text:
Rosalyn Tureck

Rosalyn Tureck (December 14, 1913 – July 17, 2003) was an American pianist and harpsichordist who was particularly associated with the music of Johann Sebastian Bach. However, she had a wide-ranging repertoire that included works by composers Ludwig van Beethoven, Johannes Brahms and Frédéric Chopin, as well as more modern composers such as David Diamond, Luigi Dallapiccola and William Schuman. Diamond's Piano Sonata No. 1 was inspired by Tureck's playing.

Biography
Tureck was born in Chicago, Illinois, the third of three daughters of Russian Jewish immigrants Samuel Tureck (né Turk; Rosalyn’s father was of Turkish descent) and Monya (Lipson) Tureck. She was the granddaughter of a cantor from Kiev. The first of her teachers to recognize her special gifts for playing the music of Bach was the Javanese-born Dutch pianist Jan Chiapusso, who gave her twice-weekly lessons in Chicago from 1929 to 1931 and also introduced her to the sounds of exotic instruments and ensembles such as the Javanese gamelan.

At Tuley High School (closed 1974), Tureck was a friend and classmate of future Nobel Prize–winning novelist Saul Bellow, who graduated in January 1932. The two remained in contact for decades.

"My technique was grounded, from my earliest years of study, in the school of Mendelssohn as passed on by Anton Rubinstein and many of his pupils, one of whom, Sophia Brilliant-Liven, was my teacher. It's essentially a finger technique, not a chordal one." Tureck reports that Brilliant-Liven was a stern teacher. "During the years I was with her, from the ages of 9 to 13, she never praised my playing." However, she made up for this, Tureck said, with a single compliment given to 13-year old Tureck after her performance in the semi-finals of a piano competition in which 80,000 young pianists participated. Brilliant-Liven told young Tureck, "If I had been listening from outside the auditorium, I would have sworn it was Anton Rubinstein himself playing." Tureck went on to the finals, and to win first prize in the competition.

She continued her musical studies in Chicago with pianist and harpsichordist Gavin Williamson. She later studied at the Juilliard School in New York City, where one of her teachers was Leon Theremin. She made her debut at Carnegie Hall playing the electronic instrument invented by Theremin, the eponymously named theremin. In 1940, Turek joined the piano faculty of the Mannes School of Music. Later in her career, she joined the faculty at Juilliard as a teacher. For a while she followed Wanda Landowska in playing Bach's keyboard music on a harpsichord but later returned to playing the piano. In 1970, Tureck performed in Boston for the Peabody Mason Concert series. She was an honorary fellow of St Hilda's College, Oxford.

Author William F. Buckley, a friend of Tureck's, when writing in his magazine "National Review" often called her "J.S. Bach's representative on Earth".

In a CBC radio special on Glenn Gould, the host told Tureck that Gould cited her as his "only" influence. She responded by stating that she was an influence and that it was very kind of him to say so.

In 1990 she served on the jury of the Paloma O'Shea Santander International Piano Competition.

During 2000 and 2001, Tureck lived in Spain teaching and practicing every day of the week, specifically in Estepona, Málaga, where she remained for a year in retirement.

Tureck was among the founders of the Music Academy of the West summer conservatory in 1947.

She died in New York City in 2003, aged 89. Her scores and recordings were given to the Music Division and the Rodgers & Hammerstein Archives of Recorded Sound, both divisions of the New York Public Library for the Performing Arts.

title: Rosalyn Tureck


text:
Golden World Records

Golden World Records was a record label owned by Eddie Wingate and Joanne Bratton (née Jackson, former wife of boxing champion Johnny Bratton). The recording studio was located in Detroit, Michigan, United States. The studio's national hits included "Oh How Happy" by Shades of Blue and "(Just Like) Romeo and Juliet" by The Reflections. The early, pre-Motown songs of Edwin Starr, such as "Agent Double-O-Soul", were recorded at the Golden World studio.

Golden World Records operated from 1962 to 1968.

The label and its subsidiaries were purchased by Berry Gordy in 1966 and folded into the Motown Record Corporation. The Golden World studio became Motown's "Studio B", working in support of the original Motown recording studio (Studio A) at Hitsville USA. Before its purchase by Gordy, the studio's recordings often included moonlighting Motown back-up musicians, including James Jamerson on bass and George McGregor on percussion.

The famous clock that hung in Golden World Records is currently owned by Melodies and Memories in Eastpointe, Michigan, and is on display there. A restored old Steinway piano that Motown inherited from Golden World is now on display at the Motown Museum.

Albums
Only one album was released on the Golden World label:

 * (Just Like) Romeo & Juliet—The Reflections—GWLPM-300—1964

title: Golden World Records


text:
Metro station

A metro station or subway station is a train station for a rapid transit system, which as a whole is usually called a "metro" or "subway". A station provides a means for passengers to purchase tickets, board trains, and evacuate the system in the case of an emergency. In the United Kingdom, they are known as underground stations, most commonly used in reference to the London Underground.

Location
The location of a metro station is carefully planned to provide easy access to important urban facilities such as roads, commercial centres, major buildings and other transport nodes.

Most stations are located underground, with entrances/exits leading up to ground or street level. The bulk of the station is typically positioned under land reserved for public thoroughfares or parks. Placing the station underground reduces the outside area occupied by the station, allowing vehicles and pedestrians to continue using the ground-level area in a similar way as before the station's construction. This is especially important where the station is serving high-density urban precincts, where ground-level spaces are already heavily utilised.

In other cases, a station may be elevated above a road, or at ground level depending on the level of the train tracks. The physical, visual and economic impact of the station and its operations will be greater. Planners will often take metro lines or parts of lines at or above ground where urban density decreases, extending the system further for less cost. Metros are most commonly used in urban cities, with great populations. Alternatively, a preexisting railway land corridor is re-purposed for rapid transit.

Facilities

At street level the logo of the metro company marks the entrances/exits of the station. Usually, signage shows the name of the station and describes the facilities of the station and the system it serves. Often there are several entrances for one station, saving pedestrians from needing to cross a street and reducing crowding.

A metro station typically provides ticket vending and ticket validating systems. The station is divided into an unpaid zone connected to the street, and a paid zone connected to the train platforms. The ticket barrier allows passengers with valid tickets to pass between these zones. The barrier may be operated by staff or more typically with automated turnstiles or gates that open when a transit pass is scanned or detected. Some metro systems dispense with paid zones and validate tickets with staff in the train carriages.

Access from the street to ticketing and the train platform is provided by stairs, concourses, escalators, elevators and tunnels. The station will be designed to minimise overcrowding and improve flow, sometimes by designating tunnels as one way. Permanent or temporary barriers may be used to manage crowds. Some metro stations have direct connections to important nearby buildings (see underground city).

Most jurisdictions mandate that people with disabilities must have unassisted use of the station. This is resolved with elevators, taking a number of people from street level to the unpaid ticketing area, and then from the paid area to the platform. In addition, there will be stringent requirements for emergencies, with backup lighting, emergency exits and alarm systems installed and maintained. Stations are a critical part of the evacuation route for passengers escaping from a disabled or troubled train.

A subway station may provide additional facilities, such as toilets, kiosks and amenities for staff and security services, such as Transit police.

Transfer stations
Some metro stations are interchanges, serving to transfer passengers between lines or transport systems. The platforms may be multi-level. Transfer stations handle more passengers than regular stations, with additional connecting tunnels and larger concourses to reduce walking times and manage crowd flows.

Platform-edge doors
In some stations, especially where trains are fully automated, the entire platform is screened from the track by a wall, typically of glass, with automatic platform-edge doors (PEDs). These open, like elevator doors, only when a train is stopped, and thus eliminate the hazard that a passenger will accidentally fall (or deliberately jump) onto the tracks and be run over or electrocuted. Control over ventilation of the platform is also improved, allowing it to be heated or cooled without having to do the same for the tunnels. The doors add cost and complexity to the system, and trains may have to approach the station more slowly so they can stop in accurate alignment with them.

Architectural design
Metro stations, more so than railway and bus stations, often have a characteristic artistic design that can identify each stop. Some have sculptures or frescoes. For example, London's Baker Street station is adorned with tiles depicting Sherlock Holmes. The tunnel for Paris' Concorde station is decorated with tiles spelling the Déclaration des Droits de l'Homme et du Citoyen. Every metro station in Valencia, Spain has a different sculpture on the ticket-hall level. Alameda station is decorated with fragments of white tile, like the dominant style of the Ciutat de les Arts i les Ciències. Each of the original four stations in the Olympic Green on Line 8 of the Beijing Subway are decorated in Olympic styles, while the downtown stations are decorated traditionally with elements of Chinese culture. On the Tyne and Wear Metro, the station at Newcastle United's home ground St James' Park is decorated in the clubs famous black and white stripes. Each station of the Red Line and Purple Line subway in Los Angeles was built with different artwork and decorating schemes, such as murals, tile artwork and sculptural benches. Every station of the Mexico City Metro is prominently identified by a unique icon in addition to its name, because the city had high illiteracy rates at the time the system was designed.

Some metro systems, such as those of Naples, Stockholm, Moscow, St. Petersburg, Tashkent, Kyiv, Montreal, Lisbon, Kaohsiung and Prague are famous for their beautiful architecture and public art. The Paris Métro is famous for its art nouveau station entrances; while the Athens Metro is known for its display of archeological relics found during construction.

However, it is not always the case that metro designers strive to make all stations artistically unique. Sir Norman Foster's new system in Bilbao, Spain uses the same modern architecture at every station to make navigation easier for the passenger, though some may argue that this is at the expense of character.

Metro stations usually feature prominent poster and video advertising, especially at locations where people are waiting, producing an alternative revenue stream for the operator.

Shallow column station
The shallow column station is a type of construction of subway stations, with the distinguishing feature being an abundance of supplementary supports for the underground cavity. Most designs employ metal columns or concrete and steel columns arranged in lines parallel to the long axis of the station.

Stations can be double-span with a single row of columns, triple-span with two rows of columns, or multi-span. The typical shallow column station in Russia is triple-span, assembled from concrete and steel, and is from 102 to 164 metres in length with a column spacing of 4–6 m. Along with the typical stations, there are also specially built stations. For example, one of the spans may be replaced with a monolithic vault (as in the Moskovskaya station of the Samara Metro or Sibirskaya of the Novosibirsk Metro). In some cases, one of the rows of columns may be replaced with a load-bearing wall. Such a dual hall, one-span station, Kashirskaya, was constructed to provide a convenient cross-platform transfer. Recently, stations have appeared with monolithic concrete and steel instead of assembled pieces, as Ploshchad Tukaya in Kazan.

The typical shallow column station has two vestibules at both ends of the station, most often combined with below-street crossings.

For many metro systems outside Russia, the typical column station is a two-span station with metal columns, as in New York City, Berlin, and others. In Chicago, underground stations of the Chicago 'L' are three-span stations if constructed with a centre platform.

In the Moscow Metro, approximately half of the stations are of shallow depth, built in the 1960s and 1970s, but in Saint Petersburg, because of the difficult soil conditions and dense building in the centre of the city this was impossible. The Saint Petersburg Metro has only five shallow-depth stations altogether, with three of them having the column design: Avtovo, Leninsky Prospekt, and Prospekt Veteranov. The first of these is less typical, as it is buried at a significant depth, and has only one surface vestibule.

Deep column station
A deep column station is a type of subway station consisting of a central hall with two side halls connected by ring-like passages between a row of columns. Depending on the type of station, the rings transmit load to the columns either by "wedged arches" or through Purlins, forming a "column-purlin complex".

The fundamental advantage of the column station is the significantly greater connection between the halls, compared with a pylon station.

The first deep column station in the world is Mayakovskaya, opened in 1938 in Moscow.

One variety of column station is the "column-wall station". In such stations, some of the spaces between the columns are replaced with walls. In this way, the resistance to earth pressure is improved in difficult ground environments. Examples of such stations in Moscow are Krestyanskaya Zastava and Dubrovka. In Saint Petersburg, Komendantsky Prospekt is an example.

Pylon station
The pylon station is a type of deep underground subway station. The basic distinguishing characteristic of the pylon station is the manner of division of the central hall from the station tunnels

The pylon station consists of three separate halls, separated from each other by a row of pylons with passages between them. The independence of the halls allows the architectural form of the central and side halls to be differentiated. This is especially characteristic in the non-metro Jerusalem–Yitzhak Navon railway station, constructed as a pylon station due to its 80-meter depth, where the platform halls are built to superficially resemble an outdoor train station.

Building stations of the pylon type is preferable in difficult geological situations, as such a station is better able to oppose earth pressure. However, the limited number of narrow passages limits the throughput between the halls.

The pylon station was the earliest type of deep underground station. One variation is the so-called London-style station. In such stations the central hall is reduced to the size of an anteroom, leading to the inclined walkway or elevators. In some cases the anteroom is also the base of the escalators. In the countries of the former USSR there is currently only one such station: Arsenalna in Kyiv. In Jerusalem, two planned underground heavy rail stations, Jerusalem–Central and Jerusalem–Khan, will be built this way. In Moscow, there were such stations, but they have since been rebuilt: Lubyanka and Chistiye Prudy are now ordinary pylon stations, and Paveletskaya-Radialnaya is now a column station.

In the Moscow Metro, typical pylon station are Kievskaya-Koltsevaya, Smolenskaya of the Arbatsko-Pokrovskaya line, Oktyabrskaya-Koltsevaya, and others.

In the Saint Petersburg Metro, pylon stations include Ploshchad Lenina, Pushkinskaya, Narvskaya, Gorkovskaya, Moskovskie Vorota, and others.

Single-vault station
The construction of a single-vault station consists of a single wide and high underground hall, in which there is only one vault (hence the name). The first single-vault stations were built in Leningrad in 1975: Politekhnicheskaya and Ploshchad Muzhestva. Not long after, the first two-level single-vault transfer stations were opened in Washington DC in 1976: L'Enfant Plaza, Metro Center and Gallery Place.

In the Moscow Metro there is only one deep underground single-vault station, Timiryazevskaya, in addition to several single-vault stations at shallow depth. In the Nizhny Novgorod Metro there are four such stations: Park Kultury, Leninskaya, Chkalovskaya and Kanavinskaya. In the Saint Petersburg Metro all single-vault stations are deep underground, for example Ozerki, Chornaya Rechka, Obukhovo, Chkalovskaya, and others. Most of the underground stations of the Washington, D.C.'s Metro system are single-vault designs, as are all the single-line vaulted stations in the Montreal Metro. In Prague Metro, there are two underground stations built as single-vault, Kobylisy and Petřiny. In the Bucharest Metro, Titan station is built in this method.

Cavern station

The cavern station is a metro station built directly inside a cavern. Many stations of the Stockholm Metro, especially on the Blue line, were built in man-made caverns; instead of being enclosed in a tunnel, these stations are built to expose the bedrock in which they are excavated. The Stockholm Metro also has a depot facility built in a cavern system.

In the Hong Kong MTR, examples of stations built into caverns include Tai Koo station on Hong Kong Island,  Other examples in the city include Sai Wan Ho, Sai Ying Pun, Hong Kong University and Lei Tung stations.

Records

 * The first and oldest extant underground station is Baker Street tube station which opened in 1863.
 * The largest and most complex metro station is the Paris Métro-RER station Châtelet-Les Halles in France, with 20 platforms serving eight (three RER commuter rail and five Métro) lines.
 * The deepest metro station is Chongqing Rail Transit's Hongyancun station located in China, the top of rail of the station's Line 9 tracks is 106 meters (347 feet 9 inches) deep, and the deepest point is about 116 meters (380 feet 7 inches) underground.
 * The highest elevated station is Chongqing Rail Transit's Hualongqiao station at 48 m above ground.
 * The northernmost metro station is the Helsinki Metro's Mellunmäki station in Finland, at 60.23917°N, 25.11083°W.
 * The southernmost metro station is the Buenos Aires Underground's Plaza de los Virreyes - Eva Perón metro station in Argentina, at -34.64303°N, -58.46161°W.

title: Metro station


text:
National security council

A national security council (NSC) is usually an executive branch governmental body responsible for coordinating policy on national security issues and advising chief executives on matters related to national security. An NSC is often headed by a national security advisor and staffed with senior-level officials from military, diplomatic, intelligence, law enforcement and other governmental bodies. The functions and responsibilities of an NSC at the strategic state level are different from those of the United Nations Security Council, which is more of a diplomatic forum.

Occasionally a nation will be ruled by a similarly named body, such as "the National Security Committee" or "Council for National Security". These bodies are often a result of the establishment or preservation of a military dictatorship (or some other national crisis), do not always have statutory approval, and are usually intended to have transitory or provisional powers. See also: coup d'état.

Some nations may have a similar body which is not formally part of the executive government. For example, the National Security Commission in China is an organ of the Chinese Communist Party (CCP), the sole ruling party, and headed by the CCP general secretary, rather than an organ of the executive government.

NSCs by country

 * Abkhazia: Security Council
 * Algeria: High Council of Security
 * Antigua and Barbuda: National Security Council
 * Armenia: Security Council of Armenia
 * Australia: National Security Committee
 * Austria: National Security Council
 * Azerbaijan: Security Council
 * Bangladesh: National Committee on Security Affairs
 * Belarus: Security Council
 * Belgium:
 * Brazil: National Defense Council
 * Canada: Canadian National Security Council
 * Chile: National Security Council (Chile)
 * China: National Security Commission of the Chinese Communist Party
 * Hong Kong: Committee for Safeguarding National Security of Hong Kong
 * Croatia: National Security Council
 * Cyprus: National Security Council
 * Czech Republic: National Security Council of the Czech Republic
 * East Germany: National Defense Council of East Germany
 * Egypt: National Security Council (Egypt)
 * Estonia: National Defence Council
 * Fiji: National Security Council
 * Finland: Ministerial Committee on Foreign and Security Policy
 * France: Defence and National Security Council
 * Georgia: National Security Council of Georgia
 * Germany: Bundessicherheitsrat (Federal Security Council), prior to 1969 Bundesverteidigungsrat (Federal Defense Council)
 * Greece: Government Council for National Security
 * Ghana: National Security Council
 * India: National Security Council
 * Iran, Islamic Republic of: Supreme National Security Council
 * Iraq: National Security Council
 * Kurdistan Region: Security Council
 * Ireland (Republic of): National Security Committee
 * Israel: Ministerial Committee on National Security Affairs (policy co-ordination functions); National Security Council (advisory functions)
 * Italy: High Council of Defence
 * Japan: National Security Council (previously Security Council)
 * Kazakhstan: Security Council of Kazakhstan
 * Korea, North: State Affairs Commission of North Korea (previously National Defense Commission)
 * Korea, South: National Security Council
 * Kyrgyzstan: Security Council
 * Lithuania: State Defence Council
 * Malaysia: National Security Division (policy co-ordination functions);National Security Council (advisory functions)
 * Moldova: Supreme Security Council
 * Mongolia: National Security Council of Mongolia
 * Myanmar: National Defence and Security Council
 * New Zealand: Cabinet National Security Committee
 * Nigeria: National Security Council (intelligence services); National Defense Council (Nigerian Armed Forces)
 * Oman: Palace Office (Oman)
 * Pakistan: National Security Council
 * Palestinian Authority: Palestinian National Security Council
 * Philippines: National Security Council
 * Poland: National Security Council (Poland)
 * Portugal: Superior Council of National Defense
 * Romania: Supreme Council of National Defense
 * Russian Federation: Security Council of the Russian Federation
 * Saudi Arabia: Council of Political and Security Affairs (previously National Security Council)
 * Serbia: National Security Council of the Republic of Serbia
 * Singapore: National Security Coordination Secretariat
 * Slovenia: National Security Council
 * Slovak Republic: Security Council
 * Spain: National Security Council
 * Sri Lanka: National Security Council of Sri Lanka
 * Sweden: National Security Council
 * Taiwan (Republic of China): National Security Council
 * Tajikistan: Security Council
 * Thailand: National Security Council
 * Turkey: National Security Council
 * Turkmenistan: State Security Council
 * Ukraine: National Security and Defense Council of Ukraine
 * United Kingdom: National Security Council
 * United States: United States National Security Council
 * Uzbekistan: National Security Council under the President of Uzbekistan
 * Vietnam: Council for National Defense and Security

title: National security council


text:
Talk:Battle of Ramillies

[Untitled]
As a Featured Article, is there a correct pronunciation for the title?--Eli Todd (talk) 19:25, 27 January 2008 (UTC)
 * Ram-ill-ees 88.110.142.252 (talk) 20:19, 27 January 2008 (UTC)

In the following sentence did you mean "vice" or "vise": In less than four hours, Villeroi’s army was utterly defeated. Marlborough's subtle moves and changes in emphasis during the battle – something the French and Bavarian commanders failed to realize until it was too late – caught his foe between a tactical vice? ANSK (talk) 00:47, 28 January 2008 (UTC)

Grammar
The first 3 paragraphs are a grammatical nightmare. I fixed them once, but Rebel Redcoat apparently thinks his grammar is superior and reverted all the edits. He chooses to use fantastic phrases, like "Nevertheless, despite" which is a redundant statement. Or that last sentence in the first paragraph: "Far from standing on the defensive, therefore, French armies in 1706 would everywhere go over to the offensive.", which makes no sense whatsoever. —Preceding unsigned comment added by Mojodaddy (talk • contribs) 18:49, 28 January 2008 (UTC)
 * I find it funny that you criticise my writing when your attempts to improve the lead are laughable and incompetent. If you want to be rude fella, bring it on. Rebel Redcoat (talk) 17:26, 16 February 2008 (UTC)

I'd agree with the previous comment. the grammer needs fixing PatrickC22 (talk)

query re "Count Schulemberg"
Would "Count Schulemberg" be the son of Jean de Schulemberg, marshal of France 1658, [died 1671], or a Prussian Schulemberg, noted at the siege of Béthune, 15 July 1710 (noted (as "Saxon") in this contemporary remark)? Or possibly an error for the duke of Schomberg, who had left France after the Revocation of the Edict of Nantes and fought for William? Perhaps both Schomberg and Schulemberg had abandoned France after 1685. --Wetman (talk) 02:32, 28 January 2008 (UTC)
 * The Count we are looking for is Matthias-Johan (or Johann) Count of Schulemberg (or Schulenberg). (1661-1747). Also seen it spelled Schulemburg. Imperial commander; also fought under Eugene at Malplaquet. Rebel Redcoat (talk) 15:48, 28 January 2008 (UTC)
 * Bethune? Yes Wetman, just found this in Chandler - ... "on 10 July they [Marlb & Eugene] advanced boldly towards Vimy . . . on 16 July [they] detached Generals Fagel and Schulenburg ... to besiege Bethune." Rebel Redcoat (talk) 16:20, 28 January 2008 (UTC)

?
I think the author should include a link to backround info on the actual war so people like me can not be clueless. Please message me with a link to the article presuming that it is an actual article.Historybuffc13 (talk) 05:04, 28 January 2008 (UTC)
 * War_of_the_Spanish_Succession Raul654 (talk) 05:16, 28 January 2008 (UTC)

Thanks for the link! Please put the link on my page next time or I won't see it!Historybuffc13 (talk) 05:22, 28 January 2008 (UTC)

Pronunciation of Ramillies
In the Irish song "Clare's Dragoons" it's pronounced, perhaps altered from to fit the meter. Is Ramillies, Belgium pronounced, ,  ,  , something else ...? jnestorius(talk) 14:12, 28 January 2008 (UTC)

 * Apparently (Google books Elements of Geography, Modern and Ancient with a Modern and an Ancient Atlas (1844), Joseph Emerson Worcester, p.127) it was formerly pronounced in Britain with the intuitively anglicized : is it still? jnestorius(talk) 14:47, 28 January 2008 (UTC)

Is that a mistake ?
The catholic Roman empire is an allied with the protistant England ? Aren't they in a long term conflict in the Midevel period ?  A M M A R   17:09, 28 January 2008 (UTC)

No, at this time the (Catholic) Holy Roman Empire (headed by Austria) was allied with (Protestant) England, Holland and Prussia, plus a number of German and Italian states (of both persuasions), in opposition to (Catholic) France and Spain plus Bavaria. The war was about King Louis XIV's ambitions to place his grandson on the throne of the Spanish Empire, thus giving him control of an autocratic 'super-state' which would dominate Europe. Guthrum (talk) 21:35, 28 January 2008 (UTC)

 * The Catholic-Protestant conflict refers to the Reformation, which dates to the 15 and 1600s. It had mostly blown over by 1706 (though I'm sure there was some discomfort). Brutannica (talk) 01:12, 29 January 2008 (UTC)

Seriously? You do know the saviour of protestantism (in this context rule and control of the states' churches by the state) were the Catholic cardinals Richilieu and Mazarin who brought France against Spain and won the 30 Years War for the protestant princes. — Preceding unsigned comment added by 108.20.195.39 (talk) 04:11, 22 November 2020 (UTC)

Problem with link
Link HMS Ramillies don't led to specific ship but to disambiguation. Can someone fix this?--Vojvodaen (talk) 17:15, 28 January 2008 (UTC)
 * Read the sentence - The battle was used as the name of several British ships; HMS Ramillies -- it's supposed to go to a disambig page listing the several ships that have born that name. Raul654 (talk) 17:52, 28 January 2008 (UTC)

 * Оk.--Vojvodaen (talk) 10:36, 29 January 2008 (UTC)

Spain in the battle
The infobox lists Bourbon Spain as having been in the battle. However, as far as I can see, the article text does not mention Spain at all. 92.99.114.108 (talk) 10:40, 14 November 2012 (UTC)

Denmark as participant?
I'm not so sure about that, anyone got source of this? Before the Great Northern War Denmark hired a lot of mercenaries from Germany in order to defeat Sweden. However after the disappointing peace in 1700 Denmark had a lot of mercenaries left but nowhere to use them, so they hired them out to other continental armies. In other words, Denmark was most likely not a participant in this war and the Danish army in this battle was surely not independent. I don't know why Denmark is included to start with. Imonoz (talk) 14:18, 4 June 2014 (UTC)

title: Talk:Battle of Ramillies


text:
American avocet

The American avocet (Recurvirostra americana) is a large wader in the avocet and stilt family, Recurvirostridae, found in North America. It spends much of its time foraging in shallow water or on mud flats, often sweeping its bill from side to side in water as it seeks crustacean and insect prey.

Taxonomy
The American avocet was formally described in 1789 by the German naturalist Johann Friedrich Gmelin in his revised and expanded edition of Carl Linnaeus's Systema Naturae. He placed it with the pied avocet in the genus Recurvirostra and coined the binomial name Recurvirostra americana. Gmelin based his description on that by the English ornithologist John Latham who in 1785 had described and illustrated the American avocet in his A General Synopsis of Birds. Latham cited the earlier publication by William Dampier and also that by Thomas Pennant. The genus name combines the Latin recurvus meaning 'bent' or 'curved backwards' with rostrum meaning 'bill'. The species is monotypic: no subspecies are recognised.

The American avocet is a member of the order Charadriiformes, which includes shorebirds, gulls, and alcids. Its family, Recurvirostridae, includes stilts and avocets. The genus Recurvirostra includes three other species: the Andean avocet, the pied avocet, and the red-necked avocet.

Description
The American avocet measures 40 - 51 cm in length, has a wingspan of 68 - 76 cm and weighs 275 - 420 g The bill is black, pointed, and curved slightly upwards towards the tip. It is long, surpassing twice the length of the avocet's small, rounded head. Like many waders, the avocet has long, slender legs and slightly webbed feet. The legs are a pastel grey-blue, giving it its colloquial name, blue shanks. The plumage is black and white on the back, with white on the underbelly. During the breeding season, the plumage is brassy orange on the head and neck, continuing somewhat down to the breast. After the breeding season, these bright feathers are swapped out for white and grey ones. The avocet preens its feathers, commonly considered to be a comfort movement.

The call has been described as both a shrill and melodic alarm bweet, which rises in inflection over time. Avocets use three distinct calls: common call, excited call, and broken wing call. The common call is a loud repeated wheep. The excited call has a similar wheep sound, but it speeds up rather than having an even rhythm. Lastly, the broken wing call is noticeably different from the other two calls. It is a distressed screech sound and is alarming rather than melodic.

Distribution and habitat
American avocets were previously found across most of the United States until extirpated from the East Coast. The breeding habitat consists of marshes, beaches, prairie ponds, and shallow lakes in the mid-west, as far north as southern Canada. These breeding grounds are largely in areas just east of the rocky mountains including parts of Alberta, Saskatchewan, Montana, Idaho, Washington, Utah, North and South Dakota, Nebraska, Colorado, and even down to parts of New Mexico, Oklahoma and Texas. Their migration route lands them in almost every state in the western United States. The avocet's wintering grounds are mainly coastal. Along the Atlantic Ocean, they are found in North and South Carolina, Georgia and Florida. There are also wintering grounds along the Gulf of Mexico in Florida, Texas, and Mexico, and along the Pacific Ocean in California and Mexico. There are resident populations in the Mexican States of Zacatecas, San Luis Potosí, Guanajuato, Hidalgo, Mexico City and Puebla, and in Central California.

Flocks ranging from 50 to 300 avocets migrate together to their breeding sites together during peak season, which is typically in the spring.

Breeding
The American avocet breeds in anything from freshwater to hypersaline wetlands in the western and mid-west United States. After reaching their breeding grounds, the avocets quickly establish territory in pairs. Throughout the breeding season, these pairs engage in a series of copulatory displays, which is initiated by either sex and often involves preening. After breeding, the birds gather in large flocks, sometimes amounting to hundreds of birds. Nesting takes place near water, often alongside black-necked stilts, usually on small islands or mucky shorelines. The avocets nest on the ground in places with little to no surrounding vegetation. They make shallow indentations in the ground for their nest and line it with grass, feathers, pebbles or other small objects. Some avocets do not line their nests at all. During the breeding season, avocets lay around 3–5 eggs, with 4 being the most common number. One case study of the 1968 and 1969 breeding seasons suggests that this number allows for the highest rate of hatching success. They incubate them over a period of 23–25 days. Both parents are involved in the incubation process, taking turns during the day (except at night, where only the female incubates the eggs). Both parents take care of the newborn American avocets after they hatch. Male parents are more present in incubation during the first 8 days, whereas female parents dominate incubation throughout the remaining 16 days. Time spent caring for newborn chicks equates to that spent incubating. Since newborns are able to walk, they are capable of feeding themselves and do not require extensive care from their parents. The young are expected to begin flying 4 to 5 weeks after hatching.

Anti-predator behavior
Each pair defends their own territory against predators and other avocets. Nests are built in areas that predators would struggle to gain access to. Predators that do try to approach the colony find themselves surrounded by several adult avocets running around, trying to distract them. While incubating however, the avocets tend only to their nest and defense is considered a lower priority, unless the eggs are in immediate danger, in which case the avocets may take more aggressive measures, calling loudly and flying at the threat.

Food and feeding
The American avocet tends to prefer habitats with fine sediments for foraging. In the winter, it feeds extensively on brine shrimp. Usually, this entails the avocet pecking while walking or wading on the shore, but it can also swim to expand foraging area. During the breeding season, avocets continue to eat brine shrimp but switch mainly to consuming brine flies. Brine flies and brine fly larvae are so abundant in avocet breeding grounds that they often blacken the surface of any exposed mud. Brine flies sustain the avocet during its breeding season. The avocet employs both visual and tactile methods of feeding. The primary visual feeding method is pecking at flies. However the avocet will also feed by plunging (wherein the head and neck of the bird are submerged), snatching (wherein the bird catches a flying insect), and bill pursuit (wherein the bird repeatedly opens and closes its bill while moving its head along the surface of the water). Methods of tactile feeding include filtering, scraping, and scythe feeding.

Threats and conservation status
Shooting and trapping of American avocets led to population decline until the 1900s. During this time, the species was extirpated from most of the East Coast of the United States. By 1918, Avocets became protected under the Migratory Bird Treaty Act (16 U.S.C. 703-712). Since that time, other threats have emerged. Contaminants and toxins such as DDT, selenium, and methylmercury, have increased in American Avocet habitats in recent years. Although the effects are highly contested, these increased concentrations have potential negative effects on American avocet breeding success, especially because Avocet chicks are more susceptible to environmental disruptions than adults. The avocet also faces habitat loss. While agricultural and industrial environments have become alternate habitat for the avocet, natural wetlands are decreasing rapidly. Additionally, increases in the salinity levels of wetlands have had adverse effects on the breeding and development of avocets. Young avocets in saline or hypersaline environments often combat weight loss and dehydration. Although American avocets face numerous environmental challenges, numbers have stayed steady, and according to the National Audubon Society, numbers have been increasing steadily in the eastern United States for the past few years. The American avocet is protected under the Migratory Bird Treaty Act of 1918.

In culture
In his famous The Birds of America, John James Audubon describes a day of stalking and spying on the avocet. He judiciously noted their foraging, nesting, defensive, and flight behaviors.

title: American avocet