Talk:Copula (probability theory)/Archive 1

The lead is probably confusing in practice, though technically correct. The main reason to use a copula is to produce a joint distribution with specified marginals. The lead gives almost the opposite impression.

Gaussian Copula

I rewrote a couple of parts of the Gaussian copula section. As it was, it was rather a mess (see this previous version | here):

1. It used implicitly a capital letter notation for random variables, but used U and V as random variables directly in the copula function, as identifiers for its variables: So the copula function became itself a random variable.
2. Related to the above, there was confusion added about how U and V was actually defined as transforms of other random variables X and Y. It tries to mix the idea behind the usefulness of the copula explained in the "The basic idea" section but completely gets it wrong. You don't define a function by defining its arguments from other arguments (especially not random variables too if the function is non random), you just define their domain. (Don't confuse usage with definition; but even then the tentative usage it tried to give was wrong). A copula is just that, a distribution function (or density) with unit cube domain. Now the reason why this might be useful is explained in "The basic idea". With regard to usage, note that it is useful not because you would want to input random variables to (thus) output another random variable from the copula (!), but because of the fact that it can express any bivariate distribution (with marginals that aren't necessarily uniform) function inside this canonical bivariate (the copula) which has as marginals uniform distributions.
3. It sometimes employed "density" when clearly identifying a cumulative distribution function.
4. Differentiating C (the copula cumulative distribution) gives c (non cap), the copula density. Thus $\Phi$ change to $\phi$ where necessary too (from normal cdf to normal pdf).
5. I kept $X$ and $Y$ for $\phi _{X,Y,\rho }(x,y)$ in $c_{\rho }(u,v)={\frac {\phi _{X,Y,\rho }(\Phi ^{-1}(u),\Phi ^{-1}(v))}{\phi (\Phi ^{-1}(u))\phi (\Phi ^{-1}(v))}}$ since it is part of the identifier of the function, not indicating variable identifiers used in it. Indeed, the copula function is built by composition of this function with $\Phi ^{-1}$. In fact it would be misleading to replace those by U and V (see point 2 above).

70.81.15.136 (talk) 09:53, 8 March 2009 (UTC)

Wrong Figure ?

The fig "Gaussian_Copula_PDF.png" is maybe wrong: Since C(u,v): [0,1]×[0,1]→[0,1] the surface cannot rise over 1. However, as you see on the figure, the surface goes almost up to 3 ...

Panhypersebastos 08:13, 1 June 2007 (UTC)Panhypersebastos

• It is okay. The depicted graph is a density of a copula (as mentioned below the picture), not a copula itself. --PeterSarkoci 10:12, 31 July 2007 (UTC)
• I changed it, so that we have a comparison between cumulative and density. I hope it is more clear since the previous one caused misunderstandings. --Matteo Zandi

I like the Figures, but is there any chance of adding some alternative views for the existing cases, so as to make things easier to interpret? Melcombe (talk) 13:31, 19 March 2008 (UTC)

• Current figures are right and they show examples of copulas, a more interpretable picture would be a comparison between the density of two multivariate distributions with different copulas but same marginals --Matteo Zandi 17:35, 12 September 2008 (UTC)
• here you go:

1) 2)

This article needs a complete rewrite :-(

I came to this article from the Correlation article, which states:

The information given by a correlation coefficient is not enough to define the dependence structure between random variables; to fully capture it we must consider a copula between them.

Unfortunately the copula article doesn't help at all with this. The first paragraph is almost unintelligible, as is becoming more and more frequent in Wikipedia. This is supposed to be an encyclopedia -- the vast majority of people looking something up are not specialists in the subject, and the articles should be aimed at them. Starting with "a multivariate joint distribution defined on the n-dimensional unit cube" followed by a bunch of mathematics effectively says "Go away; this page is not for the likes of you!".

"The articles should be aimed at them". Says who !? It's almost impossible to keep a math article useful while at the same time just keeping a "vague layman" style. Moreover I've been helped countless times by math articles on which I'm very far from being a specialist, but I was willing to track down what I needed to get to a point where I could grasp what it was talking about. Very often I could grasp the big picture and gain a general mathematical culture out of this, kind of a bird's eye view of the math scenery. Other times, it served as a useful reference. In fact, sometimes it is precisely the most popular and basic math articles that often have the sloppiest consistency precisely because of the affluence of editors eager to put their grain of salt without really understanding the subject, adding confusion to the article instead of clarity (Incidently, I just made an edit to Gaussian Copula which fixed something like that. It probably has to do with the fact that it became popular because of finance.)

Otherwise, critics to consistency (in style, in notation, in structure and flow, etc) and "integration" of an article are wholly valid, but not critics about being formal in it. I'm not against introductions and intuition, on the contrary, but only if it leads to the formal stuff and not replace it. For the rest of my rant, see my reply below to Odoketa's comment. If you have math envy, there's no shortcut to fix it. 70.81.15.136 (talk) 09:06, 8 March 2009 (UTC)

The assertion on this talk page that a copula is useful for generating a joint distribution from a specified set of marginals is intriguing. (In fact it's pretty well what I want to do.) Please can someone who knows how to do it show us how it's done, preferably with a simple numerical example we can work through. The simpler the better please; we're not all left-brained mathematical geniuses (unlike the folk who seem to generate such articles). Nevertheless, reasonably intelligent people can often understand things if they're explained in simple terms. We may be ignorant, but we're not stupid.

Put the main ideas first, and leave the gory details until near the end of the article.

I'll now have to resort to Google to find out what a copula is.

--84.9.73.5 (talk) 11:12, 23 December 2007 (UTC)

Update: This article gives a much gentler introduction.

--84.9.73.5 (talk) 14:56, 23 December 2007 (UTC)

I would agree that there is not enough discussion in this article to allow a person very familiar with math (but not necessarily statistics), as I am, to figure out what a copula is or what people do with them. What are they good for? Why would I use it? Where does it fit into the bigger picture? These questions are unanswered but needed to get any idea for the subject. The PDF above is a little better. --84.12.134.165 (talk) 01:22, 3 August 2008 (UTC)

Agreed - an introduction that talks to people who perhaps understand what a function is (an eighth grader?) and says why you should care might be more useful. Odoketa (talk) 00:48, 26 February 2009 (UTC)

Eighth grader ? So, should we write... mmm maybe, Representable_functor for 8th graders, Sheaf_cohomology for 8th graders, Skorokhod integral for 8th graders. What is it that people don't understand in the fact that mathematics is a ladder of abstraction where there is no high road to true understanding. Some concepts are inherently abstract and relate to other abstractions, dumbing it down just destroys the information content completely. I know you are talking about adding a proper introduction but as of the date of your comment, there is actually one (but I agree that the article could be improved). There is a difference between dumbing down an article and adding proper context so that it doesn't seem like a disparate collection of cut pieces plastered together on a single page.

The debate about bringing math articles (or any abstract or specialized subjects) accessible to everyone has been beaten to death here (see the village pump, or here) but has a long history so let me paraphrase a resume: There is no royal road to mathematics. 70.81.15.136 (talk) 09:06, 8 March 2009 (UTC)

Notations for Cartesian products

This appears in the article:

$B=\times _{i=1}^{n}[x_{i},y_{i}]\subseteq [0,1]^{n};$

I'm wondering if there are particular views on advantages and disadvantages of this particular notation for Cartesian products, as contrasted with this:

$B=\prod _{i=1}^{n}[x_{i},y_{i}]\subseteq [0,1]^{n};$

Michael Hardy (talk) 18:15, 9 May 2008 (UTC)

Also, the formula defining Vc(B) doesn't really make sense. The right hand side has nothing to do with B (from what it said) 131.215.220.185 (talk) 22:41, 3 March 2009 (UTC)

Impact of the Gaussian copula on the credit crunch

For 83.217.120.162: you have a point, but you wrote it down a little too harshly in my opinion for the enciclopedic spirit. Talking about the biggest financial loss caused by a mathematical model needs some backing up in terms of evidence, which is not yet fully clear. Also, in RMBS and other asset classes related to the crisis often no copula was involved. So I took the liberty of softening your statement and adding some references. Piloter (talk) 19:55, 7 January 2009 (UTC)

a wired.com article on this issue --Wongba (talk) 21:43, 24 February 2009 (UTC)

woops, it was added already. nevermind. --Wongba (talk) 21:58, 24 February 2009 (UTC)

How can applying a mathematical result cause the credit crisis? If the banks had used an alternative but equivalent modeling strategy, would the crisis not have happened? The banks also used addition and subtraction. Is addition to blame for the credit crisis? Also, the Wired article is nice entertainment but really useless if you want to understand what happened. I would remove the whole accusation at the end of the "Applications" paragraph. Cheers, Matt —Preceding unsigned comment added by 84.75.20.154 (talk) 20:11, 27 February 2009 (UTC)

Using an article from Wired on a mathematics webpage is quite odd. Particularly the one that was linked to; clearly the author is way out of his mathematical knowledge depth. While there are a lot of correct facts in the article ( David Li was the first one explicitly using copulas in finance, they were used for CDOs and MBS), saying a formula caused trouble is borderline ridiculous. Also, every time you do *anything* where a joint probability is used you are implicitly or explicitly using some copula, so it seems quite bizarre copulas (the mathematical objects) are to blame for anything. —Preceding unsigned comment added by 12.96.170.39 (talk) 22:27, 11 May 2009 (UTC)

Blaming a mathematical model for making wrong choices is like blaming hammers and screwdrivers for building bad houses. This is an entry about a mathematical concept, not a discussion on terrible architects --90.185.76.189 (talk) 20:04, 19 May 2009 (UTC)

Then the entire applications section should be removed. The fact is, copulas were used to model the risk inherent in derivative debt. Whether a misapplication or not, it didn't work well and tended to hide the true risk. I don't see that this is blaming the model. It's blaming the application of the model (hence its presence in the applications section). Perhaps modifying the language to clarify that the application to derivatives tended to artificially diminish the risk inherent, or that the misapplication to derivatives... something along those lines. It seems to me that if you can't point out that use of this model in regard to derivatives was a factor in the financial crisis, then you can't point out that use of this model is beneficial in bridge analysis. That's a civil engineering discussion, not a mathematical discussion. Gramby (talk) 02:15, 15 April 2010 (UTC)

I also added an introductory part on the idea behind copulas

I think it may be helpful to people with a basic mathematics background who want to know quickly what's up with copulas Piloter (talk) 20:20, 7 January 2009 (UTC)

Hmmm that "basic idea" section as written is not helpful for me. Angry bee (talk) 18:10, 6 March 2010 (UTC)

Why? How?

I read the article and don't understand why/how copulas are useful. I can represent the covariance of a multivariate Gaussian distribution with a covariance matrix; this is very natural. What do I gain by transforming a random variable to a multivariate random uniform and then using a copula? It seems like added complexity? What do I gain by this complexity? —Ben FrantzDale (talk) 14:02, 15 April 2009 (UTC)

They are useful for non-Gaussian distributions, and most distributions are non-Gaussian. Melcombe (talk) 15:50, 15 April 2009 (UTC)

OK. I am curious to understand how. Also, if they are most useful for non-Gaussians, why are Gaussian copulas used in finance? (Or was that the problem -- assuming Gaussian when there's really a fat tail ala The Black Swan? —Ben FrantzDale (talk) 19:49, 15 April 2009 (UTC)

Use of a Gaussian copula doesn't mean the marginal distributions are Gaussian. But I don't know the details of how these are used in finance. Michael Hardy (talk) 21:14, 19 May 2009 (UTC)

The application of Gaussian copulas is directly equivalent to the older idea of treating multivariate non-normal data by transforming each variable separately to have marginal normal distibutions and then doing the analysis as if this resulted in a multivariate normal distribution, with the final result obtained by reversing the intial transformations. The fact that "copulas" are defined in terms of transforming to a uniform rather than a normal distribution is pretty much an irrelevance. While there may be problems in fitting the marginal distributions, any problems arising from using a Gaussian copula must really arise from its adequacy, or not, in representing the joint dependence of the transformed variables. Thus the Gaussian copula model (or any other copula model) might impose a model in which too few (or too many) simultaneous occurences of large values in all variables are predicted. The older use of transforming to normal and then assuming joint normality was quite often based on convenience without too much regard to assessing the joint normality of the results, partly because there were few other procedures that could be implemented in practice. So the reason "why are Gaussian copulas used in finance?" might originate partly in the simplicity of implementing the computations, but also partly because the Gaussian copula is one where more than two variables can be considered in consistent ways (so that dropping a variable from consideration results in a model that can be deduced from the model for the full set of variables). Melcombe (talk) 16:33, 20 May 2009 (UTC)

This article is no help unless you already understand what the article supposedly explains

As a reasonably numerate and literate person I found this article to be worse than useless, it was just confusing. It is of no help to a layman seeking explanation and therefore doesn't deserve a place here. Surely it was bullshit like this that caused the problem in the first place

Cannonmc (talk) 01:01, 24 February 2011 (UTC)

I tend to agree. As you can see in my above questions, I think I understand what they do (i.e., transform unusual PDFs into well-understood ones) but I still don't understand why this is useful and how it doesn't completely mess up joint probabilities. (That is, if we are stretching a multiviariate PDF to give each axis a given distribution, it seems like that could do all sorts of things to the joint PDF... or perhaps that's the problem.) —Ben FrantzDale (talk) 13:32, 24 February 2011 (UTC)

"(That is, if we are stretching a multiviariate PDF to give each axis a given distribution, it seems like that could do all sorts of things to the joint PDF... or perhaps that's the problem.)" What? My native language is English, that isn't English Cannonmc (talk) 01:30, 25 February 2011 (UTC)

Sorry. (My native language is also English.) Let me try to rephrase. I'm picturing a 2D PDF as a 2D surface plot/histogram. My understanding of a copula is that we do a nonlinear change of variables of the axis directions to make the x and y directions both follow, e.g., a uniform distribution. That is, the projection onto x and y should be uniform. I'm having trouble coming up with an example. —Ben FrantzDale (talk) 15:08, 25 February 2011 (UTC)