It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. Right now what I'm doing is fitting the copula to the data and generate simulated observations from it, if the simulated data has proprieties which are similar to the observed data then I use the model, otherwise I don't. In general, it is indeed true that performing these gof tests can be time-consuming.
Especially if one chooses to also compare fitness of the rotated copulas, the number of candidate models can quickly go beyond types. One suggestion I can make is computing some non-linear dependence statistics such as kendall's tau etc.
Using this information you can narrow down your search to only a subset of all available families. Another way you can test is to fit all your candidate models, simulate from them and then perform a two-sample test between original and sampled data in the oringal domain.
You susequently choose the model with the best results. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. This suggests that copulas may be useful to model the dependence between random variables or between risks that affect a particular activity. Before this idea the reader may reply that the correlation coefficient is a measure of statistical dependence and that the role of copulas is performed by the correlation coefficient.
But this is true only in some cases. The correlation coefficient only measures dependence when dealing with a special class of distributions, the so-called elliptical distributions, such as normal or t multivariate distributions.
Let us imagine a bivariate normal density function with a correlation coefficient of 0. If we make an horizontal cut to the surface that represents this 3-dimensional function, we will obtain an ellipse. The point is that there are other multivariate distributions bivariate, to follow the example where the figure obtained by this procedure in no way resembles an ellipse. In these, the correlation coefficient cannot be utilized as a measure of dependence. The correlation coefficient may lead to erroneous conclusions when applied in situations where it is not suitable as a dependence measure.
Each of the following graphs shows random simulations of the losses from two risks, X and Y i. Risk X and risk Y have been simulated from the same distribution function and therefore have the same mean, variance, etc.
However, both graphs A and B, differ in something very. Is this difference due to the fact that the correlation coefficient is greater in B than in A? In both cases the coefficient is 0. The greater severity of the risks X, Y in graph B stems from the different way in which these risks depend on each other, severity that is not reflected by the correlation coefficient.
This example leads us to the conclusion that any bivariate and, in general, multivariate distribution function involves two aspects defining the statistical behavior of these random variables:. So that means we need to generate uniformly distributed data with the correlations we want.
How do we do that? We simulate from a multivariate Gaussian with the specific correlation structure, transform so that the marginals are uniform, and then transform the uniform marginals to whatever we like. So there we go, by using the uniform distribution as our lingua franca we can easily induce correlations and flexibly construct complex probability distributions. This all directly extends to higher dimensional distributions as well.
Above we used a multivariate normal which gave rise to the Gaussian copula. However, we can use other, more complex copulas as well.
For example, we might want to assume the correlation is non-symmetric which is useful in quant finance where correlations become very strong during market crashes and returns are very negative. In fact, Gaussian copulas are said to have played a key role in the Financial Crisis as tail-correlations were severely underestimated. If you've seen The Big Short , the default rates of individual mortgages among other things inside CDOs see this scene from the movie as a refresher are correlated -- if one mortgage fails, the likelihood of another failing is increased.
In the early s, the banks only knew how to model the marginals of the default rates. This infamous paper by Li then suggested to use copulas to model the correlations between those marginals.
Rating agencies relied on this model heavily, severly underestimating risk and giving false ratings. The rest, as they say, is history. Read this paper for an excellent description of Gaussian copulas and the Financial Crisis which argues that different copula choices would not have made a difference but instead the assumed correlation was way too low. It really is just a function with that property of uniform marginals.
It's really only useful though combined with another transform to get the marginals we want. We can also better understand the mathematical description of the Gaussian copula taken from Wikipedia :.
Just note that in the code above we went the opposite way to create samples from that distribution. The Gaussian copula as expressed here takes uniform 0, 1 inputs, transforms them to be Gaussian, then applies the correlation and transforms them back to uniform.
Finally, if you enjoyed this blog post, consider supporting me on Patreon which allows me to devote more time to writing new blog posts. This post is intentionally light on math. You can find that elsewhere and will hopefully be less confused as you have a strong mental model to integrate things into. I found these links helpful:. We also haven't addressed how we would actually fit a copula model.
I leave that, as well as the PyMC3 implementation, as an exercise to the motivated reader ;.
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