Questions tagged [copula]
A copula is a multivariate distribution with uniform marginal distributions. Copulas are mostly used to represent or to model the structure of dependence between random variables, separately from the marginal distributions.
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Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian?
Somebody asked me this question in a job interview and I replied that their joint distribution is always Gaussian. I thought that I can always write a bivariate Gaussian with their means and variance ...
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How to construct a multivariate Beta distribution?
What is a multidimensional generalization of the Beta distribution, in compliance with the following specification?
I am not looking for the Dirichlet distribution.
I am looking for a generalization ...
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Introductory reading on Copulas
For some time now, I have been looking for a good introductory reading on Copulas for my seminar. I am finding lots of material that talk about theoretical aspects, which is good, but before I move ...
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What are some techniques for sampling two correlated random variables?
What are some techniques for sampling two correlated random variables:
if their probability
distributions are parameterized
(e.g., log-normal)
if they have non-parametric
distributions.
The data are ...
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Difference between multivariate standard normal distribution and Gaussian copula
I wonder what the difference between multivariate standard normal distribution and Gaussian copula is since when I look at the density function they seem the same to me.
My issue is why the Gaussian ...
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Attainable correlations for lognormal random variables
Consider the lognormal random variables $X_1$ and $X_2$ with $\log(X_1)\sim \mathcal{N}(0,1)$, and $\log(X_2)\sim \mathcal{N}(0,\sigma^2)$.
I'm trying to calculate $\rho_{\max}$ and $\rho_{\min}$ for ...
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How to simulate from a Gaussian copula?
Suppose that I have two univariate marginal distributions, say $F$ and $G$, which I can simulate from. Now, construct their joint distribution using a Gaussian copula, denoted $C(F,G;\Sigma)$. All the ...
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Correlated Bernoulli trials, multivariate Bernoulli distribution?
I'm simplifying a research question that I have at work. Imagine that I have 5 coins and let's call heads a success. These are VERY biased coins with probability of success p=0.1. Now, if the coins ...
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Why don't we see Copula Models as much as Regression Models?
Is there any reason that don't see Copula Models as much as we see Regression Models (e.g. https://en.wikipedia.org/wiki/Vine_copula, https://en.wikipedia.org/wiki/Copula_(probability_theory)) ?
I ...
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Method for generating correlated non-normal data
I'm interested in finding out a method for generating correlated, non-normal data. So ideally some sort of distribution that takes in a covariance (or correlation) matrix as a parameter and generates ...
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Upper bounds for the copula density?
The Fréchet–Hoeffding upper bound applies to the copula distribution function and it is given by
$$C(u_1,...,u_d)\leq \min\{u_1,..,u_d\}.$$
Is there a similar (in the sense that it depends on the ...
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Spearman $\rho$ as a function of Pearson $r$
It is common to talk about the linear correlation, Pearson's $r$, between two random variables $\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$ as having two components: a) the copula and b) the marginal ...
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Is there a multivariate version of the Weibull distribution?
I hope this one is self-explanatory, but let me know if something is unclear: Is there a multivariate version of the Weibull distribution?
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Quantifying dependence of Cauchy random variables
Given two Cauchy random variables $\theta_1 \sim \mathrm{Cauchy}(x_0^{(1)}, \gamma^{(1)})$ and $\theta_2 \sim \mathrm{Cauchy}(x_0^{(2)}, \gamma^{(2)})$. That are not independent. The dependence ...
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Inverse CDF sampling for a mixed distribution
The out-of-context short version
Let $y$ be a random variable with CDF
$$
F(\cdot) \equiv \cases{\theta & y = 0 \\ \theta + (1-\theta) \times \text{CDF}_{\text{log-normal}}(\cdot; \mu, \sigma) &...
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How can I generate random numbers from any given copula?
Suppose that I have a 2-dim copula function C(x_1,x_2).
How can I generate bivariate numbers from this copula?
For specific types of copulas, I can use 'rCopula' function of 'copula' package in R.
...
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What is the equivalent for cdfs of MCMC for pdfs?
In conjunction with a Cross Validated question on simulating from a specific copula, that is, a multivariate cdf $C(u_1,\ldots,u_k)$ defined on $[0,1]^k$, I started wondering about the larger picture, ...
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What is an adaptive copula?
My basic question is: What is an adaptive copula?
I have slides from a presentation (unfortunately, I cannot ask the author of the slides) about adaptive copulae and I am not getting, what this means ...
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Copulas for generating uniform random variables with correlations
I want to generate uniform random variables which have a correlation structure defined by a graph i.e. a variable is only correlated with its neighbors in the graph and is uncorrelated with the rest ...
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Why is Gaussian Copula's Tail Dependence Zero?
I know that the Gaussian copula has a zero tail dependence (tail independence) due to the exponential behaviour at the tail. I am just wondering if there is a rigorous proof for this?
For simplicity ...
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Spearman's correlation as a parameter
Spearman's rank correlation for a bivariate sample $\{ (x_1, y_1), (x_2, y_2) , \ldots , (x_n, y_n) \}$ is generally defined as the correlation between the ranks of the observations, but what is the ...
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Copulas with Regression
Copulas are joint distribution of uniform marginal distributions. Traditionally I have seen examples of fitting a Copula to the data and then simulating from the data.
I haven't seen much on Copula ...
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When modeling a copula, you need to generate "pseudo observations"? Why? What is a pseudo observation? [closed]
I'm struggling with the concept of a "pseudo-observation." I can't find any material out there describing what it is in a simple, concise manner. Does it have something to do with ...
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Data transformation using copulas
I've heard about the use of copulas to transform data. For instance, supposedly it's applied to data that is non-normal to make it look more normal. However, I don't quite understand how this is done. ...
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Is there a parametric joint distribution such that $X$ and $Y$ are both uniform and $\mathbb{E}[Y \;|\; X]$ is linear?
Is there a parametric joint distribution such that $X$ and $Y$ are both uniform on $[0, 1]$ (i.e. a copula) and $\mathbb{E}[Y | X = x]$ is linear (by which I mean affine) in $x$? That is,
$$\mathbb{E}...
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Limits on conditional expectation with normal margins and specified (Pearson) correlation
I saw the following question on another forum:
"Suppose that both height and weight of adult men can be described with normal models, and that the correlation between these variables is 0.65. If a ...
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If I have a vector of $N$ correlated probabilities. How can I turn them into binary $0,1$ without destroying the correlation?
My ultimate goal is to be able to have a way to generate a vector of size $N$ of correlated Bernoulli random variables. One way I am doing this is to use the Gaussian Coupla approach. However, the ...
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Why is this representing the left tail?
In this source about the Clayton copula on page 18 they write:
It has been used to study correlated risks because it exhibits
strong left tail dependence and relatively weak right tail dependence....
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Why do copulas need the i.i.d assumption for marginal distribution?
Does anyone know if are there some assumptions for Copula method? I heard from someone that the data should be i.i.d (independent and identically distributed). Let's say, if I want to capture the ...
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Sklar’s Extension Theorem and support restrictions
This question is about an application of the Sklar's Extension Theorem, whose proof can be found in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look ...
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How can I generate random observations from a concrete copula?
Let us assume that we have two continuous random variables $X$, $Y$, with known distributions (not necessarily normal), connected/related via a concrete copula.
What is a procedure to generate random ...
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Expressing a marginal probability using copulas
Please correct me if I am wrong and kindly provide me with the correct notations. I have two questions:
We know that for the variables $(X,Y,Z)\in \mathbb{R}^3$, the marginal joint density $f(x,y)$ ...
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Sampling from conditional copula
I am having trouble finding anything on sampling from conditional copulas. I am only
interested in the bivariate case. So, if $C(u,v)$ is my copula, I want to sample from
it given a specific ...
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What is copula transformation
I have seen that copula transformation changes my sample space to the range of $[0 \; 1]^d$ where d is the number of dimensions. Can anyone explain me about copula transformation?
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Struggling with copula theory
I'm really struggling with bivariate copula's. Long story short, I can only use Gaussian copulas. I'm therefore interested in the joint PDF for which the Gaussian copula can be applied.
So for ...
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Is there a bivariate $\beta$ distribution I can fit to my data?
I am analyzing two dimensional data. After analyzing each dimension with the help of the fitdistrplus and logspline packages, they both fit the Beta distribution. Is it possible to analyze the two ...
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what does positive Gaussian copula dependency describe
Suppose I have two variables and that their dependency structure is Gaussian copula.
Suppose that the parameter of the Gaussian copula is 0.8 and the corresponding Kendall's tau is 0.7. How can I ...
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The importance of the Gaussian copula
Suppose we have a bunch of marginal distributions
$$F_1,...,F_n$$
which have some unknown joint distribution.
My understanding of the importance of copulas is twofold:
By always using the ...
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Estimating joint distributions using copula package in R
I am trying to estimate the joint distribution of stock returns using the copula package. I have read a couple of papers on copulae, but alas my lack of math ...
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What does a copula density explain about dependence of random variables?
I am studying copulas and I find it difficult to understand what a copula density tells me about the dependence of random variables.
For example, if I have a Gaussian copula density, what can I say ...
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Relation between covariance and joint distribution
This is an extremely basic question in probability theory. Namely, for any two variables $x$ and $y$, if $\mathrm{cov}(x,y)$ is not 0 (in the population), what does that imply about their joint ...
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Copula entropy: calculation is borked?
I came across a pretty cool paper whose idea makes a lot of sense to me.
Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-...
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Kendall's tau for Clayton Copula
When $\theta =-\frac{1}{2}$ the Clayton copula is given by $C(x,y)=(\sqrt{x}+\sqrt{y}-1)^2$. I've been asked to show in this case that Kendall's tau is $-\frac{1}{3}$. Using
$$\rho_{\tau}=4 \int_0^1 \...
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How to write a function for the normal copula in R?
How can I write the following function for the normal copula in R?
$$
C_\theta(u, v)=\Phi_\theta\left(\Phi^{-1}(u), \Phi^{-1}(v)\right),
$$
where $\Phi$ is the $N(0,1)$ cdf, $\Phi^{-1}$ is the ...
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Independent copula vs Student-$t$ copula with zero correlation matrix?
Suppose I have the random variables $X_1, \dots, X_n$ with the marginal distributions are not normal (in fact, unknown marginal distribution).
Will there be any difference between the assumption $X_1, ...
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An example of a bivariate pdf, where marginals are triangular distributions
What could be a form of
$$f_{X,Y}(x,y)$$
where $f_X(x)$ and $f_Y(x)$ both have the form of a triangular distribution with support $(0,1)$, but with different parameters that governs location of mode?
...
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Can you use Kendall's Tau to compute covariance matrix?
I am working with multivariate archimedean copulas, and I am wondering how I can extract a covariance matrix out of them? I can get Kendall's Tau matrix of correlation so I was thinking that maybe I ...
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Understanding tail dependence coefficients
How can I analyze the $\lambda_U$ and $\lambda_L$ results (estimated by non-parametric method)? What does higher or lower coefficients mean? Does $\lambda_U = 0.5$ mean there's some kind of linear ...
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Is the Gaussian copula (for d=2) with normal margins identical to the bivariate normal?
I am not sure about this:
In the 2 dimensional case, if I consider the Gaussian copula, is this identical to the bivariate normal distribution, in the case I choose the normal distribution for the ...
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EM algorithm is always used for mixture copula
I understand the EM algorithm and know there are other optimisation algorithms.
However, I have seen that with a mixture copula, EM algorithm always seems to be used. Is there any reason why EM ...
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