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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Conditional expectation of two identical marginal normal random variables
Let $Y_0$ and $Y_1$ be both identically (not necessarily independent)
normally distributed with mean $\mu$ and $\sigma^2$, i.e., $Y_i \sim
N(\mu, \sigma^2)$ for $i = 1, 2$. Let $\rho$ denote the ...
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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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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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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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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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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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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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Mutual information relationship to copula entropy is borked?
I have posted a related Question based on a paper, Ma, Jian, and Zengqi Sun. "Mutual information is copula entropy." Tsinghua Science & Technology 16.1 (2011): 51-54.
In the paper, they ...
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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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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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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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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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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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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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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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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 do I convert simulated values from a copula to "real" observations? R
I have managed to fit different kinds of copulas to my data in R (mostly Archimedean copulas) using the copula package.
I have no problem simulating pseudo observations (u and v), my questions are:
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Derivation of survival copula
$$ \begin{align}F(x,y) &= P(X\leq x, Y\leq y) \\
&= C(F(x), F(y)) \end{align}$$
The copula of a bivariate distribution is equal to its CDF, $C(u,v) = F(x,y)$. The density of the data's CDF ...
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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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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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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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Proof of the relation between Kendall's Tau and Pearson's Rho for the Gaussian Copula
I know in the case of the bivariate normal distribution Kendall's Tau is given by
$$
\tau=\frac{2}{\pi}\arcsin({\rho})
$$
where $\rho$ is Pearson's correlation. Can someone given a derivation of this ...
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How to forecast from GARCH-copula model?
I am reading to understand how to forecasting time-series data from the GARCH-copula model. I am looking forward to understanding the steps. From my understanding, we should follow the following steps:...
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Dependent thinning Poisson process
If $N_1$ and $N_2$ are independent Poisson processes then the superposition is a Poisson process. Is it possible to construct two dependent Poisson processes such that the superposition is a Poisson ...
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Constructing a joint distribution from pairwise bivariate marginal distributions?
It's fairly well-known that given univariate distribution functions $F_X, F_Y, F_Z$, one can construct the joint distribution $F_{(X, Y, Z)}(x, y, z) = C(F_{X}(x), F_{Y}(y), F_{Z}(z))$, where $C$ is ...
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Generating values from copula using copula package in R
I have a bunch of questions concerning the use of the copula package in R. My overall aim is to generate synthetic values using copulas. I am analyzing a hydrological data: annual peak discharge [m³/s]...
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Simulate a Gaussian Copula with t margins
The task is the following:
Given is $Z_1,...Z_{50}$ different hypothetical assets.
Each $Z_k \sim t_3$ with standard deviation $\sigma=0.01$ and $\tau(Z_i,Z_k)=0.4$ for $j\neq k$.
I want to ...
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Sampling from copula given a particular value of marginal
I am working in the bivariate case with copulas as follows:
I have two marginal Gamma distributions $f_1=Ga(\alpha_1, 1)$ and $f_2=Ga(\alpha_2, 1)$ that are bound by a copula Frank copula $C$, with ...
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Does assumption of normality of each mixture components implies that each margins is normal
I just would like to understand some information about the joint normality and the margins. I read that the normal joint distribution almost always implies that the univariate margins are all normal.
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What is the expectation of the following joint CDF?
Yesterday, I asked the following question regarding copulas:
"Let's say $X=(X_1,X_2)′$, where $X\in \mathbb R^2$. What is the expectation of the copula function $C(F_{X_1}(x_1),F_{X_2}(x_2))$ - i.e....
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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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Inverse CDF sampling for a mixed distribution
The out-of-context short version
Let $y$ be a random variable with CDF
$$
F(\cdot) \equiv \cases{\theta & y = 0 \\ \theta + (1-\theta) \times \text{CDF}_{\text{log-normal}}(\cdot; \mu, \sigma) &...
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When modeling a copula, you need to generate "pseudo observations"? Why? What is a pseudo observation? [closed]
I'm struggling with the concept of a "pseudo-observation." I can't find any material out there describing what it is in a simple, concise manner. Does it have something to do with ...
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Why do copulas need the i.i.d assumption for marginal distribution?
Does anyone know if are there some assumptions for Copula method? I heard from someone that the data should be i.i.d (independent and identically distributed). Let's say, if I want to capture the ...
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Estimating joint distributions using copula package in R
I am trying to estimate the joint distribution of stock returns using the copula package. I have read a couple of papers on copulae, but alas my lack of math ...
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what does positive Gaussian copula dependency describe
Suppose I have two variables and that their dependency structure is Gaussian copula.
Suppose that the parameter of the Gaussian copula is 0.8 and the corresponding Kendall's tau is 0.7. How can I ...
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How can I generate random observations from a concrete copula?
Let us assume that we have two continuous random variables $X$, $Y$, with known distributions (not necessarily normal), connected/related via a concrete copula.
What is a procedure to generate random ...
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The importance of the Gaussian copula
Suppose we have a bunch of marginal distributions
$$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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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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Copula is not unique if the margins in not continuous
The copula is a very interesting tool to describe the dependence structure. However, I read that if the margins are continuous then copula is unique.
However, if margins are discrete then copula is ...
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showing that a function is a copula
In general, is there an easier way of showing that a function is a copula than showing that:
$C(u_1,\dots,u_d) =P(U_1\le u_1,\dots,U_d \le u_d) \quad$is nondecreasing in each $u_i \in [0,1] $
$C(1,\...
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An example of a bivariate distribution with normal marginals and a nonlinear conditional mean curve? [duplicate]
That is, an example of a bivariate distribution with normal marginals for which a linear regression is inappropriate.
Does an asymmetric copula always produce this?
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Computing probability distribution function for uniform random variables and Y=1-X
X is uniform random variable in [0,1] and Y=1-X. How do I calculate the distribution function F(X,Y)? I can see that Y is also uniformly distributed and can draw the intervals. But I am unable to ...
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Copula to ensure one team wins and the other loses (Bernoulli margins)
Team $1$ has a historical win percentage of $p_1$.
Team $2$ has a historical win percentage of $p_2$.
The upcoming game features team $1$ against team $2$ and cannot end in a tie (one team wins, and ...
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Why is the Gaussian Copula $C_P(u_1, \ldots, u_d) = \boldsymbol{\Phi}_P(\Phi^{-1}(u_1), \ldots, \Phi^{-1}(u_d))$ with $\Phi^{-1}$ instead of $\Phi$?
From Wikipedia, Gaussian Copula,
it states that a Gaussian Copula can be defined as:
$$
C_P(u_1, \ldots, u_d) = \boldsymbol{\Phi}_P(\Phi^{-1}(u_1), \ldots, \Phi^{-1}(u_d)),
$$
where $\boldsymbol{\...
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How to simulate from a log-copula function?
Does anybody know how to simulate from a log-copula function?
I'm trying to simulate $(u,v)$ from a log-copula function with the CDF:
$$
C(u,v, a) = \exp\bigg(1-\big[(1 - \ln u)^a + (1 - \ln v)^a - 1\...
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How to fit a copula when zeros abound?
I am modelling a joint distribution for two random variables: $F(x,y)$. I observe $n$ data points $(x^{}_{i},y^{}_{i})^{N}_{i=1}$. I would like to model $F$ as the product of its marginals and a ...
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Appropriate number of degrees of freedom in t-Copula
In a consultation paper (EBA/CP/2014/08) the European Banking Authority (EBA) wrote: “it is proposed […] that Gaussian or Normal like Copulas are not to be used for operational risk modelling. For ...
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