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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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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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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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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What is the difference between copulas and normalizing flows?
The goal of normalizing flows is to produce arbitrarily complex probability-distributions from a simple distribution (usually the Normal distribution) through learning an invertible transform.
Copulas ...
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Difference between the degree of dependence, and structure of dependence?
Correlation measures the degree of dependence between two variables,
while the
copula function defines the degree of dependence as well as their structure
of dependence.
How does the concept of ...
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How to sample from a multivariate empirical distribution
Recently, I’m working on the multivariate conditional estimation issue. Considering $2n$ variables:
$$\{X_{1},X_{2},\dots,X_{n},Y_{1},Y_{2},\dots,Y_{n}\}$$
where each follows an empirical ...
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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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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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Correlation of the sample as the estimation of parameter of Gaussian copula
Given 3 variables $X, Y, Z$ and I assume that the multivariate distribution of them is a Gaussian copula.
Now I need to estimate the correlation matrix of the Gaussian copula. As in many textbooks, i ...
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Trying to wrap my arms around copulas
This topic is dense with notation that makes things a bit confusing. But is this the correct interpretation?
Suppose we have two jointly distributed random variables – $X$ and $Y$ – of arbitrary (...
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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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Copula-based Value-at-risk in R
I'm working on a value-at-risk calculation using copulas on different stock market indices. I know how to fit the copula, but I can't figure out how to apply the VaR approach in the next step. The ...
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How to specify the parameter df in the goodness-of-fit test for a t-copula?
I have read the Q&A, and Consultation Paper, Article 26, p. 47-48.
I tried to fit a t-copula. I have found the coefficients rho.1 and ...
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How do we select the best copula family?
Suppose that I have bivariate data, and I need to model the bivariate dependence structure using a copula. Suppose further, I do not know what the best-fit copula family to my data is. Hence, I can ...
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How many observations to estimate a parameter of an Archimedean copula?
Let's consider for example the bivariate Gumbel copula.
$$C(u_1, u_2)=\exp \left[-\left(\left(-\ln\left(u_1\right)\right)^{\theta}+\left(-\ln\left(u_2\right)\right)^{\theta}\right)^{\frac{1}{\theta}}\...
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Geometric construction of copula - question regarding C-volume
I am learning about copula's, using Nelsen's book, and more specifically about the geometric method of constructing copula's. The problem is replicated in the following link: http://www.stat.ubc.ca/...
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Generating samples from Copula in R
Suppose I want to model dependence between $d$ r.v.´s $Y_1,...,Y_d$ with the copula $C_\theta$, where $\theta$ are the corresponding parameters of that copula. I've also determined the correlation ...
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Deriving conditional distribution using Gaussian copula
This question shows how to derive an analytical expression for the conditional distribution from a multivariate normal. I am curious how well this extends to when there's a Gaussian copula, but ...
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Had statisticians predicted 2008 financial crisis?
Are there any statistical or econometric studies before 2008 that predicted 2008 financial crisis?
Note that there are some publications that attemp to predict contagion between markets using copula ...
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Are there margins such that, while the "correlation" parameters of a Gaussian copula are positive, the correlations between the margins are negative?
Let there be a multivariate distribution $F$ with margins $F_1,\dots,F_n$ and a Gaussian copula with "correlation" matrix $\Sigma$. Let the off-diagonal elements of $\sigma$ be positive.
Let ...
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Fitting Vine Copula tree by tree
I am using the R programming language. I want to manually fit the D-vine copula for tree level 2 using BiCopHfunc(). Still, I ...
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Forecasting with empirical copulas
I estimated the beta copula with 3 variable time series. Now I'd like to make forecasts to evaluate the out-of-sample performance of my model. I know 2 of the 3 variables and I have the dependence ...
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Interpretable General Measure of Dependence
I am looking for an interpretable measure between two random variables $X$ and $Y$ which quantifies the dependence between the two but does not assume linearity.
Essentially, I am looking for a ...
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R: Calculating the convolution of two (multivariate) functions using FFT
I'm looking for a way to calculate:
$$(f\ast g)(x) = \int_{\mathbb{R}^d}f(y)g(x-y)dy$$
in R. I have solved this problem using Monte-Carlo integration. However, ...
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Question about using Bayesian rule as a classification for continuous data set
Please note that my question is not about coding.
I am now learning Bayesian classification and I think I understand it in a discrete case. I have trouble understanding it for multivariate continuous ...
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Finding expression of $n$-th derivative, when $n$ is large
For completeness, assume $C$ is an Archimedean copula with some generator function $\varphi$, which is usually assumed to have nice properties. It is known that $$ C(u_1, u_2, \ldots, u_n)=\varphi^{-1}...
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How to calculate conditional probability when only marginals are known?
Say $X, Y$ are standard normals with correlation $\rho$.
How do I calculate conditional probabilities such as $P(X \le x \mid Y \le y)$?
I don't have any assumptions on their joint distribution, ...
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Why does rank correlation only depend on copula, and not on the margins, if the margins are continuous
I read that for continuous margins, then Rank correlation only depend on copula and not on the margins. However, it is not the case for the non-continuous margins. Is that because the Kendall's tau (...
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Conditional correlation, copula, portfolio optimization and diversification
I have a data set which consists of > 500 hedge funds, their historical monthly returns, and their benchmark (index) monthly returns. The number of data points (# of monthly returns) differs from a ...
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Model groups of variables and their interactions separately
I want to decompose a multivariate pdf $p(x_1, x_2, x_3)$, where each $x_i$ is a collection of one or more random variables, into its "marginals" $p(x_1), p(x_2), p(x_3)$:
$p(x_1, x_2, x_3) = p(x_1)p(...
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Using a copula to generate correlated data draws; attenuation bias in result correlation
I would like to generate correlated data draws across two unrelated distributions (or across two empirical datasets that are from unknown distributions). The correct way to do this is to use a copula. ...
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Question about E step in Em algorithm, a challenge part, any help please?
I am new to EM algorithm and copula. I was reading a paper in mixture pair-copula. The authors use $u=(u_r, u_s) = (u_r^t,u_s^t), (t= 1,...,T)$ to indictae to the vector of copula data. Then, they ...
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A confidence area for an Archimedean's copula family
I'm reading the paper by Lourme A. et al (2016) and tried to plot $2D$ confidence areas like on the Fig.2. Discussion about confidence areas of normal distributed data and some results are here.
Let'...
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Copula Behaviour : Gaussian vs Student T -Numerical Stability
I would like to get your opinion on the following topic:
I am comparing the behaviour of Gaussian and Student-t Copulas.
I employ the follwing procedure:
Simulate N=100,000 samples from a Student ...
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Empirical multivariate probability integral transform
Is there a 'simple' way to obtain a non-parametric empirical multivariate probability integral transform?
Univariate case
The probability integral transform relates to the transform of any random ...
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Simulating from empirical copula density estimate
I went through these excellent slides by Prof. Arthur Charpentier on the use of Kernel density estimators (KDEs) with boundary correction to estimate Copulas in a nonparametric fashion.
As explained ...
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Fitting a Copula to Two Stochastically Dependent Variables in R
I have two sets of observed data, and I would like to model their joint distribution using a copula in R. I have transformed each set into a uniform distribution using their CDFs, and so now I have ...
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Copula compatibility problem
Suppose I have a 3-Copula which I would like to construct with two 2-Copula's, as the following construct:
$$ C_2(u, C_1(v,w)) = C(u,v,w) $$
My question is, if $C$ is known to be a valid copula (i.e....
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Sampling an N-dimensional copula via N independent uniforms
In order to draw a sample from an N-dimensional Gaussian copula, we draw N independent standard Gaussian random variables, form a vector, and multiply it by an appropriate matrix (Cholesky and such). ...
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How to find a conditional probability using copula-based Markov process?
I have a monthly time series of a water quality parameter. I used copula-based Markov process of C(Y(t), Y(t-1) and I forecasted the mean behavior of Yt by following equation:
Now, I need to find ...
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Generate samples from multivariate correlated data which have non-parametric cumulative distribution functions
I have 40 samples that contain information about 6 variables (hence a 40x6 data matrix). Each variable (column) has a cumulative distribution function (marginal distribution) based on the 40 values, ...
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What is this copula?
I have a bivariate sample in the [0,1] square for which I am trying to find the copula that best describes it. (I am new to copulas.)
So far, I have tried all classes in the "copula" R ...
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Uniqueness of a Latent Representation Under Monotonicity Condition?
Suppose that I observe a bi-variate joint distribution over two random variables, $(X_1,X_2)$. I want to represent this joint distribution as arising from a function $F$ applied to i.i.d. uniform ...
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Interpretation of basis functions in a logistic regression: can we test for univariate and multivariate/copula differences between the categories?
O'Brien (1988) has shown that a strong method for doing multivariate testing is to reverse the problem. That is, instead of seeing if the category impacts the measured values, see how the measured ...
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How do I derive a pair-copula decomposition for a joint density function?
In Section 4.1 of Analyzing Dependent Data with Vine Copulas, the author decomposes a three-dimensional joint density function into bivariate copula densities and marginal density functions. I’m ...
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integral related to a general bivariate copula C(u,v) of |u-v|
I'm trying to compute the following integral over the unit square $I^2=[0,1]^2$:
$$
\int_0^1\int_0^1 |u-v|dC(u,v),
$$
where $C(u,v)$ is a generic bivariate copula, which should be equal to
$$
1-2\...
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Is a bivariate copula relevant in this physics setting manifesting uniform univariate marginals--and, if so, how can it be constructed?
To quickly place our probabilistic (copula) question in its subject matter setting, we note that a fundamental concept in quantum theory is that of entanglement QuantumEntanglement.
The states of ...
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How can a copula be seen as a characterization of association?
On the post A formal definition of a “measure of association” @kjetil b halvorsen commented the following:
A copula could be seen as a characterization of association, so maybe a "measure of ...
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Should I say "copula" or "copula function"? Is the latter superfluous?
I am writing a paper involving copulas and have been thinking of whether I should say "copula function" or just "copula", as I am not sure whether the term "copula" ...
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What is the meaning of maximum level in copula density plot?
The copula density of 3 different pairs of variables is shown below. Why do the colorbar meters on the right-hand side of each plot have the same range, and what does the value $4.8$ mean exactly, ...
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