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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Structural equation model and latent instrumental variable approach with copulas
Consider the following structural equations:
$store sales := f(store visits, x_1, x_2)$
$store visits := f(x_1, x_2)$
Where as $x_1, x_2$ denotes promotional activity spending.
Note that $x_1, x_2$ ...
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Differentiating a copula joint distribution
I am trying to derive the differentiation of joint copula from this paper http://www.nicksun.fun/assets/ms_references/madsen2009.pdf, which is done in equation (4.3). To summarize
I fail to ...
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Using Copulas to find mutual information
I have two multidimensional datasets $X, Y$ of dimensions $m \times n$. Here $m$ is the successive measurements and $n$ is the data collected during each measurement. We can say each of $m$ are ...
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How to prove an asymptotical converge estimations to the MLE estimations?
To estimate the parameters of copulas, both the classical maximum likelihood method (MLE) and alternative methods can be used, for example, the natural sampler and the efficient importance sampler.
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Are two marginal distributions of a student-t copula equivalent to using two independent uniform distributions?
I am trying to figure out if these two are the same:
Using the marginal uniform distributions of a student-t copula
Using independent uniform distributions
I have generated SAS code to figure this ...
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Does the larger tail dependence implies a larger positive dependence?
For Archimedean copula such as Clayton and Gumbel, their positive dependence can be measured by Kendall's $\tau$ that $\tau_\textrm{Clayton} = \frac{\theta}{\theta + 2}, \tau_\textrm{Gumbel} = 1-\...
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Conditional CDF given one dimension equals derivative of joint CDF towards that dimension divided by the density at that dimension?
So I am familiar with the following:
$$P\left(X<x|Y=y\right) =\int_{-\infty}^{x}f\left(X=u|Y=y\right)du=\frac{1}{f\left(Y=y\right)}\cdot\int_{-\infty}^{x}f\left(X=u,Y=y\right)du$$
But during a ...
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Copula Invariance Principle
I don't get why equation 7 is true, can someone explain me why? This is part of the proof of the invariance principle in copula theory.
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Should the data transformation to the uniform [0,1] always be performed in copula modeling, even for Archimedean copula families?
Is the data transformation to the uniform [0,1] always required in copula modeling, even for Archimedean copula? I have read some sources stating that the first step in copula modeling involves ...
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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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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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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 to forecast using copula
Previously I had tried to do forecasting using Copula where the Copula chosen was Copula Frank. I generate random variate as data simulation. I use the random variates for forecasting. I transform the ...
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Are there families of known parametric copulas for non-standard marginal normal distributions?
I know that a family of Gaussian copulas generates a standard bivariate normal distribution if and only if the marginal ones are standard normal. This characterizes the Gaussian copulas, where I have ...
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Gaussian copula: how to scale data back to get target covariance matrix (not correlation)
I would like to use a Gaussian Copula to simulate data with a given covariance matrix and given marginal distributions. I understand that the input to the copula cannot be the covariance matrix $\...
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Generate nonnegative variates with mean 1 and specified variance-covariance
Problem
In several applications in surveys, it would be helpful to be able to generate a set of $R$ $n$-dimensional variates with the following properties:
Has mean vector $1$
Has a specified ...
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statistics linking McFadden's $R^2$ to the relationship between two binary variables, akin to correlation (Copula with Bernoulli margins?)
My goal is to create a visualization of the strength of the McFadden's $R^2$ of a (multinomial) logistic regression, where McFadden's $R^2$ is $1-\dfrac{LL(M_1)}{LL(M_0)}$, involving the ratio of the ...
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Calculating conditional probability using VineCopula in R
I have a dataframe X (with columns x1 and x2) and would like to calculate conditional probability, something like P(x1<0.5|x2&...
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Uniqueness of copula when marginals are continuous
I have a basic question about copula. I am not an expert in statistics myself but use statistics for modelling and data analysis a lot.
I have read in multiple sources and also in Wikipedia that:
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derivatives and distribution of a 3-dimensional copula in R
I am looking for a way to calculate in the R software, the distribution, the density and the derivatives (of order 1, 2) partial of a Gaussian copula of dimension 3.
Indeed, I have three variables (u1,...
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Creating multivariate distribution using marginal: can I use a copula?
I am modeling a multivariate distribution, where I already have the distributions of the marginals. Let's call the marginals, $f_{i}(x)$ for every $i$ marginal distribution, where $i=1,2,3$. We ...
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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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Methods to estimate bivariate survival function under *bivariate* censoring
I am looking at the relation between two time-to-event variables subject to censoring. The seminal work from Lin and Ying is unfortunately paywalled (https://www.jstor.org/stable/2337178), but I ...
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Which parametric family could describe these asymmetric copulas?
In my recent project I keep encountering asymmetric copulas like this:
However, most of the common parametric copulas I could find are symmetric. While there are some resources on asymmetric copulas ...
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Sum of Multivariate Distribution with Normal Copula
When summing multivariate distributions that uses a normal copula, is it true that the normal copula part of the distribution remains unchanged?
The R code below simulates from a two-dimensional ...
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Correlation bounds on transformations of two random variables
Suppose I have 3 real-valued random variables, $X, Y, Z$ and their pairwise correlations, $\rho_{XY}, \rho_{YZ}, \rho_{XZ}$, and I am tasked with finding a 2d function $f$ such that $f(X,Y)$ has ...
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Why use a copula to generate synthetic data?
For class, I am tasked to generate synthetic stock data using the copula R package. The step-by-step process is picking 2 stocks (i.e., Amazon & Apple), fit their marginal distributions (I am ...
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Generate a random variable with a defined correlation to an existing variable(s) considering copula
In this question it has been described how to generate a random variable with a defined correlation to an existing variable. My question is what happens if I want to have a specific pattern of ...
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Sampling from multivariate Bernoulli
Suppose you have a vector p drawn from a multivariate Beta distribution (not a Dirichlet), such as the one described here ( How to construct a multivariate Beta distribution? ) with a Gaussian copula.
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How to find the likelihood that a multivariate observation could have been generated by a given copula?
I can take a set of multivariate observations belonging to a class, and use them to generate a copula. This can then be used to generate synthetic multivariate data which are statistically compatible ...
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For a given Copula what distribution is the resulting joint density?
Say you have 2 random variables $x, y$ and a copula $C$ to model their interdependence. The two distributions are made uniform and the copula has some form (e.g., Gaussian, Clayton).
Given that, does ...
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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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Fitting a copula vs. directly fitting a multivariate distribution
I understand that the joint density of two random variables $f(x,y)$ can be decomposed as the product of its marginals and a copula: $f(x,y) = g(x)k(y) \times c(G(x),K(y))$. Alternatively this may be ...
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Copula with one property
We know that a multivariate distribution $F(x,y)$ can be expressed as copula via $C(F_1(x),F_2(y))$ with $F_1,F_2$ being the marginal distribution.
I am trying to construct a multivariate distribution ...
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What does evalue the density of a fitted model at specific points mean?
I have read a description of R package and find the following:
"Evaluate the density of the fitted model at (2.747, 0.1467, 0.13, 0.05334)".
I do not understand what the author mean by ...
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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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Are multivariate and cumulative exchangable in copula?
In copula model, some researchers, identify it as a multivariate distribution function, while other present it as a cumulative distribution function. I believe multivariate differs of cumulative. But ...
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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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Obtain minimum-variance hedge ratio from a copula-GARCH model
Let $r_{s, t}$ and $r_{f, t}$ be the return rates of the spot and futures of a commodity at time $t$. The hedging ratio based on variance minimization is calculated by finding the minimum of the ...
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Copula Application
Among other things, I am working on dependency structures for the reliability analysis of river embankments and use Copulas for this purpose.
I have 3 questions about this:
Strucure
I am not yet ...
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Forecasting using Copula GARCH methods
I need to replicate what Huang and al (2009)* did without using built-in functions in R. What I'm struggling with is how to forecast returns for my two data samples. I've found the GARCH specs and ...
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t-Copula Simulations of Bivariate Random Variables: Assigned Correlation != Calculated Linear Correlation
I have an n x n correlation matrix of random variables. Consider the following example:
What I would like to do is study the behavior of the joint distributions using a t-Copula with, say, 5 degrees ...
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Why is a 3-copula better than a 2-copula?
Suppose that I have known that $X$ and $Y$ have high dependency, $Y$ and $Z$ have high dependency, and $Z$ and $X$ also have high dependency through three different 2-copulas.
Suppose I fit one 3-...
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Joint Exceedance Probability for Multivariate Distribution
I was reading this paper:
https://nhess.copernicus.org/preprints/nhess-2020-28/nhess-2020-28.pdf
Where the notion of "joint exceedance probability" is discussed for the bivariate case. That ...
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Copula for 3 variables
I want to make Copula analysis for 3 variables to understand the dependency. I applied following code to obtain $c_{1,2}$ and $c_{1,3}$
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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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From marginal distribution to joint distribution with independence
Consider a random vector $(X,Y,Z)$, Let $f_X, f_Y, f_Z$ be the probability distributions of each component.
Question: Does there always exist a distribution $f$ for the whole vector $(X,Y,Z)$ such ...
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How to compute the combined probability of loss for 2 time series (consisting of historical stock prices)?
May I please ask the community's support with the following problem?
I have 2 time series, with approximately 1000 observations each (same number of observations for both). They represent the daily ...
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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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What are examples of symmetric copulas $f(x,y)=f(y,x)$ having relative minima for $f(x,x)$?
In a previous posting on this site RepulsiveBehavior I attempted to detail
a quantum-information-theoretic separability/entanglement problem I am pursuing. Detailed issues of sampling sizes for a data ...
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