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 Copula Modeling Suitable for Accounting for Temporal Dynamics in Olive Plantation Data?
I am working on a project analyzing olive plantation data, where I aim to simulate the relationship between investment costs (Costs), revenues (...
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Optimal Coverage Sets of a Sum of Random Variables with Known Copula
Let $X, Y$ be continuous random variables with known copula $C$. Let us have access to sets (intervals) $S_x(\alpha),S_y(\alpha), \alpha\in(0,1)$ such that $$P(X\in S_x(\alpha))\geq \alpha,$$ $$P(Y\in ...
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How Do Copulas Provide Insight Into Dependence Between Random Variables?
I am trying to understand how copulas help us analyze the dependence structure between random variables. From Sklar's theorem, I know that given any joint distribution, you can extract the associated ...
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Copulas: obtaining uniform marginal distribution via pseudo observation or via CDF?
I'm studying copulas for a Time Series Analysis exam, and, to practice, our professor gave us an exercise on financial data.
One point of the exercise is to fit an elliptical copula to estimate the ...
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Question on Clayton Copula
The Clayton Copula is defined, for $u=F_X(x)$ and $v=F_Y(y)$ distribution functions, say of two continuous rv's,
$$C_{\rm Clayton} = \Big[\max\{u^{-\theta} + v^{-\theta}-1;\,0\}\Big]^{-1/\theta},\quad ...
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Copula density estimation and plotting using orthogonal Legendre polynomials
I have been unsuccessfully trying to replicate a copula density plot based on the following steps:
Use a uniform measure on I=[0,1]
Use an orthonormal basis of shifted Legendre polynomials with the ...
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Similarity measure between copulas
I have five 1,000,000$\times$30 matrices labeled as $M_1, M_2, M_3, M_4, M_5$. Each column in these matrices consists of random uniforms $\in[0, 1]$. The five $30\times 30$ covariance matrices are ...
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How to derive the Copula of $(X,Y)$ for X continuous random variable and $Y=X^{-1}$?
I didn't solve them in the exam and I still can't figure out how to solve it correctly.
For $X\sim U(-1,1)$ and $Y=\frac{1}{X}$, derive the Copula of $(X,Y)$ and what is its uniqueness.
For $X\sim N(...
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Fitting copula and simulating from it in R [closed]
Let’s say we have log returns of three assets: APPLE, NETFLIX and GOOGLE and I want to optimise my investment portfolio. In order to do this I can create mathematical programming problem on top of it. ...
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Generate Quasi Random Numbers for a Multivariate Distribution Using Copulas
Consider a sequence $U_1, U_2, ..., U_n$ of $\mathbb R^d$-valued quasi random vectors (e.g., Halton). Given a multivariate distribution $F$, I want to generate samples from $F$. In a question I asked ...
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Gap between the given correlation parameter and the empirical correlation in (Gaussian) copula simulation
Now I am trying to simulate normal copula with initial parameters being a correlation matrix of my wish. I found that the empirical correlations are generally lower than what is entered as the initial ...
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Degrees of freedom in a likelihood ratio test - multivariate normal vs univariate normal and Archimedean copula
Hopefully the title is self explanatory! To be more specific, I have three datasets. First, I fit them to a multivariate normal distribution, and calculate the log-likelihood. Then, I fit normal ...
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Is conditional expectation evaluated by the copula strictly increasing when the correlation coefficient is positive and vice versa?
I used the copula to evaluate the $\mathbb{E}[Y|X]$ and from my experiments on some copulas, I observed that when the random variables have positive correlation coefficient, $\mathbb{E}[Y|X]$ is ...
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Why does the multivariate data generated by a copula in R not exhibit the prespecified correlation?
I am using the package copula in R to generate a bivariate sample. The marginal distributions are binomial with p=0.5 and ...
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How to prove this relation for Kendall's distribution function (or Kendall's measure)
Kendall Distribution Function (Nelsen, 2006, p. 163) Or Kendall Measure (Salvadori et al., 2007, p. 148) Or Kendall Function (Joe, 2014, pp. 419–422) is the cumulative distribution function (CDF) of ...
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Steps for Forecasting with known copula's parameters
I want to calculate the Mean absolute percentage error (MAPE) for my copula model. I am stuck at the forecasting step. I am not specifying the copula here for different data pairs.
I have two time ...
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How many degrees of freedom in a T-copula is commonly used to model financial data?
I'm looking to fit some T-copulas to my financial return data. Unfortunately, the software I'm using can only fit a T-copula with a fixed degree of freedom parameter.
So what parameter should I use? ...
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Suppose $(X,Y)$ have copula $C(u,v)$, does $(aX,aY)$ have the same copula for $a>0$?
Suppose $(X,Y)$ have copula $c(u,v)$ in the sense of $Pr(X\leq x,Y\leq y)=Pr(F_X(X)\leq F_X(x),F_Y(Y)\leq F_Y(y))=Pr(U\leq u, V\leq v)=c(u,v)$,
where $u\equiv F_X(x)$ and $v\equiv F_Y(y)$ and $c(u,v)$ ...
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Should I not use copula if there is no significant dependence?
I have two variables, and I would like to get the joint distribution of those. I want to use copulas for that. However, when I checked for the dependence between those, I found no significant ...
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What is the formula for the conditional inverse function for the Ali-Mikhail-Haq and the Farlie-Gumbel-Morgenstern Copulas?
I am trying to do a Monte Carlo simulation and want to define a function for the conditional inverse function for the the Ali-Mikhail-Haq and the Farlie-Gumbel-Morgenstern Copula. Here is an example ...
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Correlation of a Gaussian copula
Suppose I have a 2D Gaussian copula with correlation matrix
$$
R = \begin{bmatrix}
1 & \rho \\
\rho & 1
\end{bmatrix}
$$
for $\rho\in[-1,1]$. The Copula is
$$
C_R(u)=\Phi_R\big(\Phi^{-1}(u_1),\...
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How to calibrate and simulate a copula-garch model in R using rmgarch package
So I have been trying to calibrate and simulate cryptocurrencies for VaR and ES analysis using the rmgarch package in R. I have been using the t-copula, my reason being that cryptocurrencies' ...
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How to write a function for the normal copula in R?
How can I write the following function for the normal copula in R?
$$
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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Return period of vine copula
In the formula for "OR" joint return period of 3-dimensional copula modeling, we need to find C(u,v,w) but when we model using vine, we are only able to obtain the copula C(uv|w). So, how to ...
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What are the convincing examples of copulas uncovering not obvious statistical dependences (or the lack of them)?
What may be a good, strong and convincing example demonstrating the power of copulas by uncovering some not obvious statistical dependencies?
I am especially interested in the example contrasting ...
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Expression of estimated standart deviation in GAMLSS model
I'm trying to replicate the results of the North Carolina Birth Data Analysis in the paper A Bivariate Copula Additive Model for Location, Scale and Shape by Giamperro Marra and Rosalba Radice (https:/...
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Grid search for estimation of degrees of freedom parameters in likelihood function
In the script below I attempt to estimate parameters for Apple and Amazon using a Gaussian Copula with t-Student marginals for the purpose of this exercice.
When executing the script, I notice at each ...
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Endogeneity Analysis without the access of raw data?
I currently have the correlation/covariance matrix for a set of variables, as well as the output from a regression analysis, but lack access to the underlying raw dataset. Given these constraints, ...
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How are the joint distribution and dependency related? [closed]
Here are some notes about copula functions,
Copula is a probability model that represents a multivariate uniform distribution, which examines the association or dependence between many variables. Put ...
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More than one possibility for the generator function of a Clayton's copula?
This question is about the bivariate family of Clayton's copulas, defined as:
$$
C(u,v)=\left(u^{-\theta} + v^{-\theta}\right)^{-1/\theta} \; \text{,} \quad \quad \text{(1)}
$$
with $\theta \in [-1, +\...
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Fitting a Copula from Scratch
I am trying to learn about how to work with Copulas. I find that I am often getting lost in the notations and distributions, and wanted to try and solidify my understanding.
As it stands, here is my ...
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How to draw samples from two correlated Negative Binomial variables?
Data and problem description: my data is the number of corner kicks of home team, away team, total corner kicks and corner kicks difference. Below is code for data and the plots (assuming the number ...
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Advantages of using Vine Copulas over Regular Copulas?
I am new to Copulas and am trying to conceptually understand that differences between the two main types of Copulas: Regular Copulas and Vine Copulas. Both are used to simulate data from correlated ...
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best strategy to test bivariate data
Here's a revised version of your text:
I have two sets of data: Intensity and Duration. For each set, I possess both observation data and model data, denoted as (I_obs, I_mod) and (D_obs, D_mod) ...
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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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Parametric copulas with marginals that are regressions
In Dependence Modeling with Copulas (Harry Joe) I'm struggling to interpret the meaning of a statement. In Chaper 5.1, it is stated:
Parametric inference for copulas
For dependence modeling with ...
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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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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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Are precipitation and soil moisture time-independent variables?
I was studying about Copula functions from here. It says:
Basically, copula is a set of mathematical tools that have the ability to connect two or more time-independent variables (Nelsen, $2003$)
As ...
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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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Is it possible to sample a copula in the original units?
I have a fairly high dimensional dataset that is not mvnormal. I used a copula to model the data and it fits well. How can I go about generating random samples from that copula that are in the ...
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Transforming data with a fitted distribution function
I have a bivariate dataset on $[0,1]^2$ in which I am interested in fitting a joint distribution. I fit a Gaussian copula but am unsure how to judge if it's a good fit. I tried transforming my data ...
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Estimate multivariate distribution with several variables on real data (continuous and categoricals) and sample from it
I have a complex dataset, collected through a survey, with both continuous (such as Age, Body mass index, etc..) and categorical variables (i.e. Gender, Education, etc..). I want to estimate their ...
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How to calculate the covariance-standardized residuals from a joint distribution estimated with the copula density function
I have issues when reading a paper when calculating the eta parameter in the following equation
where the authors of the paper describe eta as the vector of the covariance-standardized residuals, ...
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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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Does an algorithm exist that generate copula when marginal distributions are available and stable distributed and correlation is not simple?
I have simulated data of a 4-dimensional random variable $(X_1,X_2,X_3,X_4)$. The individual pdfs of these random variables, i.e., $X_i$ where $i\in\{1,2,3,4\}$ turns out to be stable distributed with ...
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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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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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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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