## R code for fitting a model to longitudinal data with a copula

The R code below demonstrates how to fit a model to longitudinal data by means of a copula. Longitudinal data is also referred to as panel, or repeated measures data.

The R code also shows how to create forecasts for longitudinal data, and how to compute prediction intervals for these forecasts. Continue reading R code for fitting a model to longitudinal data with a copula

## R code for fitting a quantile regression model to censored data by means of a copula

In my previous blog post I showed how to fit a copula to censored data. For the ease of use, I’m going to call these fitted copulas censored copulas.

The following R code demonstrates how these censored copulas in turn can be used for fitting a quantile regression model to censored data.

A more detailed description of the method employed for fitting the quantile regression model can be found in this blog post. Continue reading R code for fitting a quantile regression model to censored data by means of a copula

## R code for fitting a multiple (nonlinear) quantile regression model by means of a copula

In my previous blog post I demonstrated how to fit a simple (nonlinear) quantile regression model using a bivariate copula. In these simple quantile regression models, we have one independent and one dependent variable.

The R code below may be used for fitting a multiple (nonlinear) quantile regression model. These multiple (nonlinear) quantile regression models have two or more independent variables (but only one dependent variable). The R code fits these multiple (nonlinear) quantile regression models by means of a multivariate (Archimedean or elliptical) copula.

## R code for performing quantile regression using bivariate copulas

Nelsen explained in his 1999 book An introduction to copulas how to fit a (nonlinear) quantile regression model by means of a bivariate copula (pp. 175-176).

In short, Nelsen’s method for fitting a (nonlinear) quantile regression model is as follows:

1. Take the partial derivative of the copula function C(u, v) with respect to u, where u and v are both defined in [0, 1]. Denote this partial derivative by cu(u, v), and note that cu(u, v) = P{V ≤ v | U = u}.
2. For fitting the quantile regression model, set cu(u, v) = p, where p is defined in [0, 1].
3. For regressing v on u, evaluate cu(u, v) at u, and subsequently solve cu(u, v) = p for v (or, similarly, solve 0 = cu(u, v) – p for v).
4. Replace u by Fx-1(u) and v by Fy-1(v), where Fx-1(·) and Fy-1(·) are the quantile functions for x (=independent variable) and y (=dependent variable), respectively.

The following R code implements this copula method proposed by Nelsen for fitting a (nonlinear) quantile regression model. In addition, the R code may also compute confidence intervals for the fitted quantiles using a Monte-Carlo method. Continue reading R code for performing quantile regression using bivariate copulas

## R code for constructing confidence areas around the level curves of bivariate copulas

In his 2013 paper called An uncertain journey around the tails of multivariate hydrological distributions Serinaldi discusses the problem of constructing confidence areas for the level curves of bivariate copulas. A level curve at a specific p-value (also referred to as a p-level curve) may be used for estimating the p-th quantiles.

The R code below implements a nonparametric bootstrap method for computing such confidence areas for p-level curves. Continue reading R code for constructing confidence areas around the level curves of bivariate copulas