# Example usage of the vinereg package

#### September 2017

This file contains the source code of an exemplary application of the D-vine copula based quantile regression approach implemented in the R-package vinereg and presented in Kraus and Czado (2017): D-vine copula based quantile regression, Computational Statistics and Data Analysis, 110, 1-18. Please, feel free to address questions to daniel.kraus@tum.de.

library(vinereg)

pkgs_required <- c("ggplot2", "dplyr", "tidyr", "AppliedPredictiveModeling")
pkgs_available <- sapply(pkgs_required, require)

# Data analysis

## set.seed(5)

We consider the data set abalone from the UCI Machine Learning Repository (https://archive.ics.uci.edu/ml/datasets/abalone) and focus on the female sub-population. In a first application we only consider continuous variables and fit models to predict the quantiles of weight (whole) given the predictors length, diameter, and height.

## data(abalone, package = "AppliedPredictiveModeling")
## abalone_f <- abalone %>%
##     dplyr::filter(sex == "F") %>%        # select female abalones
##     dplyr::select(-id, -sex) %>%         # remove id and sex variables
##     dplyr::filter(height < max(height))  # remove height outlier
## pairs(abalone_f, pch = ".")

# D-vine regression models

## Parametric D-vine quantile regression

We consider the female subset and fit a parametric regression D-vine for the response weight given the covariates len, diameter and height (ignoring the discreteness of some of the variables). The D-vine based model is selected sequentially by maximizing the conditional log-likelihood of the response given the covariates. Covariates are only added if they increase the (possibly AIC- or BIC-corrected) conditional log-likelihood.

We use the function vinereg() to fit a regression D-vine for predicting the response weight given the covariates length, diameter, and height. The argument family_set determines how the pair-copulas are estimated. We will only use one-parameter families and the t copula here. The selcrit argument specifies the penalty type for the conditional log-likelihood criterion for variable selection.

## fit_vine_par <- vinereg(
##   whole ~ length + diameter + height,
##   data = abalone_f,
##   family_set = c("onepar", "t"),
##   selcrit = "aic"
## )

The result has a field order that shows the selected covariates and their ranking order in the D-vine.

## fit_vine_par$order The field vine contains the fitted D-vine, where the first variable corresponds to the response. The object is of class "vinecop_dist" so we can use rvineocpulib’s functionality to summarize the model ## summary(fit_vine_par$vine)

We can also plot the contours of the fitted pair-copulas.

## contour(fit_vine_par$vine) ## Estimation of corresponding conditional quantiles In order to visualize the predicted influence of the covariates, we plot the estimated quantiles arising from the D-vine model at levels 0.1, 0.5 and 0.9 against each of the covariates. ## # quantile levels ## alpha_vec <- c(0.1, 0.5, 0.9) We call the fitted() function on fit_vine_par to extract the fitted values for multiple quantile levels. This is equivalent to predicting the quantile at the training data. The latter function is more useful for out-of-sample predictions. ## pred_vine_par <- fitted(fit_vine_par, alpha = alpha_vec) ## # equivalent to: ## # predict(fit_vine_par, newdata = abalone.f, alpha = alpha_vec) ## head(pred_vine_par) To examine the effect of the individual variables, we will plot the predicted quantiles against each of the variables. To visualize the relationship more clearly, we add a smoothed line for each quantile level. This gives an estimate of the expected effect of a variable (taking expectation with respect to all other variables). ## plot_effects(fit_vine_par) The fitted quantile curves suggest a non-linear effect of all three variables. ## Comparison to the benchmark model: linear quantile regression This can be compared to linear quantile regression: ## pred_lqr <- pred_vine_par ## for (a in seq_along(alpha_vec)) { ## my.rq <- quantreg::rq( ## whole ~ length + diameter + height, ## tau = alpha_vec[a], ## data = abalone_f ## ) ## pred_lqr[, a] <- quantreg::predict.rq(my.rq) ## } ## ## plot_marginal_effects <- function(covs, preds) { ## cbind(covs, preds) %>% ## tidyr::gather(alpha, prediction, -seq_len(NCOL(covs))) %>% ## dplyr::mutate(prediction = as.numeric(prediction)) %>% ## tidyr::gather(variable, value, -(alpha:prediction)) %>% ## ggplot(aes(value, prediction, color = alpha)) + ## geom_point(alpha = 0.15) + ## geom_smooth(method = "gam", formula = y ~ s(x, bs = "cs"), se = FALSE) + ## facet_wrap(~ variable, scale = "free_x") + ## ylab(quote(q(y* "|" * x[1] * ",...," * x[p]))) + ## xlab(quote(x[k])) + ## theme(legend.position = "bottom") ## } ## plot_marginal_effects(abalone_f[, 1:3], pred_lqr) ## Nonparametric D-vine quantile regression We also want to check whether these results change, when we estimate the pair-copulas nonparametrically. ## fit_vine_np <- vinereg( ## whole ~ length + diameter + height, ## data = abalone_f, ## family_set = "nonpar", ## selcrit = "aic" ## ) ## fit_vine_np ## contour(fit_vine_np$vine)

Now only the length and height variables are selected as predictors. Let’s have a look at the marginal effects.

## plot_effects(fit_vine_np, var = c("diameter", "height", "length"))

The effects look similar to the parametric one, but slightly more wiggly. Note that even the diameter was not selected as a covariate, it’s marginal effect is captured by the model. It just does not provide additional information when height and length are already accounted for.

## Discrete D-vine quantile regression

To deal with discrete variables, we use the methodology of Schallhorn et al. (2017). For estimating nonparametric pair-copulas with discrete variable(s), jittering is used (Nagler, 2017).

We let vinereg() know that a variable is discrete by declaring it ordered.

## abalone_f$rings <- as.ordered(abalone_f$rings)
## fit_disc <- vinereg(rings ~ ., data = abalone_f, selcrit = "aic")
## fit_disc
## plot_effects(fit_disc)

# References

Kraus and Czado (2017), D-vine copula based quantile regression, Computational Statistics and Data Analysis, 110, 1-18

Nagler (2017), A generic approach to nonparametric function estimation with mixed data, Statistics & Probability Letters, 137:326–330

Schallhorn, Kraus, Nagler and Czado (2017), D-vine quantile regression with discrete variables, arXiv preprint