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.

# Load required packages

    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`.

## Load and clean data

    ## 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