Author: Daniel Kraus and Claudia Czado

# Example usage of the vinereg package

#### Daniel Kraus and Claudia Czado

#### 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.

# Load required packages

    library(vinereg)
    require(ggplot2)
    require(dplyr)
    require(tidyr)
    require(AppliedPredictiveModeling)

# 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")
    colnames(abalone) <- c(
      "sex", "length", "diameter", "height", "whole",
      "shucked", "viscera", "shell", "rings"
    )
    abalone_f <- abalone %>%
        dplyr::filter(sex == "F") %>%        # select female abalones
        dplyr::select(-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

    ## [1] "length"   "height"   "diameter"

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)

    ## # A data.frame: 6 x 11
    ##  tree edge conditioned conditioning var_types family rotation  parameters df
    ##     1    1        1, 2                    c,c gumbel      180         5.2  1
    ##     1    2        2, 4                    c,c gumbel      180         2.4  1
    ##     1    3        4, 3                    c,c gumbel      180         2.4  1
    ##     2    1        1, 4            2       c,c      t        0 0.45, 20.20  2
    ##     2    2        2, 3            4       c,c      t        0  0.91, 4.68  2
    ##     3    1        1, 3         4, 2       c,c      t        0  0.31, 7.39  2
    ##   tau loglik
    ##  0.81   1629
    ##  0.58    672
    ##  0.59    709
    ##  0.30    147
    ##  0.73   1193
    ##  0.20     72

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)

    ##         0.1       0.5       0.9
    ## 1 0.6719522 0.7666997 0.8715491
    ## 2 0.6969903 0.7903960 0.9012124
    ## 3 0.6794323 0.7842671 0.8916937
    ## 4 0.7755592 0.8800229 0.9974089
    ## 5 0.5952851 0.6997763 0.8272583
    ## 6 0.6757838 0.7729635 0.8858529

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)

    ## `geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = "cs")'

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

    ## D-vine regression model: whole | length, height, diameter
    ## nobs = 1306, edf = 95.67, cll = 1205.54, caic = -2219.73, cbic = -1724.64

    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"))

    ## `geom_smooth()` using method = 'gam' and formula = 'y ~ s(x, bs = "cs")'

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

    ## D-vine regression model: rings | shell, shucked, whole, height, viscera, length, diameter
    ## nobs = 1306, edf = 9, cll = -2781.56, caic = 5581.12, cbic = 5627.7

    plot_effects(fit_disc)

    ## `geom_smooth()` using method = 'loess' and formula = 'y ~ x'

# 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