Load and clean data

data(abalone, package = "PivotalR")
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 = ".")

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, 
    selcrit = "aic"
))

## D-vine regression model: whole | length, height, diameter 
## nobs = 1306, edf = 20.73, cll = 1092.08, caic = -2142.69, cbic = -2035.41

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 9 
##   tree edge conditioned conditioning family rotation  parameters df  tau
## 1    1    1        1, 2              gumbel      180         5.1  1 0.80
## 2    1    2        2, 4              gumbel      180         2.3  1 0.57
## 3    1    3        4, 3              gumbel      180         2.4  1 0.58
## 4    2    1        1, 4            2      t        0 0.45, 15.23  2 0.30
## 5    2    2        2, 3            4      t        0  0.91, 4.59  2 0.74
## 6    3    1        1, 3         4, 2      t        0  0.32, 8.26  2 0.20

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 0.6566616 0.7600738 0.8732819
## 2 0.6826392 0.7860813 0.9030168
## 3 0.6619671 0.7761635 0.8924578
## 4 0.7623786 0.8732481 0.9929738
## 5 0.5812824 0.6890220 0.8231798
## 6 0.6600508 0.7663987 0.8852566

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"
))

## D-vine regression model: whole | length, height 
## nobs = 1306, edf = 144.38, cll = 1136.86, caic = -1984.97, cbic = -1237.86

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 theory which is developed in Nagler (2017) and applied to D-vine quantile regression in Schallhorn et al. (2017). For the estimation the discrete variable(s) are transformed to continuous ones by jittering.

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,
    family_set = "nonpar", 
    selcrit = "aic"
))

## D-vine regression model: rings | shell, shucked 
## nobs = 1306, edf = 114.32, cll = -2774.44, caic = 5777.53, cbic = 6369.12

plot_effects(fit_disc)

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