Author: Dani Kraus and Thomas Nagler
Date: 2017-11-08

# D-vine quantile regression with discrete variables: analysis of bike rental data

#### Dani Kraus and Thomas Nagler

#### November 8, 2017

#### Required packages

    library(vinereg)
    require(ggplot2)
    require(dplyr)
    require(purrr)
    require(scales)
    require(quantreg)

#### Plot function for marginal effects

    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)) %>%
            dplyr::mutate(value = as.numeric(value)) %>%
            ggplot(aes(value, prediction, color = alpha)) +
            geom_point(alpha = 0.15) +
            geom_smooth(span = 0.5, se = FALSE) +
            facet_wrap(~ variable, scale = "free_x") +
            theme(legend.position = "none") +
            theme(plot.margin = unit(c(0, 0, 0, 0), "mm")) +
            xlab("")
    }

## Data preparation

#### Load data

    bikedata <- read.csv("day.csv")
    bikedata[, 2] <- as.Date(bikedata[, 2])
    head(bikedata)

    ##   instant     dteday season yr mnth holiday weekday workingday weathersit
    ## 1       1 2011-01-01      1  0    1       0       6          0          2
    ## 2       2 2011-01-02      1  0    1       0       0          0          2
    ## 3       3 2011-01-03      1  0    1       0       1          1          1
    ## 4       4 2011-01-04      1  0    1       0       2          1          1
    ## 5       5 2011-01-05      1  0    1       0       3          1          1
    ## 6       6 2011-01-06      1  0    1       0       4          1          1
    ##       temp    atemp      hum windspeed casual registered  cnt
    ## 1 0.344167 0.363625 0.805833 0.1604460    331        654  985
    ## 2 0.363478 0.353739 0.696087 0.2485390    131        670  801
    ## 3 0.196364 0.189405 0.437273 0.2483090    120       1229 1349
    ## 4 0.200000 0.212122 0.590435 0.1602960    108       1454 1562
    ## 5 0.226957 0.229270 0.436957 0.1869000     82       1518 1600
    ## 6 0.204348 0.233209 0.518261 0.0895652     88       1518 1606

#### Rename variables

    bikedata  <- bikedata %>%
        rename(
            temperature = atemp,
            month = mnth,
            weathersituation = weathersit,
            humidity = hum,
            count = cnt
        )

#### Un-normalize variables

See variable description on UCI web page.

    bikedata <- bikedata %>%
        mutate(
            temperature = 66 * temperature + 16,
            windspeed = 67 * windspeed,
            humidity = 100 * humidity
        )

#### Show trend

    ggplot(bikedata, aes(dteday, count)) +
        geom_line() +
        scale_x_date(labels = scales::date_format("%b %y")) +
        xlab("date") +
        ylab("rental count") +
        stat_smooth(method = "lm", se = FALSE, linetype = "dashed") +
        theme(plot.title = element_text(lineheight = 0.8, size = 20)) +
        theme(text = element_text(size = 18))

#### Remove trend

    lm_trend <- lm(count ~ instant, data = bikedata)
    trend <- predict(lm_trend)
    bikedata <- mutate(bikedata, count = count / trend)
    ggplot(bikedata, aes(dteday, count)) +
        geom_line() +
        scale_x_date(labels = scales::date_format("%b %y")) +
        xlab("date") +
        ylab("detrended rental count") +
        theme(plot.title = element_text(lineheight = 0.8, size = 20)) +
        theme(text = element_text(size = 18))

#### Drop useless variables

    bikedata <- bikedata %>%
        select(-instant, -dteday, -yr) %>%  # time indices
        select(-casual, -registered) %>%    # casual + registered = count
        select(-holiday) %>%                # we use 'workingday' instead
        select(-temp)                       # we use 'temperature' (feeling temperature)

#### Declare discrete variables as `ordered`

    disc_vars <- c("season", "month", "weekday", "workingday", "weathersituation")
    bikedata <- bikedata %>%
        mutate(weekday = ifelse(weekday == 0, 7, weekday)) %>%  # sun at end of week
        purrr::modify_at(disc_vars, as.ordered)

## D-vine regression model

#### Fit model

    fit <- vinereg(
      count ~ .,
      data = bikedata,
      family = c("onepar", "tll"),
      selcrit = "aic"
    )
    fit

    ## D-vine regression model: count | temperature, humidity, windspeed, month, season, weathersituation, weekday, workingday
    ## nobs = 731, edf = 83.93, cll = 451.69, caic = -735.53, cbic = -349.94

    summary(fit)

    ##                var       edf         cll       caic        cbic       p_value
    ## 1            count  8.127625 -198.076002  412.40726  449.748927            NA
    ## 2      temperature 21.962102  415.807777 -787.69135 -686.788370 1.062642e-161
    ## 3         humidity 17.921054  118.868746 -201.89538 -119.558653  2.267548e-40
    ## 4        windspeed  1.000000   22.820159  -43.64032  -39.045905  1.420866e-11
    ## 5            month 16.094496   28.514118  -24.83924   49.105524  1.754541e-06
    ## 6           season  1.000000   13.665350  -25.33070  -20.736286  1.714673e-07
    ## 7 weathersituation  1.000000   11.830673  -21.66135  -17.066932  1.148667e-06
    ## 8          weekday 15.821312   30.592167  -29.54171   43.147942  2.900363e-07
    ## 9       workingday  1.000000    7.669975  -13.33995   -8.745536  8.979736e-05

#### In-sample predictions

    alpha_vec <- c(0.1, 0.5, 0.9)
    pred <- fitted(fit, alpha_vec)

### Marginal effects

    plot_marginal_effects(
        covs = select(bikedata, temperature),
        preds = pred
    )

    plot_marginal_effects(covs = select(bikedata, humidity), preds = pred) +
        xlim(c(25, 100))

    plot_marginal_effects(covs = select(bikedata, windspeed), preds = pred)

    month_labs <- c("Jan","", "Mar", "", "May", "", "Jul", "", "Sep", "", "Nov", "")
    plot_marginal_effects(covs = select(bikedata, month), preds = pred) +
            scale_x_discrete(limits = 1:12, labels = month_labs)

    plot_marginal_effects(covs = select(bikedata, weathersituation),
                          preds = pred) +
        scale_x_discrete(limits = 1:3,labels = c("good", "medium", "bad"))

    weekday_labs <- c("Mon", "Tue", "Wed", "Thu", "Fri", "Sat", "Sun")
    plot_marginal_effects(covs = select(bikedata, weekday), preds = pred) +
        scale_x_discrete(limits = 1:7, labels = weekday_labs)

    plot_marginal_effects(covs = select(bikedata, workingday), preds = pred) +
        scale_x_discrete(limits = 0:1, labels = c("no", "yes")) +
        geom_smooth(method = "lm", se = FALSE) +
        xlim(c(0, 1))

    season_labs <- c("spring", "summer", "fall", "winter")
    plot_marginal_effects(covs = select(bikedata, season), preds = pred) +
        scale_x_discrete(limits = 1:4, labels = season_labs)