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+
-

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 .

-
-

-Load required packages

+

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 .

+
-
-

-Data analysis

+
+

Data analysis +

-

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

+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(
+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
+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 = ".")
+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.

+
+

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"),
+  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.

+

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

+
## [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$vine)
-
## # A data.frame: 6 x 11 
-##  tree edge conditioned conditioning var_types family rotation  parameters df
-##     1    1        1, 2                    c,c gumbel      180         5.2  1
-##     1    2        2, 4                    c,c gumbel      180         2.4  1
-##     1    3        4, 3                    c,c gumbel      180         2.4  1
-##     2    1        1, 4            2       c,c      t        0 0.45, 16.94  2
-##     2    2        2, 3            4       c,c      t        0  0.91, 4.68  2
-##     3    1        1, 3         4, 2       c,c      t        0  0.31, 7.36  2
-##   tau loglik
-##  0.81   1629
-##  0.58    672
-##  0.59    709
-##  0.30    147
-##  0.73   1193
-##  0.20     72
+summary(fit_vine_par$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$vine)
+contour(fit_vine_par$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.

+
+

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.

+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)
+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 0.6723651 0.7666475 0.8713561
-## 2 0.6972505 0.7904207 0.9005937
-## 3 0.6793975 0.7840285 0.8923618
-## 4 0.7759436 0.8799531 0.9967233
-## 5 0.5958959 0.6999980 0.8270238
-## 6 0.6762859 0.7730820 0.8854436
-

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).

+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")'
+
## `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.

+

The fitted quantile curves suggest a non-linear effect of all three +variables.

-
-

-Comparison to the benchmark model: linear quantile regression

+
+

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(
+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)
+    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")
+    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.

+
+

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,
@@ -244,41 +280,54 @@ 

selcrit = "aic" ) fit_vine_np

-
## D-vine regression model: whole | length, height, diameter 
-## nobs = 1306, edf = 95.15, cll = 1209.48, caic = -2228.66, cbic = -1736.29
+
## D-vine regression model: whole | length, height, diameter 
+## nobs = 1306, edf = 95.59, cll = 1208.61, caic = -2226.04, cbic = -1731.39
-contour(fit_vine_np$vine)
+contour(fit_vine_np$vine)

-

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

+

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")'
+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.

+

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.

+
+

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)
+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 = -2782.44, caic = 5582.88, cbic = 5629.45
+
## D-vine regression model: rings | shell, shucked, whole, height, viscera, length, diameter 
+## nobs = 1306, edf = 9, cll = -2782.06, caic = 5582.12, cbic = 5628.69
 plot_effects(fit_disc)
-
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
+
## `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

+
+

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

@@ -293,11 +342,14 @@

-

Site built with pkgdown 1.6.1.

+

+

Site built with pkgdown +2.0.1.

@@ -306,5 +358,7 @@

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    +
    - -
    -

    -Plot function for marginal effects

    +
    +

    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("")
    +    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

    +
    +

    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
    +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

    +
    +

    Rename variables +

    -bikedata  <- bikedata %>%
    -    rename(
    +bikedata  <- bikedata %>%
    +    rename(
             temperature = atemp, 
             month = mnth,
             weathersituation = weathersit,
    @@ -165,146 +163,145 @@ 

    count = cnt )

    -
    -

    -Un-normalize variables

    +
    +

    Un-normalize variables +

    See variable description on UCI web page.

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

    -Show trend

    +
    +

    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))
    +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

    +
    +

    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))
    +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

    +
    +

    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)
    +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 +
    +

    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)
    +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

    +
    +

    D-vine regression model +

    +
    +

    Fit model +

     fit <- vinereg(
       count ~ ., 
       data = bikedata, 
    -  family = c("onepar", "tll"), 
    +  family = c("onepar", "tll"), 
       selcrit = "aic"
     )
     fit
    -
    ## D-vine regression model: count | temperature, humidity, windspeed, month, season, weathersituation, weekday, workingday 
    -## nobs = 731, edf = 72.82, cll = 436.36, caic = -727.09, cbic = -392.54
    +
    ## D-vine regression model: count | temperature, humidity, windspeed, month, season, weathersituation, workingday, weekday 
    +## nobs = 731, edf = 55.49, cll = 410.19, caic = -709.39, cbic = -454.45
    -summary(fit)
    -
    ##                var      edf         cll       caic       cbic       p_value
    -## 1            count  9.59683 -198.076002  415.34567  459.43747            NA
    -## 2      temperature 21.98331  415.786154 -787.60568 -686.60525 1.128529e-161
    -## 3         humidity 17.93515  118.945781 -202.02127 -119.61979  2.149975e-40
    -## 4        windspeed  1.00000   22.843501  -43.68700  -39.09259  1.387404e-11
    -## 5            month  1.00000   13.846220  -25.69244  -21.09803  1.422175e-07
    -## 6           season  1.00000   12.324711  -22.64942  -18.05501  6.876540e-07
    -## 7 weathersituation  1.00000   12.737383  -23.47477  -18.88035  4.482071e-07
    -## 8          weekday 18.30101   29.457650  -22.31328   61.76912  3.708185e-06
    -## 9       workingday  1.00000    8.494936  -14.98987  -10.39546  3.757975e-05
    +summary(fit)
    +
    ##                var      edf         cll        caic        cbic       p_value
    +## 1            count  9.59683 -198.076002  415.345665  459.437472            NA
    +## 2      temperature 21.96569  415.796680 -787.661982 -686.742527 1.081331e-161
    +## 3         humidity 17.92777  118.896648 -201.937762 -119.570188  2.228987e-40
    +## 4        windspeed  1.00000   22.818931  -43.637862  -39.043448  1.422649e-11
    +## 5            month  1.00000   13.494037  -24.988073  -20.393660  2.047147e-07
    +## 6           season  1.00000   11.793281  -21.586562  -16.992149  1.194186e-06
    +## 7 weathersituation  1.00000   11.308234  -20.616468  -16.022054  1.977821e-06
    +## 8       workingday  1.00000   10.001404  -18.002808  -13.408395  7.732851e-06
    +## 9          weekday  1.00000    4.152908   -6.305817   -1.711403  3.951831e-03
    -
    -

    -In-sample predictions

    +
    +

    In-sample predictions +

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

    -Marginal effects

    +
    +

    Marginal effects +

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

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

    @@ -321,11 +318,14 @@

    -

    Site built with pkgdown 1.6.1.

    +

    +

    Site built with pkgdown +2.0.1.

    @@ -334,5 +334,7 @@

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