diff --git a/M1/General Linear Models/Projet/GLM Code - DANJOU & DUROUSSEAU.rmd b/M1/General Linear Models/Projet/GLM Code - DANJOU & DUROUSSEAU.rmd
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+---
+title: "Generalized Linear Models - Project by Arthur DANJOU"
+always_allow_html: true
+output:
+html_document:
+toc: true
+toc_depth: 4
+fig_caption: true
+---
+
+## Data Analysis
+
+```{r}
+setwd('/Users/arthurdanjou/Workspace/studies/M1/General Linear Models/Projet')
+```
+
+```{r}
+library(MASS)
+library(AER)
+library(rmarkdown)
+library(car)
+library(corrplot)
+library(carData)
+library(ggfortify)
+library(ggplot2)
+library(gridExtra)
+library(caret)
+```
+
+### Data Preprocessing
+```{r}
+data <- read.csv("./projet.csv", header = TRUE, sep = ",", dec = ".")
+
+# Factors
+data$saison <- as.factor(data$saison)
+data$mois <- as.factor(data$mois)
+data$jour_mois <- as.factor(data$jour_mois)
+data$jour_semaine <- as.factor(data$jour_semaine)
+data$horaire <- as.factor(data$horaire)
+data$jour_travail <- as.factor(data$jour_travail)
+data$vacances <- as.factor(data$vacances)
+data$meteo <- as.factor(data$meteo)
+
+# Quantitative variables
+data$humidite_sqrt <- sqrt(data$humidite)
+data$humidite_square <- data$humidite^2
+
+data$temperature1_square <- data$temperature1^2
+data$temperature2_square <- data$temperature2^2
+
+data$vent_square <- data$vent^2
+data$vent_sqrt <- sqrt(data$vent)
+
+# Remove obs column
+rownames(data) <- data$obs
+data <- data[, -1]
+paged_table(data)
+```
+
+```{r}
+colSums(is.na(data))
+sum(duplicated(data))
+```
+
+```{r}
+str(data)
+summary(data)
+```
+
+### Study of the Quantitative Variables
+
+#### Distribution of Variables
+
+```{r}
+plot_velos <- ggplot(data, aes(x = velos)) +
+ geom_histogram(bins = 25, fill = "blue") +
+ labs(title = "Distribution du nombre de vélos loués", x = "Nombre de locations de vélos")
+
+plot_temperature1 <- ggplot(data, aes(x = temperature1)) +
+ geom_histogram(bins = 30, fill = "green") +
+ labs(title = "Distribution de la température1", x = "Température moyenne mesuré (°C)")
+
+plot_temperature2 <- ggplot(data, aes(x = temperature2)) +
+ geom_histogram(bins = 30, fill = "green") +
+ labs(title = "Distribution de la température2", x = "Température moyenne ressentie (°C)")
+
+plot_humidite <- ggplot(data, aes(x = humidite)) +
+ geom_histogram(bins = 30, fill = "green") +
+ labs(title = "Distribution de la humidité", x = "Pourcentage d'humidité")
+
+plot_vent <- ggplot(data, aes(x = vent)) +
+ geom_histogram(bins = 30, fill = "green") +
+ labs(title = "Distribution de la vent", x = "Vitesse du vent (Km/h)")
+
+grid.arrange(plot_velos, plot_temperature1, plot_temperature2, plot_humidite, plot_vent, ncol = 3)
+```
+
+#### Correlation between Quantitatives Variables
+
+```{r}
+#detach(package:arm) # If error, uncomment this line due to duplicate function 'corrplot'
+corr_matrix <- cor(data[, sapply(data, is.numeric)])
+corrplot(corr_matrix, type = "lower", tl.col = "black", tl.srt = 45)
+pairs(data[, sapply(data, is.numeric)])
+cor.test(data$temperature1, data$temperature2)
+```
+
+### Study of the Qualitative Variables
+
+```{r}
+plot_saison <- ggplot(data, aes(x = saison, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par saison", x = "Saison", y = "Nombre de vélos loués")
+
+plot_jour_semaine <- ggplot(data, aes(x = jour_semaine, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par jour de la semaine", x = "Jour de la semaine", y = "Nombre de vélos loués")
+
+plot_mois <- ggplot(data, aes(x = mois, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par mois de l'année", x = "Mois de l'année", y = "Nombre de vélos loués")
+
+plot_jour_mois <- ggplot(data, aes(x = jour_mois, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par jour du mois", x = "Jour du mois", y = "Nombre de vélos loués")
+
+plot_horaire <- ggplot(data, aes(x = horaire, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par horaire de la journée", x = "Horaire de la journée", y = "Nombre de vélos loués")
+
+plot_jour_travail <- ggplot(data, aes(x = jour_travail, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par jour travaillé", x = "Jour travaillé", y = "Nombre de vélos loués")
+
+plot_vacances <- ggplot(data, aes(x = vacances, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par vacances", x = "Vacances", y = "Nombre de vélos loués")
+
+plot_meteo <- ggplot(data, aes(x = meteo, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par météo", x = "Météo", y = "Nombre de vélos loués")
+
+grid.arrange(plot_horaire, plot_jour_semaine, plot_jour_mois, plot_mois, plot_saison, plot_jour_travail, plot_vacances, plot_meteo, ncol = 3)
+```
+
+```{r}
+chisq.test(data$mois, data$saison)
+chisq.test(data$meteo, data$saison)
+chisq.test(data$meteo, data$mois)
+```
+
+### Outliers Detection
+
+```{r}
+boxplot_velos <- ggplot(data, aes(x = "", y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de vélos", y = "Nombre de vélos")
+
+boxplot_temperature1 <- ggplot(data, aes(x = "", y = temperature1)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de température1", y = "Température moyenne mesuré (°C)")
+
+boxplot_temperature1_sq <- ggplot(data, aes(x = "", y = temperature1_square)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de température1^2", y = "Température moyenne mesuré (°C^2)")
+
+boxplot_temperature2 <- ggplot(data, aes(x = "", y = temperature2)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de température2", y = "Température moyenne ressentie (°C)")
+
+boxplot_temperature2_sq <- ggplot(data, aes(x = "", y = temperature2_square)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de température2^2", y = "Température moyenne ressentie (°C^2)")
+
+boxplot_humidite <- ggplot(data, aes(x = "", y = humidite)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de humidité", y = "Pourcentage d'humidité")
+
+boxplot_humidite_sqrt <- ggplot(data, aes(x = "", y = humidite_sqrt)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de sqrt(humidité)", y = "Pourcentage d'humidité (sqrt(%))")
+
+boxplot_humidite_square <- ggplot(data, aes(x = "", y = humidite_square)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de humidité^2", y = "Pourcentage d'humidité (%^2)")
+
+boxplot_vent <- ggplot(data, aes(x = "", y = vent)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de vent", y = "Vitesse du vent (Km/h)")
+
+boxplot_vent_sqrt <- ggplot(data, aes(x = "", y = vent_sqrt)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de sqrt(vent)", y = "Vitesse du vent sqrt(Km/h)")
+
+boxplot_vent_square <- ggplot(data, aes(x = "", y = vent_square)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de vent^2", y = "Vitesse du vent (Km/h)^2")
+
+grid.arrange(boxplot_velos, boxplot_temperature1, boxplot_temperature1_sq, boxplot_temperature2, boxplot_temperature2_sq, boxplot_humidite, boxplot_humidite_sqrt, boxplot_humidite_square, boxplot_vent, boxplot_vent_sqrt, boxplot_vent_square, ncol = 4)
+```
+
+```{r}
+length_data <- nrow(data)
+numeric_vars <- sapply(data, is.numeric)
+total_outliers <- 0
+
+for (var in names(data)[numeric_vars]) {
+ data_outlier <- data[[var]]
+ Q1 <- quantile(data_outlier, 0.25)
+ Q3 <- quantile(data_outlier, 0.75)
+ IQR <- Q3 - Q1
+
+ lower_limit <- Q1 - 1.5 * IQR
+ upper_limit <- Q3 + 1.5 * IQR
+
+ outliers <- data[data_outlier < lower_limit | data_outlier > upper_limit,]
+ cat("Number of outliers for the variable", var, ":", nrow(outliers), "\n")
+ data <- data[data_outlier >= lower_limit & data_outlier <= upper_limit,]
+}
+
+cat("Number of outliers removed :", length_data - nrow(data), "\n")
+cat("Data length after removing outliers :", nrow(data), "\n")
+```
+
+## Model Creation and Comparison
+
+### Data Split
+
+```{r}
+set.seed(123)
+data_split <- rsample::initial_split(data, prop = 0.8)
+data_train <- rsample::training(data_split)
+data_test <- rsample::testing(data_split)
+```
+
+### Choice of the Distribution *vélos*
+
+```{r}
+model_poisson <- glm(velos ~ ., family = poisson, data = data_train)
+model_nb <- glm.nb(velos ~ ., data = data_train)
+model_gaussian <- glm(velos ~ ., family = gaussian, data = data_train)
+t(AIC(model_poisson, model_nb, model_gaussian)[2])
+
+dispersiontest(model_poisson)
+
+mean_velos <- mean(data_train$velos)
+var_velos <- var(data_train$velos)
+cat("Mean :", mean_velos, "Variance :", var_velos, "\n")
+```
+
+### Model Selection
+
+```{r}
+model_full_quantitative <- glm.nb(
+ velos ~ vent +
+ vent_square +
+ vent_sqrt +
+
+ humidite +
+ humidite_square +
+ humidite_sqrt +
+
+ temperature1_square +
+ temperature1 +
+
+ temperature2 +
+ temperature2_square
+ , data = data_train)
+summary(model_full_quantitative)
+```
+
+```{r}
+anova(model_full_quantitative)
+```
+
+```{r}
+model_quantitative_quali <- glm.nb(
+ velos ~
+ vent +
+ vent_square +
+ vent_sqrt +
+ humidite +
+ humidite_square +
+ humidite_sqrt +
+ temperature1 +
+ temperature1_square +
+
+ horaire +
+ saison +
+ meteo +
+ mois +
+ vacances +
+ jour_travail +
+ jour_semaine +
+ jour_mois
+ , data = data_train)
+summary(model_quantitative_quali)
+```
+
+```{r}
+anova(model_quantitative_quali)
+```
+
+```{r}
+model_quanti_quali_final <- glm.nb(
+ velos ~
+ vent +
+ vent_square + #
+ humidite +
+ humidite_square +
+ humidite_sqrt + #
+ temperature1 +
+ temperature1_square +
+
+ horaire +
+ saison +
+ meteo +
+ mois +
+ vacances
+ , data = data_train)
+summary(model_quanti_quali_final)
+```
+
+```{r}
+model_0 <- glm.nb(velos ~ 1, data = data_train)
+model_forward <- stepAIC(
+ model_0,
+ vélos ~ vent +
+ humidite +
+ humidite_square +
+ temperature1 +
+ temperature1_square +
+
+ horaire +
+ saison +
+ meteo +
+ mois +
+ vacances,
+ data = data_train,
+ trace = FALSE,
+ direction = "forward"
+)
+model_backward <- stepAIC(
+ model_quanti_quali_final,
+ ~1,
+ trace = FALSE,
+ direction = "backward"
+)
+model_both <- stepAIC(
+ model_0,
+ vélos ~ vent +
+ humidite +
+ humidite_square +
+ temperature1 +
+ temperature1_square +
+
+ horaire +
+ saison +
+ meteo +
+ mois +
+ vacances,
+ data = data_train,
+ trace = FALSE,
+ direction = "both"
+)
+AIC(model_forward, model_both, model_backward)
+summary(model_forward)
+```
+
+```{r}
+model_final_without_interaction <- glm.nb(
+ velos ~ horaire +
+ mois +
+ meteo +
+ temperature1 +
+ saison +
+ temperature1_square +
+ humidite_square +
+ vacances +
+ humidite +
+ vent
+ , data = data_train)
+summary(model_final_without_interaction)
+```
+
+### Final Model
+
+#### Choice and Validation of the Model
+
+```{r}
+model_final <- glm.nb(
+ velos ~ horaire +
+ mois +
+ meteo +
+ temperature1 +
+ saison +
+ temperature1_square +
+ humidite_square +
+ vacances +
+ humidite +
+ vent +
+
+ horaire:temperature1 +
+ temperature1_square:mois
+ , data = data_train
+)
+summary(model_final)
+```
+
+```{r}
+anova(model_final, test = "Chisq")
+```
+
+```{r}
+lrtest(model_final, model_final_without_interaction)
+```
+
+```{r}
+dispersion_ratio <- sum(residuals(model_final, type = "pearson")^2) / df.residual(model_final)
+print(paste("Dispersion Ratio:", round(dispersion_ratio, 2)))
+
+hist(residuals(model_final, type = "deviance"), breaks = 30, freq = FALSE, col = "lightblue", main = "Histogram of Deviance Residuals")
+x <- seq(-5, 5, length = 100)
+curve(dnorm(x), col = "darkblue", lwd = 2, add = TRUE)
+
+autoplot(model_final, 1:4)
+```
+
+```{r}
+library(arm)
+binnedplot(fitted(model_final), residuals(model_final, type = "deviance"), col.int = "blue", col.pts = 2)
+```
+
+## Performance and Limitations of the Final Model
+
+### Predictions and *Mean Square Error*
+
+```{r}
+data_test$predictions <- predict(model_final, newdata = data_test, type = "response")
+data_train$predictions <- predict(model_final, newdata = data_train, type = "response")
+
+predictions_ci <- predict(
+ model_final,
+ newdata = data_test,
+ type = "link",
+ se.fit = TRUE
+)
+
+data_test$lwr <- exp(predictions_ci$fit - 1.96 * predictions_ci$se.fit)
+data_test$upr <- exp(predictions_ci$fit + 1.96 * predictions_ci$se.fit)
+
+MSE <- mean((data_test$velos - data_test$predictions)^2)
+cat("MSE :", MSE, "\n")
+cat("RMSE train :", sqrt(mean((data_train$velos - data_train$predictions)^2)), "\n")
+cat('RMSE :', sqrt(MSE), '\n')
+```
+
+### Evaluation of Model Performance
+
+```{r}
+bounds <- c(200, 650)
+
+cat("Observations vs Prédictions\n")
+cat(nrow(data_test[data_test$velos < bounds[1],]), "vs", sum(data_test$predictions < bounds[1]), "\n")
+cat(nrow(data_test[data_test$velos > bounds[1] & data_test$velos < bounds[2],]), "vs", sum(data_test$predictions > bounds[1] & data_test$predictions < bounds[2]), "\n")
+cat(nrow(data_test[data_test$velos > bounds[2],]), "vs", sum(data_test$predictions > bounds[2]), "\n")
+cat('\n')
+
+categories <- c(0, bounds, Inf)
+categories_label <- c("Low", "Mid", "High")
+
+data_test$velos_cat <- cut(data_test$velos,
+ breaks = categories,
+ labels = categories_label,
+ include.lowest = TRUE)
+data_test$predictions_cat <- cut(data_test$predictions,
+ breaks = categories,
+ labels = categories_label,
+ include.lowest = TRUE)
+conf_matrix <- confusionMatrix(data_test$velos_cat, data_test$predictions_cat)
+conf_matrix
+```
+
+```{r}
+ggplot(data_test, aes(x = velos, y = predictions)) +
+ geom_point(color = "blue", alpha = 0.6, size = 2) +
+ geom_abline(slope = 1, intercept = 0, color = "red", linetype = "dashed") +
+ geom_ribbon(aes(ymin = lwr, ymax = upr), alpha = 0.2, fill = "grey") +
+ labs(
+ title = "Comparison of Observed and Predicted Bikes",
+ x = "Number of Observed Bikes",
+ y = "Number of Predicted Bikes"
+ ) +
+ theme_minimal()
+
+ggplot(data_test, aes(x = velos_cat, fill = predictions_cat)) +
+ geom_bar(position = "dodge") +
+ labs(title = "Predictions vs Observations (by categories)",
+ x = "Observed categories", y = "Number of predictions")
+
+df_plot <- data.frame(observed = data_test$velos, predicted = data_test$predictions)
+
+# Créer l'histogramme avec la courbe de densité
+ggplot(df_plot, aes(x = observed)) +
+ geom_histogram(aes(y = ..density..), binwidth = 50, fill = "lightblue", color = "black", alpha = 0.7) + # Histogramme des données observées
+ geom_density(aes(x = predicted), color = "red", size = 1) + # Courbe de densité des valeurs prédites
+ labs(title = "Observed distribution vs. Predicted distribution",
+ x = "Number of vélos",
+ y = "Density") +
+ theme_bw()
+
+```
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Data Analysis
+
setwd('/Users/arthurdanjou/Workspace/studies/M1/General Linear Models/Projet')
+
library(MASS)
+library(AER)
+
## Loading required package: car
+
## Loading required package: carData
+
## Loading required package: lmtest
+
## Loading required package: zoo
+
##
+## Attaching package: 'zoo'
+
## The following objects are masked from 'package:base':
+##
+## as.Date, as.Date.numeric
+
## Loading required package: sandwich
+
## Loading required package: survival
+
library(rmarkdown)
+library(car)
+library(corrplot)
+
## corrplot 0.95 loaded
+
library(carData)
+library(ggfortify)
+
## Loading required package: ggplot2
+
library(ggplot2)
+library(gridExtra)
+library(caret)
+
## Loading required package: lattice
+
##
+## Attaching package: 'caret'
+
## The following object is masked from 'package:survival':
+##
+## cluster
+
+
Data Preprocessing
+
data <- read.csv("./projet.csv", header = TRUE, sep = ",", dec = ".")
+
+# Factors
+data$saison <- as.factor(data$saison)
+data$mois <- as.factor(data$mois)
+data$jour_mois <- as.factor(data$jour_mois)
+data$jour_semaine <- as.factor(data$jour_semaine)
+data$horaire <- as.factor(data$horaire)
+data$jour_travail <- as.factor(data$jour_travail)
+data$vacances <- as.factor(data$vacances)
+data$meteo <- as.factor(data$meteo)
+
+# Quantitative variables
+data$humidite_sqrt <- sqrt(data$humidite)
+data$humidite_square <- data$humidite^2
+
+data$temperature1_square <- data$temperature1^2
+data$temperature2_square <- data$temperature2^2
+
+data$vent_square <- data$vent^2
+data$vent_sqrt <- sqrt(data$vent)
+
+# Remove obs column
+rownames(data) <- data$obs
+data <- data[, -1]
+paged_table(data)
+
+
+
+
colSums(is.na(data))
+
## saison mois jour_mois jour_semaine
+## 0 0 0 0
+## horaire jour_travail vacances meteo
+## 0 0 0 0
+## velos humidite vent temperature1
+## 0 0 0 0
+## temperature2 humidite_sqrt humidite_square temperature1_square
+## 0 0 0 0
+## temperature2_square vent_square vent_sqrt
+## 0 0 0
+
sum(duplicated(data))
+
## [1] 0
+
str(data)
+
## 'data.frame': 1817 obs. of 19 variables:
+## $ saison : Factor w/ 4 levels "1","2","3","4": 1 1 1 1 1 1 1 1 1 1 ...
+## $ mois : Factor w/ 12 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
+## $ jour_mois : Factor w/ 31 levels "1","2","3","4",..: 1 1 1 1 1 2 2 2 2 2 ...
+## $ jour_semaine : Factor w/ 7 levels "1","2","3","4",..: 7 7 7 7 7 1 1 1 1 1 ...
+## $ horaire : Factor w/ 5 levels "1","2","3","4",..: 1 2 3 4 5 1 2 3 4 5 ...
+## $ jour_travail : Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
+## $ vacances : Factor w/ 2 levels "1","2": 1 1 1 1 1 1 1 1 1 1 ...
+## $ meteo : Factor w/ 3 levels "1","2","3": 1 1 1 2 2 2 2 2 3 1 ...
+## $ velos : int 105 61 340 305 174 54 82 297 268 100 ...
+## $ humidite : num 78 78.2 75.5 82.2 88.8 ...
+## $ vent : num 1.38 6.84 30.98 30.58 26.39 ...
+## $ temperature1 : num 2.88 5.39 11.98 12.21 11.55 ...
+## $ temperature2 : num 2.29 5.25 11.5 12.5 11.8 ...
+## $ humidite_sqrt : num 8.83 8.85 8.69 9.07 9.42 ...
+## $ humidite_square : num 6084 6123 5700 6765 7885 ...
+## $ temperature1_square: num 8.29 29.05 143.52 149.08 133.4 ...
+## $ temperature2_square: num 5.24 27.56 132.25 156.25 139.24 ...
+## $ vent_square : num 1.9 46.8 959.8 935.1 696.4 ...
+## $ vent_sqrt : num 1.17 2.62 5.57 5.53 5.14 ...
+
summary(data)
+
## saison mois jour_mois jour_semaine horaire jour_travail
+## 1:444 3 :155 1 : 60 1:259 1:362 1: 573
+## 2:460 5 :155 2 : 60 2:260 2:363 2:1244
+## 3:468 7 :155 3 : 60 3:258 3:364
+## 4:445 10 :155 4 : 60 4:259 4:365
+## 12 :155 5 : 60 5:257 5:363
+## 8 :153 6 : 60 6:260
+## (Other):889 (Other):1457 7:264
+## vacances meteo velos humidite vent
+## 1:1767 1:1192 Min. : 7.0 Min. : 0.00 Min. : 0.00
+## 2: 50 2: 464 1st Qu.: 275.0 1st Qu.: 48.40 1st Qu.:12.47
+## 3: 161 Median : 619.0 Median : 62.75 Median :20.00
+## Mean : 684.2 Mean : 62.94 Mean :21.23
+## 3rd Qu.:1041.0 3rd Qu.: 78.00 3rd Qu.:28.28
+## Max. :2036.0 Max. :100.00 Max. :68.66
+##
+## temperature1 temperature2 humidite_sqrt humidite_square
+## Min. :-6.59 Min. :-14.50 Min. : 0.000 Min. : 0
+## 1st Qu.: 7.51 1st Qu.: 5.25 1st Qu.: 6.957 1st Qu.: 2343
+## Median :15.31 Median : 15.80 Median : 7.921 Median : 3938
+## Mean :15.24 Mean : 15.25 Mean : 7.828 Mean : 4323
+## 3rd Qu.:23.15 3rd Qu.: 25.00 3rd Qu.: 8.832 3rd Qu.: 6084
+## Max. :36.65 Max. : 47.25 Max. :10.000 Max. :10000
+##
+## temperature1_square temperature2_square vent_square vent_sqrt
+## Min. : 0.0001 Min. : 0.00 Min. : 0.0 Min. :0.000
+## 1st Qu.: 56.4001 1st Qu.: 39.06 1st Qu.: 155.5 1st Qu.:3.531
+## Median : 234.3961 Median : 249.64 Median : 400.0 Median :4.472
+## Mean : 319.6898 Mean : 369.70 Mean : 585.5 Mean :4.400
+## 3rd Qu.: 535.9225 3rd Qu.: 625.00 3rd Qu.: 799.8 3rd Qu.:5.318
+## Max. :1343.2225 Max. :2232.56 Max. :4714.2 Max. :8.286
+##
+
+
+
Study of the Quantitative Variables
+
+
Distribution of Variables
+
plot_velos <- ggplot(data, aes(x = velos)) +
+ geom_histogram(bins = 25, fill = "blue") +
+ labs(title = "Distribution du nombre de vélos loués", x = "Nombre de locations de vélos")
+
+plot_temperature1 <- ggplot(data, aes(x = temperature1)) +
+ geom_histogram(bins = 30, fill = "green") +
+ labs(title = "Distribution de la température1", x = "Température moyenne mesuré (°C)")
+
+plot_temperature2 <- ggplot(data, aes(x = temperature2)) +
+ geom_histogram(bins = 30, fill = "green") +
+ labs(title = "Distribution de la température2", x = "Température moyenne ressentie (°C)")
+
+plot_humidite <- ggplot(data, aes(x = humidite)) +
+ geom_histogram(bins = 30, fill = "green") +
+ labs(title = "Distribution de la humidité", x = "Pourcentage d'humidité")
+
+plot_vent <- ggplot(data, aes(x = vent)) +
+ geom_histogram(bins = 30, fill = "green") +
+ labs(title = "Distribution de la vent", x = "Vitesse du vent (Km/h)")
+
+grid.arrange(plot_velos, plot_temperature1, plot_temperature2, plot_humidite, plot_vent, ncol = 3)
+

+
+
+
Correlation between Quantitatives Variables
+
#detach(package:arm) # If error, uncomment this line due to duplicate function 'corrplot'
+corr_matrix <- cor(data[, sapply(data, is.numeric)])
+corrplot(corr_matrix, type = "lower", tl.col = "black", tl.srt = 45)
+

+
pairs(data[, sapply(data, is.numeric)])
+

+
cor.test(data$temperature1, data$temperature2)
+
##
+## Pearson's product-moment correlation
+##
+## data: data$temperature1 and data$temperature2
+## t = 384.61, df = 1815, p-value < 2.2e-16
+## alternative hypothesis: true correlation is not equal to 0
+## 95 percent confidence interval:
+## 0.9933370 0.9944541
+## sample estimates:
+## cor
+## 0.993921
+
+
+
+
Study of the Qualitative Variables
+
plot_saison <- ggplot(data, aes(x = saison, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par saison", x = "Saison", y = "Nombre de vélos loués")
+
+plot_jour_semaine <- ggplot(data, aes(x = jour_semaine, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par jour de la semaine", x = "Jour de la semaine", y = "Nombre de vélos loués")
+
+plot_mois <- ggplot(data, aes(x = mois, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par mois de l'année", x = "Mois de l'année", y = "Nombre de vélos loués")
+
+plot_jour_mois <- ggplot(data, aes(x = jour_mois, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par jour du mois", x = "Jour du mois", y = "Nombre de vélos loués")
+
+plot_horaire <- ggplot(data, aes(x = horaire, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par horaire de la journée", x = "Horaire de la journée", y = "Nombre de vélos loués")
+
+plot_jour_travail <- ggplot(data, aes(x = jour_travail, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par jour travaillé", x = "Jour travaillé", y = "Nombre de vélos loués")
+
+plot_vacances <- ggplot(data, aes(x = vacances, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par vacances", x = "Vacances", y = "Nombre de vélos loués")
+
+plot_meteo <- ggplot(data, aes(x = meteo, y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Nombre de vélos par météo", x = "Météo", y = "Nombre de vélos loués")
+
+grid.arrange(plot_horaire, plot_jour_semaine, plot_jour_mois, plot_mois, plot_saison, plot_jour_travail, plot_vacances, plot_meteo, ncol = 3)
+

+
chisq.test(data$mois, data$saison)
+
##
+## Pearson's Chi-squared test
+##
+## data: data$mois and data$saison
+## X-squared = 4380.8, df = 33, p-value < 2.2e-16
+
chisq.test(data$meteo, data$saison)
+
##
+## Pearson's Chi-squared test
+##
+## data: data$meteo and data$saison
+## X-squared = 20.78, df = 6, p-value = 0.002009
+
chisq.test(data$meteo, data$mois)
+
##
+## Pearson's Chi-squared test
+##
+## data: data$meteo and data$mois
+## X-squared = 91.211, df = 22, p-value = 2.13e-10
+
+
+
Outliers Detection
+
boxplot_velos <- ggplot(data, aes(x = "", y = velos)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de vélos", y = "Nombre de vélos")
+
+boxplot_temperature1 <- ggplot(data, aes(x = "", y = temperature1)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de température1", y = "Température moyenne mesuré (°C)")
+
+boxplot_temperature1_sq <- ggplot(data, aes(x = "", y = temperature1_square)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de température1^2", y = "Température moyenne mesuré (°C^2)")
+
+boxplot_temperature2 <- ggplot(data, aes(x = "", y = temperature2)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de température2", y = "Température moyenne ressentie (°C)")
+
+boxplot_temperature2_sq <- ggplot(data, aes(x = "", y = temperature2_square)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de température2^2", y = "Température moyenne ressentie (°C^2)")
+
+boxplot_humidite <- ggplot(data, aes(x = "", y = humidite)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de humidité", y = "Pourcentage d'humidité")
+
+boxplot_humidite_sqrt <- ggplot(data, aes(x = "", y = humidite_sqrt)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de sqrt(humidité)", y = "Pourcentage d'humidité (sqrt(%))")
+
+boxplot_humidite_square <- ggplot(data, aes(x = "", y = humidite_square)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de humidité^2", y = "Pourcentage d'humidité (%^2)")
+
+boxplot_vent <- ggplot(data, aes(x = "", y = vent)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de vent", y = "Vitesse du vent (Km/h)")
+
+boxplot_vent_sqrt <- ggplot(data, aes(x = "", y = vent_sqrt)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de sqrt(vent)", y = "Vitesse du vent sqrt(Km/h)")
+
+boxplot_vent_square <- ggplot(data, aes(x = "", y = vent_square)) +
+ geom_boxplot(fill = "lightblue") +
+ labs(title = "Boxplot de vent^2", y = "Vitesse du vent (Km/h)^2")
+
+grid.arrange(boxplot_velos, boxplot_temperature1, boxplot_temperature1_sq, boxplot_temperature2, boxplot_temperature2_sq, boxplot_humidite, boxplot_humidite_sqrt, boxplot_humidite_square, boxplot_vent, boxplot_vent_sqrt, boxplot_vent_square, ncol = 4)
+

+
length_data <- nrow(data)
+numeric_vars <- sapply(data, is.numeric)
+total_outliers <- 0
+
+for (var in names(data)[numeric_vars]) {
+ data_outlier <- data[[var]]
+ Q1 <- quantile(data_outlier, 0.25)
+ Q3 <- quantile(data_outlier, 0.75)
+ IQR <- Q3 - Q1
+
+ lower_limit <- Q1 - 1.5 * IQR
+ upper_limit <- Q3 + 1.5 * IQR
+
+ outliers <- data[data_outlier < lower_limit | data_outlier > upper_limit,]
+ cat("Number of outliers for the variable", var, ":", nrow(outliers), "\n")
+ data <- data[data_outlier >= lower_limit & data_outlier <= upper_limit,]
+}
+
## Number of outliers for the variable velos : 0
+## Number of outliers for the variable humidite : 5
+## Number of outliers for the variable vent : 24
+## Number of outliers for the variable temperature1 : 0
+## Number of outliers for the variable temperature2 : 0
+## Number of outliers for the variable humidite_sqrt : 1
+## Number of outliers for the variable humidite_square : 0
+## Number of outliers for the variable temperature1_square : 4
+## Number of outliers for the variable temperature2_square : 11
+## Number of outliers for the variable vent_square : 80
+## Number of outliers for the variable vent_sqrt : 27
+
cat("Number of outliers removed :", length_data - nrow(data), "\n")
+
## Number of outliers removed : 152
+
cat("Data length after removing outliers :", nrow(data), "\n")
+
## Data length after removing outliers : 1665
+
+
+
+
Model Creation and Comparison
+
+
Data Split
+
set.seed(123)
+data_split <- rsample::initial_split(data, prop = 0.8)
+data_train <- rsample::training(data_split)
+data_test <- rsample::testing(data_split)
+
+
+
Choice of the Distribution vélos
+
model_poisson <- glm(velos ~ ., family = poisson, data = data_train)
+model_nb <- glm.nb(velos ~ ., data = data_train)
+model_gaussian <- glm(velos ~ ., family = gaussian, data = data_train)
+t(AIC(model_poisson, model_nb, model_gaussian)[2])
+
## model_poisson model_nb model_gaussian
+## AIC 82988.75 17532.5 18420.81
+
dispersiontest(model_poisson)
+
##
+## Overdispersion test
+##
+## data: model_poisson
+## z = 20.793, p-value < 2.2e-16
+## alternative hypothesis: true dispersion is greater than 1
+## sample estimates:
+## dispersion
+## 54.85188
+
mean_velos <- mean(data_train$velos)
+var_velos <- var(data_train$velos)
+cat("Mean :", mean_velos, "Variance :", var_velos, "\n")
+
## Mean : 693.4317 Variance : 214261.1
+
+
+
Model Selection
+
model_full_quantitative <- glm.nb(
+ velos ~ vent +
+ vent_square +
+ vent_sqrt +
+
+ humidite +
+ humidite_square +
+ humidite_sqrt +
+
+ temperature1_square +
+ temperature1 +
+
+ temperature2 +
+ temperature2_square
+ , data = data_train)
+summary(model_full_quantitative)
+
##
+## Call:
+## glm.nb(formula = velos ~ vent + vent_square + vent_sqrt + humidite +
+## humidite_square + humidite_sqrt + temperature1_square + temperature1 +
+## temperature2 + temperature2_square, data = data_train, init.theta = 2.91504998,
+## link = log)
+##
+## Coefficients:
+## Estimate Std. Error z value Pr(>|z|)
+## (Intercept) 8.2125765 2.2378964 3.670 0.000243 ***
+## vent 0.0281498 0.0499554 0.563 0.573096
+## vent_square -0.0002027 0.0004951 -0.409 0.682254
+## vent_sqrt -0.1673324 0.2546295 -0.657 0.511078
+## humidite 0.1061905 0.0816427 1.301 0.193370
+## humidite_square -0.0004965 0.0002360 -2.104 0.035388 *
+## humidite_sqrt -0.8882801 0.8081472 -1.099 0.271700
+## temperature1_square -0.0010461 0.0009065 -1.154 0.248518
+## temperature1 0.0702146 0.0379386 1.851 0.064206 .
+## temperature2 0.0260605 0.0252172 1.033 0.301397
+## temperature2_square -0.0006738 0.0005534 -1.218 0.223397
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+##
+## (Dispersion parameter for Negative Binomial(2.915) family taken to be 1)
+##
+## Null deviance: 2358.2 on 1331 degrees of freedom
+## Residual deviance: 1408.9 on 1321 degrees of freedom
+## AIC: 19155
+##
+## Number of Fisher Scoring iterations: 1
+##
+##
+## Theta: 2.915
+## Std. Err.: 0.108
+##
+## 2 x log-likelihood: -19131.246
+
anova(model_full_quantitative)
+
## Warning in anova.negbin(model_full_quantitative): tests made without
+## re-estimating 'theta'
+
## Analysis of Deviance Table
+##
+## Model: Negative Binomial(2.915), link: log
+##
+## Response: velos
+##
+## Terms added sequentially (first to last)
+##
+##
+## Df Deviance Resid. Df Resid. Dev Pr(>Chi)
+## NULL 1331 2358.2
+## vent 1 8.84 1330 2349.3 0.002946 **
+## vent_square 1 4.04 1329 2345.3 0.044442 *
+## vent_sqrt 1 0.33 1328 2345.0 0.568601
+## humidite 1 183.86 1327 2161.1 < 2.2e-16 ***
+## humidite_square 1 48.09 1326 2113.0 4.067e-12 ***
+## humidite_sqrt 1 5.51 1325 2107.5 0.018912 *
+## temperature1_square 1 431.26 1324 1676.2 < 2.2e-16 ***
+## temperature1 1 265.92 1323 1410.3 < 2.2e-16 ***
+## temperature2 1 0.02 1322 1410.3 0.898079
+## temperature2_square 1 1.44 1321 1408.9 0.229603
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
model_quantitative_quali <- glm.nb(
+ velos ~
+ vent +
+ vent_square +
+ vent_sqrt +
+ humidite +
+ humidite_square +
+ humidite_sqrt +
+ temperature1 +
+ temperature1_square +
+
+ horaire +
+ saison +
+ meteo +
+ mois +
+ vacances +
+ jour_travail +
+ jour_semaine +
+ jour_mois
+ , data = data_train)
+summary(model_quantitative_quali)
+
##
+## Call:
+## glm.nb(formula = velos ~ vent + vent_square + vent_sqrt + humidite +
+## humidite_square + humidite_sqrt + temperature1 + temperature1_square +
+## horaire + saison + meteo + mois + vacances + jour_travail +
+## jour_semaine + jour_mois, data = data_train, init.theta = 10.11407919,
+## link = log)
+##
+## Coefficients: (1 not defined because of singularities)
+## Estimate Std. Error z value Pr(>|z|)
+## (Intercept) 5.7578159 1.2735469 4.521 6.15e-06 ***
+## vent 0.0065391 0.0277120 0.236 0.813457
+## vent_square 0.0000256 0.0002747 0.093 0.925750
+## vent_sqrt -0.0872814 0.1411176 -0.619 0.536245
+## humidite 0.0718573 0.0465415 1.544 0.122603
+## humidite_square -0.0003074 0.0001352 -2.273 0.023026 *
+## humidite_sqrt -0.5879792 0.4597808 -1.279 0.200959
+## temperature1 0.0538688 0.0050283 10.713 < 2e-16 ***
+## temperature1_square -0.0009887 0.0001523 -6.490 8.59e-11 ***
+## horaire2 1.4014996 0.0282339 49.639 < 2e-16 ***
+## horaire3 1.2441366 0.0342452 36.330 < 2e-16 ***
+## horaire4 1.6669509 0.0340096 49.014 < 2e-16 ***
+## horaire5 1.3009922 0.0289632 44.919 < 2e-16 ***
+## saison2 0.2113832 0.0569855 3.709 0.000208 ***
+## saison3 0.3238081 0.0647808 4.999 5.78e-07 ***
+## saison4 0.5120136 0.0557309 9.187 < 2e-16 ***
+## meteo2 -0.0388243 0.0227839 -1.704 0.088376 .
+## meteo3 -0.4183265 0.0423535 -9.877 < 2e-16 ***
+## mois2 0.1673029 0.0489754 3.416 0.000635 ***
+## mois3 0.1690126 0.0550596 3.070 0.002143 **
+## mois4 0.2656971 0.0844978 3.144 0.001664 **
+## mois5 0.5093488 0.0894762 5.693 1.25e-08 ***
+## mois6 0.4535267 0.0938529 4.832 1.35e-06 ***
+## mois7 0.3494709 0.1036055 3.373 0.000743 ***
+## mois8 0.3217628 0.1007131 3.195 0.001399 **
+## mois9 0.3638295 0.0911294 3.992 6.54e-05 ***
+## mois10 0.2067528 0.0832300 2.484 0.012987 *
+## mois11 0.1452113 0.0794983 1.827 0.067760 .
+## mois12 0.1929111 0.0622809 3.097 0.001952 **
+## vacances2 -0.1803626 0.0653453 -2.760 0.005778 **
+## jour_travail2 0.0234146 0.0328769 0.712 0.476347
+## jour_semaine2 -0.0435844 0.0340130 -1.281 0.200052
+## jour_semaine3 -0.0365816 0.0336789 -1.086 0.277397
+## jour_semaine4 -0.0390571 0.0331463 -1.178 0.238668
+## jour_semaine5 -0.0133017 0.0336982 -0.395 0.693043
+## jour_semaine6 NA NA NA NA
+## jour_semaine7 0.0302082 0.0329428 0.917 0.359148
+## jour_mois2 0.0413982 0.0679635 0.609 0.542442
+## jour_mois3 0.0756509 0.0680085 1.112 0.265978
+## jour_mois4 0.1520220 0.0659987 2.303 0.021256 *
+## jour_mois5 0.1174765 0.0679582 1.729 0.083870 .
+## jour_mois6 0.0194055 0.0678443 0.286 0.774855
+## jour_mois7 0.0515925 0.0695970 0.741 0.458510
+## jour_mois8 0.0088355 0.0679031 0.130 0.896472
+## jour_mois9 0.0763066 0.0669394 1.140 0.254314
+## jour_mois10 0.1225251 0.0661751 1.852 0.064094 .
+## jour_mois11 0.1082326 0.0713126 1.518 0.129085
+## jour_mois12 0.0648527 0.0675506 0.960 0.337024
+## jour_mois13 0.0099991 0.0692432 0.144 0.885181
+## jour_mois14 0.0894572 0.0685495 1.305 0.191892
+## jour_mois15 0.0621761 0.0702483 0.885 0.376108
+## jour_mois16 0.0553908 0.0653521 0.848 0.396674
+## jour_mois17 0.1266634 0.0669505 1.892 0.058505 .
+## jour_mois18 0.0767661 0.0652818 1.176 0.239628
+## jour_mois19 0.0961955 0.0675449 1.424 0.154397
+## jour_mois20 0.1152297 0.0705546 1.633 0.102427
+## jour_mois21 0.1368864 0.0705549 1.940 0.052362 .
+## jour_mois22 0.0723416 0.0677257 1.068 0.285450
+## jour_mois23 0.0672894 0.0678375 0.992 0.321236
+## jour_mois24 -0.0371423 0.0678872 -0.547 0.584298
+## jour_mois25 0.0099320 0.0686310 0.145 0.884935
+## jour_mois26 -0.0009005 0.0681434 -0.013 0.989456
+## jour_mois27 -0.0409743 0.0713894 -0.574 0.565999
+## jour_mois28 0.0344889 0.0671148 0.514 0.607336
+## jour_mois29 0.0385154 0.0673973 0.571 0.567683
+## jour_mois30 0.1637766 0.0670492 2.443 0.014580 *
+## jour_mois31 0.1191179 0.0777575 1.532 0.125544
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+##
+## (Dispersion parameter for Negative Binomial(10.1141) family taken to be 1)
+##
+## Null deviance: 8029.0 on 1331 degrees of freedom
+## Residual deviance: 1356.7 on 1266 degrees of freedom
+## AIC: 17531
+##
+## Number of Fisher Scoring iterations: 1
+##
+##
+## Theta: 10.114
+## Std. Err.: 0.398
+##
+## 2 x log-likelihood: -17396.536
+
anova(model_quantitative_quali)
+
## Warning in anova.negbin(model_quantitative_quali): tests made without
+## re-estimating 'theta'
+
## Analysis of Deviance Table
+##
+## Model: Negative Binomial(10.1141), link: log
+##
+## Response: velos
+##
+## Terms added sequentially (first to last)
+##
+##
+## Df Deviance Resid. Df Resid. Dev Pr(>Chi)
+## NULL 1331 8029.0
+## vent 1 30.35 1330 7998.6 3.609e-08 ***
+## vent_square 1 13.88 1329 7984.8 0.0001951 ***
+## vent_sqrt 1 1.12 1328 7983.6 0.2905323
+## humidite 1 630.37 1327 7353.3 < 2.2e-16 ***
+## humidite_square 1 164.51 1326 7188.8 < 2.2e-16 ***
+## humidite_sqrt 1 18.83 1325 7169.9 1.426e-05 ***
+## temperature1 1 2047.02 1324 5122.9 < 2.2e-16 ***
+## temperature1_square 1 340.16 1323 4782.7 < 2.2e-16 ***
+## horaire 4 2909.21 1319 1873.5 < 2.2e-16 ***
+## saison 3 285.86 1316 1587.7 < 2.2e-16 ***
+## meteo 2 114.28 1314 1473.4 < 2.2e-16 ***
+## mois 11 66.77 1303 1406.6 4.995e-10 ***
+## vacances 1 10.42 1302 1396.2 0.0012449 **
+## jour_travail 1 1.30 1301 1394.9 0.2541936
+## jour_semaine 5 4.18 1296 1390.7 0.5231791
+## jour_mois 30 34.01 1266 1356.7 0.2803690
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
model_quanti_quali_final <- glm.nb(
+ velos ~
+ vent +
+ vent_square + #
+ humidite +
+ humidite_square +
+ humidite_sqrt + #
+ temperature1 +
+ temperature1_square +
+
+ horaire +
+ saison +
+ meteo +
+ mois +
+ vacances
+ , data = data_train)
+summary(model_quanti_quali_final)
+
##
+## Call:
+## glm.nb(formula = velos ~ vent + vent_square + humidite + humidite_square +
+## humidite_sqrt + temperature1 + temperature1_square + horaire +
+## saison + meteo + mois + vacances, data = data_train, init.theta = 9.822854355,
+## link = log)
+##
+## Coefficients:
+## Estimate Std. Error z value Pr(>|z|)
+## (Intercept) 5.373e+00 1.248e+00 4.304 1.67e-05 ***
+## vent -9.504e-03 4.036e-03 -2.355 0.018529 *
+## vent_square 1.630e-04 9.481e-05 1.719 0.085660 .
+## humidite 5.968e-02 4.615e-02 1.293 0.195913
+## humidite_square -2.771e-04 1.341e-04 -2.067 0.038757 *
+## humidite_sqrt -4.645e-01 4.559e-01 -1.019 0.308254
+## temperature1 5.358e-02 4.882e-03 10.976 < 2e-16 ***
+## temperature1_square -1.033e-03 1.469e-04 -7.034 2.00e-12 ***
+## horaire2 1.398e+00 2.847e-02 49.083 < 2e-16 ***
+## horaire3 1.248e+00 3.442e-02 36.244 < 2e-16 ***
+## horaire4 1.668e+00 3.414e-02 48.867 < 2e-16 ***
+## horaire5 1.298e+00 2.919e-02 44.454 < 2e-16 ***
+## saison2 2.114e-01 5.714e-02 3.700 0.000216 ***
+## saison3 3.146e-01 6.485e-02 4.851 1.23e-06 ***
+## saison4 4.924e-01 5.589e-02 8.810 < 2e-16 ***
+## meteo2 -3.463e-02 2.274e-02 -1.523 0.127822
+## meteo3 -4.184e-01 4.210e-02 -9.938 < 2e-16 ***
+## mois2 1.657e-01 4.924e-02 3.366 0.000762 ***
+## mois3 1.718e-01 5.498e-02 3.125 0.001777 **
+## mois4 2.794e-01 8.413e-02 3.321 0.000898 ***
+## mois5 5.298e-01 8.927e-02 5.934 2.95e-09 ***
+## mois6 4.858e-01 9.293e-02 5.227 1.72e-07 ***
+## mois7 3.945e-01 1.026e-01 3.846 0.000120 ***
+## mois8 3.642e-01 9.993e-02 3.644 0.000268 ***
+## mois9 4.037e-01 9.013e-02 4.479 7.49e-06 ***
+## mois10 2.384e-01 8.283e-02 2.878 0.004005 **
+## mois11 1.710e-01 7.934e-02 2.155 0.031136 *
+## mois12 2.175e-01 6.227e-02 3.493 0.000478 ***
+## vacances2 -1.746e-01 5.379e-02 -3.246 0.001170 **
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+##
+## (Dispersion parameter for Negative Binomial(9.8229) family taken to be 1)
+##
+## Null deviance: 7803.6 on 1331 degrees of freedom
+## Residual deviance: 1357.7 on 1303 degrees of freedom
+## AIC: 17496
+##
+## Number of Fisher Scoring iterations: 1
+##
+##
+## Theta: 9.823
+## Std. Err.: 0.386
+##
+## 2 x log-likelihood: -17436.000
+
model_0 <- glm.nb(velos ~ 1, data = data_train)
+model_forward <- stepAIC(
+ model_0,
+ vélos ~ vent +
+ humidite +
+ humidite_square +
+ temperature1 +
+ temperature1_square +
+
+ horaire +
+ saison +
+ meteo +
+ mois +
+ vacances,
+ data = data_train,
+ trace = FALSE,
+ direction = "forward"
+)
+model_backward <- stepAIC(
+ model_quanti_quali_final,
+ ~1,
+ trace = FALSE,
+ direction = "backward"
+)
+model_both <- stepAIC(
+ model_0,
+ vélos ~ vent +
+ humidite +
+ humidite_square +
+ temperature1 +
+ temperature1_square +
+
+ horaire +
+ saison +
+ meteo +
+ mois +
+ vacances,
+ data = data_train,
+ trace = FALSE,
+ direction = "both"
+)
+AIC(model_forward, model_both, model_backward)
+
## df AIC
+## model_forward 28 17496.08
+## model_both 28 17496.08
+## model_backward 29 17495.00
+
summary(model_forward)
+
##
+## Call:
+## glm.nb(formula = velos ~ horaire + mois + meteo + temperature1 +
+## saison + humidite_square + temperature1_square + humidite +
+## vacances + vent, data = data_train, init.theta = 9.793225092,
+## link = log)
+##
+## Coefficients:
+## Estimate Std. Error z value Pr(>|z|)
+## (Intercept) 4.073e+00 1.134e-01 35.913 < 2e-16 ***
+## horaire2 1.397e+00 2.851e-02 49.005 < 2e-16 ***
+## horaire3 1.247e+00 3.442e-02 36.232 < 2e-16 ***
+## horaire4 1.668e+00 3.419e-02 48.794 < 2e-16 ***
+## horaire5 1.296e+00 2.921e-02 44.351 < 2e-16 ***
+## mois2 1.675e-01 4.927e-02 3.399 0.000676 ***
+## mois3 1.740e-01 5.504e-02 3.161 0.001573 **
+## mois4 2.850e-01 8.407e-02 3.390 0.000700 ***
+## mois5 5.331e-01 8.929e-02 5.971 2.36e-09 ***
+## mois6 4.899e-01 9.302e-02 5.266 1.39e-07 ***
+## mois7 4.001e-01 1.027e-01 3.896 9.77e-05 ***
+## mois8 3.725e-01 9.997e-02 3.726 0.000195 ***
+## mois9 4.072e-01 9.024e-02 4.512 6.42e-06 ***
+## mois10 2.420e-01 8.290e-02 2.919 0.003510 **
+## mois11 1.733e-01 7.945e-02 2.181 0.029176 *
+## mois12 2.213e-01 6.234e-02 3.550 0.000385 ***
+## meteo2 -3.797e-02 2.272e-02 -1.672 0.094590 .
+## meteo3 -4.297e-01 4.158e-02 -10.334 < 2e-16 ***
+## temperature1 5.379e-02 4.887e-03 11.007 < 2e-16 ***
+## saison2 2.070e-01 5.679e-02 3.644 0.000268 ***
+## saison3 3.105e-01 6.468e-02 4.801 1.58e-06 ***
+## saison4 4.910e-01 5.597e-02 8.773 < 2e-16 ***
+## humidite_square -1.372e-04 2.661e-05 -5.157 2.51e-07 ***
+## temperature1_square -1.045e-03 1.468e-04 -7.113 1.13e-12 ***
+## humidite 1.206e-02 3.387e-03 3.560 0.000371 ***
+## vacances2 -1.718e-01 5.383e-02 -3.192 0.001414 **
+## vent -2.818e-03 1.028e-03 -2.741 0.006118 **
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+##
+## (Dispersion parameter for Negative Binomial(9.7932) family taken to be 1)
+##
+## Null deviance: 7780.7 on 1331 degrees of freedom
+## Residual deviance: 1357.8 on 1305 degrees of freedom
+## AIC: 17496
+##
+## Number of Fisher Scoring iterations: 1
+##
+##
+## Theta: 9.793
+## Std. Err.: 0.385
+##
+## 2 x log-likelihood: -17440.082
+
model_final_without_interaction <- glm.nb(
+ velos ~ horaire +
+ mois +
+ meteo +
+ temperature1 +
+ saison +
+ temperature1_square +
+ humidite_square +
+ vacances +
+ humidite +
+ vent
+ , data = data_train)
+summary(model_final_without_interaction)
+
##
+## Call:
+## glm.nb(formula = velos ~ horaire + mois + meteo + temperature1 +
+## saison + temperature1_square + humidite_square + vacances +
+## humidite + vent, data = data_train, init.theta = 9.793225092,
+## link = log)
+##
+## Coefficients:
+## Estimate Std. Error z value Pr(>|z|)
+## (Intercept) 4.073e+00 1.134e-01 35.913 < 2e-16 ***
+## horaire2 1.397e+00 2.851e-02 49.005 < 2e-16 ***
+## horaire3 1.247e+00 3.442e-02 36.232 < 2e-16 ***
+## horaire4 1.668e+00 3.419e-02 48.794 < 2e-16 ***
+## horaire5 1.296e+00 2.921e-02 44.351 < 2e-16 ***
+## mois2 1.675e-01 4.927e-02 3.399 0.000676 ***
+## mois3 1.740e-01 5.504e-02 3.161 0.001573 **
+## mois4 2.850e-01 8.407e-02 3.390 0.000700 ***
+## mois5 5.331e-01 8.929e-02 5.971 2.36e-09 ***
+## mois6 4.899e-01 9.302e-02 5.266 1.39e-07 ***
+## mois7 4.001e-01 1.027e-01 3.896 9.77e-05 ***
+## mois8 3.725e-01 9.997e-02 3.726 0.000195 ***
+## mois9 4.072e-01 9.024e-02 4.512 6.42e-06 ***
+## mois10 2.420e-01 8.290e-02 2.919 0.003510 **
+## mois11 1.733e-01 7.945e-02 2.181 0.029176 *
+## mois12 2.213e-01 6.234e-02 3.550 0.000385 ***
+## meteo2 -3.797e-02 2.272e-02 -1.672 0.094590 .
+## meteo3 -4.297e-01 4.158e-02 -10.334 < 2e-16 ***
+## temperature1 5.379e-02 4.887e-03 11.007 < 2e-16 ***
+## saison2 2.070e-01 5.679e-02 3.644 0.000268 ***
+## saison3 3.105e-01 6.468e-02 4.801 1.58e-06 ***
+## saison4 4.910e-01 5.597e-02 8.773 < 2e-16 ***
+## temperature1_square -1.045e-03 1.468e-04 -7.113 1.13e-12 ***
+## humidite_square -1.372e-04 2.661e-05 -5.157 2.51e-07 ***
+## vacances2 -1.718e-01 5.383e-02 -3.192 0.001414 **
+## humidite 1.206e-02 3.387e-03 3.560 0.000371 ***
+## vent -2.818e-03 1.028e-03 -2.741 0.006118 **
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+##
+## (Dispersion parameter for Negative Binomial(9.7932) family taken to be 1)
+##
+## Null deviance: 7780.7 on 1331 degrees of freedom
+## Residual deviance: 1357.8 on 1305 degrees of freedom
+## AIC: 17496
+##
+## Number of Fisher Scoring iterations: 1
+##
+##
+## Theta: 9.793
+## Std. Err.: 0.385
+##
+## 2 x log-likelihood: -17440.082
+
+
+
Final Model
+
+
Choice and Validation of the Model
+
model_final <- glm.nb(
+ velos ~ horaire +
+ mois +
+ meteo +
+ temperature1 +
+ saison +
+ temperature1_square +
+ humidite_square +
+ vacances +
+ humidite +
+ vent +
+
+ horaire:temperature1 +
+ temperature1_square:mois
+ , data = data_train
+)
+summary(model_final)
+
##
+## Call:
+## glm.nb(formula = velos ~ horaire + mois + meteo + temperature1 +
+## saison + temperature1_square + humidite_square + vacances +
+## humidite + vent + horaire:temperature1 + temperature1_square:mois,
+## data = data_train, init.theta = 10.64651881, link = log)
+##
+## Coefficients:
+## Estimate Std. Error z value Pr(>|z|)
+## (Intercept) 4.1688082 0.1195673 34.866 < 2e-16 ***
+## horaire2 1.6136959 0.0494090 32.660 < 2e-16 ***
+## horaire3 1.3248624 0.0622321 21.289 < 2e-16 ***
+## horaire4 1.6124217 0.0611669 26.361 < 2e-16 ***
+## horaire5 1.1846038 0.0526976 22.479 < 2e-16 ***
+## mois2 0.1166036 0.0554402 2.103 0.035446 *
+## mois3 0.0947222 0.0664671 1.425 0.154128
+## mois4 0.0604694 0.0986704 0.613 0.539980
+## mois5 0.5272220 0.1174380 4.489 7.14e-06 ***
+## mois6 0.6533283 0.1542792 4.235 2.29e-05 ***
+## mois7 0.7377272 0.1852322 3.983 6.81e-05 ***
+## mois8 0.4846330 0.1807033 2.682 0.007320 **
+## mois9 0.4051305 0.1386855 2.921 0.003487 **
+## mois10 0.1218501 0.1020733 1.194 0.232575
+## mois11 0.2258393 0.0942154 2.397 0.016528 *
+## mois12 0.1426387 0.0728088 1.959 0.050103 .
+## meteo2 -0.0120673 0.0220598 -0.547 0.584360
+## meteo3 -0.4136117 0.0404725 -10.220 < 2e-16 ***
+## temperature1 0.0509424 0.0074086 6.876 6.15e-12 ***
+## saison2 0.2240149 0.0552593 4.054 5.04e-05 ***
+## saison3 0.3385659 0.0625326 5.414 6.16e-08 ***
+## saison4 0.5050795 0.0538878 9.373 < 2e-16 ***
+## temperature1_square -0.0036795 0.0012369 -2.975 0.002931 **
+## humidite_square -0.0001265 0.0000264 -4.792 1.65e-06 ***
+## vacances2 -0.1925533 0.0524129 -3.674 0.000239 ***
+## humidite 0.0101473 0.0033687 3.012 0.002593 **
+## vent -0.0033359 0.0010029 -3.326 0.000880 ***
+## horaire2:temperature1 -0.0157180 0.0031003 -5.070 3.98e-07 ***
+## horaire3:temperature1 -0.0047919 0.0035916 -1.334 0.182146
+## horaire4:temperature1 0.0028109 0.0035586 0.790 0.429588
+## horaire5:temperature1 0.0067708 0.0031848 2.126 0.033508 *
+## mois2:temperature1_square 0.0027785 0.0011641 2.387 0.016998 *
+## mois3:temperature1_square 0.0032223 0.0011660 2.764 0.005716 **
+## mois4:temperature1_square 0.0036563 0.0011573 3.159 0.001581 **
+## mois5:temperature1_square 0.0027471 0.0011715 2.345 0.019033 *
+## mois6:temperature1_square 0.0024540 0.0011820 2.076 0.037883 *
+## mois7:temperature1_square 0.0022636 0.0011880 1.905 0.056727 .
+## mois8:temperature1_square 0.0025486 0.0011881 2.145 0.031948 *
+## mois9:temperature1_square 0.0027241 0.0011786 2.311 0.020818 *
+## mois10:temperature1_square 0.0032170 0.0011638 2.764 0.005706 **
+## mois11:temperature1_square 0.0022141 0.0011684 1.895 0.058106 .
+## mois12:temperature1_square 0.0034507 0.0012066 2.860 0.004237 **
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+##
+## (Dispersion parameter for Negative Binomial(10.6465) family taken to be 1)
+##
+## Null deviance: 8440.1 on 1331 degrees of freedom
+## Residual deviance: 1357.4 on 1290 degrees of freedom
+## AIC: 17416
+##
+## Number of Fisher Scoring iterations: 1
+##
+##
+## Theta: 10.647
+## Std. Err.: 0.420
+##
+## 2 x log-likelihood: -17329.890
+
anova(model_final, test = "Chisq")
+
## Warning in anova.negbin(model_final, test = "Chisq"): tests made without
+## re-estimating 'theta'
+
## Analysis of Deviance Table
+##
+## Model: Negative Binomial(10.6465), link: log
+##
+## Response: velos
+##
+## Terms added sequentially (first to last)
+##
+##
+## Df Deviance Resid. Df Resid. Dev Pr(>Chi)
+## NULL 1331 8440.1
+## horaire 4 4252.9 1327 4187.3 < 2.2e-16 ***
+## mois 11 2126.6 1316 2060.7 < 2.2e-16 ***
+## meteo 2 306.4 1314 1754.2 < 2.2e-16 ***
+## temperature1 1 78.8 1313 1675.4 < 2.2e-16 ***
+## saison 3 72.3 1310 1603.2 1.385e-15 ***
+## temperature1_square 1 40.2 1309 1563.0 2.294e-10 ***
+## humidite_square 1 60.0 1308 1503.0 9.541e-15 ***
+## vacances 1 9.2 1307 1493.8 0.0024119 **
+## humidite 1 13.4 1306 1480.4 0.0002541 ***
+## vent 1 8.1 1305 1472.3 0.0043736 **
+## horaire:temperature1 4 71.0 1301 1401.2 1.371e-14 ***
+## mois:temperature1_square 11 43.8 1290 1357.4 7.908e-06 ***
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
lrtest(model_final, model_final_without_interaction)
+
## Likelihood ratio test
+##
+## Model 1: velos ~ horaire + mois + meteo + temperature1 + saison + temperature1_square +
+## humidite_square + vacances + humidite + vent + horaire:temperature1 +
+## temperature1_square:mois
+## Model 2: velos ~ horaire + mois + meteo + temperature1 + saison + temperature1_square +
+## humidite_square + vacances + humidite + vent
+## #Df LogLik Df Chisq Pr(>Chisq)
+## 1 43 -8664.9
+## 2 28 -8720.0 -15 110.19 < 2.2e-16 ***
+## ---
+## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
+
dispersion_ratio <- sum(residuals(model_final, type = "pearson")^2) / df.residual(model_final)
+print(paste("Dispersion Ratio:", round(dispersion_ratio, 2)))
+
## [1] "Dispersion Ratio: 0.97"
+
hist(residuals(model_final, type = "deviance"), breaks = 30, freq = FALSE, col = "lightblue", main = "Histogram of Deviance Residuals")
+x <- seq(-5, 5, length = 100)
+curve(dnorm(x), col = "darkblue", lwd = 2, add = TRUE)
+

+
autoplot(model_final, 1:4)
+

+
library(arm)
+
## Loading required package: Matrix
+
## Loading required package: lme4
+
##
+## arm (Version 1.14-4, built: 2024-4-1)
+
## Working directory is /Users/arthurdanjou/Workspace/studies/M1/General Linear Models/Projet
+
##
+## Attaching package: 'arm'
+
## The following object is masked from 'package:corrplot':
+##
+## corrplot
+
## The following object is masked from 'package:car':
+##
+## logit
+
binnedplot(fitted(model_final), residuals(model_final, type = "deviance"), col.int = "blue", col.pts = 2)
+

+
+
+
+
+
Performance and Limitations of the Final Model
+
+
Predictions and Mean Square Error
+
data_test$predictions <- predict(model_final, newdata = data_test, type = "response")
+data_train$predictions <- predict(model_final, newdata = data_train, type = "response")
+
+predictions_ci <- predict(
+ model_final,
+ newdata = data_test,
+ type = "link",
+ se.fit = TRUE
+)
+
+data_test$lwr <- exp(predictions_ci$fit - 1.96 * predictions_ci$se.fit)
+data_test$upr <- exp(predictions_ci$fit + 1.96 * predictions_ci$se.fit)
+
+MSE <- mean((data_test$velos - data_test$predictions)^2)
+cat("MSE :", MSE, "\n")
+
## MSE : 43331
+
cat("RMSE train :", sqrt(mean((data_train$velos - data_train$predictions)^2)), "\n")
+
## RMSE train : 195.0359
+
cat('RMSE :', sqrt(MSE), '\n')
+
## RMSE : 208.161
+
+
+
Evaluation of Model Performance
+
bounds <- c(200, 650)
+
+cat("Observations vs Prédictions\n")
+
## Observations vs Prédictions
+
cat(nrow(data_test[data_test$velos < bounds[1],]), "vs", sum(data_test$predictions < bounds[1]), "\n")
+
## 50 vs 39
+
cat(nrow(data_test[data_test$velos > bounds[1] & data_test$velos < bounds[2],]), "vs", sum(data_test$predictions > bounds[1] & data_test$predictions < bounds[2]), "\n")
+
## 124 vs 128
+
cat(nrow(data_test[data_test$velos > bounds[2],]), "vs", sum(data_test$predictions > bounds[2]), "\n")
+
## 159 vs 166
+
cat('\n')
+
categories <- c(0, bounds, Inf)
+categories_label <- c("Low", "Mid", "High")
+
+data_test$velos_cat <- cut(data_test$velos,
+ breaks = categories,
+ labels = categories_label,
+ include.lowest = TRUE)
+data_test$predictions_cat <- cut(data_test$predictions,
+ breaks = categories,
+ labels = categories_label,
+ include.lowest = TRUE)
+conf_matrix <- confusionMatrix(data_test$velos_cat, data_test$predictions_cat)
+conf_matrix
+
## Confusion Matrix and Statistics
+##
+## Reference
+## Prediction Low Mid High
+## Low 36 14 0
+## Mid 3 95 26
+## High 0 19 140
+##
+## Overall Statistics
+##
+## Accuracy : 0.8138
+## 95% CI : (0.7678, 0.8542)
+## No Information Rate : 0.4985
+## P-Value [Acc > NIR] : < 2.2e-16
+##
+## Kappa : 0.6903
+##
+## Mcnemar's Test P-Value : NA
+##
+## Statistics by Class:
+##
+## Class: Low Class: Mid Class: High
+## Sensitivity 0.9231 0.7422 0.8434
+## Specificity 0.9524 0.8585 0.8862
+## Pos Pred Value 0.7200 0.7661 0.8805
+## Neg Pred Value 0.9894 0.8421 0.8506
+## Prevalence 0.1171 0.3844 0.4985
+## Detection Rate 0.1081 0.2853 0.4204
+## Detection Prevalence 0.1502 0.3724 0.4775
+## Balanced Accuracy 0.9377 0.8004 0.8648
+
ggplot(data_test, aes(x = velos, y = predictions)) +
+ geom_point(color = "blue", alpha = 0.6, size = 2) +
+ geom_abline(slope = 1, intercept = 0, color = "red", linetype = "dashed") +
+ geom_ribbon(aes(ymin = lwr, ymax = upr), alpha = 0.2, fill = "grey") +
+ labs(
+ title = "Comparison of Observed and Predicted Bikes",
+ x = "Number of Observed Bikes",
+ y = "Number of Predicted Bikes"
+ ) +
+ theme_minimal()
+

+
ggplot(data_test, aes(x = velos_cat, fill = predictions_cat)) +
+ geom_bar(position = "dodge") +
+ labs(title = "Predictions vs Observations (by categories)",
+ x = "Observed categories", y = "Number of predictions")
+

+
df_plot <- data.frame(observed = data_test$velos, predicted = data_test$predictions)
+
+# Créer l'histogramme avec la courbe de densité
+ggplot(df_plot, aes(x = observed)) +
+ geom_histogram(aes(y = ..density..), binwidth = 50, fill = "lightblue", color = "black", alpha = 0.7) + # Histogramme des données observées
+ geom_density(aes(x = predicted), color = "red", size = 1) + # Courbe de densité des valeurs prédites
+ labs(title = "Observed distribution vs. Predicted distribution",
+ x = "Number of vélos",
+ y = "Density") +
+ theme_bw()
+
## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
+## ℹ Please use `linewidth` instead.
+## This warning is displayed once every 8 hours.
+## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
+## generated.
+
## Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
+## ℹ Please use `after_stat(density)` instead.
+## This warning is displayed once every 8 hours.
+## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
+## generated.
+

+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
diff --git a/M1/General Linear Models/Projet/projet.csv b/M1/General Linear Models/Projet/projet.csv
new file mode 100644
index 0000000..a728809
--- /dev/null
+++ b/M1/General Linear Models/Projet/projet.csv
@@ -0,0 +1,1818 @@
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\ No newline at end of file