Cars

Example of solution

Preprocessing

Load data and change variable types

pacman::p_load(ggplot2, prettydoc)

cars <- read.csv("cars.csv")

Exploratory analysis

In the initial exploration you can already estimate an outlier

plot(cars$speed.of.car, cars$distance.of.car)

Check for distributions

box_plot <- boxplot(cars[, c("distance.of.car","speed.of.car")])

Exclude outlier

cars <- cars[which(cars$distance.of.car != box_plot$out),]

Modeling

Split data

set.seed(314)
train_size <- round(nrow(cars)*0.7)
test_size <- nrow(cars)-train_size
training_indices <- sample(seq_len(nrow(cars)), size = train_size)
train_set <- cars[training_indices,]

test_set <- cars[-training_indices,]

Train a model

lm <- lm(distance.of.car~ speed.of.car,train_set)

Pred_dist <- predict(lm,test_set)

test_set$Pred_dist <- Pred_dist

Plot model

ggplot(cars, aes(x=speed.of.car, y=distance.of.car)) + geom_point() + 
geom_abline(aes(intercept=lm$coefficients[1], slope=lm$coefficients[2]), colour='red')+
  labs(title = paste("Model with normal intercept\n","Adj R2 = ",signif(summary(lm)$adj.r.squared, 5),
                     "Intercept =",signif(lm$coef[[1]],5 ),
                     " Slope =",signif(lm$coef[[2]], 5),
                     " P =",signif(summary(lm)$coef[2,4], 5)))

The intercept in the model above is at negative levels, which doesn’t make logical sense. The car should be driving at a speed of -25 in order to achieve a distance of zero. Instead, we will try to fix the intercept at zero. Below, we can see the fit becomes worse when we fix the intercept at zero and keep the model linear.

lm_intercept <- lm(distance.of.car~ 0+speed.of.car,train_set)
Pred_dist_intercept <- predict(lm_intercept,test_set)


test_set <- cbind(test_set, Pred_dist_intercept)

# Model with intercept = 0
ggplot(cars, aes(x=speed.of.car, y=distance.of.car)) + geom_point() + xlim(-2, 30) +
geom_abline(aes(intercept=0, slope=lm_intercept$coefficients[1]), colour='turquoise2')  +  
  labs(title = paste("Model with intercept = 0\n","Adj R2 =" ,signif(summary(lm_intercept)$adj.r.squared, 5),
                     "Intercept = 0 ",
                     " Slope =",signif(lm_intercept$coef[[1]], 5),
                     " P =",signif(summary(lm_intercept)$coef[1,3], 4)))

Let’s compare both models

# Comparing both linear regressions

ggplot(cars, aes(x=speed.of.car, y=distance.of.car)) + geom_point() + 
geom_abline(aes(intercept=lm$coefficients[1], slope=lm$coefficients[2], colour='LM')) +
geom_abline(aes(intercept=0, slope=lm_intercept$coefficients[1], colour='LM with intercept = 0 '))  

test_set$error_lm <- test_set$distance.of.car - test_set$Pred_dist
test_set$abs_error_lm <- abs(test_set$distance.of.car - test_set$Pred_dist)
test_set$rel_error_lm <- test_set$abs_error_lm/test_set$distance.of.car


test_set$abs_error_model_intercept <- abs(test_set$distance.of.car - test_set$Pred_dist_intercept)
test_set$rel_error_model_intercept <- test_set$abs_error_model_intercept/test_set$distance.of.car

MAE_lm <- mean(test_set$abs_error_lm) 

MAE_model_intercept <- mean(test_set$abs_error_model_intercept)

Log model

We can try to log it:

lm_log <- lm(log(distance.of.car) ~ speed.of.car,train_set)
test_set$pred_dist_log <- predict(lm_log,test_set)

test_set$pred_dist_log <- exp(test_set$pred_dist_log)


ggplot(cars, aes(x=speed.of.car, y=log(distance.of.car))) + geom_point() + 
geom_abline(aes(intercept=lm_log$coefficients[1], slope=lm_log$coefficients[2]), colour='red')+
  labs(title = paste("Model with log \n","Adj R2 = ",signif(summary(lm_log)$adj.r.squared, 5),
                     "Intercept =",signif(lm_log$coef[[1]],5 ),
                     " Slope =",signif(lm_log$coef[[2]], 5),
                     " P =",signif(summary(lm_log)$coef[2,4], 5)))

But we see that the best model is the quadratic one:

lm_sqr <- lm(distance.of.car ~ I(speed.of.car^2),train_set)
test_set$pred_dist_sqr <- predict(lm_sqr,test_set)


ggplot(cars, aes(x=I(speed.of.car^2), y=distance.of.car)) + geom_point() + 
geom_abline(aes(intercept=lm_sqr$coefficients[1], slope=lm_sqr$coefficients[2]), colour='red')+
  labs(title = paste("Model with squared x\n","Adj R2 = ",signif(summary(lm_sqr)$adj.r.squared, 5),
                     "Intercept =",signif(lm_sqr$coef[[1]],5 ),
                     " Slope =",signif(lm_sqr$coef[[2]], 5),
                     " P =",signif(summary(lm_sqr)$coef[2,4], 5)))

test_set$error_quad <-  test_set$pred_dist_sqr - test_set$distance.of.car

Error Analysis

error

ggplot(test_set, aes(speed.of.car,error_quad)) + geom_point(aes(colour = factor(name.of.car))) + geom_hline(yintercept=0 , linetype="dashed", color = "red")

08/04/2019