# Lecture 1

# Working with scalar, vector and matrix
# Scalar
a <- 1
b <- 2

# Operation
a+b
a*b

# Vector
a <- c(1,2)
a[2] #accessing to the element of a vector

b <- c(2,4)

a*b # elmentwise multiplication

a%*%b # scalar product

# Matrix 
A <- matrix(c(25,7, 19, 90), nrow = 2, ncol = 2)
B <- matrix(c(25,7, 19, 90), nrow = 2, ncol = 2, byrow = T) # fill by row

A*B #elementwise multiplication
A%*%B #matrix product (pay attention, the matrices must be conformable)

### 
# install and load the package 
#install.packages("ISLR")
library(ISLR)
data("Wage")

# See the help 
help(Wage) 

# Exampleting working with a data frame
Wage[1:2, ]         # extracting first two rows
Wage[c(1,100), ]    # extracting first and 100 row

Wage[, 1:2]         # first two columns 
Wage[, c("year", "age")] # first two columns, but by name 

Wage$year # extract a variablle using the $ syntax

# Structure of a dataset
str(Wage)

# (Some) Numerical summary
summary(Wage$wage)
sd(Wage$wage)
IQR(Wage$wage)
quantile(Wage$wage, p = 0.25)

# Exploratory Analysis 
# Univariate 
boxplot(Wage$wage) # boxplot
hist(Wage$wage)    # histogram

# Bivariate 
plot(Wage$age, Wage$wage, cex = 0.2) #Scatterplot
boxplot(wage ~ education, data = Wage,
        cex.axis = 0.5) # Boxplot Y|X 

# Fit linear model 
lmFit <- lm(wage ~ age + year + education, data = Wage)
lmFit
summary(lmFit)

## get the quantities by hand ----
# The OLS estimate
X <- model.matrix(wage ~ age + year + education, data = Wage)
y <- Wage$wage
beta_OLS <- solve(t(X)%*%X)%*%t(X)%*%y
beta_OLS

lmFit$coefficients

# Std. error
p <- ncol(X)
hat_s2 <- sum(residuals(lmFit)^2)/(nrow(Wage) - p)
hat_Vb <- hat_s2*solve(t(X)%*%X)
sqrt(diag(hat_Vb))

# Predicted values and residuals 
head(residuals(lmFit))

head(Wage$wage - predict(lmFit))

summary(residuals(lmFit))

summary(lmFit)$sigma
sqrt(sum(residuals(lmFit)^2)/(nrow(Wage)-ncol(X)))


# Exercise: obtain the R^2 measures by hand 
# (using the formulas reported on the slides)
summary(lmFit)$r.squared
summary(lmFit)$adj.r.squared