1 - SLR.md

SLR

Simple empirical formuals of the SLR Model

SLR Model and assumptions

Model: $Y_{i} = \beta_{0} + \beta_{1}X_{i} + \varepsilon_{i}$

Data Loading

# Define the data
x <- dataset$predictor
y <- dataset$target
n <- lenght(x)

Least Squares Estimates - 1

We can obtain the optimal result through the following:

s2x <- sum((x-mean(x))^2)/n
s2y <- sum((y-mean(y))^2)/n
covxy <- cov(x,y) 
rxy <- cor(x,y)
mx <- mean(x)
my <- mean(y)

# Parameters 
(beta1 <- rxy * sqrt(s2y/s2x))
(beta0 <- my - beta1 *mx)

# Estimated Values
yhat <- beta0 +  beta1 * x 

# Empirical MSE
mse_hat <- sum((y-yhat)^2)

lm Class

The lm class follows exactly the same approach we used above but it automates it.

# Define model
fit <- lm(formula = y ~ x, data = dataset)

coef(fit)
summary(fit)
fitted(fit)
residuals(fit)

---

SLR Gaussian

---

Single Parameter Tests

---

Confidence Intervals

Parameter

Multivariate Parameters