Class 11

library(magrittr)
x <- rnorm(1000, 3)
y <- 45 - 6 * x + rnorm(1000, sd = 3)
#lm(y~x) %>% summary

X <- cbind(1, x)

solve(t(X)%*%X)%*%t(X)%*%y

       [,1]
  44.872832
x -5.975382

sse <- function(x, y, betas){
  sum((y-(betas[1] + betas[2] * x))^2)
}

sse(x,y,c(0,0))

[1] 782680.6

optim(c(0,0), sse, x = x, y=y)

$par
[1] 44.866147 -5.975136

$value
[1] 7399.005

$counts
function gradient 
      91       NA 

$convergence
[1] 0

$message
NULL

para <- \(params, a, b){
  (params[1]-3)^2/(a^2) + (params[2]+4)^2/(b^2)
}

para(c(1,2), 3, 5)

[1] 1.884444

optim(c(0,0), para, a=3, b=5)

$par
[1]  2.999949 -4.000284

$value
[1] 3.509462e-09

$counts
function gradient 
      79       NA 

$convergence
[1] 0

$message
NULL