library(ISLR2)
attach(Wage)
?Wage

agelims <- range(age)+c(-5,5)
age.grid <- seq(from = agelims[1], to = agelims[2])
plot(age, wage, xlim = agelims, cex = .5, col = "darkgrey")

### Lokalna regresija
## U praksi se velicina okoline tipicno zadaje preko 
## tzv. span parametra.
## To je postotak tocaka iz skupa za trening 
## koje koristimo za prilagodbu lin. modela.
## U nasoj notaciji to dobijemo ako stavimo
## h_lambda(x)=udaljenost 
## od x do span*n najblizeg susjeda od x u {x^(i):i=1,...,n}
## npr. u kNN metodi span = k/n -> span*n=k

plot(age, wage, xlim = agelims, cex = .5, col = "darkgrey")
title("Local Regression")
fit <- loess(wage ~ age, span = 0.05, data = Wage, degree = 1) 
## moze i kvadratna funkcija umjesto linearne, 
## a za degree = 0 dobijemo NW procjenitelj
fit2 <- loess(wage ~ age, span = 0.2, data = Wage, degree = 1)
fit3 <- loess(wage ~ age, span = 0.7, data = Wage, degree = 1)
lines(age.grid, predict(fit, data.frame(age = age.grid)),
      col = "darkred", lwd = 1.5)
lines(age.grid, predict(fit2, data.frame(age = age.grid)),
      col = "darkblue", lwd = 1.5)
lines(age.grid, predict(fit3, data.frame(age = age.grid)),
      col = "darkorange", lwd = 1.5)

legend("topright", legend = c("Span = 0.05", "Span = 0.2", "Span= 0.7"),
       col = c("darkred", "darkblue", "darkorange"), lty = 1, lwd = 2, cex = .5)

## gore se koristi tricube jezgra, 
## tj. K_lambda(x,t)=D((x-t)/h_lambda(x)) za D: 
tricubic=function(t){
  (1 - abs(t)^3)^3 * (abs(t)<=1)
}
## za fiksni x=10, t |-> K_lambda(x,t) je:
t=seq(6,14, by=0.01)
plot(t, tricubic((10-t)/3), type="l", ylab="") ## h_lambda(x)=3 
lines(t, tricubic((10-t)/1), lty="dashed") ## h_lambda(x)=1



### usporedba s NW procjeniteljem 
### (problem pristranosti na rubovima domene)

set.seed(1)
x<- runif(50, max = 3)
y <- 9 - x^2 + rnorm(50, sd = 0.3)
plot(x, y)

## stvarna regresijska funkcija
curve(9 - x^2, col = "grey", add = TRUE, lwd = 3) 

grid.x <- seq(from = 0, to = 3, length.out = 300)
lok_lin <- loess(y ~ x, span=0.2, degree=1) ## 0.2*50=10
lines(grid.x, predict(lok_lin, newdata = grid.x), col="darkblue")
NW <- loess(y ~ x, span=0.2, degree=0) 
lines(grid.x, predict(NW, newdata = grid.x), col="darkorange")
## lok.lin. regresija tipicno je glada od NW procjenitelja 
## te ima manje problema s pristranosti na rubu (vidi desni rub!)

## probaj span=0.1 za NW -> manji problem na rubu, 
## ali veci u unutrasnjosti
NW <- loess(y ~ x, span=0.1, degree=0) 
plot(x, y)
curve(9 - x^2, col = "grey", add = TRUE, lwd = 3) 
lines(grid.x, predict(NW, newdata = grid.x), col="darkorange")