The dataset \((y_i, x_i)\) \((i = 1, \cdots, 100)\) is generated by R:

    set.seed(1)
    x <- sort(rnorm(100))
    y<- 3+x^2 + rnorm(100)
    plot(x, y, pch = 20)

• We ca fit the regression line(model) by R
• The fitting is implemented by lm() function.

fit = lm(y~x)
fit$coefficient

## (Intercept)           x 
##   3.7565367   0.1488341

• The result (outputs) of lm() is restored in the variable (list object) fit.
• fit has a nod with name of coefficient.
• fit$coefficient

plot(x, y, pch=20)
abline(a=fit$coefficients[1], b=fit$coefficients[2], col='blue')
yTrueMean = 3+x^2
lines(x, yTrueMean , lty=2, col='black')

• k-nearest algorithm

• \(\hat y(x)\): estimates of \(y\) for a given \(x\)

• \(N_k(x)\): the index set of \(x_i\) that are k-nearest to \(x\).

• \(N_k(x)\) is computed by knnx.indexin FNN library.

library(FNN)
knnx.index(x, 0, k = 10)

##      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,]   47   46   48   45   44   43   42   41   49    50

idx <- c(knnx.index(x, 0, k = 10))
idx

##  [1] 47 46 48 45 44 43 42 41 49 50

• \(x = 0\)

yhat <- mean( y[idx] ) 
yhat 

## [1] 2.446989

eval.point = 0
plot(x, y, pch = 20)
abline( v = 0, col = 'black')
idx<- c( knnx.index(x, eval.point, k = 10) )
points( x[idx], y[idx], col = 'red', pch = 20)
abline(h = mean(y[idx]), lty = 2, col = 'red')
mean(y[idx])

## [1] 2.446989

• Display \(\hat y\) on \(-1\) and \(1\).

• Using for statment we can display \(\hat y\) on [-3,3].

eval.n = 100
eval.point = seq(-3,3, length= eval.n)
plot(x, y, pch = 20)
idx.mat<- knnx.index(x, eval.point , k = 10)
yhat = rep(0,eval.n)
for (i in 1:eval.n)   yhat[i]<-mean(y[idx.mat[i,]])
lines(eval.point , yhat, type= 'l', lty = 2, col = 'red')

library(scatterplot3d)
attach(mtcars)
scatterplot3d(wt, disp, mpg, main="3D Scatterplot")

library(rgl)
attach(mtcars)
plot3d(wt, disp, mpg, col="red", size=3)

• assign colors by factor

mypal = c('blue', 'red', 'green')
class(mtcars$cyl)
factor(mtcars$cyl)
mypal[factor(mtcars$cyl)]
plot3d(wt, disp, mpg, col= a[factor(mtcars$cyl)], size=10)

Note that

c("A","B","C")[c(2,2,3,1)]

## [1] "B" "B" "C" "A"

library(rgl)
z = 2 * volcano        # Exaggerate the relief
x = 10 * (1:nrow(z))   # 10 meter spacing (S to N)
y = 10 * (1:ncol(z))   # 10 meter spacing (E to W)
## Don't draw the grid lines :  border = NA
par(bg = "slategray")
persp(x, y, z, theta = 135, phi = 30, col = "green3", scale = FALSE,
      ltheta = -120, shade = 0.75, border = NA, box = FALSE)

• persp() draws the surfaces in the 3d dimensional space by heigh informaiton.

persp3d(x, y, z, col = "green3")