lecture1.R

### Regression Example

f = function(x) {
  0.1 * cos(x)^2 + 0.25 * sin(x)^(3)
}


regressionData = data.frame(rnorm(5000), sd = 5)
regressionData[,2] = f(regressionData[,1]) + rnorm(5000, sd = 0.1)
names(regressionData) = c("x", "y")

plot(x = regressionData$x, y = regressionData$y, pch = 19, cex = 0.25)
curve(f, n = 10000, ylim = c(-5, 5), col = "red", add = T, lwd = 2.25)

linearModel = lm(y ~ x, data = regressionData)
splineModel = smooth.spline(x = regressionData$x, y = regressionData$y)

points(x = regressionData$x, y = linearModel$fitted.values, col = "blue", pch = 19, cex = 0.15)
points(x = splineModel$x, y = splineModel$y, col = "green", pch = 19, cex = 0.15)

### Classification Example

zeros = data.frame(
  rnorm(650, mean = 1.5, sd = 2), 
  rnorm(650, mean = 1, sd = 3), 
  rep(0, 650))
names(zeros) = c("x1", "x2", "y")
ones = data.frame(
  rnorm(350, mean = -2.5, sd = 4),
  rnorm(350, mean = -5, sd = 2),
  rep(1, 350))
names(ones) = c("x1", "x2", "y")
classificationData = rbind(zeros, ones)[sample(1:1000, size = 1000),]

plot(x = classificationData$x1, y = classificationData$x2, cex = 0)
points(
  x = classificationData[which(classificationData$y == 1),]$x1, 
  y = classificationData[which(classificationData$y == 1),]$x2,
  col = "green", pch = 19, cex = 0.75)
points(
  x = classificationData[which(classificationData$y == 0),]$x1, 
  y = classificationData[which(classificationData$y == 0),]$x2,
  col = "red", pch = 19, cex = 0.75)

library(MASS)

ldaModel = lda(y ~ x1 + x2, data = classificationData)
meanZero = ldaModel$means[1,]
meanOne = ldaModel$means[2,]
invCov = ginv(cov(as.matrix(classificationData[,-3])))
a0  = log(ldaModel$prior[1] / ldaModel$prior[2]) - 
  1/2 * ((meanZero + meanOne) %*% invCov %*% (meanZero - meanOne))
aS = invCov %*% (meanZero - meanOne)

g = Vectorize(function(x) {
  (-aS[1]/aS[2])*x - a0 / aS[2]
})
curve(g, col = "blue", lwd = 1.2, add = T)

# Inference example.

zeros = classificationData[which(classificationData$y == 0),]
mean(zeros$x1)
mean(zeros$x2)
cov(zeros[, -3])

### Clustering example.

plot(x= classificationData$x1, y = classificationData$x2, cex = 0.5, pch = 19)
clusters = kmeans(x = classificationData[, -3], centers = 2)$cluster

predZeros = which(clusters == 1)
predOnes = which(clusters == 2)
points(
  x = classificationData[predZeros,]$x1, 
  y = classificationData[predZeros,]$x2, 
  col = "red", pch = 19, cex = 0.25)
points(
  x = classificationData[predOnes,]$x1, 
  y = classificationData[predOnes,]$x2, 
  col = "green", pch = 19, cex = 0.25)