Machine Learning Ex3 - Multivariate Linear Regression

Part 1. Finding alpha.
The first question to resolve in Exercise 3 is to pick a good learning rate alpha.

This require making an initial selection, running gradient descent and observing the cost function.

I test alpha range from 0.01 to 1.

?View Code RSPLUS
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##preparing data input.
x <- read.table("ex3x.dat", header=F)
y <- read.table("ex3y.dat", header=F)
 
#normalize features using Z-score.
x[,1] <- (x[,1] - mean(x[,1]))/sd(x[,1])
x[,2] <- (x[,2] - mean(x[,2]))/sd(x[,2])
 
x <- cbind(x0=rep(1, nrow(x)), x)
x <- as.matrix(x)
 
##gradient descent algorithm.
gradDescent_internal <- function(theta, x, y, m, alpha) {
  h <- sapply(1:nrow(x), function(i) t(theta) %*% x[i,])
	j <- t(h-y) %*% x
	grad <- 1/m * j
  theta <- t(theta) - alpha * grad 
	theta <- t(theta)
  return(theta)
}
 
## cost function.
J <- function(theta, x, y, m) {
  h <- sapply(1:nrow(x), function(i) t(theta) %*% x[i,])
  j <- 2*sum((h-y)^2)/m
  return(j)
}
 
## calculate cost function J for every iteration at specific alpha value.
testLearningRate <- function(x,y, alpha, niter=50) {
  j <- rep(0, niter)
  m <- nrow(x)
  theta <- matrix(rep(0, ncol(x)), ncol=1)
  for (i in 1:niter) {
    theta <- gradDescent_internal(theta,x,y,m, alpha)
    j[i] <- J(theta, x, y, m)
  }  
  return(j)
}
 
 
## test learning rate.
alpha=c(0.01, 0.03, 0.1, 0.3, 1)
xxx=sapply(alpha, testLearningRate, x=x, y=y)
colnames(xxx) <- as.character(alpha)
 
require(ggplot2)
xxx <- melt(xxx)
names(xxx) <- c("niter", "alpha", "J")
p <- ggplot(xxx, aes(x=niter, y=J))
p+geom_line(aes(colour=factor(alpha))) +xlab("Number of iteractions") +ylab("Cost J")


alpha = 1 seems to be the best.

Part 2. Normal Equations.
The cost function:
\( J(\theta) = \frac{1}{2m} \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)})^2 \)

can be written in matrix notation.
\( \theta=\left(X^{T}X\right)^{-1}X^{T}\vec{y} \)

This function estimate how big is the error of our model VS the data.

To minimize it, we can calculate its derivative, set it to 0 and find the value of theta:
\( \frac{\delta}{\delta \theta_j} J(\theta_j) = 0 \)

Then, the value of theta will be obtained with:
\( \theta = (X^T X)^{-1} (X^T y) \)

That can be easily implemented by:

##loading data...
x <- read.table("ex3x.dat", header=F)
y <- read.table("ex3y.dat", header=F)
x <- cbind(x0=rep(1, nrow(x)), x)
x <- as.matrix(x)
y <- y[,1]

## using normal equation to calculate theta.
theta <- solve(t(x) %*% x) %*% t(x) %*% y

that is :

> theta
              [,1]
theta_0 89597.9095
theta_1   139.2107
theta_2 -8738.0191

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