# Machine Learning Ex3 – Multivariate Linear Regression

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**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 = 0.3 seems to be the best.

**Part 2. Normal Equations.**

to be continued…

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