Linear regression by gradient descent

linear_regression_by_gradient_descent.R

##
## Linear regression by gradient descent
##
## A learning exercise to help build intuition about gradient descent.
## J. Christopher Bare, 2012
##

# generate random data in which y is a noisy function of x
x <- runif(1000, -5, 5)
y <- x + rnorm(1000) + 3

# fit a linear model
res <- lm( y ~ x )

# plot the data and the model
plot(x,y, col=rgb(0.2,0.4,0.6,0.4), main='Linear regression')
abline(res, col='blue')

# squared error cost function
cost <- function(X, y, theta) {
  sum( (X %*% theta - y)^2 ) / (2*length(y))
}

# learning rate and iteration limit
alpha <- 0.01
num_iters <- 1000

# keep history
cost_history <- double(num_iters)
theta_history <- list(num_iters)

# initialize coefficients
theta <- matrix(c(0,0), nrow=2)

# add a column of 1's for the intercept coefficient
X <- cbind(1, matrix(x))

# gradient descent
for (i in 1:num_iters) {
  error <- (X %*% theta - y)
  delta <- t(X) %*% error / length(y)
  theta <- theta - alpha * delta
  cost_history[i] <- cost(X, y, theta)
  theta_history[[i]] <- theta
}

# plot data and converging fit
plot(x,y, col=rgb(0.2,0.4,0.6,0.4), main='Linear regression by gradient descent')
for (i in c(1,3,6,10,14,seq(20,num_iters,by=10))) {
  abline(coef=theta_history[[i]], col=rgb(0.8,0,0,0.3))
}
abline(coef=theta, col="blue")

# check out the trajectory of the cost function
cost_history[seq(1,num_iters, by=100)]
plot(cost_history, type='l', col='blue', lwd=2, main='Cost function', ylab='cost', xlab='Iterations')

perfect

i have doubt in this ,

squared error cost function

cost <- function(X, y, theta) {
sum( (X %% theta - y)^2 ) / (2length(y))
}
what is the meaning of %*% in this and how it will work ,
please help me in this

plot data and converging fit

plot(x,y, col=rgb(0.2,0.4,0.6,0.4), main='Linear regression by gradient descent')
for (i in c(1,3,6,10,14,seq(20,num_iters,by=10))) {
abline(coef=theta_history[[i]], col=rgb(0.8,0,0,0.3))
}
abline(coef=theta, col="blue")
please explain this code also