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May 5, 2016 08:54
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Linear regression by gradient descent
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## | |
## 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') |
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