Creating multiple Bias-Variance Tradeoff plots in R - r

Question

I am relatively new to R. I would like to know how to create the following graphic. I have been stuck for over two hours.
Suppose the red line - the true relationship - is y = x^2.
Suppose I want to fit 100 linear models to 100 random samples (blue lines).
How would I do this? So far, this is what I have:
# create the true relationship
f <- function(x) x^2 # true model
x <- seq(0, 1, by = 0.01)
y <- f(x)
# plot the true function
plot(x, y, type = "l", col = "red", ylim = c(-0.2, 1.2), lwd = 4)
# fit 100 models
set.seed(1)
for (i in 1:100)
{
errors <- rnorm(n, 0, sigma) # random errors, have standard deviation sigma
obs_y <- f(obs_x) + errors # observed y = true_model + error
model <- lm(obs_y ~ obs_x) # fit a linear model to the observed values
points(obs_x[i], mean(obs_y[i]), col = "green") # mean values
abline(model, col = "purple") # plot the fitted model
}
which creates this:
in which the green dots are definitely off...
and I don't have the black dots...
Thanks!

Answer 1

Here's your code after several adjustments:
f <- function(x) x^2
x <- seq(0, 1, by = 0.05)
n <- length(x)
sigma <- 0.05
y <- f(x)
plot(x, y, type = "l", col = "red", ylim = c(-0.2, 1.2), lwd = 2)
fitted <- ys <- matrix(0, ncol = n, nrow = 100)
set.seed(1)
for (i in 1:100)
{
errors <- rnorm(n, 0, sigma)
ys[i, ] <- obs_y <- f(x) + errors
model <- lm(obs_y ~ x)
fitted[i, ] <- fitted(model)
abline(model, col = "purple", lwd = 0.1)
}
points(x = rep(x, each = 100), y = ys, cex = 0.1)
points(x = x, y = colMeans(fitted), col = 'green', cex = 0.3)

Answer 2

With ggplot, e.g.
library(ggplot2)
x <- seq(0, 1, 0.1)
y <- x^2
dat <- as.data.frame(do.call(rbind, lapply(1:100, function(i){
y_err <- y + rnorm(1, 0, 0.06)
l <- lm(y_err ~ x)$coefficients
cbind(samp = i, intercept = l[1], slope = l[2], t(x * l[2] + l[1]), t(y_err))
})), row.names = 1:100)
ggplot() +
geom_abline(aes(intercept = dat$intercept, slope = dat$slope)) +
geom_point(aes(x = rep(x, each = 100), y = unlist(dat[, 15:25])), alpha = 0.5) +
geom_line(aes(x = x, y = y), color = "red", lwd = 2) +
geom_point(aes(x = x, y = colMeans(dat[, 4:14])), color = "green")