Fitting various random lines to the housing data to get an intuition about the loss function.

intuitive_regression

# Example of randomly chosen lines
plot(housing)
abline(0, 5, col="red")
abline(-50, 10, col="blue")

# Create the loss function
loss <- function(intercept, slope) sum(((intercept + (slope * housing[, "num.rooms"])) - housing[, "median.values"])^2)/2

# Create some data for a given line and compute the loss
loss(0, 5)
loss(-30, 10)

# Test a few different slopes with different intercepts
x <- -50:50
y <- -10:10
z <- sapply(x, function(intercept) (sapply(y, function(slope, intercept) loss(intercept, slope), intercept=intercept)))
rownames(z) <- y
colnames(z) <- x

# 3D plot of loss function
library(lattice)

wireframe(z, shade=TRUE, xlab="theta0", ylab="theta1", zlab="loss function", aspect =  c(61/87, 0.4), light.source = c(10,0,10))

# Contour plot
library(reshape)
library(ggplot2)

loss.values <- as.data.frame(melt(z))
names(loss.values) <- c("slope", "intercept", "loss")

v <- ggplot(loss.values, aes(intercept, slope, z = loss)) 
v + geom_tile(aes(fill = loss)) + stat_contour()