Reproduce Fisher linear discriminant figure

Question

Many books illustrate the idea of Fisher linear discriminant analysis using the following figure (this particular is from Pattern Recognition and Machine Learning, p. 188)

I wonder how to reproduce this figure in R (or in any other language). Pasted below is my initial effort in R. I simulate two groups of data and draw linear discriminant using abline() function. Any suggestions are welcome.

set.seed(2014)
library(MASS)
library(DiscriMiner) # For scatter matrices

# Simulate bivariate normal distribution with 2 classes
mu1 <- c(2, -4)
mu2 <- c(2, 6)
rho <- 0.8
s1 <- 1
s2 <- 3
Sigma <- matrix(c(s1^2, rho * s1 * s2, rho * s1 * s2, s2^2), byrow = TRUE, nrow = 2)
n <- 50
X1 <- mvrnorm(n, mu = mu1, Sigma = Sigma)
X2 <- mvrnorm(n, mu = mu2, Sigma = Sigma)
y <- rep(c(0, 1), each = n)
X <- rbind(x1 = X1, x2 = X2)
X <- scale(X)

# Scatter matrices
B <- betweenCov(variables = X, group = y)
W <- withinCov(variables = X, group = y)

# Eigenvectors
ev <- eigen(solve(W) %*% B)$vectors
slope <- - ev[1,1] / ev[2,1]
intercept <- ev[2,1]

par(pty = "s")
plot(X, col = y + 1, pch = 16)
abline(a = slope, b = intercept, lwd = 2, lty = 2)

MY (UNFINISHED) WORK

I pasted my current solution below. The main question is how to rotate (and move) the density plot according to decision boundary. Any suggestions are still welcome.

require(ggplot2)
library(grid)
library(MASS)

# Simulation parameters
mu1 <- c(5, -9)
mu2 <- c(4, 9)
rho <- 0.5
s1 <- 1
s2 <- 3
Sigma <- matrix(c(s1^2, rho * s1 * s2, rho * s1 * s2, s2^2), byrow = TRUE, nrow = 2)
n <- 50
# Multivariate normal sampling
X1 <- mvrnorm(n, mu = mu1, Sigma = Sigma)
X2 <- mvrnorm(n, mu = mu2, Sigma = Sigma)
# Combine into data frame
y <- rep(c(0, 1), each = n)
X <- rbind(x1 = X1, x2 = X2)
X <- scale(X)
X <- data.frame(X, class = y)

# Apply lda()
m1 <- lda(class ~ X1 + X2, data = X)
m1.pred <- predict(m1)
# Compute intercept and slope for abline
gmean <- m1$prior %*% m1$means
const <- as.numeric(gmean %*% m1$scaling)
z <- as.matrix(X[, 1:2]) %*% m1$scaling - const
slope <- - m1$scaling[1] / m1$scaling[2]
intercept <- const / m1$scaling[2]

# Projected values
LD <- data.frame(predict(m1)$x, class = y)

# Scatterplot
p1 <- ggplot(X, aes(X1, X2, color=as.factor(class))) + 
  geom_point() +
  theme_bw() +
  theme(legend.position = "none") +
  scale_x_continuous(limits=c(-5, 5)) + 
  scale_y_continuous(limits=c(-5, 5)) +
  geom_abline(intecept = intercept, slope = slope)

# Density plot 
p2 <- ggplot(LD, aes(x = LD1)) +
  geom_density(aes(fill = as.factor(class), y = ..scaled..)) +
  theme_bw() +
  theme(legend.position = "none")

grid.newpage()
print(p1)
vp <- viewport(width = .7, height = 0.6, x = 0.5, y = 0.3, just = c("centre"))
pushViewport(vp)
print(p2, vp = vp)

Answer 1

Basically you need to project the data along the direction of the classifier, plot a histogram for each class, and then rotate the histogram so its x axis is parallel to the classifier. Some trial-and-error with scaling the histogram is needed in order to get a nice result. Here's an example of how to do it in Matlab, for the naive classifier (difference of class' means). For the Fisher classifier it is of course similar, you just use a different classifier w. I changed the parameters from your code so the plot is more similar to the one you gave.

rng('default')
n = 1000;
mu1 = [1,3]';
mu2 = [4,1]';
rho = 0.3;
s1 = .8;
s2  = .5;
Sigma = [s1^2,rho*s1*s1;rho*s1*s1, s2^2];
X1 = mvnrnd(mu1,Sigma,n);
X2 = mvnrnd(mu2,Sigma,n);
X = [X1; X2];
Y = [zeros(n,1);ones(n,1)];
scatter(X1(:,1), X1(:,2), [], 'b' );
hold on
scatter(X2(:,1), X2(:,2), [], 'r' );
axis equal
m1 = mean(X(1:n,:))';
m2 = mean(X(n+1:end,:))';
plot(m1(1),m1(2),'bx','markersize',18)
plot(m2(1),m2(2),'rx','markersize',18)
plot([m1(1),m2(1)], [m1(2),m2(2)],'g')
%% classifier taking only means into account
w = m2 - m1; 
w = w / norm(w);
% project data onto w
X1_projected = X1 * w;
X2_projected = X2 * w;
% plot histogram and rotate it
angle = 180/pi * atan(w(2)/w(1));
[hy1, hx1] = hist(X1_projected);
[hy2, hx2] = hist(X2_projected);
hy1 = hy1 / sum(hy1); % normalize
hy2 = hy2 / sum(hy2); % normalize
scale = 4; % set manually
h1 = bar(hx1, scale*hy1,'b');
h2 = bar(hx2, scale*hy2,'r');
set([h1, h2],'ShowBaseLine','off')
% rotate around the origin
rotate(get(h1,'children'),[0,0,1], angle, [0,0,0])
rotate(get(h2,'children'),[0,0,1], angle, [0,0,0])