LDA vs difference vector

library(tidyverse)
library(MASS)
library(magrittr)

set.seed(41)

1 Generate data

theta <- rnorm(1) # angle
A <- matrix(rnorm(100 * 2), 100, 2) # rotation vector
A <- A %*% diag(c(3, 1)) %*% matrix(c(cos(theta), sin(theta), -sin(theta), cos(theta)), 2, 2)
Sigma <- var(A)
Sigma

         [,1]     [,2]
[1,] 5.238469 4.280309
[2,] 4.280309 5.060916

Bias vectors

ma <- c(2, 0)
mb <- c(5, 2)

Generate two multivariate gaussians

a <- mvrnorm(n = 1000, ma, Sigma)
b <- mvrnorm(n = 1000, mb, Sigma)

Print covariance matrices

a %>% as.data.frame %>% cov

         V1       V2
V1 5.193249 4.379969
V2 4.379969 5.332480

b %>% as.data.frame %>% cov

         V1       V2
V1 5.195009 4.128270
V2 4.128270 4.779776

2 LDA

Compute difference vector and LDA vector

mdiff <- mb - ma
mdiff_lda <- solve(Sigma, mdiff)

3 Display

Create data frame for the vectors

mdiff_df <- data_frame(x2 = mdiff[1], y2 = mdiff[2], x1 = 0, y1 = 0)
mdiff_lda_df <- data_frame(x2 = mdiff_lda[1], y2 = mdiff_lda[2], x1 = 0, y1 = 0)

df <- 
  bind_rows(
    as.data.frame(a) %>% mutate(class = "a"),
    as.data.frame(b) %>% mutate(class = "b")
    )

Plot data

ggplot() + 
  geom_point(data = df, aes(V1, V2, color = class), alpha = .5) + 
  coord_equal()

Plot data with difference vector (black) and LDA vector (blue)

ggplot() + 
  geom_point(data = df, aes(V1, V2, color = class), alpha = .5) + 
  geom_segment(data = mdiff_df, aes(x = x1, y = y1, xend = x2, yend = y2), arrow = arrow()) +
  geom_segment(data = mdiff_lda_df, aes(x = x1, y = y1, xend = x2, yend = y2), arrow = arrow(), color = "blue") +
  coord_equal() + 
  facet_wrap(~class) +
  ggtitle("Difference vector (black) and LDA vector (blue)")

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