PCoA of distances generated from points in Euclidean space

knitr::opts_chunk$set(
  message = F
)
library(tidyverse)
library(ggrepel)
library(ape) # for pcoa()

Original points in \(\mathbb{R}^2\) are generated:

set.seed(2019)
n <- 15
d0 <- data.frame(ID = 1:n, x = rnorm(n), y = rnorm(n))
g <- ggplot(d0, aes(x, y)) +
  geom_point() +
  geom_text_repel(aes(label = ID), size = 3, segment.color = "grey") +
  theme(aspect.ratio = 1)
print(g)

DistMat <- d0 %>%
  select(-ID) %>%
  dist()
PCoA_Result <- pcoa(DistMat)
d <- PCoA_Result$vectors %>%
  as.data.frame() %>%
  mutate(ID = 1:n)
g <- ggplot(d, aes(Axis.1, Axis.2)) +
  geom_point() +
  geom_text_repel(aes(label = ID), size = 3, segment.color = "grey") +
  theme(aspect.ratio = 1)
print(g)

So when the usual \(\ell_2\) distance is used, PCoA produces coordinates that are the same as original. What if \(\ell_\infty\) norm is used?

DistMat <- d0 %>%
  select(-ID) %>%
  dist(method = "maximum")
PCoA_Result <- pcoa(DistMat)
d <- PCoA_Result$vectors %>%
  as.data.frame() %>%
  mutate(ID = 1:n)
g <- ggplot(d, aes(Axis.1, Axis.2)) +
  geom_point() +
  geom_text_repel(aes(label = ID), size = 3, segment.color = "grey") +
  theme(aspect.ratio = 1)
print(g)

What about \(\ell_1\)?

DistMat <- d0 %>%
  select(-ID) %>%
  dist(method = "manhattan")
PCoA_Result <- pcoa(DistMat)
d <- PCoA_Result$vectors %>%
  as.data.frame() %>%
  mutate(ID = 1:n)
g <- ggplot(d, aes(Axis.1, Axis.2)) +
  geom_point() +
  geom_text_repel(aes(label = ID), size = 3, segment.color = "grey") +
  theme(aspect.ratio = 1)
print(g)