Cross-correlations and Pearson similarities

The message here is: don’t confuse (a) measuring similarities between feature vectors using Pearson correlation with (b) measuring correlation among features

suppressPackageStartupMessages(library(magrittr))
suppressPackageStartupMessages(library(tidyverse))

1 Generate data

n <- 100 # observatins
d <- 10 # dimensions
eps <- .5 # standard deviation

Function to compute vector of cross-correlation values

corvec <- function(m) {cm <- cor(m); cm[upper.tri(cm)]}

Generate data

x <- rnorm(d) # bias

m <- matrix(rnorm(n*d), n, d) * eps # sample from d-dim gaussion

mx <- sweep(m, 2, x, "+") # add bias to sample

mxn <- sweep(mx, 2, colMeans(mx), "-") # remove mean from biased data

2 Pearson similarity

Create data frame of pearson similarities between samples of the 3 noise, biased data, mean-subtracted data

df <- data.frame(
  noise = corvec(t(m)),
  signal = corvec(t(mx)),
  meansub = corvec(t(mxn))
  ) 

dim(df) 

[1] 4950    3

df %<>% 
  pivot_longer(everything())

Plot data

ggplot(df, aes(x=value, y=..scaled.., fill = name)) + geom_density(alpha = 0.5)

ggplot(df, aes(x=name, y=value)) + geom_violin()

3 Cross-correlation

Create data frame of cross-correlations of noise, biased data, mean-subtracted data

df <- data.frame(
  noise = corvec((m)),
  signal = corvec((mx)),
  meansub = corvec((mxn))
  ) 

dim(df) 

[1] 45  3

df %<>% 
  pivot_longer(everything()) 

Plot data

ggplot(df, aes(x=value, y=..scaled.., fill = name)) + geom_density(alpha = 0.5)

ggplot(df, aes(x=name, y=value)) + geom_violin()