non_robust_correlations.R

cor_-1_to_1.R

n <- 100000L

x <- rnorm(n, 0, 1)
y <- -x

cor(x, y)

for (z in c(10, 100, 1000, 10000, 100000)) {
    rho <- cor(c(-z, x, z), c(-z, y, z))
    print(paste0("z = ", z, "; rho = ", rho))
}

# [1] "z = 10; rho = -0.996002903776044"
# [1] "z = 100; rho = -0.666312532886097"
# [1] "z = 1000; rho = 0.904877424566746"
# [1] "z = 10000; rho = 0.999001772006972"
# [1] "z = 1e+05; rho = 0.999990012785159"

cor_0_to_1.R

n <- 100000L

x <- rnorm(n, 0, 1)
y <- rnorm(n, 0, 1)

cor(x, y)

for (z in c(10, 100, 1000, 10000, 100000)) {
    rho <- cor(c(-z, x, z), c(-z, y, z))
    print(paste0("z = ", z, "; rho = ", rho))
}

# [1] "z = 10; rho = 0.00376657938871808"
# [1] "z = 100; rho = 0.168606844273084"
# [1] "z = 1000; rho = 0.952616981340804"
# [1] "z = 10000; rho = 0.999502805310329"
# [1] "z = 1e+05; rho = 0.999995025600232"

uncorrelated_betas.jl

using Distributions

function two_views(a, b)
    rho = cor(a, b)

    n1 = sum( (a .> mean(a)) & (b .> mean(b)) )
    n2 = sum( (a .> mean(a)) )
    p = n1 / n2

    return rho, p
end

n = 10_000

a = rand(Beta(1.0, 1.0), n);
b = rand(Beta(1.0, 1.0), n);

two_views(a, b)
# (0.015660398444906496,0.5133239831697055)

a = rand(Beta(0.01, 10000.0), n);
b = rand(Beta(0.01, 10000.0), n);

two_views(a, b)
# (-0.0007933918288009543,0.0379746835443038)

a = rand(Beta(10000.0, 0.01), n);
b = rand(Beta(10000.0, 0.01), n);

two_views(a, b)
# (-0.00812362128868224,0.9585589337775927)