Lecture 1 Quick Demo: Pearson vs Spearman vs Kendall

1) Linear relationship (baseline)

We choose a linear model with noise and record correlations.

n <- 400
x <- rnorm(n)

# Linear signal + noise
y_lin_signal <- 2 * x
y_lin <- y_lin_signal + rnorm(n, sd=0.9)

corr_lin <- corr_table(x, y_lin)
corr_lin

plot(x, y_lin, pch=16, cex=0.6,
     main="Linear: y = 2x + noise",
     xlab="x", ylab="y")
abline(lm(y_lin ~ x), lwd=2)

2) Monotone nonlinear relationship

2.1) (bounded) with matched Spearman

Use a monotone nonlinear transform: tanh(a x), which saturates.

a <- 2.0
y_non_signal <- tanh(a * x)   # bounded in (-1, 1), strictly increasing in x

# Tune noise so Spearman(x, y_non) is close to Spearman(x, y_lin)
target_s <- cor(x, y_lin, method="spearman")

tuned <- tune_noise_for_spearman(x, y_non_signal, target=target_s, grid=seq(0, 1.5, length.out=61))

y_non <- tuned$y

corr_non <- corr_table(x, y_non)
corr_non

plot(x, y_non, pch=16, cex=0.6,
     main=paste0("Monotone nonlinear: y = tanh(",a,"x) + noise (tuned)"),
     xlab="x", ylab="y")

rbind(
  Linear = corr_lin,
  Nonlinear_Monotone = corr_non
)

2.2) A more extreme example

Another monotone but nonlinear relationship is \(y = \exp(kx)\).

k_values <- c(1, 2, 3, 5)

results <- do.call(rbind, lapply(k_values, function(k){
  y <- exp(k*x)  # monotone transform, no extra noise
  cbind(k = k, corr_table(x, y))
}))

results

Pearson correlation can drop arbitrarily which Spearman and Kendall’s tau are kept at 1.

If we add some noise: \(y = \exp(2x) + \text{noise}\)

k <- 2
eps <- rnorm(n, sd = 0.05 * sd(exp(k*x)))
y_noisy <- exp(k*x + eps)

corr_table(x, y_noisy)

plot(x, y_noisy, pch=16, cex=0.7,
     main="Monotone nonlinear + noise: y = exp(2x) + noise",
     xlab="x", ylab="y", log = "y")

3) U-shaped (non-monotone comparison)

x3 <- rnorm(n)
y3 <- (x3^2) + rnorm(n, sd = 0.5)

corr_table(x3, y3)

plot(x3, y3, pch=16, cex=0.6,
     main="U-shaped: y = x^2 + noise",
     xlab="x", ylab="y")