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Copula Invariance Explorer

Separate a joint distribution into its margins and its dependence structure, and see which statistics survive monotone transformation.

  • dependence
  • copula
  • rank statistics
  • transformation
Pearson r (raw Y)
0.583
Pearson r (transformed)
0.583
changes
Spearman ρ (raw Y)
0.655
Spearman ρ (transformed)
0.655
invariant

Observed feature space

-3-2-10123020406080100120140160180Feature X (margin applied)Feature Y (transform applied)

Copula space (ranks on the unit square)

00.10.20.30.40.50.60.70.80.9100.20.40.60.81U = F_X(X)V = F_Y(Y)
What just happened
The right-hand panel is identical no matter which margin or transform you choose — it is the copula, the pure dependence structure. The left panel, and with it Pearson's r, is reshaped by every monotone map. Rank-based statistics read the right panel; product-moment statistics read the left.