Formalize Kendall's tau (#225)

* docs(dependence_measures): Improve explanations and implement interfaces for obtaining Kendall's tau and Spearman's Rho.

* feat: Added functions to calculate bivariate Kendall taus and Spearman rhos.

* feat: Add corkendall function to EmpiricalCopula in fit method

docs/src/assets/references.bib

@@ -864,4 +864,14 @@ @incollection{lindskog2003kendall
   pages={149--156},
   year={2003},
   publisher={Springer}
+}
+
+@article{blier2022stochastic,
+  title={Stochastic representation of FGM copulas using multivariate Bernoulli random variables},
+  author={Blier-Wong, Christopher and Cossette, H{\'e}l{\`e}ne and Marceau, Etienne},
+  journal={Computational Statistics \& Data Analysis},
+  volume={173},
+  pages={107506},
+  year={2022},
+  publisher={Elsevier}
 }

src/Copula.jl

@@ -41,20 +41,22 @@ function τ(C::Copula)
 end
 function StatsBase.corkendall(C::Copula{d}) where d
     # returns the matrix of bivariate kendall taus. 
-    K = zeros(d,d)
+    K = ones(d,d)
     for i in 1:d
-        for j in 1:d
-            K[i,j] = i==j ? 1 : τ(SubsetCopula(C::Copula{d},(i,j)))
+        for j in i+1:d
+            K[i,j] = τ(SubsetCopula(C::Copula{d},(i,j)))
+            K[j,i] = K[i,j]
         end
     end
     return K
 end
 function StatsBase.corspearman(C::Copula{d}) where d
     # returns the matrix of bivariate spearman rhos. 
-    K = zeros(d,d)
+    K = ones(d,d)
     for i in 1:d
-        for j in 1:d
-            K[i,j] = i==j ? 1 : ρ(SubsetCopula(C::Copula{d},(i,j)))
+        for j in i+1:d
+            K[i,j] = ρ(SubsetCopula(C::Copula{d},(i,j)))
+            K[j,i] = K[i,j]
         end
     end
     return K

src/MiscellaneousCopulas/EmpiricalCopula.jl

@@ -44,4 +44,4 @@ end
 function Distributions.fit(::Type{CT},u) where {CT <: EmpiricalCopula}
     return EmpiricalCopula(u)
 end
-StatsBase.corkendall(C::EmpiricalCopula) = StatsBase.corkendall(C.u)
+StatsBase.corkendall(C::EmpiricalCopula) = StatsBase.corkendall(C.u')