# Case Study: Checkerboard Copula Defined by Sperman Rho Coefficients (entropyr)

Back to main page

Case study background and problem formulations

Instructions for optimization with PSG Run-File, PSG MATLAB Toolbox, and PSG R.

PROBLEM1: Problem_Copula_Defined_by_Correlation_Coefficients
Maximize Entropy
subject to
Linearmulti  = correlation coefficients
Linearmulti  = 1  (constraints defining  5-dimensional hyper–matrix)
Box constraints
——————————————————————–
Box constraints = Lower bounds of elements of hyper–matrix
——————————————————————–

n=4

Dataset 1 1024 N/A 15.57941 0.03 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Matlab Code Data R R Code Data

n=6

Dataset 2 7776 N/A 34.58157 0.3 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Matlab Code Data R R Code Data

n=8

Dataset 3 32768 N/A 55.82529 1.91 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Matlab Code Data R R Code Data

n=10

Dataset 4 1000000 N/A 78.977 7.66 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Matlab Code Data R R Code Data

CASE STUDY SUMMARY

This case study considers optimization problem statement for building a checkerboard copula of a joint distribution, using a prior information about Spearman Rho rank correlation coefficients. An m-dimensional copula where m≥2, is a continuous, m -increasing, probability distribution function C:[0,1]^m→ [0,1] on the unit m-dimensional hyper-cube with uniform marginal distributions. A checkerboard copula is a distribution with a corresponding density c:[0,1]^m→ [0,∞) defined almost everywhere by a step function on an m-uniform subdivision of the hyper-cube. I.e., the checkerboard copula is a distribution defined by subdividing the hyper-cube into n^m identical small hyper-cubes with constant density on each one. We  found checkerboard copulas with known Spearmans Rho coefficients (m=5 and n=4, 8,10 ).