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

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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 Entropynsubject to
Linearmulti = correlation coefficients
Linearmulti = 1 (constraints defining 5-dimensional hyper–matrix) Box constraints ——————————————————————– Box constraints = Lower bounds of elements of hyper–matrix ——————————————————————– nn=4n

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

=6n

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

=8n

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

=10n

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 ).