** 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

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Box constraints = Lower bounds of elements of hyper–matrix

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n=4

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset 1 | 1024 | N/A | 15.57941 | 0.03 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

n=6

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset 2 | 7776 | N/A | 34.58157 | 0.3 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

n=8

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset 3 | 32768 | N/A | 55.82529 | 1.91 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

n=10

# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset 4 | 1000000 | N/A | 78.977 | 7.66 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

Matlab Toolbox | Data | ||||||

Matlab | Matlab Code | Data | |||||

R | R Code | Data |

CASE STUDY SUMMARYCASE 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 ).