** Case study background and problem formulations**

**PROBLEM 1: Checkerboard_Copula_with_Mean-Abs-Err_Objective**

Minimize Meanabs_err

subject to

Linearmulti = 1 (constraints defining multi-stochastic matrix)

Box constraints (Lower bounds of elements of multi-stochastic matrix)

——————————————————————–

Meanabs_err = Mean Absolute Error

——————————————————————–

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

Dataset 1 | 1000 | 16 | 0.003125 | 1.45 | |||
---|---|---|---|---|---|---|---|

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**PROBLEM 2: Checkerboard_Copula_with_Mean-Square-Err_Objective **

Minimize Measquare_err

subject to

Linearmulti = 1 (constraints defining multi-stochastic matrix)

Box constraints (Lower bounds of elements of multi-stochastic matrix)

——————————————————————–

Measquare_err = Mean Square Error

——————————————————————–

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

Dataset 1 | 1000 | 16 | 0.00015625 | 5.57 | |||
---|---|---|---|---|---|---|---|

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**PROBLEM 3: Checkerboard_Copula_with_CVaR-Abs-Err_Objective**

Minimize CVaR_abs_err

subject to

Linearmulti = 1 (constraints defining multi-stochastic matrix)

Box constraints (Lower bounds of elements of multi-stochastic matrix)

——————————————————————–

CVaR_abs_err = CVaR Absolute Error

——————————————————————–

**alpha=0.9**

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

Dataset 1 | 1000 | 16 | 0.03125 | 1.42 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**alpha=0.99**

Dataset 2 | 1000 | 16 | 0.05 | 0.75 | |||
---|---|---|---|---|---|---|---|

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**PROBLEM 4: Checkerboard_Copula_with_Weighted_Sum_of_Mean_Absolute_Errors **

Minimize Weighted Sum of Meanabs_err_s

subject to

Linearmulti = 1 (constraints defining multi-stochastic matrix)

Box constraints (Lower bounds of elements of multi-stochastic matrix)

——————————————————————–

Meanabs_err = Mean Absolute Error

——————————————————————–

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

Dataset 1 | 1000 | 48 | 0.02060244 | 1.81 | |||
---|---|---|---|---|---|---|---|

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**PROBLEM 5: Checkerboard_Copula_with_Weighted_Sum_of_Mean_Square_Errors **

Minimize Weighted Sum of Measquare_err_s

subject to

Linearmulti = 1 (constraints defining multi-stochastic matrix)

Box constraints (Lower bounds of elements of multi-stochastic matrix)

——————————————————————–

Measquare_err = Mean Square Error

——————————————————————–

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

Dataset 1 | 1000 | 48 | 0.001014874 | 1.98 | |||
---|---|---|---|---|---|---|---|

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**PROBLEM 6: Checkerboard_Copula_with_Weighted_Sum_of_CVaR_Absolute_Errors **

Minimize Weighted Sum of CVaR_abs_err_s

subject to

Linearmulti = 1 (constraints defining multi-stochastic matrix)

Box constraints (Lower bounds of elements of multi-stochastic matrix)

——————————————————————–

CVaR_abs_err = CVaR Absolute Error

——————————————————————–

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

Dataset 1: alpha=0.9 | 1000 | 48 | 0.06406186 | 1.95 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

Dataset 2: alpha=0.99 | 1000 | 48 | 0.07499925 | 1.14 | |||
---|---|---|---|---|---|---|---|

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

CASE STUDY SUMMARY

CASE STUDY SUMMARY

This case study builds a 3-dimensional, m = 3, checkerboard copula with n×n×n grid, where n =10. For 3 random variables W,X,Y, cumulative distribution functions F_W (w), F_X (x), F_Y (y) based on 1000 observations are available. The problem is to find a checkerboard copula based on available information.

**Case 1 (Optimization Problems 1-3).**

Additionally, it is available the empirical distribution F_Z (z) of the random value Z=W+ X+Y with K=16 observations, z_1,…,z_(16 ). We suppose that these observations are equally probable and the distribution function F_Z (z) takes K values 1/K, 2/K,…, K/K. To find copula we minimized error functions: Mean Squared Error, Mean Absolute Error, and CVaR Absolute Error (with confidence levels 0.9 and 0.99).

**Case 2 (Optimization Problems 4-6).**

Additionally, three random values Z_1=W+ X, Z_2=W+ Y, Z_3=X+ Y, and their empirical probability distributions F_1(z), F_2(z), F_3(z) are available. We have K=16 observations from every distribution F_1, F_2, F_3 . We suppose that these observations are equally probable and every distribution function takes K values 1/K, 2/K,…, K/K. To find copula we minimized the weighted average (over residuals corresponding random variables Z_1, Z_2, Z_3) of error functions: Mean Squared Error, Mean Absolute Error, and CVaR Absolute Error (with confidence levels 0.9 and 0.99).