# Case Study: Checkerboard Copula Defined by Sums of Random Variables

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Case study background and problem formulations

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

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
——————————————————————–

Dataset 1 1000 16 0.003125 1.45 # 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

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
——————————————————————–

Dataset 1 1000 16 0.00015625 5.57 # 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

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

Dataset 1 1000 16 0.03125 1.42 # 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

alpha=0.99

Dataset 2 1000 16 0.05 0.75
Environments
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
——————————————————————–

Dataset 1 1000 48 0.02060244 1.81 # 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

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
——————————————————————–

Dataset 1 1000 48 0.001014874 1.98 # 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

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
——————————————————————–

Dataset 1: alpha=0.9 1000 48 0.06406186 1.95 # 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
Dataset 2: alpha=0.99 1000 48 0.07499925 1.14
Environments
Run-File Problem Statement Data Solution
Matlab Toolbox Data
Matlab Matlab Code Data
R R Code Data

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).