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_errnsubject 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_errnsubject 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_errnsubject to
Linearmulti = 1 (constraints defining multi-stochastic matrix) Box constraints (Lower bounds of elements of multi-stochastic matrix) ——————————————————————– CVaR_abs_err = CVaR Absolute Error ——————————————————————– nalpha=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	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_snsubject 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_snsubject 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_snsubject 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	0.07499925	1.14
Environments
Run-File	Problem Statement	Data	Solution
Matlab Toolbox		Data
Matlab	Matlab Code	Data
R	R Code	Data

CASE STUDY SUMMARYn This case study builds a 3-dimensional, m = 3, checkerboard copula with ×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).