# Case Study: Estimating Probability Distributions with Quantile Regression

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

PROBLEM 1, PROBLEM 2 and PROBLEM 3 use the same m-file “sum_of_kb_err_example_figure.m” which runs problem-specific Matlab Subroutines.

Problem 1: problem_kb_err
Minimize kb_err (minimizing Koenker and Basset error in cycle with different confidence levels)
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KB_err = Koenker and Basset error function
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Run-File and Matlab Toolbox files contain problem formulation and data for calculation of quantiles (VaRs) for multiple confidence levels in cycle (for 49 confidence levels) for the same design matrix. Dimension of the design matrix is (# of Variables)*(# of Scenarios). Values presented in columns “Objective Value” and “Solving Time, PC 3.14GHz (sec)” correspond to results of optimization with the first confidence level. Matlab Subroutines: 1) Link “Matlab Code” contains zip file with m-file subroutine for minimizing KB_err error function with multiple confidence levels without constraints “one_kb_err_function.m” and m-file “sum_of_kb_err_example_figure.m”; 2) Link “Data” contains zipped data files “matrix_new_data.mat” (Dataset1) and “matrix_design.mat” (Dataset2).

Dataset1 3 90 0.25982154535 0.01 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data Solution

Dataset2 7 102 0.006043634180 0.01 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data Solution

Problem 2: problem_sum_kb_err
Minimize sum of(kb_err) (minimizing sum of Koenker and Basset error functions with different confidence levels)
subject to
Linearmulti ≤ 0 (monotonicity constraints on quantiles for one point)
———————————————————————————
KB_err = Koenker and Basset error function
Linearmulti = Linearmulti
———————————————————————————
Run-File and Matlab Toolbox files contain problem formulation and data for calculation of quantiles (VaRs) for multiple confidence levels (for 49 confidence levels) by minimizing sum of Koenker and Basset error functions with different confidence levels under monotonicity constraints on quantiles for one point.
Matlab Subroutines: 1) Link “Matlab Code” contains zip file with m-file subroutine for minimizing sum of KB_err error functions with multiple confidence levels “sum_kb_err_function.m” and m-file “sum_of_kb_err_example_figure.m”; 2) Link “Data” contains zipped data files “matrix_new_data.mat” (Dataset1) and “matrix_design.mat” (Dataset2).

Dataset1 147 90 87.2628707232 2.69 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data Solution

Dataset2 343 102 1.60762360745 13.34 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data Solution
Problem 3: problem_sum_kb_err
Minimize sum of(kb_err) (minimizing sum of Koenker and Basset error functions with different confidence levels)
subject to
Linearmulti ≤ 0 (monotonicity constraints on quantiles for multiple points)
———————————————————————————
KB_err = Koenker and Basset error function
Linearmulti = Linearmulti
———————————————————————————
Run-File and Matlab Toolbox files contain problem formulation and data for calculation of quantiles (VaRs) for multiple confidence levels (for 49 confidence levels) by minimizing sum of Koenker and Basset error functions with different confidence levels under monotonicity constraints on quantiles for multiple points.
Matlab Subroutines: 1) Link “Matlab Code” contains zip file with m-file subroutine for minimizing sum of KB_err error functions with multiple confidence levels “sum_kb_err_function.m” and m-file “sum_of_kb_err_example_figure.m”; 2) Link “Data” contains zipped data files “matrix_new_data.mat” (Dataset1) and “matrix_design.mat” (Dataset2).

Dataset1 147 90 87.217775187 1.70 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Environments Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data Solution

Dataset2 343 102 1.606999147 18.85 # of Variables # of Scenarios Objective Value Solving Time, PC 3.14GHz (sec) Run-File Problem Statement Data Solution Matlab Toolbox Data Matlab Subroutines Matlab Code Data Solution

CASE STUDY SUMMARY
Quntile regressions are done for a grid of confidence levels. Also, quantile regressions are done for multiple confidence levels in one optimization problem with constraints assuring monotonicity of quntile estimates.