# Case Study: Estimating Probability Distributions with Quantile Regression

Back to main page

** Case study background and problem formulations**

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 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)

———————————————————————————

KB_err = Koenker and Basset error function

———————————————————————————

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).

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

Dataset1 | 3 | 90 | 0.25982154535 | 0.01 | ||||
---|---|---|---|---|---|---|---|---|

Environments |
||||||||

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

Matlab Toolbox | Data | |||||||

Matlab Subroutines | Matlab Code | Data | Solution |

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

Dataset2 | 7 | 102 | 0.006043634180 | 0.01 | ||||
---|---|---|---|---|---|---|---|---|

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).

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

Dataset1 | 147 | 90 | 87.2628707232 | 2.69 | ||||
---|---|---|---|---|---|---|---|---|

Environments |
||||||||

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

Matlab Toolbox | Data | |||||||

Matlab Subroutines | Matlab Code | Data | Solution |

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

Dataset2 | 343 | 102 | 1.60762360745 | 13.34 | ||||
---|---|---|---|---|---|---|---|---|

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).

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

Dataset1 | 147 | 90 | 87.217775187 | 1.70 | ||||
---|---|---|---|---|---|---|---|---|

Environments |
||||||||

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

Matlab Toolbox | Data | |||||||

Matlab Subroutines | Matlab Code | Data | Solution |

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

Dataset2 | 343 | 102 | 1.606999147 | 18.85 | ||||
---|---|---|---|---|---|---|---|---|

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.