# Case Study: Fitting mixture models with CVaR constraints

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** Case study background and formulation of problems**

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

**PROBLEM1: problem_fitting_mixture (Without Cardinality)**

Minimize KS_distance (KS distance between empirical CDF and normal mixture CDF)

Subject to

Linear = 1 (weights of the mixture sum to 1)

wcvar_ni[j] >= x[j] (CVaRs of the mixture are larger than the CVaRs of the empirical distribution)

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

Dataset | 5 | 10,000 | 0.03091 | 0.03 | |||
---|---|---|---|---|---|---|---|

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**PROBLEM2: problem_fitting_mixture_cardn (With Cardinality)**

Minimize KS_distance (KS distance between empirical CDF and normal mixture CDF)

Subject to

Linear = 1 (weights of the mixture sum to 1)

wcvar_ni[j] >= x[j] (CVaRs of the mixture are larger than the CVaRs of the empirical distribution)

cardn_pos<= z (number of nonzero weights is less than z)

——————————————————————–——————

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

Dataset | 5 | 10,000 | 0.02986 | 2.08 | |||
---|---|---|---|---|---|---|---|

Environments |
|||||||

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

Matlab Toolbox | Data | ||||||

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

R | R Code | Data |

**CASE STUDY SUMMARY**

This case study demonstrates the example of fitting a mixture of normal distributions (to the empirical distribution) with Conditional Value at Risk (CVaR) constraints on the mixture distribution. The initial approximation without constraint was obtained with the Expectation Maximization (EM) algorithm. Further weights of the mixture were corrected to guarantee the minimal CVaR values (estimating fatness of tails). CVaR distance was used to measure closeness of the mixture model to the empirical distribution. Minimization of CVaR distance subject to CVaR constraints on the mixtures weights is a convex optimization problem.