Application Area: Advanced Statistics

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Case Study Functions Methodology
Fitting Mixture Models with CVaR Constraints KS distance (ksm_cvar_ni), Mixture CVaR (wcvar_ni), Cardinality (cardn_pos) A mixture of Gaussian distributions is fitted with CVaR and cardinality constraints. Step 1. EM algorithm finds the means and variances of Gaussian distributions in the mixture. Step2. CVaR distance is minimized for finding optimal weights satisfying CVaR and cardinality constrains.
CVaR Norm Regression CVaR (Cvar_risk) , CVaR Max Risk (cvar_max_risk) Linear Regression with CVaR Norm Error function. Two alternative implementations, where CVaR Norm is calculated with:
1. CVaR Risk function with doubled design matrix.
2. CVaR Max Risk function, which calculates CVaR of the maximum of loss and gain on every scenario.
Optimal Hedging of CDO Book (PSG and MIP formulation) Mean Absolute Penalty (meanabs_pen), Polynomial Absolute (polynom_abs), Cardinality (cardn), Linear (linear), Linearmulti (linearmulti) This case study explains how to formulate and solve some nonstandard Linear Regression problems with additional constraints. Such linear regression is used, for instance, to hedge Collateralized Debt Obligation (CDO) with Credit Default Swaps (CDSs).
Mortgage Pipeline Hedging CVaR Deviation (Cvar_dev), Mean Absolute Deviation (Meanabs_dev), Standard Deviation (St_dev), VaR Deviation (Var_dev) Standard versus tail-targeted linear regression problems. Optimal mortgage pipeline hedging strategy with five different deviation measures: Standard Deviation, Mean Absolute Deviation, CVaR Deviation, two-tailed VaR75, and two-tailed VaR90.
Relative Entropy Minimization Relative Entropy (Entropyr) Optimization problem for minimizing Relative Entropy with linear constraints. Relative Entropy is used to find a probability distribution which is the most close to some “prior” probability distribution subject to available information about the distribution. For instance, moments of a distribution are known and we find the “best” distribution accounting for this information.
Distribution Approximation by Maximizing Entropy with Second-Order Stochastic Dominance and Moment Constraints Relative Entropy (Entropyr), CVaR for Discrete Distribution as a Function of Atom Probabilities (pCvar), Linear (Linear) Discrete distribution is approximated by maximizing entropy with the second-order stochastic dominance constraints and moment constraints. The discrete version of Boltzmann theorem implies that the optimal solution has a piecewise-Gaussian form.
Style Classification with Quantile Regression Koenker and Bassett error function (kb_err), Partial Moment Penalty (Pm_pen), Partial Moment Penalty for Gain (Pm_pen_g) Percentile regression for the return-based style classification of a mutual fund. The procedure regresses fund return by several indices as explanatory variables. The estimated coefficients represent the fund’s loads on the indices.
Estimating Probability Distributions with Quantile Regression Koenker and Bassett error function (kb_err), Multiple Linear (linearmulti) Quntile regressions are done for a grid of confidence levels. Also, quintile regressions are done for multiple confidence levels in one optimization problem with constraints assuring monotonicity of quntile estimates.
Estimation of CVaR through Explanatory Factors with Mixed Quantile Regression Partial Moment Penalty (Pm_pen), Partial Moment Penalty for Gain (Pm_pen_g) Mixed Percentile Regression for the estimation of Conditional Value-at-Risk (CVaR) of return distribution of a mutual fund. The estimated coefficients represent the fund’s style with respect to some indices, and therefore the procedure is called “style classification.” We regresses CVaR of the return distribution of the Fidelity Magellan Fund on indices RUJ, RLV, RUO and RLG. CVaR with confidence level 0.9 is approximated by the weighted average of four Value-at-Risks (VaRs) with confidence levels 0.92, 0.94, 0.96, 0.98 .
Estimation of CVaR through Explanatory Factors with CVaR (Superquantile) Regression CVaR (Superquantile) Error Function (cvar2_err), CVaR (Superquantile) Deviation (cvar2_dev), Rockafellar Error Function (ro_err), CVaR Deviation (Cvar_dev), CVaR (Cvar_risk) Estimation of CVaR with CVaR (Superquantile) regression is done by minimizing CVaR (Superquantile) Error. Alternatively, CVaR regression is done in two steps: Step 1) Minimization of Deviation from CVaR (Superquantile) Quadrangle with the residual depending only upon loading factors; Step 2) Intercept = CVaR for the residual from Step1. Equivalently, CVaR regression is also done with Rockafellar Error and Mixed CVaR deviation from the Mixed-Quantile-Based Quadrangle.
Sparse Signal Reconstruction: a Cardinality Approach Cardinality (Cardn), Mean Absolute Penalty (Meanabs_pen), Polynomial Absolute (Polynom_abs) This case study suggests an upproch for Signal Sparse Reconstruction using nonconvex formulations with cardinality functions counting number of nonzero elements in a vector. Three formulated problems are special cases of a broad family of approaches known as Compressive Sensing. Problem 1 minimizes L1-error of regression subject to a constraint on cardinality of the solution vector; Problem 2 minimizes cardinality of the solution vector subject to a constraint on L1-error of regression; Problem 3 minimizes L1-error of regression subject to constraint on the sum of absolute values of the solution vector.
Sparse reconstruction problems from SPARCO toolbox Mean Absolute Penalty (Meanabs_pen), Polynomial Absolute (Polynom_abs) This case study presents problem formulations and its solutions for a set of sparse reconstruction problems taken from SPARCO toolbox.
The objective of Sparse Reconstruction is to find a decision vector which has a small number of non-zero components and satisfies exactly or almost exactly a system of linear equations. There are many variants of optimization formulations of such problems.
Many problems included in SPARCO toolbox were solved in so called “L1Relaxed D” formulation. “L1Relaxed D” minimizes L1-error of regression with one linear inequality on the sum of decision vector components; the decision vector components are nonnegative (number of decision variables is doubled to achieve non-negativity). The non-negativity of variables is quite important because an optimal vector contains many zero variables. To investigate property of solution we solved several problems with different values of upper bound in the linear inequality and calculated cardinality and max functions in optimal points.
Some problems were solved in so called “L1Relaxed” formulation with original set of variables (without doubling the number of variables to achieve non-negativity). Variables are bounded by box constraints in this formulation. For these problems “L1Relaxed” formulation is more effective compared to “L1Relaxed D” formulation.
Additionally, many problems were solved in so called “L2 D” or LASSO formulation which also has double set of variables but does not have constraints.
Sum of decision variables multiplied by some coefficient is used as regularization term in the objective function. This problem can be easy solved by methods for unconstrained optimization.
We used SPARCO toolbox software to extract data for the considered problems.
SPARCO toolbox provides a set of operators to deal with data.
We converted the problems data to PSG format and solved them in PSG Run-File envoronment.
Sparse reconstruction problems from SPARCO toolbox in MATLAB Environment (fnext) Quadratic Function (Quadratic) This case study solves sparse reconstruction problems from SPARCO toolbox using External Functions Tool of PSG and operator given by SPARCO toolbox for implicit working with matrices.
SPARCO is a suite of problems for testing and benchmarking algorithms for sparse signal reconstruction (see, [1,2]). It is also an environment for creating new test problems. Also, a suite of standard linear operators is provided from which new problems can be assembled. SPARCO is implemented entirely in MATLAB and is self contained.
Problems included in the SPARCO toolbox were initially considered by different authors in different application areas: imaging, compressed sensing, geophysics, information compressing, etc. Relevant references can be found in the SPARCO toolbox.
The objective of Sparse Reconstruction is to find a decision vector which has a small number of non-zero components and satisfies exactly or almost exactly a system of linear equations. There are many variants of optimization formulations of such problems.
We solved problems included in SPARCO toolbox in so called “LASSO-O” formulation. “LASSO-O” minimizes L2-error of regression with adding to objective a regularization linear term which is equal to the sum of absolute values of variables. The regularization term is intended to “suppress” components with small values. To investigate property of solution we solved every problem with different weight of regularization linear term and calculated cardinality and max functions in optimal points. These problems can be easy solved by methods for unconstrained optimization.
SPARCO toolbox provides a set of operators to deal with data. Problems were solved in PSG MATLAB Environment with the PSG External Function subroutine to avoid generating full matrix and to save time and memory.
Calibrating Risk Preferences Mean Absolute Penalty (Meanabs_pen) This case study extracts risk preferences of investors by solving a linear regression model with linear constraints on coefficients. “Risk preferences” are expressed by a risk functional (a deviation measure), which is used by an investor for measuring risk and solving portfolio optimization problems. Contrary to the classical Markowitz portfolio theory, where investors measure risk by standard deviation, this case study assumes that the unknown deviation measure belongs to a class of Mixed CVaR Deviations. In particular, we consider the case when the Mixed CVaR Deviation is a weighted average of the following five CvaR Deviation terms with confidence levels 50%, 75%, 85%, 95%, and 99%.
The Mixed CVaR Deviation has five weighting parameters (lambdas), which are nonnegative and sum up to 1. These lambda coefficients are estimated by matching the market option prices with prices expressed via generalized CAPM pricing relations. Matching is done by minimizing a L1 (the error term is sum of absolute values of the differences between market and calculated prices).
Support Vector Machines Based on Tail Risk Measures Quadratic Function (Quadratic), CVaR (Cvar_risk), Maximum of CVaR (Max_cvar_risk) This case study illustrates the application of the CVaR methodology to the Support Vector Machine (SVM) classification problem.
Given a training data , where are features and are class labels, the basic idea of SVM is to find an optimal separating hyper-plane (in the features space) maximizing a margin between two classes. Cortes et al. (1995) proposed to solve SVM classification problem using quadratic programming. An alternative formulation, known as nu-SVM, was suggested by Scholkopf, et al. (2000). Takeda and Sugiyama (2008) proposed to use the CVaR risk measure in classification and formulated the SVM learning problem as a CVaR minimization problem. Wang (2009) proposed robust nu -Support Vector Machine based on worst-case CVaR Minimization.
Case study contains two problem formulations: 1) regularized CVaR and 2) regularized robust CVaR minimization. Both problems include additional quadratic regularization term.
Logistic Regression and Regularized Logistics Regression Applied to Estimating the Probability of Cesarean Section Logarithms Exponents Sum (Logexp_sum), Polynomial Absolute (Polynom_abs) Two optimization formulations of the logistic regression problem: 1) Maximization of the log-likelihood function (“plain vanilla” logistic regression); 2) Maximization of the log-likelihood function minus additional regularization term (regularized logistic regression). The first formulation is implemented by maximizing the log-likelihood PSG function “logexp_sum”. The regularization term in the second formulation was subtracted from the log-likelihood function to improve the out-of-sample performance of the regression model. For regularization we used PSG “polynom_abs” function (which is the sum of weighted absolute values of factors).
Projection on Polyhedron with Various Norms CVaR Component Absolute (Cvar_comp_abs), Polynomial Absolute (Polynom_abs), Maximum Component Absolute (max_comp_abs), quadratic (quadratic) Projection problems with various norms on a polyhedron set given by a system of linear inequalities. Computational experiments for and spaces. Several polyhedron sets with different number of hyperplanes, such that . The projection of 0 on is solved with CVaR Absolute Norm (Problem 1), norm (Problem 3), and the weighted average of and norms (Problem 2).
Spline Approximation Spline Sum (spline_sum), Standard Penalty (st_pen), Mean Absolute Penalty (meanabs_pen), Logarithms Exponents Sum (Logexp_sum) Splines are calibrated to approximate one dimension observation data. Input data for building a spline are vectors containing data of independent and dependent variables and parameters defining number of knots and smoothing degree of the spline. The splines are calibrated by minimizing various error functions, such as mean square error, mean absolute error, and maximum likelihood logistic regression function (PSG functions: st_pen, meanabs_pen, and logexp_sum, accordingly).
Spline Regression Spline Sum (spline_sum), Logarithms Exponents Sum (Logexp_sum), Polynomial Absolute (polinom_abs) Logistic regression with Sum of Splines for approximation of the multi-dimension observation data. Input data are observations of independent and dependent variables,parameters of every spline (number of knots, power and smoothing degree of the spline, upper bounds for range of variation for individual spline in knots, positions of knots – optionally). Likelihood logistic regression function is maximized (PSG function: logexp_sum). The k-fold Cross Validation technique is used. The number of independent input variables is reduced by including in optimization problem boolean variables. Values of splines at knot points are bounded.
Data Envelopment Analysis Linear (Linear), Multiple Linear (linearmulti), Maximum Risk (max_risk) Comparing relative managerial efficiency of five companies by applying the CCR Model from Data Envelopment Analysis (DEA). The model maximize the ratio of the weighted outputs and the weighted inputs of the company for every company, subject to constraints prohibiting that the ratio of the other companies to be higher than 1. The optimization problem is solved with two equivalent formulations utilizing the Multilinear and Max_Risk PSG functions
Classification by Maximizing Area Under ROC Curve (AUC) Probability of Exceedance (pr_pen), Difference of Two Loss Functions This case study maximizes AUC (Area Under ROC Curve) by minimizing PSG probability function (pr_pen). Two equivalent variants are considered: 1) using difference of two independent random linear functions presented by two different matrices of scenarios (with the same column headers); 2) using one matrix of scenarios which is manually generated by taking differences of linear functions from two different matrices (it is possible only for small dimensions because the number of rows in the resulting matrix equals the product of number of rows in the first and in the second matrix).
Classification by Buffered AUC (bAUC) Maximization Partial Moment Penalty (pm_pen), Partial Moment Penalty for Gain (pm_pen_g) This case study considers buffered version (bAUC) of classification criteria Area Under the Receiver Operating Characteristic Curve (AUC). Two optimization settings: 1) maximizing bAUC and then finding intercept by minimizing some cost function; 2) Construction of “Efficient Frontier” by minimizing cost function with several constraints on bAUC.
Linear Regression in Tranche ltranche, Mean Squared Error (meansquare_err), L1 Error (meanabs_pen) This case study solves a linear regression problem for building an optimal reinsurance contract. The contract specifies loading coefficients for losses and attachment and detachment points of a tranche. The insurance company wants to find an optimal linear combination of losses in the contract specification to get a good protection in the specified tranche.
Minimization of Kantorovich-Rubinstein distance between two distributions (Kantorovich Distance between Two Distributions (kantor), Average Kolmogorov-Smirnov Distance between Two Distributions (ksm_avg), Cardinality (cardn), Linear (linear) This case study demonstrates how approximate fixed distribution by variable distribution minimizing Kantorovich-Rubinstein distance or Average Kolmogorov-Smirnov distance between distributions and obtain the same rezult. The case study provides two Problem statements for minimization Kantorovich-Rubinstein distance and two Problem statements for minimization Average Kolmogorov-Smirnov distance.
Approximation of a Discrete Distribution by other Discrete Distribution Using CVaR Distance.
Error Type Control via Buffered Neyman-Pearson Classification and Spline-based Feature Transformations  Minimizing bpoe_g with constraint on cvar_g.  This case study demonstrates how to solve binary classification with error type control via solving the Buffered Neyman-Pearson (bNP) classification problem where features are first transformed via splines in a way that is optimal for the bNP classification task, then the transformed features are fed into the final bNP classification problem.