Notations

 

= number of points (observations) of vector of factors and corresponding independent variable;

= number of independent factors;

= vector of independent factors at point ;

= value of factor i in scenario j ;

= point of dependent variable corresponding to the point ;

= decision variable; coefficient of degree in polynomial piece for factor n,;

= Gain Functions with zero scenario benchmark for factor n at point . The piece number depends on the factor n and point number ;

= joint vector of decision variables; vector of coefficients of polynomial pieces;

= sum of Gain Functions with zero scenario benchmark at point ;

= Loss Functions at point ;

= degree of spline of factor n, , integer;

= number of pieces for factor n, , integer;

= smoothing degree of a spline of factor n, , integer;

= variable for range of variation for individual spline in knots of factor n;

= upper bounds for range of variation for individual spline in knots of factor n;

= vector of polynomial degrees;

= vector of polynomial piece numbers;

= vector of polynomial smoothness;

= vector of upper bounds for splines knots ranges;

 

= PSG function Spline_sum generating a set of loss scenarios using initial data and a smoothing constraints (assuring smoothness according to specification);
 

Logarithms Exponents Sum (logexp_sum) function (log-likelihood function in logistic regression):

.

 

       

= PSG Maximum Likelihood for Logistics Regression function Logarithms Exponents Sum applied to Spline_sum function. All should be 0 or 1;

 

Note. Function evaluates probability of outcome 1 for every scenario i.

 

Cardinality function (cardn) function counts the number of scaled nonzero elements of a decision vector with precision :

,

 

 

Optimization Problem 1

 

maximizing log-likelihood for building spline

                  (CS1)

 

calculation of Logistic on built spline

 

     

 

 

Optimization Problem 2

maximizing log-likelihood

                                                    (CS.2)

calculation of Logistic function

 

 

Optimization Problem 3

maximizing log-likelihood

                                                        (CS.3)

subject to

constraint on cardinality

                                                            (CS.4)

 

calculation of Logistic function

 

 

Optimization Problem 4

CrossValidation

           maximizing log-likelihood

                                        (CS.5)

           calculation of Logistic function