Multiple Independent Factors

Case of the Spline_Sum Operator for One Independent Factor may be expanded to the case of Multiple Independent Factors as follows.

Notations

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

= number of independent factors;

= vector of independent factors at point ;

= 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;

= vector of polynomial degrees;

= vector of polynomial piece numbers;

= vector of polynomial smoothness;

= PSG function Spline_sum generating a set of loss scenarios using initial data and according to specification.