In case of one independent factor the Spline_Sum Operator is defined as follows.

 

Notations

 

= independent factor;

= number of values (points) of independent factor ordered as follows: ;

= index of points, ;

= observation in the point , (dependent value);

= j-th data point, ;

= vector of the factor values;

= vector of dependent values;

= degree of a piecewise polynomial (spline) that interpolates data points ;

= number of polynomial pieces in spline, > 0 ;

= index of polynomial pieces, ;

 = k-th polynomial piece, ;

= decision variable = unknown d-th coefficient in k-th polynomial piece, ;

= vector of decision variables;

= number of unknown coefficients of all polynomial pieces in spline;

= set of points (knots) partitioning segment into sub-segments , ;

= index of polynomial pieces, ;

= sub-set of indexes of points which are included into sub-segment .  The set is a partition of set {1,...,J}. It partitions data points between polynomial pieces of a spline;

If a set of knots is not specified as an input data then it is generate so that pairs are partitioned evenly between polynomial pieces, i.e. the difference between numbers of indexes for segments  is not greater than 1, i.e. , .

In case of using spline function in logexp_sum function a set of knots is generated so that significant components of vector be partitioned evenly. In this case values of vector’s components should be 0 or 1. Values 0 and 1 partition components on two subsets. Significant components are components included in subset with less cardinality.

 

= smoothing degree of a spline, . This parameter is required to support smoothness of piecewise polynomial function in knots ;

= Gain Functions with zero scenario benchmark at point j, ;

= Loss Functions at point j, .

 

PSG function Spline_sum with one factor is defined as follows:

.

It generates a set of loss scenarios according to specification..