An Algorithm for Restoring a Function from Different Functionals for Predicting Rare Events in the Economy

Abstract

This paper aims to restore some parameters of functionals using cubic splines to forecast rare events in finance and economics. The article considers the mathematical method for recovering an unknown function from many different functionals, such as the value of a function, the value of its first derivative, second derivative, as well as a definite integral over a certain interval. Moreover, all observations can occur with an error. Therefore, the author uses a method of recovering a function from different functionals observed with an error. The function is restored in the form of a cubic spline, which has a value-second derivative representation. The optimization problem consists in minimizing several sums of squares of the deviation at once, for ordinary values, for the first derivatives, for the second derivatives, for integrals, and for roughness penalty. For greater flexibility, weights have been introduced both for each group of observations and for each individual observation separately. The article shows in detail how the elements of each corresponding matrix are filled in. The appendix provides an implementation of the method as a FunctionalSmoothingSpline function in R language. Examples of using the method for the analysis and forecasting of rare (discrete) events in the economy are given. Formulas for calculating the cross-validation score CV (α) for the automatic procedure for determining the smoothing parameter α from the data in our problem of recovering a function by many functionals are shown. The paper concludes that the presented method makes it possible to analyze and predict rare events, which will allow you to prepare for such future events, get some benefit from this, or reduce possible risks or losses.