Algorithm xxx: Restoration of function by integrals with cubic integral smoothing spline in R

In this paper, a cubic integral smoothing spline with roughness penalty for restoring a function by integrals is described. A mathematical method for building such a spline is described in detail. The method is based on cubic integral spline with a penalty function, which minimizes the sum of squares of the difference between the observed integrals of the unknown function and the integrals of the spline being constructed, plus an additional penalty for the nonlinearity (roughness) of the spline. This method has a matrix form, and this paper shows in detail how to fill in each matrix. The parameter α governs the desired smoothness of the restored function. Spline knots can be chosen independently of observations, and a weight can be defined for each observation for more control over the resulting spline shape. An implementation in the R language as function int_spline is given. The function int_spline is easy to use, with all arguments completely described and corresponding examples given. An example of the application of the method in rare event analysis and forecasting is given.