On a Test for Trends in Piecewise Monotonic Data Approximation Method - Vassiliou, E Vassiliou; Demetriou, I C (Lecture notes in engineering and computer science, 2021)
We consider the numerical results of the piecewise monotonic approximation method combined with a test for trends of the residuals that is applied to a data set which is considered difficult to be fitted. If n are the data and k−1 is the number of positions of the joins of the monotonic sections, the method requires O(n k−1 ) combinations of positions in order to find an optimal one. The test attempts to distinguish genuine trends from data errors in order to provide automatically an adequate number of monotonic sections. This problem is hard to solve, due to both the inherent combinatorial nature of the piecewise monotonic approximation method and the need to compare the goodness of competing fits. We show that the test for trends by controlling an upper bound of the variance of the fit allows progressively improved approximations. Our results expose some critical questions and suggest a subject for future research.