Topic: Contributed on Theory and methods for big data Title: Piecewise monotonic approximation for estimating the extreme values of a function from large sets of noisy measurements Authors:  Ioannis Demetriou - University of Athens (Greece) [presenting]
Abstract: We consider the problem of estimating the extrema of a univariate function from noisy measurements of its values by the least squares piecewise monotonic data approximation method. This method makes the smallest change to $n$ data so that the piecewise linear interpolant to the smoothed values consists of $k$ monotonic sections, where $k$ is a prescribed positive integer. The positions of the joins of the monotonic sections are unknowns of the optimization process whose optimal values are determined automatically, which is a combinatorial problem that can have $O(n^k)$ local minima. However, the method gives a global solution in $O(n^2+kn\log_2n)$ computer operations, which is far less than the number of local minima that can occur. We show the efficiency of the method on large sets of noisy observations. Our results suggest some subjects for future research, as for example in automatic peak finding, which is a subject of continuous interest in spectroscopy and chromatography. Other examples arise from financial mathematics, from medical image processing and data science for instance.