The problem of noise removal from signals (a signal is just a sequence of measurements of some quantity like a wind speed or stock price) which are secretly composed of steps is surprisingly common and important, and arises in a host of disciplines, including analysis of drill hole data in exploration geophysics, detecting DNA copy-number ratios in genomics, and separating out molecular dynamics from background noise. In each case one seeks to filter away the fluctuations in one's signal to leave behind the steps one believes are contained within.
Our recent papers with the glamourous titles Generalized methods and solvers for noise removal from piecewise constant signals I. Background theory and II New Methods (articles free online), which appeared in the Proceedings of the Royal Society A, introduced a new mathematical framework for the problem, and, as special cases of this framework, developed some new signal processing algorithms to address the problem. The study of this task has been very fragmented across disciplines (from geoscience to physics to biology) so one of our major goals was to present a synthesis of existing work which would expose natural developments. For example having performed our synthesis we presented a particularly simple new method called "robust jump penalization", that exhaustively tests for the location of a new step when the noise is not from a normal or Gaussian distribution while subject to the constraint that the number of jumps should be small. Max and Nick
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