Identification of step pattern in ordered data sets using the Walsh transform algorithm

Abstract

This work is an attempt to recognize discontinuities within ordered data series by the use of the Walsh functions. Walsh functions are a system of orthogonal functions used in showing how the energy in a given signal is distributed among these rectangular wave components revealing the existent boundaries. This paper presents the capacity of Walsh transform on determining the statistical significance of discontinuities within ordered marine ecological data. Marine ecosystems present high vertical patterns; dataseries on particulate matter concentrations (measured by the Underwater Video Profiler (UVP)) were used to recognize homogeneity limits in water columns. Firstly, Walsh functions are generated in a compact form. Then, a spectral decomposition is performed on the ecological signal to obtain a stepped Walsh version; the step width at this stage is unvarying, picking constant vertical sections. To avoid this limitation, the estimated series is then smoothed to recognize changing step widths by merging successive blocks presenting statistical non-significant difference; here the nonparametric Kolmogorov–Smirnov test is used. This Walsh transform algorithm provides a fast, simple yet accurate means of separating ordered ecological data into groups of observations corresponding to different marine water masses. This method is applied to four profiles measured at the same site during 2 days. The temporal evolution of depth transitions are first discussed and then used for the validation of the transform model.