The Logarithmic Derivative in Scientific Data Analysis - Ruediger Grunwald (Encyclopedia, 2025)
The logarithmic derivative has been shown to be a useful tool for data analysis in applied sciences because of either simplifying mathematical procedures or enabling an improved understanding and visualization of structural relationships and dynamic processes. In particular, spatial and temporal variations in signal amplitudes can be described independently of their sign by one and the same compact quantity, the inverse logarithmic derivative. In the special case of a single exponential decay function, this quantity becomes directly identical to the decay time constant. When generalized, the logarithmic derivative enables local gradients of system parameters to be flexibly described by using exponential behavior as a meaningful reference. It can be applied to complex maps of data containing multiple superimposed and alternating ramping or decay functions. Selected examples of experimental and simulated data from time-resolved plasma spectroscopy, multiphoton excitation, and spectroscopy are analyzed in detail, together with reminiscences of early activities in the field. The results demonstrate the capability of the approach to extract specific information on physical processes. Further emerging applications are addressed.