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Peer Reviewed
The Logarithmic Derivative in Scientific Data Analysis
This is a peer-reviewed entry published in Encyclopedia Journal, full entry can be found at 10.3390/encyclopedia5020044
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.
spectroscopy data analysis logarithmic derivative temporal decay nonlinear optics nonlinear order multiphoton processes pulsed lasers signal processing overlapping spectral lines
The spectroscopic analysis of materials or the study of the dynamics of excitation processes require the efficient processing of large amounts of data at high speed. Therefore, a significant data reduction and high feature selectivity by appropriate mathematical procedures are of increasing importance for modern fields like materials research, analytical chemistry, pharmacy, medicine, plasma dynamics, laser physics, or nonlinear spectroscopy. It is well known that characteristic properties of spectral data can be filtered out by mathematical derivatives [1][2]. Logarithmic differentiation is of particular interest in natural sciences and engineering because many processes follow exponential rules or contain exponential components. Prominent examples are the decay of radioactive isotopes, the spontaneous light emission of excited atoms, the discharge of capacitors, the growth of populations [3], or heat conduction [4], to mention but a few. Therefore, logarithmic derivatives often directly reveal the key parameters of functions of interest and deserve closer consideration. For exponentially growing or decaying spatio-temporal gradients, the logarithmic derivative is constant and can be used as a characteristic parameter. In dielectric relaxation spectroscopy, for example, the logarithmic derivative is applied to calculate the real part of the complex permittivity function of multiphase or polymeric materials [5][6], or liquids [7]. Similar mathematical methods help to analyze the conduction properties of superconducting materials [8]. For the temporal analysis of diffusion [9] or for the analysis of cell growth in biology [10], the logarithmic derivative is also a useful tool. In semiconductor physics, an inverse logarithmic derivative method is used to determine energy gaps and types of electronic transitions [11] or to simulate inelastic scattering [12]. Moreover, a logarithmic derivative lemma is applied to the solution of partial differential equations, which are related to complex-valued functions with isolated pole points [13]. Another application in mathematics is to compute Bessel functions of arbitrary order [14]. In the field of mathematical statistics, which has an essential impact on genetics, psychology, finance, etc., the score function based on the logarithmic derivative plays a central role [15][16]. The score function is the slope of the logarithmic likelihood function, the variance of which is well-known as the Fisher information [17]. The relevance of the Fisher information for fundamental physics is to link statistics to extremal principles, e.g., in diffraction optics [17][18] and quantum mechanics. The limit to the standard deviation is given by the Cramér–Rao lower bound, which is related to the inverse of the Fisher information matrix [19]. The theoretical description of nuclear collisions and interactions between atoms and molecules was improved by introducing the logarithmic derivative of the wave function [20][21]. Plenty of examples give proof of the capabilities of the mathematical approach.
Moreover, the approach of logarithmic differentiation offers specific advantages from the point of view of computation techniques. It simplifies the determination of derivatives of multifactorial products or quotients by transforming them into easy-to-handle sums or differences [22]. This also promises to open up new prospects for efficient separation and filtering procedures. In deep machine learning, data mining, astronomical image processing and many other sectors, the logarithmic derivative is a frequently used method for clustering [23] or feature selection [24]. The price one has to pay for the simplification of the mathematical procedures, however, is that derivative and logarithm operators respond sensitively to noise. To improve the analysis of noisy signals, different strategies like interpolation, the pre-filtering of data, and the introduction of thresholds have been studied. Most probable distributions can be selected by additional Bayesian analysis [25]. For applications in hydraulics, another regularization approach was developed, which is based on the minimization of an adapted functional that penalizes fluctuations and leads to a diffusion-like differential equation in the logarithmic derivatives [26].
If the logarithmic derivative approach is carefully applied, it proves to be an efficient and stimulating technique that extends the capabilities of data processing. Here, the application of the logarithmic derivative and related functions will be demonstrated for selected examples addressing temporal decay processes, the nonlinear order of multiphoton dissociation, and spectroscopy.

References

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Ruediger Grunwald
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Online Date: 14 Apr 2025
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