The Logarithmic Derivative in Scientific Data Analysis

1. Introduction

The spectroscopic analysis of materials or the study of the dynamics of excitation processes require to efficiently process 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 a closer consideration. For exponentially growing or decaying spatio-temporal gradients, the logarithmic derivative is constant and can be used as characteristic parameter. In dielectric relaxation spectroscopy, e.g., 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 a useful tool as well. 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 essential impact on genetics, psychology, finance etc., the score function which is based on 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 point of view of computation techniques. It simplifies the determination of derivatives of multifactorial products or quotients by transforming them into easier-to-handle sums or differences [22]. This also promises to open 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 [23,24], or feature selection [25]. The price one has to pay for the simplification of the mathematical procedures, however, is that a differentiation sensitively responds on noise. To improve the analysis of noisy signals, most probable distributions can be selected by additional Bayesian analysis [26].

If the approach of logarithmic derivative 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.

2. Logarithmic Derivative and a Generalized Decay Constant

The variation of a signal function f(t) in time domain can be described by appropriate time constants. For example, a single exponential relaxation process (e.g., by spontaneous emission) in a simplistic model is fully characterized by a decay time τ:

$$f\left(t\right)=f_0\exp \left(-t/\tau \right)$$

(1)

To also involve rising parts of the signal (e.g., by populating a radiative level via optical or electrical pumping), eq. (1) has to be generalized. This is possible by using the temporal logarithmic derivative which is defined as

$$\mathcal{L}\left(t\right)={\displaystyle \frac{\partial \left[\ln \left(f\left(t\right)\right)\right]}{\partial t}}={\displaystyle \frac{f^{\prime }\left(t\right)}{f\left(t\right)}}$$

(2)

with

$$f^{\prime }\left(t\right)={\displaystyle \frac{\partial f\left(t\right)}{\partial t}}$$

(3)

The relationship between $\mathcal{L}\left(t\right)$ and f’ enables to differentiate functions by differentiating their logarithm:

$$f^{\prime }\left(t\right)=f\left(t\right){\displaystyle \frac{\partial \left[\ln \left(f\left(t\right)\right)\right]}{\partial t}}$$

(4)

The negative inverse of the logarithmic derivative represents a characteristic time τ^* which is well suited for a specific description of signals in time domain except of the zeros of the f’(t) where the function is not defined and singularities appear:

$${\tau }^{*}\left(t\right)=-{\displaystyle \frac{1}{\mathcal{L}\left(t\right)}}=-{\displaystyle \frac{f\left(t\right)}{f^{\prime }\left(t\right)}};f\text{’}\left(\mathrm{t}\right)\ne 0.$$

(5)

In the special case of an exponential decay according to eq. (1), τ^* is found to be a constant and identical to τ. In all other cases, however, the time dependent signal can also be described by τ^* (t), independently on the sign of the time derivation (i.e., including increasing signal amplitudes). This essential extension makes the negative inverse logarithmic derivative a particularly valuable tool for the analysis of arbitrary temporal functions with exponential components.

The formalism is not restricted to the temporal domain and enables in the same way the determination of characteristic parameters of spatial, spectral or arbitrary other distribution functions of physical quantities where the time coordinate has to be replaced by corresponding parameters, respectively.

3. Examples

3.1. Time-Dependent Spontaneous Emission of Molecular Nitrogen

To understand the complex population dynamics of gas discharges, laser-induced plasma filaments, or gas lasers, relevant time resolved pump and relaxation processes have to be detected. In this case, the logarithmic derivative can be used to describe the temporal evolution of the population of a characteristic molecular level involving both population and depopulation paths. In the following we will demonstrate how the logarithmic derivate can be applied to analyze the temporal development of a diffuse volume plasma in a low-pressure nitrogen laser excited by a high-voltage electrical-discharge.

At the time of pioneering laser studies during the sixties of last century [27], nitrogen lasers were an attractive short-wavelength source because of enabling to easily construct home-made, cost-effective, highly compact devices [28,29]. Lasing at a UV wavelength of 337.1 nm corresponds to the (0,0) transition C³π_u → B²πg of diatomic N₂ molecules in the second positive system [30]. Further N₂ laser vibrational bands were reported for transitions at 357.69 nm (0,1), 371.05 nm (2,4), 315.93 nm (1,0), 380.49 nm (0,2), 405.94 nm (0,3), 375.54 nm (1,3), 399.84 nm (1,4), and 353.67 nm (1,2) [31].

When high-power excimer lasers entered the market and solid state lasers were efficiently frequency converted, the application of nitrogen-lasers was mainly focused on pumping of dye lasers or to provide portable low-power sources for fluorescence spectroscopy and low-power materials processing [29] with repetition rates of up to some kHz and pulse durations in nanosecond or picosecond range. With the availability of femtosecond lasers, however, the remote optical excitation of air filaments moved into the spotlight [32,33,34,35,36,37] which are promising candidates for THz and MIR generation [38], as well as low-jitter laser-triggered switches [39]. Furthermore, nitrogen discharges in capillaries enable to generate x-ray lasing [40]. Motivated by such investigations, nitrogen plasma spectroscopy experienced a remarkable renaissance [41,42]. In diffuse gas discharges, the energy transfer of electrons to N₂-molecules strongly depends on nonstationary plasma parameters which are tuned by gas pressure, gas mixture, time-dependent voltage, pre-ionization, electrode profile and distance, electrode material etc. [43,44]. The discharge follows the oscillation of the resonant electrical circuit formed by the capacity, inductivity and plasma resistance [45,46,47]. Simultaneous emission of up to seven laser bands in the second positive band was observed depending on discharge parameters [48].

A direct and highly resolved measurement of the time-dependent current density profile is challenging. Measurements with antennas or Rogowski coils [49] may cause distortions of the discharge or integrate over large volume. If the measurements are performed at sufficiently low gas pressure, the lifetime of the spontaneous emission of the upper level represents, in good approximation, the collision-free case and therefore should follow the upper level population. To properly sample the temporal behavior, the time constant of the detecting system has to be small compared to the characteristic time constants of discharge pumping and radiative transitions, and the electronic jitter has to be negligible as well.

Allow me a short excursion to the history. In the laboratory of Prof. Johannes Heinrich Hertz at the Central Institute for Optics and Spectroscopy (ZOS) of the Academy of Sciences in Berlin who would have had his 100th birthday in 2024, spectroscopic experiments with low-pressure nitrogen lasers were performed during the early 1980s [50]. To optimize the laser efficiency, the energy transfer from an oscillating discharge circuit to the population inversion of the laser plasma had to be analyzed. Lasing in nitrogen was of particular interest, not only because of the relative ease of construction of nitrogen lasers as pump sources for dye lasers but also because of being a cheap and available medium for testing the pre-ionization and switching dynamics of excimer lasers [51]. With thyratron-switched transversal discharges, UV pulses of 0.9 MW were obtained. To optimize the laser parameters, the time dependence of selected spectral bands was used to monitor plasma excitation and decay processes at gas pressures down to 0.7 kPa and peak voltages < 24 kV [52,53]. Figure 1 shows the time delay between the oscillating signals of voltage and spontaneous emission for a selected gas pressure of 4 kPa. Spontaneous emission was measured in small signal regime with a calibrated photomultiplier (time resolution 500 ps, detection wavelength 337.1 nm, amplification factor 10⁷).

The detection wavelength corresponds to the laser transition is representative for the most relevant energy transfer channel for this type of laser. Amplified spontaneous emission (ASE) was suppressed to directly derive time-dependent small signal gain of (0,0) transition of the second positive system (C³π_u → B²πg) of the N₂ molecules. The corresponding parameters τ*(t) according to eq. (5) are compared in Figure 2.

One can see that, on the basis of the specific mathematical approach, the dynamics of the population inversion can directly be derived from the rising and falling parts of the curves of the plasma fluorescence. The interplay of electronic excitation (depending on electron density and temperature), optical transitions and fluorescence quenching (depending on pressure dependent collisions with molecules and atoms) is very complex. The influence of amplified spontaneous emission (ASE) was minimized by optically dividing the plasma in short sections. Because of low gas pressures, re-absorption was minimized as well. In this way, it was possible to determine the parameter τ* in the small signal regime. The generalized parameter describes the optical response resulting from the counteracting amplification and absorption processes in the non-thermal laser plasma in a compact manner.

In the example presented here, the effective time constants for upper laser level population are reduced down to about 10 ns which is significantly faster compared to the undisturbed lifetime of about 40 ns [54,55,56,57]. In general, replacing “time constants” by time-variant physical quantities enables for the improved simulation and optimization of time-dependent systems (e.g., pulsed gas lasers) on the basis of appropriate theoretical models.

It should be mentioned that the logarithmic derivative in time-domain was also applied to the analysis of time-dependent photoluminescence processes in semiconductors [58]. It was found that different contributions to the carrier relaxation can be separated by analyzing the logarithmic derivative in a confined polynomial model.

3.2. Nonlinear Order of Multiphoton Processes

To The intensity dependence of the multiphoton ionization (MPI) or multiphoton dissociation (MPD) processes in gas under simplifying assumptions (quasi-monochromatic excitation, exact resonance, square-wave pulses, intensity along the interaction zone, collision-free conditions) can be determined by solving the related population balance equations. At sufficiently low intensity far from saturation, with negligible background signal, and if the conversion rate is simply proportional to the intensity, the product concentration C can be assumed to depend exponentially on the excitation intensity I_exc:

$$C={I_{exc}}^n$$

(6)

where n is the nonlinear order (or exponential index). In general, n may also depend on the intensity and can be described by a modified form of a logarithmic derivation [59,60]

$$n={\displaystyle \frac{\partial \ln C}{\partial \ln I_{exc}}}$$

(7)

A particular example for a nonlinear photolytical process is presented in Figure 3 by the gas-phase multiphoton dissociation of Mo(CO)₆ (molybdenum hexacarbonyl) at XeCl-laser wavelengths around 308 nm and a pulse duration of 7.5 ns. The gas pressure was stabilized at 20 Pa without additional buffer gas. The intensity dependent emission line of the molybdenum atom transition z⁵P⁰-a⁵S (J₂=3 to J₁=2, wavelength 550.649 nm [61]) is depicted in a double-logarithmic plot [62]. The maximum slope of the quasi-linear part corresponds to a nonlinear exponent of about n_max = 3.3 between laser intensities of 2.5 x 10⁸ W/cm² and 3.3 x 10⁸ W/cm² which indicates a sequential multiphoton excitation mechanism involving at least 3 photons. At intensities above 3.3 x 10⁸ W/cm², saturation behavior is observed. The intensity dependent nonlinear exponents n(I_L) along the entire curve can be determined by applying eq. (7). Assuming a dissociation energy of 9.44 eV for a complete release of all ligands [63,64,65] and a 3-step excitation with photon-energies of 4.03 eV, an excess energy of up to 2.59 eV can be expected (in contrast to the values reported in [62] which were calculated for a different photon energy). The experimentally detected nonlinear order widely confirms this model and an additional contribution of a 4-photon excitation channel seems to play a role.

A more detailed information on the excitation mechanisms is provided by determining the nonlinear order as a function of excitation energy (Figure 4).

The indicated nonlinear path to neutral Mo atoms and similar findings for the generation of neutral C₂-radicals by the multiphoton dissociation of aromatic compounds [62] lead to the conclusion that atomic or diatomic neutral photolysis products can be obtained in appropriate parameter windows via sequential UV-multiphoton super-excitation of molecules. The parameter windows (intensity, photon energy) for the formation of such reactive and energetic non-ionic fragments are of particular interest for applications in laser-induced thin-film deposition [65,66] or MPD-pumped dissociation lasers [67]. Furthermore, the nonlinear intensity dependence on the basis of the logarithmic deviation can be exploited to determine laser parameters by analyzing the geometry of focal zones via spatial profiles of fragment fluorescence (photochemical power meter) [68,69]. The quantitative interpretation of MPD-induced fluorescence requires a careful spatio-temporal calibration.

3.3. Separation of Overlapping Spectral Peaks

The accurate analysis of spectra is essential because of identifying the chemical signature of substances, e.g., in chromatography, or to monitor chemical reactions. Often this requires a disentanglement of the contributions of overlapping spectral intensity peaks. For this purpose, numerous methods were developed which are based on approximating functions of the line shape (Gaussian, Lorentzian, Pseudo-Voigt, B-spline, Exponentially Modified Gaussian, Poisson, etc.). A comprehensive review of these techniques would exceed the frame of this paper. In many cases, however, only the positions of overlapping peaks have to be determined for a sufficiently meaningful analysis. To process spectra with idealized Gaussian or Lorentzian peaks, derivative functions from first to fourth order were applied [70,71,72,73,74,75,76,77]. It was found that the reliability of these methods is limited so that spectral peaks remain undetected or incorrect positions may be calculated [78,79]. An alternative approach for a more accurate determination of spectral peak coordinates was developed on the basis of the logarithmic derivative [80]. In this paper, the authors demonstrated that the used assumption of Gaussian shape functions (which is the necessary precondition for the mathematical procedure) enables to reliably identify peak positions independently on Lorentzian contributions.

Close to a single spectral peak, the detected spectral intensity S(λ) can be approximated by a Gaussian distribution function (probability density function):

$$S\left(\lambda \right)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}·\exp \left[-{\displaystyle \frac{1}{2}}·{\left({\displaystyle \frac{\lambda -\text{µ}}{\sigma }}\right)}^2\right]$$

(8)

(λ = wavelength, σ = standard deviation, µ = maximum position). The parameter σ is proportional to the half width at half maximum (HWHM)

$$HWHM=\sigma ·\sqrt{2\ln 2}$$

(9)

The derivative of the natural logarithm of the assumed spectral profile according to eq. (8) enables to linearize the problem:

$${\displaystyle \frac{d(\ln S\left(\lambda \right))}{d\lambda }}=-\left({\displaystyle \frac{\lambda -\text{µ}}{{\sigma }^2}}\right)$$

(10)

In ref. [79] it is shown how this procedure can easily be extended to the case of Lorentzian line shapes and approximating Gaussian distribution functions can be applied. In the case of multiple overlapping peaks, one can assume

$$S_i\left(\lambda \right)={\displaystyle \frac{1}{{\sigma }_i\sqrt{2\pi }}}·\exp \left[-{\displaystyle \frac{1}{2}}·{\left({\displaystyle \frac{\lambda -{\text{µ}}_i}{{\sigma }_i}}\right)}^2\right]$$

(11)

where σ_i and µ_i are the HWHM and the positions of the individual peaks, respectively. The superposition of the peaks results in the spectral intensity function

$$S\left(\lambda \right)={\displaystyle \frac{1}{\sqrt{2\pi }}}{{\displaystyle \sum }}_i^N{\displaystyle \frac{1}{{\sigma }_i}}·\exp \left[-{\displaystyle \frac{1}{2}}·{\left({\displaystyle \frac{\lambda -{\text{µ}}_i}{{\sigma }_i}}\right)}^2\right]$$

(12)

The logarithmic derivative of this sum with respect to the wavelength can be written as

$${\displaystyle \frac{d(\ln S\left(\lambda \right))}{d\lambda }}={\displaystyle \frac{{{\displaystyle \sum }}_i^N\left(\frac{\lambda -{\text{µ}}_i}{{\sigma }_i}\right)S_i\left(\lambda \right)}{{{\displaystyle \sum }}_i^N{S}_i\left(\lambda \right)}}$$

(13)

The approach linearizes the problem. Figure 5, Figure 6 and Figure 7 visualize the principle of linearization schematically for two adjacent Gaussian peaks of small spectral distance and equal maximum intensity. The existence of two peaks (Figure 5) is clearly indicated by the different slopes (Figure 6).

Eq. (13) provides the total spectral intensity of the two superposed Gaussian contributions as

$${\displaystyle \frac{d(\ln S\left(\lambda \right))}{d\lambda }}=\left({\displaystyle \frac{\lambda -{\text{µ}}_1}{{\sigma }_1}}\right)S_1\left(\lambda \right)+\left({\displaystyle \frac{\lambda -{\text{µ}}_2}{{\sigma }_2}}\right)S_2\left(\lambda \right)$$

(14)

The further mathematical procedure for a reliable determination of the peak positions, however, is more complicated and has to be adapted to the ratios of individual HWHMs and distances of the peaks to each other (for the details, see ref. [80]). The error of spectral resolution strongly depends on these parameters. For the example peaks in Figure 5, Figure 6 and Figure 7 it can be shown that the statistical error is too large for a reliable determination of the peak positions for realistic values of experimental noise.

In the following, the frequently occurring case of two overlapping peaks of significantly different HWHMs σ₁ and σ₂ will be discussed. For the sake of simplicity, it is assumed that S(µ₂) = S(µ₁). One expects a spectral shift of narrower peaks towards larger peaks because of the asymmetric background (Figure 8 and Figure 9). The trivial solution for a correction would be to subtract one peak to exactly determine the position of the other one. Unfortunately, the information is not a priori available in most cases. The solution can be found on the basis of the linearization by logarithmic derivation.

To determine the original peak positions, one has to solve the complete system of equations which describes the mutual influences by all neighboring spectral lines [80]. In the case of only two overlapping peaks, these equations are

$$S\left({\text{µ}}_1\right)={\displaystyle \frac{1}{{\sigma }_1\sqrt{2\pi }}}+{\displaystyle \frac{1}{{\sigma }_2\sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{1}{2}}·{\left({\displaystyle \frac{{\text{µ}}_1-{\text{µ}}_2}{{\sigma }_2}}\right)}^2\right]$$

(15)

$$S\left({\text{µ}}_2\right)={\displaystyle \frac{1}{{\sigma }_2\sqrt{2\pi }}}+{\displaystyle \frac{1}{{\sigma }_1\sqrt{2\pi }}}\exp \left[-{\displaystyle \frac{1}{2}}·{\left({\displaystyle \frac{{\text{µ}}_2-{\text{µ}}_1}{{\sigma }_1}}\right)}^2\right]$$

(16)

and the related matrix can be written as

$$\left[\begin{array}{ccc}S\left({\text{µ}}_1\right)& 1& X_{1,2}\\ S\left({\text{µ}}_2\right)& X_{2,1}& 1\end{array}\right]$$

(17)

with the matrix elements for the exponential contributions

$$X_{1,2}=\exp \left[-{\displaystyle \frac{1}{2}}·{\left({\displaystyle \frac{{\text{µ}}_1-{\text{µ}}_2}{{\sigma }_2}}\right)}^2\right]$$

(18)

$$X_{2,1}=\exp \left[-{\displaystyle \frac{1}{2}}·{\left({\displaystyle \frac{{\text{µ}}_2-{\text{µ}}_1}{{\sigma }_1}}\right)}^2\right]$$

(19)

Solving matrix (17) enables for determining the factors ${\displaystyle \frac{1}{{\sigma }_i\sqrt{2\pi }}}$ (see eq. (11)) which contain the unknown HWHMs. By subtracting the contributions of the neighboring peaks, one can calculate the original spectral peak positions µ₁ and µ₂, respectively.

4. Conclusions

In summary, the logarithmic derivative was proven to be a simple and efficient tool for transforming scientific data to extract essential properties if the basic processes have a relationship to exponential structures in spatial, temporal or spectral domain. This was demonstrated by a few simple experimental examples. In the language of image processing, data processing with the logarithmic derivative can be interpreted as a specific kind of contrast amplification. In the particular case of temporal data, the negative inverse of the logarithmic derivative is a universal and compact descriptor for both decreasing as well as increasing functions. Here, the mathematical approach supports an unambiguous physical interpretation. A limitation of the inverse function, however, is the appearance of singularities at minima and maxima of signal functions.

In general, processing data with the logarithmic derivative procedure can be applied to any multi-dimensional data set. If this is meaningful, however, depends on the mathematical structure of the related dynamics or relationships and has to be evaluated from case to case. In physical or chemical context, some prior information on relaxation-, growth- or diffusion-type processes is often available and enables to decide if an application is appropriate. The approach could also be interesting for further tasks of signal processing like decoding of phase information in wave trains or super-resolution microscopy if a Gaussian decomposition is possible.

Additionally, it is worth to be mentioned that the logarithmic derivative can also play an important role in fundamental mathematical research, for example in analytical number theory, cryptology or quantum state estimation [81,82,83]. One can surely expect more surprising applications in future.