Applications of NIMRAD in Electrochemistry

Abstract

This chapter shows how to combine the methods outlined in the previous chapters for application to electrochemistry. The first adopted method, which can be considered as original, is based on the ideas of Yu. Babenko, who generalised the Pythagorean theorem for a wide class of geometrical figures with high symmetry. By applying this method is possible to find some discrete geometrical invariants (DGI) in random sequences to identify deterministic and quantitative parameters inside the measured data, which can represent a universal platform for comparing random sequences, one with each other. An example illustrates the procedure for the treatment of electrochemical measurements, which can be generalised for the quantitative reading of other random functions. Another approach derives from the combination of the modified Fourier transform and the generalised principal component analysis (GPCA), which enables, in some cases, a more detailed data analysis. The third method is associated with the generalisation of the quasi-periodic processes on fractal objects. If one considers the self-similar/fractal processes, then it is possible to create the fractal theory of percolation, which can find confirmation from the analysis on real data. The reader will find this theory rather efficient for application to other fractal objects where is possible to observe similar percolation phenomena.

Appendices

8.A Appendix

In this Appendix, it is justified the selection of the scaling parameter ξ in the general fitting formula (8.57). Suppose that instead of the scaling factor ξⁿ it is given the product ξ₁ξ₂…ξ_n generated by the random structure of the percolation cluster. It is also supposed that these random scaling factors have small deviations with respect to the mean value ξ_i = 〈ξ〉 + δ_i, |δ_i| <  < 〈ξ〉. By substituting these factors into the product gives

$$ {\displaystyle \begin{array}{l}\prod \limits_{i=1}^n{\upxi}_i=\exp \left[\left(\sum \limits_{i=1}^n\ln \left(\left\langle \upxi \right\rangle +{\updelta}_i\right)\right)\right]={\left\langle \upxi \right\rangle}^n\exp \left[\left(\sum \limits_{i=1}^n\ln \left(1+\frac{\updelta_i}{\left\langle \upxi \right\rangle}\right)\right)\right]\\ {}\cong {\left\langle \upxi \right\rangle}^n\exp {\left(\frac{\left\langle \updelta \right\rangle }{\left\langle \upxi \right\rangle }-\frac{1}{2}\frac{\left\langle {\updelta}^2\right\rangle }{{\left\langle \upxi \right\rangle}^2}+\dots \right)}^n,\kern0.36em \left\langle {\updelta}^s\right\rangle =\frac{1}{n}\sum \limits_{i=1}^n{\left({\updelta}_i\right)}^s.\end{array}} $$

(8.A1)

This useful relationship shows that it is possible to replace the set of the random scaling parameters by one averaged parameter under the relationship

$$ \upxi \to \sqrt[n]{\left(\prod \limits_{i=1}^n{\upxi}_i\right)}=\left\langle \upxi \right\rangle \exp \left(\frac{\left\langle \updelta \right\rangle }{\left\langle \upxi \right\rangle }-\frac{1}{2}\frac{\left\langle {\updelta}^2\right\rangle }{{\left\langle \upxi \right\rangle}^2}+\frac{1}{3}\frac{\left\langle {\updelta}^3\right\rangle }{{\left\langle \upxi \right\rangle}^3}+\dots \right). $$

(8.A2)

Therefore, this parameter is intended in the averaged sense and evaluated on real data with the help of the fitting procedure.

8.B Questions for Self-Testing

• 1. What is the DGI? Why is it useful for the comparison of two random curves?

• 2. Reproduce all functional results that are given in Sect. 8.1. In particular, test expressions (8.11), (8.17) and (8.18) as the most important.

• 3. There is another invariant of the fourth order that can be used for comparison of two random sequences. It admits the separation of variables as well. It has the following form:

$$ {L}_k={A}_{31}{\left(x-{x}_k\right)}^3\cdot \left(y-{y}_k\right)+{A}_{13}\left(x-{x}_k\right)\cdot {\left(y-{y}_k\right)}^3-2{A}_{22}{\left(x-{x}_k\right)}^2\cdot {\left(y-{y}_k\right)}^2 $$

Prove that it has the same form as before (see expressions (8.17) and (8.20)) however, the statistical parameters are different:

$$ {\displaystyle \begin{array}{l}{\upsigma}_x=\frac{2\left[3{\left(\left\langle \Delta x\cdot {\left(\Delta y\right)}^2\right\rangle \right)}^2-\left\langle {\left(\Delta x\right)}^2\Delta y\right\rangle \cdot \left\langle {\left(\Delta y\right)}^3\right\rangle \right]}{\left\langle {\left(\Delta x\right)}^3\right\rangle \left\langle {\left(\Delta y\right)}^3\right\rangle -9\left\langle {\left(\Delta x\right)}^2\Delta y\right\rangle \cdot \left\langle \Delta x\cdot {\left(\Delta y\right)}^2\right\rangle },\\ {}{\upsigma}_y=\frac{2\left[3{\left(\left\langle {\left(\Delta x\right)}^2\cdot \Delta y\right\rangle \right)}^2-\left\langle \Delta x\cdot {\left(\Delta y\right)}^2\right\rangle \cdot \left\langle {\left(\Delta x\right)}^3\right\rangle \right]}{\left\langle {\left(\Delta x\right)}^3\right\rangle \left\langle {\left(\Delta y\right)}^3\right\rangle -9\left\langle {\left(\Delta x\right)}^2\Delta y\right\rangle \cdot \left\langle \Delta x\cdot {\left(\Delta y\right)}^2\right\rangle },\\ {}I={\upsigma}_x\left\langle {\left(\Delta x\right)}^3\Delta y\right\rangle +{\upsigma}_Y\left\langle \Delta x\cdot {\left(\Delta y\right)}^3\right\rangle +\left\langle {\left(\Delta x\right)}^2\cdot {\left(\Delta y\right)}^2\right\rangle .\end{array}} $$

These parameters modify also the quadratic and the fourth forms

$$ {\displaystyle \begin{array}{l}K\left(X,Y\right)={K}_4\left(X,Y\right)+{K}_2\left(X,Y\right)=I\\ {}{K}_4\left(X,Y\right)={\upsigma}_X{X}^3Y+{\upsigma}_Y{XY}^3-2{X}^2{Y}^2\\ {}{K}_2\left(X,Y\right)=\left(3{\upsigma}_X\left\langle \Delta x\Delta y\right\rangle -2\left\langle {\left(\Delta y\right)}^2\right\rangle \right){X}^2+\\ {}+\left(3{\upsigma}_Y\left\langle \Delta x\Delta y\right\rangle -2\left\langle {\left(\Delta x\right)}^2\right\rangle \right){Y}^2+\\ {}+\left(3{\upsigma}_X\left\langle {\left(\Delta x\right)}^2\right\rangle +3{\upsigma}_Y\left\langle {\left(\Delta y\right)}^2\right\rangle -8\left\langle \Delta x\Delta y\right\rangle \right) XY.\end{array}} $$

• 4. In what cases the GPCA is applicable? Try to formulate the limits of applicability of this approach.

• 5. What are the limits of applicability of the percolation model, outlined in Sect. 8.3?

• 6. For a better understanding of the proposed approaches, try to derive the key formulas figuring in Sects.8.2 and 8.3.