Complex numbers in chemometrics: Examples from multivariate impedance measurements on lipid monolayers
Introduction
Chemometrics has been very successful in applying multivariate data analysis, but almost exclusively by filling vectors, matrices and multiway arrays with real numbers. In electrochemistry however, some of the data are given as complex numbers. The questions that can be asked about this situation are: (1) Do complex numbers fit in with the methods of chemometrics? (2) Should the methods of chemometrics be modified or completely changed when complex numbers are used? (3) Do complex number data lead to different interpretations when used in the complex form? The paper tries to introduce the basic concepts and to give partial answers to the above questions. The examples used are based on multivariate impedance measurements on synthetic biomembranes, also called lipid monolayers, deposited on a mercury electrode. The nomenclature and symbols used in the paper come from three different fields: electrochemistry, complex number theory and linear algebra. Where necessary, ways of avoiding naming and indexing confusion are used.
Two examples are used: (1) a 16 × 50 complex matrix of impedance data for the interaction of different concentrations of magainin with a monomolecular lipid layer deposited on a mercury electrode [1] and (2) a 38 × 50 complex matrix of impedance data using different concentrations in two different electrolyte media of gramicidin A used with the same electrode system. Different aspects of the gramicidin example are given elsewhere [2].
Section snippets
Complex numbers
Complex numbers are given as:or aswhere x is complex number x; a the real part of x; b the imaginary part of x; j: j2 = −1, usually called i outside electrochemistry; |x|: (a2 + b2)1/2 the amplitude; t is the phase angle, tan(t) = b/a.
Properties of complex numbers are described in the mathematics literature [3], [4], [5], [6]. Eqs. (1) and (2) are in the standard notation used in many mathematics books.
An important graphical device is the Argand diagram, sometimes called Wessel plot in
Basic physics
An electrode system in solution conducting a sinusoidal current may be seen as a capacitor. For an ideal capacitor in vacuum the capacitance equation is:The capacitance C is proportional to the area A and inversely proportional to the distance between the plates d. Capacitance is measured in Farad (F). The permittivity of vacuum ɛ0 is 8.85 × 10−12 F m−1.
When the capacitor is filled with a non-conducting (dielectric) material, the equation is changed:where k is the relative
Data analysis
With multivariate measurements, the nomenclature definitions from linear algebra are used: x is a scalar, x is a vector, and X is a matrix. All vectors are column vectors and a row vector is the transpose of a column vector. A transposed vector xT is a row vector. xH is the combined transpose and complex conjugate [11]. For real number vectors: xH = xT. From now on only the above notation will be used. For clarity, the size of a matrix will be indicated between parentheses e.g. Z (I × K) means a
Data pretreatment
In chemometrics, data are sometimes pretreated before a multivariate analysis is carried out. Basic pretreatments are mean-centering over the columns and scaling of each column by its standard deviation. For complex number data, these operations are slightly different from those for real numbers.
The mean z (K × 1) of a complex matrix Z (I × K) is a vector of complex mean values:where 1 is a vector of ones (I × 1) mean-centering of Z is done as follows:where Zc is a mean-centered matrix
Experimental
A monomolecular layer of dioleoyl phophatidylcholine (DOPC) can be deposited on a mercury drop electrode and impedance measurements are made at 50 frequencies 0.1 Hz, 0.13, 0.17, 0.23, …, 49,463 and 65,000 Hz (each frequency is the previous one multiplied by 1.3141) at 5 mV rms and −400 mV DC. The reference electrode was Ag/AgCl in 3.5 M KCl and the counter electrode was Pt. The measurement cell was at 25 °C in an Argon flow. Current generating and recording electronics was from Autolab. Current and
Results and discussion
The magainin data were analyzed by PCA, without any pretreatment. For the 16 × 100 real matrix Y, components 1 and 2 explained 97% and 2.6% of the total sum of squares, respectively. For the 16 × 50 complex matrix Z, components 1 and 2 explained 99.3% and 0.5% of the total sum of squares. One may conclude for this data set that the complex number PCA is better at concentrating the information in the first components.
Fig. 1 shows the Argand diagram of the complex first component. It can be seen that
Conclusion
Complex numbers occur often in chemistry. All Fourier transform based methods produce complex numbers: infrared and Raman spectroscopy, near infrared spectroscopy, nuclear magnetic resonance. In all these methods the Fourier transform is used to transform an interferogram to a spectrum and the spectrum is basically one of complex numbers. There is however a difference with electrochemistry in that the spectra are expected to be amplitude spectra and all effort is made to remove the phase part.
Acknowledgements
Many thanks to Conor Whitehouse and Kathryn Bradley (SOMS, Leeds) for carrying out the experiments and obtaining the data on which this paper is based. The laboratory work was funded by EPSRC Grant No. GR/R67/439/01. The sixth framework EU project BBMO is acknowledged for financial support.
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