Estimation of diffusion coefficients from voltammetric signals by support vector and gaussian process regression
- Martin Bogdan^{1, 3},
- Dominik Brugger^{1},
- Wolfgang Rosenstiel^{1} and
- Bernd Speiser^{2}Email author
DOI: 10.1186/1758-2946-6-30
© Bogdan et al.; licensee Chemistry Central Ltd. 2014
Received: 5 December 2013
Accepted: 24 April 2014
Published: 28 May 2014
Abstract
Background
Support vector regression (SVR) and Gaussian process regression (GPR) were used for the analysis of electroanalytical experimental data to estimate diffusion coefficients.
Results
For simulated cyclic voltammograms based on the EC, E_{qr}, and E_{qr}C mechanisms these regression algorithms in combination with nonlinear kernel/covariance functions yielded diffusion coefficients with higher accuracy as compared to the standard approach of calculating diffusion coefficients relying on the Nicholson-Shain equation. The level of accuracy achieved by SVR and GPR is virtually independent of the rate constants governing the respective reaction steps. Further, the reduction of high-dimensional voltammetric signals by manual selection of typical voltammetric peak features decreased the performance of both regression algorithms compared to a reduction by downsampling or principal component analysis. After training on simulated data sets, diffusion coefficients were estimated by the regression algorithms for experimental data comprising voltammetric signals for three organometallic complexes.
Conclusions
Estimated diffusion coefficients closely matched the values determined by the parameter fitting method, but reduced the required computational time considerably for one of the reaction mechanisms. The automated processing of voltammograms according to the regression algorithms yields better results than the conventional analysis of peak-related data.
Keywords
Support vector regression Gaussian process regression Diffusion coefficient Principal component analysis Voltammetry Reaction mechanismBackground
Voltammetric signals are measurements of the current flowing through an electrode as a function of an externally controlled electrode potential. For example, in a simple case for an initial oxidation, during a single cycle in cyclic voltammetry the electrode potential first increases linearly with time and, upon reaching the switching potential, decreases linearly back to the starting potential [1, 2]. It has been argued that voltammetric techniques have found widespread use due to their high sensitivity, adequate selectivity, and ready availability of instrumentation [3]. Measurements of cyclic voltammetric signals provide detailed information about reactions which include, or are coupled to, electron transfer steps, and thus enable the analysis of the underlying mechanisms [4]. In a special context, these measurements are used, for example, to study the release of neurotransmitters [5], and to characterize the electrochemical properties of recording and stimulation microelectrodes in neuroscience research [6].
Automated acquisition of experimental data [7, 8] and computer simulations of electrochemical systems [9, 10] play an important role in modern electrochemistry. Due to the wide applicability and high speed of voltammetric experiments [3], data analysis methods are required to aid electrochemists in extracting knowledge about electrochemical systems [11–14]. Recently proposed data analysis methods include, for example, multi-parameter estimation from hypersurface models [15, 16], artificial neural networks for classifying voltammetric signals by reaction mechanism [17], and bootstrap resampling to extract system parameters and their error distributions [18].
The diffusion coefficient D is an important physical parameter of the species involved in an electrochemical reaction, that describes diffusional transport. Since Nicholson and Shain’s classical treatment [1], diffusion coefficients are directly extracted from voltammetric signals based on theoretical relations (Randles-Sevčik equation), valid for particular electrode reaction mechanisms. Recently analytical solutions for calculating the diffusion coefficient from flux data have also been proposed [19, 20], but are restricted to pure diffusive and diffusive-convective conditions. Semiintegral analysis provides a “linearization” method that allows D to be determined for single electron transfers without kinetic complications [21]. As an alternative, fitting of simulated voltammetric features to experimental data [11, 15, 16, 22], or full current/potential curves [23, 24] may provide values for D. Both approaches have limitations: Theoretical relationships are only valid for certain reaction mechanisms and kinetic schemes, while the fitting of simulated data requires formulation of a reasonable mechanistic hypothesis, substantial computation time and is very sensitive to the initialization of the electrochemical system parameters [15]. Non-electrochemical approaches to determine D include PGSE-NMR spectroscopy [25, 26]. However, these require expensive instrumentation and considerable additional expertise.
To overcome such limitations, we investigate the estimation of diffusion coefficients from experimental cyclic voltammograms by means of two function estimation techniques, support vector regression (SVR) and Gaussian process regression (GPR) [27, 28]. Support vector machines, as a tool for both regression and classification, have recently gained popularity across different application fields such as genetics [29], neuroscience [30, 31], quantum chemistry [32], spectroscopy [33–35], and electrochemistry [36]. Similar to support vector machines, Gaussian processes have lately seen a revival of interest due to their combination with covariance kernels [28] and were successfully applied to problems in (bio)chemistry and robotics concerning micro-array analysis [37], and decoding of spike trains [38].
Methods
In the following, f will denote a scalar function, mapping vectors $x\in {\mathbb{R}}^{n}$ to a scalar $y\in \mathbb{R}$. Then, the estimation of diffusion coefficients from voltammetric signals is equivalent to estimating the unknown function f(x) ↦ y, where x is a cyclic voltammogram (CV) and $y\in \mathbb{R}$ the diffusion coefficient D. Function f hence describes the relationship between experimentally acquired data (CVs) and an unknown physical property (D) of the electrochemical species. The following Sections “Support vector regression” and “Gaussian processes” introduce two different techniques for estimating function f.
Support vector regression
In equation (1), the sum of all (f(x_{ i }) − y_{ i }))^{2} is minimized with respect to the weight vector w and offset b. After finding w and b, diffusion coefficients are estimated for previously unseen cyclic voltammograms by evaluating f. In general, function f relating voltammograms and diffusion coefficients will not be linear and we will describe the extension to estimating nonlinear functions later in this paragraph.
where K_{ i j } = k(x_{ i },x_{ j }) is the kernel matrix and K_{ i } denotes its i-th row. Similar to the linear case, the objective function in (7) contains a regularization term, $\parallel \phantom{\rule{0.3em}{0ex}}f{\parallel}_{\mathcal{H}}^{2}={\beta}^{T}K\beta $, and a loss function term, ${l}_{\epsilon}^{2}({K}_{i}\beta +b-{y}_{i})$. As discussed above for the linear case, parameter C controls the complexity of the estimated function.
Kernel functions
Type | Function |
---|---|
Linear | k(x_{ i },x_{ j }) = 〈x_{ i },x_{ j }〉 |
RBF | k(x_{ i },x_{ j }) = exp(−γ∥x_{ i }−x_{ j }∥^{2}) |
Gaussian processes
A Gaussian process is defined as a collection of random variables, any finite number of which have consistent joint Gaussian distributions [28]. A Gaussian process generalizes the concept of the Gaussian distribution over vectors to a distribution over functions and is fully defined by its mean function $\stackrel{\u0304}{m}\left(x\right)$ and covariance function k(x,x^{′}). In order to draw samples from a Gaussian process one first evaluates the mean and covariance function at a finite set of data points to obtain a mean vector ${\mu}_{i}=\stackrel{\u0304}{m}\left({x}_{i}\right)\in {\mathbb{R}}^{m}$ and covariance matrix ${\Sigma}_{\mathit{\text{ij}}}=k({x}_{i},{x}_{j})\in {\mathbb{R}}^{m\times m}$, and subsequently draws a vector of function values $f\sim \mathcal{N}(\mu ,\Sigma )$ where $\mathcal{N}(\mu ,\Sigma )$ denotes a multi-dimensional Gaussian distribution with mean vector μ and covariance matrix Σ. Specifying the mean and covariance function thus reflects prior knowledge about the properties, for example, the smoothness of the estimated function.
Thus calculating the distribution of f_{∗} just requires evaluation of the mean vectors and covariance matrices, and the inversion of the training set covariance matrix by a Cholesky decomposition [47].
Covariance functions
Type | Function |
---|---|
Linear | k(x,x^{′}) = σ^{2}(1 + 〈x,x^{′}〉) |
Squared exp. | k(x,x^{′}) = σ^{2} exp(-∥x-x^{′}∥^{2}/2l^{2}) |
After calculating the partial derivative of Equation 10 with respect to θ one can use a conjugate gradients algorithm to optimize the parameters. It should be noted that the first term in the objective function (10) regularizes the solution, while the second term measures the quality of the data fit, and the third term is a constant independent of the data. In contrast to the SVR algorithm (Section “Support vector regression”) there is no regularization parameter C that needs to be set, since there is an implicit trade-off between function complexity and data fit. For the Gaussian process regression we used the freely available GPML toolbox for MATLAB^{®}; [28].
Nicholson-Shain equation approach
Simulations
Common simulation parameters for all mechanisms
Parameter and unit | Value |
---|---|
Scan rate v (V s ^{-1}) | 0.2 |
Potential step size Δ E (mV) | 1 |
Initial concentration c_{0} (mmol/L) | 0.4 |
Temperature T (°K) | 293.15 |
Electrode area A (cm ^{2}) | 0.064 |
Symmetry factor α | 0.5 |
Fitting of simulation parameters
Fitting simulation parameters by globally minimizing the sum of squared errors between experimental and simulated cyclic voltammograms was used to identify the formal potential E^{0}, the heterogeneous electron transfer rate constant k_{s}, and D for the E _{qr} and E _{qr}C mechanisms, as well as the homogeneous chemical rate constant k_{1} for the E _{qr}C mechanism from the experimental cyclic voltammograms. The resulting D were used as approximations to the real value. To achieve a homogeneous fit across all experimental voltammograms and avoid large deviations for small-amplitude voltammograms, the currents of simulated and experimental voltammograms were scaled to the interval [-1,1], prior to computing the objective function. The minimization of the sum of squared errors measure was carried out by an interior point algorithm [50] as implemented in the KNITRO software library [51]. Values for the diffusion coefficients obtained by this approach served as a reference for judging the accuracy of coefficients estimated by SVR and GPR for the experimental cyclic voltammograms of the organometallic complexes (Section “Estimations from experimental data”).
Results and discussion
In a first step (Section “Estimation from simulated data”) the approach based on the Nicholson-Shain equation and the regression algorithms SVR and GPR were used to estimate diffusion coefficients for simulated cyclic voltammograms with known diffusion coefficients. This allowed us to compare the performance of the different methods in terms of accuracy of the estimated diffusion coefficients. Furthermore, the simulated data helped to analyze the dependence of accuracy on the rate constants of the underlying reaction mechanism. In a second step (Section “Estimations from experimental data”) the regression algorithms, trained on the simulated data, were used to estimate D for experimental cyclic voltammograms with unknown diffusion coefficients.
Estimation from simulated data
Simulation parameters for the EC, E _{ qr } , and E _{ qr } C mechanism
EC: A$\stackrel{\pm e}{\mathit{\rightleftharpoons}}$ B$\stackrel{{k}_{1}}{\to}$ C | |
---|---|
k_{1} (s ^{-1}) | 0.001, 0.01, 0.1, 1, 10, 100, 1000 |
D (cm ^{2} s ^{-1}) | 1 ·10^{-6}, 1.5 ·10^{-6}, …, 5 ·10^{-5}, 5.05 ·10^{-5} |
E^{0} (V) | 0.3 |
E_{start} (V) | 0 |
E_{rev} (V) | 0.7 |
E _{ qr } : A $\stackrel{\pm e}{\mathit{\rightleftharpoons}}$ B | |
k_{s} (cm s ^{-1}) | 0.001, 0.005, 0.01, 0.02, …, 0.1, 0.5, 1 |
D (cm ^{2} s ^{-1}) | as EC |
E^{0} (V) | 0.2108 |
E_{start} (V) | 0 |
E_{rev} (V) | 0.5 |
E _{ qr } C: A $\stackrel{\pm e}{\mathit{\rightleftharpoons}}$ B $\stackrel{{k}_{1}}{\to}$ C | |
k_{1} (s ^{-1}) | as EC |
k_{s} (cm s ^{-1}) | as E_{qr} |
D (cm ^{2} s ^{-1}) | as EC |
E^{0} (V) | 0.2775 |
E_{start} (V) | 0 |
E_{rev} (V) | 0.6 |
For each mechanism one combination of diffusion coefficient and rate constant(s) was used per simulation run (Table 4). The resulting simulated data set comprised a total of 700 simulated voltammograms for the EC mechanism, 1400 for the E _{qr} mechanism, and 2800 for the E _{qr}C mechanism. This full data set was randomly partitioned into training and test data sets, each containing 50% of the simulated cyclic voltammograms. Only the training data set was used for the function estimation by SVR and GPR, while the performance of each algorithm was assessed on the test data set.
In the Nicholson-Shain Equation 12 the diffusion coefficient is a quadratic function of the forward peak current ${i}_{\text{p}}^{\text{for}}$. It is therefore not surprising that the nonlinear functions estimated by SVR with RBF kernel and GPR with the squared exponential covariance function are better suited to describe the relationship between cyclic voltammogram and diffusion coefficient for all investigated mechanisms. There is a significant difference between the means of the error distributions of SVR with linear/RBF kernel, and GPR with linear/squared exponential covariance function, as shown in Figure 4. In addition, the nonlinear functions estimated by SVR and GPR consistently yield lower errors on average than the Nicholson-Shain equation approach for all the reaction mechanisms. Please note that the broad range of errors induced by the Nicholson-Shain equation based approach is not surprising, due to the non-constant dimensionless peak current χ_{p} in the test voltammograms, although this method assumes a constant value (Figure 3).
As shown in Figure 5 the manual preprocessing method yields the lowest accuracy of the estimated diffusion coefficients for both regression algorithms and all reaction mechanisms. This indicates that, albeit being helpful for a human observer, the manually extracted features discard too much of the information contained in the full cyclic voltammogram. The performance differences between the PCA and downsampling method are small, yet PCA works best for the E _{qr}C mechanism, while there is no difference between the preprocessing methods on the EC and E _{qr} mechanism in conjunction with the SVR algorithm. For the GPR algorithm PCA is slightly better for the EC mechanism, while downsampling is better for the E _{qr} mechanism. We used PCA preprocessing for both regression algorithms when estimating diffusion coefficients from real data, as it allows to judge the quality of the data reduction depending on the amount of explained variance.
EC mechanism — dependence on k_{1}
This behaviour of the results from the Nicholson-Shain equation based approach is expected due to the dependence of the dimensionless peak current $\sqrt{\pi}{\chi}_{\text{p}}$ on the dimensionless rate constant κ_{1} described in Section “Nicholson-Shain equation approach”. The black bars on the abscissa of Figures 3 and 6 mark the region where the dimensionless peak current does not deviate significantly from the constant asymptotic value of 0.4463. It should be noted that the scales on the abscissa in both, Figures 3 and 6, are equivalent apart from a constant offset since, for n = 1, log(κ_{1}) = log(k_{1}/s^{-1})- log(a/s^{-1}) and log(a/s^{-1}) ≈ 0.9. The quality of the diffusion coefficients calculated by the Nicholson-Shain equation for rate constants in this range (log(k_{1}/s^{-1})∈(-∞,-1]) is even better than the coefficient values estimated by the SVR algorithm with RBF kernel (Figure 4). Since the exact value of the rate constant is often not known in practice, however, it seems to be better to resort to one of the regression algorithms for finding the diffusion coefficient in general.
E_{qr} mechanism — dependence on k_{s}
The error of the Nicholson-Shain equation approach, on the other hand, increases from 10^{-7} to 10^{-5} for electron transfer rates log(k_{s}/cm s^{-1}) in the range [-3,-2] and thus shows a stronger dependence of diffusion coefficient accuracy on the rate constant. The absolute error approaches the order of magnitude of the values of D. Overall, the regression algorithms SVR and GPR yield a more accurate estimate of the diffusion coefficient for simulated E _{qr} voltammograms in comparison to the Nicholson-Shain equation and to table look-up.
E_{qr}C mechanism — dependence on k_{1} and k_{s}
Estimations from experimental data
Parameter values yielding the best fit between simulated and experimental cyclic voltammograms for the three metal complexes
Parameter | 1 | 2a | 2b |
---|---|---|---|
E^{0} (V) | 0.2767 | 0.2084 | 0.1938 |
k_{s} (cm s ^{-1}) | 0.0232 | 0.0199 | 0.0118 |
k_{1} (s ^{-1}) | 0.1473 | ||
D (cm ^{2} s ^{-1}) | 1.5846e-5 | 1.0535e-5 | 1.0824e-5 |
Based on the results with simulated data (Section“Estimation from simulated data”) we used SVR with RBF kernel and GPR with squared exponential covariance function in conjunction with the PCA preprocessing method to estimate diffusion coefficients for the experimental data sets. For complex 1, the training data consisted of all 2800 simulated cyclic voltammograms created for the E _{qr}C mechanism (Section “E_{qr}C mechanism — dependence on k_{1} and k_{s}”), while 1200 simulated cyclic voltammograms for the E _{qr} mechanism served as training data for 2a/2b. In order to have the voltammograms on a comparable scale the current was normalized by multiplying the signal with the factor ${\left({c}_{0}\sqrt{v}\right)}^{-1}$.
Diffusion coefficients in 10 ^{ -5 } cm ^{ 2 } s ^{ -1 } determined by different methods for the experimental cyclic voltammograms; bold values: best matches with respect to parameter fitting results
1 | 2a | 2b | |
---|---|---|---|
Parameter-Fit | 1.58 | 1.05 | 1.08 |
Nicholson-Shain | 1.07 | 0.84 | 0.82 |
SVR | 2.32 | 1.07 | 1.09 |
GPR | 1.54 | 1.10 | 1.13 |
For 1 the diffusion coefficient estimated by GPR is the best match with respect to the fitted coefficient value. Although there is only a small difference in the estimates of SVR and GPR, the best diffusion coefficient estimates for 2a/2b are provided by SVR. In contrast to the regression algorithms, the Nicholson-Shain equation consistently underestimates the diffusion coefficient value on all data sets.
Average absolute error of currents in μ A between simulated and experimental cyclic voltammograms
1 | 2a | 2b | |
---|---|---|---|
Parameter-Fit | 3.23 | 0.75 | 1.09 |
Nicholson-Shain | 3.74 | 1.41 | 1.74 |
SVR | 4.66 | 0.76 | 1.10 |
GPR | 3.19 | 0.803 | 1.14 |
Experimental
Voltammetric signals in each data set in Section “Estimations from experimental data” were acquired twice for ten scan rates of 0.02, 0.05, 0.1, 0.2, 0.5, 1.003, 2.007, 5.120, 10.240, and 20.480 V s ^{-1}, and four different initial concentrations c_{0} of 0.2, 0.4, 0.6, 0.8 mmol L ^{-1} in a dichloromethane electrolyte with 0.1 M tetra-n-butylammonium hexafluorophosphate as supporting electrolyte at a Pt electrode (for further experimental details, see [22, 54]). The scanning potential varied between 0 and 0.6 V for 1, and between 0 and 0.5 V for 2a/2b with an increment of 1 mV in each case.
Conclusion
The results presented in this work show the feasibility of estimating diffusion coefficients from experimental cyclic voltammograms by regression algorithms trained on simulated data. This approach is generic in the sense that it is not restricted to a particular reaction mechanism and range of rate constants, as demonstrated by the results obtained on simulated data for the EC, E _{qr}, and E _{qr}C mechanisms. On simulated data the accuracy of diffusion coefficients estimated by SVR with RBF kernel and GPR with squared exponential covariance function is higher as compared to the Nicholson-Shain equation approach over a wide range of rate constants. The best preprocessing method for estimating D with the regression algorithms turned out to be the principal component projection of the cyclic voltammograms. Projecting the data to the subspace spanned by the first five principal components apparently retains important shape information that is discarded by the manual extraction of prominent peak features. This indicates that the commonly used evaluation of the limited set of human recognizable features related to voltammetric peaks might not be optimal for data evaluation in all cases. For the three experimental data sets, estimation with GPR yielded diffusion coefficients that closely matched the values determined by the classical parameter fitting approach, whereas SVR showed comparable performance only for 2a/2b. These results indicate that GPR with a squared exponential covariance function is better suited than SVR to reliably determine diffusion coefficients from experimental data. Furthermore the GPR based determination of the diffusion coefficient requires less computational time in contrast to the parameter fitting approach.
Declarations
Acknowledgements
We thank the Deutsche Forschungsgemeinschaft, Bonn-Bad Godesberg, Germany, for financial support of this work within the Graduiertenkolleg 441 “Chemie in Interphasen”. We are grateful to Filip Novak, Institut für Organische Chemie, Universität Tübingen, for preparing and providing the experimental data sets. We acknowledge support by the Deutsche Forschungsgemeinschaft and the Open Access Publishing Fund of the Universität Tübingen to cover the processing charges of the article. This paper is part 3 of the series “Chemical Information from Electrochemical Data”; for part 2, see [15].
Authors’ Affiliations
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