The mathematical origins of the kinetic compensation effect: 1. The effect of random experimental errors

The kinetic compensation effect states that there is a linear relationship between Arrhenius parameters ln A and E for a family of related processes. It is a widely observed phenomenon in many areas of science, notably heterogeneous catalysis. This paper explores one of the mathematical, rather than physicochemical, explanations for the compensation effect and for the isokinetic relationship. It is demonstrated, both theoretically and by numerical simulations, that random errors in kinetic data generate an apparent compensation effect (sometimes termed the statistical compensation effect) when the true Arrhenius parameters are constant. Expressions for the gradient of data points on a plot of ln A against E are derived when experimental kinetic data are analysed by linear regression, by non-linear regression and by weighted linear regression. It is shown that the most appropriate analysis technique depends critically on the error structure of the kinetic data. Whenever data points on a plot of ln A against E are in a straight line with a gradient close to 1/RT, then confidence ellipses should be calculated for each data point to investigate whether the apparent compensation effect arises from random errors in the kinetic measurements or has some other origin.     

D-MORPH regression for modeling with fewer unknown parameters than observation data

D-MORPH regression is a procedure for the treatment of a model prescribed as a linear superposition of basis functions with less observation data than the number of expansion parameters. In this case, there is an infinite number of solutions exactly fitting the data. D-MORPH regression provides a practical systematic means to search over the solutions seeking one with desired ancillary properties while preserving fitting accuracy. This paper extends D-MORPH regression to consider the common case where there is more observation data than unknown parameters. This situation is treated by utilizing a proper subset of the normal equation of least-squares regression to judiciously reduce the number of linear algebraic equations to be less than the number of unknown parameters, thereby permitting application of D-MORPH regression. As a result, no restrictions are placed on model complexity, and the model with the best prediction accuracy can be automatically and efficiently identified. Ignition data for a H ₂/air combustion model as well as laboratory data for quantum-control-mechanism identification are used to illustrate the method.     

Information content of data and variables and types of weighting in least-squares regression methods

Algorithms are given for evaluating the relative amount of useful information related to a particular parameter which is carried by individual data points and intervals of the variables. The algorithms provide an efficient means of using the information contained in a set of data. Applications to the optimization of weighting in regression methods are described. Several informational and combined informational-statistical types of weighting are studied as a means of improving the accuracy and precision of the parameters obtained by non-linear regression.     

Mathematical modeling of dispersion in single interface flow analysis

This work describes the optimization of the recently proposed fluid management methodology single interface flow analysis (SIFA) using chemometrics modelling. The influence of the most important physical and hydrodynamic flow parameters of SIFA systems on the axial dispersion coefficients estimated with the axially dispersed plug-flow model, was evaluated with chemometrics linear (multivariate linear regression) and non-linear (simple multiplicative and feed-forward neural networks) models. A D-optimal experimental design built with three reaction coil properties (length, configuration and internal diameter), flow-cell volume and flow rate, was adopted to generate the experimental data. Bromocresol green was used as the dye solution and the analytical signals were monitored by spectrophotometric detection at 614 nm. Results demonstrate that, independent of the model type, the statistically relevant parameters were the reactor coil length and internal diameter and the flow rate. The linear and non-linear multiplicative models were able to estimate the axial dispersion coefficient with validation r² = 0.86. Artificial neural networks estimated the same parameter with an increased accuracy (r² = 0.93), demonstrating that relations between the physical parameters and the dispersion phenomena are highly non-linear. The analysis of the response surface control charts simulated with the developed models allowed the interpretation of the relationships between the physical parameters and the dispersion processes.     

A hybrid genetic algorithm for estimating the equilibrium potential of an ion-selective electrode

Non-linear equations can be used to model the measured potential of ion-selective electrodes (ISEs) as a function of time. This can be done by using non-linear least squares regression to fit parameters of non-linear equations to an ISE response curve. In iterative non-linear least squares regression (which can be considered as local optimisers), the determination of starting parameter estimates that yield convergence to the global optimum can be difficult. Starting values away from the global optimum can lead to either abortive divergence or convergence to a local optimum. To address this issue, a global optimisation technique was used to find initial parameter estimates near the global optimum for subsequent further refinement to the absolute optimum. A genetic algorithm has been applied to two non-linear equations relating the measured potential from selected ISEs to time. The parameter estimates found from the genetic algorithm were used as starting values for non-linear least squares regression, and subsequent refinement to the absolute optimum. This approach was successfully used for both expressions with measured data from three different ISEs; namely, calcium, chloride and lead ISEs.     

Two-dimensional and three-dimensional fitting of enzyme kinetic data with the kalman filter

The availability of instrumentation which is capable of collecting multidimensional data has put increased demands on the data-processing methods utilized to obtain information about reaction kinetics. An enzyme-catalyzed reaction, the hydrolysis of p-nitrophenyl phosphate to p-nitrophenol, is examined so that various data-processing methods and data-collection formats can be examined and compared. The extended Kalman filter is used to obtain rate constants and values for the initial substrate concentration for three-dimensional data, and for two-dimensional kinetically perturbed data. In addition, non-linear least-squares regression with the simplex algorithm, and linear least-squares regression methods are used to analyze absorbance/time curves for this reaction. These results are compared to the results from a two-point kinetic method, and the accuracy and precision of each of the methods is evaluated. It is found that the methods based on the Kalman filter yielded results which were equivalent to or better than the results obtained from the other approaches.     

Assessment of quality performance parameters for straight line calibration curves related to the spread of the abscissa values around their mean

In validation of quantitative analysis methods, knowledge of the response function is essential as it describes, within the range of application, the existing relationship between the response (the measurement signal) and the concentration or quantity of the analyte in the sample. The most common response function used is obtained by simple linear regression, estimating the regression parameters slope and intercept by the least squares method as general fitting method. The assumption in this fitting is that the response variance is a constant, whatever the concentrations within the range examined.The straight calibration line may perform unacceptably due to the presence of outliers or unexpected curvature of the line. Checking the suitability of calibration lines might be performed by calculation of a well-defined quality coefficient based on a constant standard deviation.The concentration value for a test sample calculated by interpolation from the least squares line is of little value unless it is accompanied by an estimate of its random variation expressed by a confidence interval. This confidence interval results from the uncertainty in the measurement signal, combined with the confidence interval for the regression line at that measurement signal and is characterized by a standard deviation s_x0 calculated by an approximate equation. This approximate equation is only valid when the mathematical function, calculating a characteristic value g from specific regression line parameters as the slope, the standard error of the estimate and the spread of the abscissa values around their mean, is below a critical value as described in literature.It is mathematically demonstrated that with respect to this critical limit value for g, the proposed value for the quality coefficient applied as a suitability check for the linear regression line as calibration function, depends only on the number of calibration points and the spread of the abscissa values around their mean.     

Factors affecting the precisions of analyses,by potentiometric titrimetry,of solutions containing two weak acids

This paper deals with potentiometric titrations in which mixtures of two monobasic weak acids are titrated with a strong base, and in which weighted non-linear regression analysis is used to find the concentrations and values of pK_a for both acids. The precisions of the resulting values of the concentrations depend on the difference between the values of pka, the ratio of the initial concentrations, and on the standard errors of measurement of both the pH and the volume of base. For any given values of the ratio of concentrations and the standard errors of measurement, the precision with which the concentration of the stronger acid can be evaluated is, in general, poorest when the difference between the pK_a values (Δ) is approximately 1.5, and improve if Δ is either larger or smaller than that value. Even if Δ is as small as 0.1, the precision that is attainable is very much better than it is generally believed to be. Surprisingly, there are conditions under which the concentrations of both acids can be determined with fair precision even though titrimetric data will not reveal that two acids are present in the sample.     

Microcomputer application of non-linear regression analysis to metal-ligand equilibria

A non-linear least-squares regression program is described which is suitable for PC-compatible microcomputers. The program is written in GWBASIC, but compiled to run with the Intel 8087 fast numeric processor. Subroutines which simulate functions are compiled separately from the main program. Parameters are optimized by a Gauss-Newton-Marquardt algorithm which can be provided with either analytically or numerically calculated partial derivatives. Multi-component potentiometric titrations are simulated and parameters optimized by using analytical derivatives. Spectrophotometric titrations are also simulated, but absorptivities are optimized by linear regression while stability constants are optimized non-linearly by using numerical derivatives. Provision is made for "global analysis" of parameters. The experimental points can be displayed on screen, along with the "best" fit and the speciation. The program is demonstrated here by the determination of the pK(a) values and stability constants of a hydroxypyridinone ligand and its complexes with Fe(III).     

The effects of errors in measuring the independent variable in least-squares regression analysis

It is well known that linear regression analysis gives unbiased estimates of the slope and intercept of a straight line if the dependent variable y is subject to random errors of measurement while the independent variable x is not. It is much less well known that, if x is also subject to random errors of measurement, linear regression analysis yields an underestimate of the slope and a correspondingly biased estimate of the intercept. These errors cannot be removed by weighted regression. Similar errors arise in non-linear regression when the independent variable is afflicted by random errors of measurement.     

Kinetic Modeling of Ethanol Production by Scheffersomyces stipitis from Xylose

This work focuses on the kinetics of ethanol production by Scheffersomyces stipitis on xylose with the development of a mathematical model considering the effect of substrate and product concentrations on growth rate. Experiments were carried out in batch and continuous modes, with substrate concentration varying from 7.2 to 145 g L^?1. Inhibitory effects on cell growth, substrate uptake, and ethanol production rates were found to be considerable. Kinetic parameters were obtained through linear and non-linear regression methods. Experiments in continuous mode were performed at different dilution rates to evaluate the inhibitory effect of ethanol. A mixed mathematical model which combined Andrews and Levenspiel's models, combining substrate and product inhibition, was used. A quasi-Newton routine was applied to obtain a more accurate fitting of kinetic parameters. The parameters such as cell to product factor (Y _P/X) and limiting cell yield (Y _X) were shown to be dependent on substrate concentration. The kinetic model fitted satisfactorily the experimental data.     

The quality coefficient as performance assessment parameter of straight line calibration curves in relationship with the number of calibration points

Knowledge of the response function (y?=?f(x)) is essential in the validation of quantitative analysis methods as it describes the mathematical relationship between measurable responses and the concentrations or quantities of the analyte in the sample within a suitable range. The most common response function used is a straight line obtained by ordinary least squares (OLS) regression. Suitability of calibration lines obtained by OLS regression might be verified by calculation of a quality coefficient (QC_mean). Mathematical modelling performed previously showed that with respect to critical limit values for g, which controls the symmetry of the prediction interval of the abscissa value obtained from the confidence intervals around the OLS calibration curve, a corresponding quality coefficient value exists as a quality performance parameter which is related to the spread of the abscissa values around their mean. In this paper, new mathematical models are developed to demonstrate to which extend also the number n of calibration points (x _i ,y _i ) defines the required value for the quality coefficient (QC_mean) for different values of g. From these models, it could be established that the attribution of a critical limit value to QC_mean as a performance parameter for straight line calibration cannot be arbitrary chosen but has to rely on the mathematical model relating QC_mean, the g-value, the number n of calibration points and the spread of the x _i -values around their mean. Practical measures for analysts are provided which tend to lower the g-value of straight calibration lines beneath critical values and enable to improve the quality of the calibration line applied for analysis, as demonstrated in an elaborated example.     

Characterization of the Acid-Base Properties of Humic Substances by Chemometric Analysis of Synchronous Fluorescence and pH Potentiometric Data

Abstract

A procedure for the analysis of the acid-base characteristics of humic substances based on a self-modeling analysis of synchronous fluorescence spectra, collected at varying pH, and on a non-linear least squares adjustment of potentiometic pH data, is described. The data analysis methodology consists of two steps: first, the number of acid-base systems and the corresponding spectra and distribution diagrams are calculated by evolving factor analysis (EFA) with concentration constraints of the spectroscopic data; second, the potentiometric data is analyzed by a standard non-linear least square procedure using as fixed parameters the number of components and their pKas, determined in the first step of the analysis. As an example, for a sample of marine fulvic acids studied at pH between 2 and 11, four acid-base systems were found with average pKas: 3.1, 4.8, 8.0 and 10.0. The concentrations of the corresponding systems were: 2.55(5), 1.95(7), 0.14(4) and 1.8(3) meq/g.     

Application of non-linear regression methods for the estimation of common kinetic parameters from several thermoanalytical curves

Novel methods of unified evaluation of two (or more) thermogravimetric curves have been worked out on the basis of known non-linear parameter estimating procedures (Gauss-Newton-Marquardt-type regression and the direct integral method of Valkó and Vajda were adapted). Their ability to provide estimate for common kinetic parameters of several TG (m?T) or DTG (dm/dt-T) curves were tested for pairs of curves of different heating rates, and for repeated curves of the same heating rate, obtained for the decomposition of CaCO₃ in open crucible. In these cases the Arrhenius terms and then-th order model functions were assumed. The fitting ability of estimations made for single curves and for pairs of curves sharing different number of parameters, was judged on the base of residual deviations (S _res ) and compared to the standard deviation of the measurements. In the case of different heating rates, the two curves could not be described with the assumption of three common parameters, because of the minimum residual deviation was very high. However, sharing of activation energy and preexponential term only, and applying different exponents for the two curves, provided a satisfactory fit by our methods. Whilst in the case of repeated curves, we could find such a common three-parameter set, which has a residual deviation comparable with the standard deviation of the measurements. Because of their flexibility (taking into account the variable number of common parameters and the versatile forms of model equations), these methods seem to be promising means for unified evaluation of several related thermoanalytical curves.     

Qsar Models for Predicting the Acute Toxicity of Selected Organic Chemicals with Diverse Structures to Aquatic Non-Vertebrates and Humans

Abstract

The linear and non-linear relationships of acute toxicity (as determined on five aquatic non-vertebrates and humans) to molecular structure have been investigated on 38 structurally-diverse chemicals. The compounds selected are the organic chemicals from the 50 priority chemicals prescribed by the Multicentre Evaluation of In Vitro Cytotoxicity (MEIC) programme. The models used for the evaluations are the best combination of physico-chemical properties that could be obtained so far for each organism, using the partial least squares projection to latent structures (PLS) regression method and backpropagated neural networks (BPN). Non-linear models, whether derived from PLS regression or backpropagated neural networks, appear to be better than linear models for describing the relationship between acute toxicity and molecular structure. BPN models, in turn, outperform non-linear models obtained from PLS regression. The predictive power of BPN models for the crustacean test species are better than the model for humans (based on human lethal concentration). The physico-chemical properties found to be important to predict both human acute toxicity and the toxicity to aquatic non-vertebrates are the n?octanol water partition coefficient (Pow) and heat of formation (HF). Aside from the two former properties, the contribution of parameters that reflect size and electronic properties of the molecule to the model is also high, but the type of physico-chemical properties differs from one model to another. In all of the best BPN models, some of the principal component analysis (PCA) scores of the ¹³C-NMR spectrum, with electron withdrawing/accepting capacity (LUMO, HOMO and IP) are molecular size/volume (VDW or MS1) parameters are relevant. The chemical deviating from the QSAR models include non-pesticides as well as some of the pesticides tested. The latter type of chemical fits in a number of the QSAR models. Outliers for one species may be different from those of other test organisms.     

Multivariate monitoring of fermentation processes with non-linear modelling methods

Multiway principal components analysis (MPCA) and parallel factor analysis (PARAFAC) are widely used in exploratory data analysis and multivariate statistical process control (MSPC). These models are linear in nature, thus, limited when non-linear relations are present in the data. Principal component analysis (PCA) can be extended to non-linear principal components analysis using autoassociative neural networks. In this paper, the network’s bottleneck layer outputs (non-linear components) were made orthogonal. A method to estimate confidence limits based on a kernel probability density function was proposed since these limits do not assume that the non-linear scores are normally distributed. A measure for the non-linear scores (D_NL) was presented here to monitor on-line the process replacing the well known Hotelling’s T² statistic. One hundred and two industrial fermentation runs were used to evaluate the performance of a non-linear technique for multivariate process statistical monitoring. Three process runs with faults were used to compare the error detection performance using a statistic for the non-linear scores and the residuals statistic (SPE).     

A program for the processing of analytical data (DPP)

Computer-assisted interactive data presentation and analysis facilities are needed to handle the vast amount of information produced by automated instruments. The data processing program, DPP, presented here in a FORTRAN-77 program designed to solve this problem. The program is equipped with a leading verb command language for input and job scheduling, thus providing an efficient and user-friendly operator/program interface, and with a data-base organization that accommodates a wide variety of data structures. Data presentation and analysis procedures include tabulation and plotting, regression analysis, non-linear least-squares fitting, polynomial fitting, principal component analysis, hierarchal clustering and non-hierarchal clustering. Data matrices with up to 10 000 data points, distributed over a maximum of 3000 variables and 3000 samples, can be examined. Because of the open-ended structure of the program, it is straightforward to incorporate additional data analysis procedures when they are needed. Recent applications are discussed.     

On the application of open circuit relaxation spectroelectrochemistry to the diagnosis and evaluation of the kinetics of succeeding second order chemical reactions

The spectroelectrochemical technique of open circuit relaxation (OCR) has found limited application in the kinetic characterization of systems wherein the electrogenated absorbing species is consumed in a second or higher order reaction. This limited usage arises from the inapplicability of conventional homogeneous kinetic data analysis techniques to the absorbance-time transients observed in the OCR experiment. Treatment of such transients by conventional kinetic expressions (e.g., 1/A vs. t for second order reactions) results in non-linear plots which neither serve as diagnostic criteria for the assignment of kinetic order nor provide meaningful rate constants. This paper presents an empirical method for the treatment of spectroelectrochemical OCR data arising from post-electron transfer kinetic processes which exhibit second order dependence on the concentration of the electrogenerated absorbing species. This procedure, which takes into account the inhomogeneous distribution of the reactant species, affords reduced data plots which not only are linear, hence diagnostic of kinetic order, but also provide a valid measure of the bimolecular rate parameters characteristic of this type of dynamic system. The procedure has been applied to the treatment of OCR data for the reaction of 9,10-diphenylanthracene cation radical with 4-cyanopyridine in acetonitrile. Excellent agreement between the resulting kinetic parameters and those obtained using stopped-flow techniques has been demonstrated.