Comparison of the extended kalman filter and marquardt algorithms for processing kinetic data

The extended Kalman filter and Marquardt's gradient expansion algorithm for nonlinear least squares are compared with respect to accuracy and precision of parameter extimates, computational burden, sensitivity to initial parameter estimates and ability to indicate model errors. Fits of synthetic first-order data and combined first- and zero-order data produce estimates of equivalent precision and accuracy in most cases. Similar results were obtained for both simulated and experimental data for combined zero-order/first-order processes. However, for the simulated zero-order/first-order data with small zero-order components processed over two half-lives of the first-order process, the Kalman filter overestimated the zero-order rate constant by a substantially larger amount than the Marquardt algorithm. Significant differences in computational burden and sensitivity to initial parameter estimates are demonstrated; however, neither algorithm has a significant advantage over the other for the detection of model error.     

Estimation of first-order kinetic parameters by using the extended kalman filter

A model for first-order kinetics is derived for spectra obtained while a reaction is taking place. A technique for nonlinear-regression analysis known as the extended Kalman filter is used to estimate the initial concentrations of the reactants and the rate constant from the spectral data. The effects of the magnitude of the rate constant and the identity of the absorbing species are examined for synthetic spectra containing overlapped responses. The technique is used successfully to obtain the rate constant for the dissociation reaction of a praseodymium complex. The filter is also shown to be useful for the detection of erros in the kinetic model employed to fit the data. The extended Kalman filter can be used to fit kinetic models other than the one discussed here, and may prove to be a valuable technique for estimation of kinetic parameters.     

Estimation of ester hydrolysis parameters by using fourier-transform infrared spectroscopy and the extended kalman filter

Fourier-transform (FTIR) spectroscopy has found a wide range of applications. Attempts have been made to make FTIR spectroscopy a reliable quantiative technique for obtaining chemical information foru complex systems, but it is often difficult to obtain even qualitative information about a complex system because of matrix, interaction, and background effects on the infrared on the spectra. In this study, FTIR spectroscopy was used in conjuction with the extended Kalaman filter, a recursive digital filter, to investigate the simple neutral hydrolysis of two esters in aqueous solutions. Interactions between the solvent, reactants and products of the hyrolysis perturb the spectrum, making a single-wavelength kinetic analysis impractical. With the extended Kalman filter, a range of frequencies where severe perturbation does not occur is processed to obtain not only the rate constant but also the initial reactant concentrations.     

Multicomponent determinations in flow-injection systems with square-wave voltammetric detection using the kalman filter

Electrochemical detection in flow-injection systems has become increasingly popular. Scanning electrochemical methods of detection have received less attention. The use of rapid square-wave voltammetric detection for flow injection is described. Experimental considerations for the use of square-wave voltammetric detection are discussed. Results of the application of this combination are shown for dopamine in 0.28 M sulfuric acid, and for lead(II) and thallium(I) in 0.9 M nitric acid. A linear, recursive estimator known as the Kalman filter is used to resolve overlapping responses. Empirical models consisting of the square-wave voltammetric responses of single species are used. Results are shown for mixtures of lead(II) and thallium(I) in 0.9 M nitric acid.     

On the estimation of stability parameters from heat-induced conformational transition curves of proteins

A method is suggested to determine valid and authentic values of thermodynamic stability parameters of proteins from their heat-induced conformational transition curves. We show (a) that the estimate of ΔH_m ^van, the enthalpy change on denaturation at T_m, the midpoint of denaturation, is significantly less than ΔH_m ^cal, the value obtained by the calorimetric measurements, if the analysis of the conformational transition curve uses the conventional method which assumes a linear temperature-dependence of the pre- and post-transition baselines; and (b) that there exists an excellent agreement between ΔH_m ^van and ΔH_m ^cal values of proteins, if the analysis of thermal denaturation curves assumes that the temperature-dependence of pre- and post-transition baselines is described by a parabolic function. The latter analysis is supported by our observations that the temperaturedependencies of the absorption and circular dichroism properties of protein groups are indeed nonlinear. It is observed that the estimate of ΔC_p, the constant-pressure heat capacity change is independent of the model used to describe the temperaturedependence of the pre- and post-transition baselines. An important conclusion is that for proteins which exhibit a two-state character, all stability parameters are measured with the same error as that observed with a calorimeter.     

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.     

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.     

超微盘、微半球电极上准稳态电流公式及其实验验证

经过严格的数学推导及合理的数学近似, 提出了超微盘及微半球电极上准稳态可逆波、准可逆波及不可逆波电流方程式。根据导出的准稳态电流方程, 提出了一个测定动力学参数的新方法。对理论公式也进行了实验验证。     

First- and second-order chemical-electrochemical mechanisms: Extraction of standard potential,equilibrium and rate constants from linear sweep voltammetric curves

Procedures are described for extracting the thermodynamic and kinetic key parameters from cyclic voltammetric data in the case of mechanisms involving a chemical reaction preceding the electron transfer process (CE mechanisms). Both first-order and second-order kinetics conditions for the preceding reactions are considered. These procedures are devised for practical applications mostly in the field of coordination electrochemistry, where CE mechanisms are very frequently encountered.     

Analytical procedures for the simultaneous voltammetric determination of heavy metals in meals

An analytical procedure regarding the determination of copper(II), lead(II), cadmium(II), zinc(II) and antimony(III) in matrices involved in foods and food chain as wholemeal, wheat and maize meal is proposed. The digestion of each matrix was carried out using concentrated HCl suprapure at 130 °C for 3 h. Differential pulse anodic stripping voltammetry (DPASV) was employed for simultaneously determining all the elements, using a conventional three-electrode cell and 0.5 M HCl as supporting electrolyte. The analytical procedure has been verified on the reference standard materials Wholemeal BCR-CRM 189, Wheat Flour NIST-SRM 1567a and Rice Flour NIST-SRM 1568a. For all the elements in the certified matrix, the precision as repeatability, expressed as relative standard deviation (s_r), and the accuracy, expressed as relative error (e), were of the order of 3 to 6%. The limits of detection were in the range 0.009–0.096 μg/g.     

Kinetic parameters obtained from TG and DTG curves of acrylamide-maleic anhydride copolymers

The thermal behaviour of acrylamide-maleic anhydride copolymers was studied by thermogravimetric (TG and DTG) analysis. The obtained data permitted the calculation of activation energies and reaction orders of the decomposition steps by the Coats-Redfern and Freeman-Carroll methods.Thermal analysis shows four distinct peaks in the case of polyacrylamide and AAMA (71) copolymer and only three for AAMA (11) copolymer.In AA:MA (11) copolymers intermolecular imidization occurs only and thermal degradation is influenced more by the anhydride groups which are equal in number with the amide ones.

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Zusammenfassung Das thermische Verhalten von Acrylamid-Maleinsäureanhydrid-Kopolymeren wurde durch thermogravimetrische Analyse (TG und DTG) untersucht. Aus den erhaltenen Daten wurden die Aktivierungsenergien und Reaktionsordnungen der Zersetzungsschritte nach den Methoden von Coats-Redfern und Freeman-Carroll berechnet. Durch thermische Analyse können im Falle von Polyacrylamid und AAMA (71 (-Kopolymeren 4 Peaks und bei AA:MA (11) -Kopolymeren nur 3 Peaks unterschieden werden. Bei AAMA (1 1)-Kopolymeren verläuft nur eine intermolekulare Imidisierung und der thermische Abbau wird mehr durch die in gleicher Zahl wie die Amidgruppen vorliegenden Anhydridgruppen beeinflußt.

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On the method of bae for the determination of kinetic parameters from DTA curves

The mathematical error in the method proposed by Bae for the determination of kinetic parameters from DTA curves has been corrected. The proposed equation does not contain thermal constants of the apparatus, and can be applied to DTA curves by an iterative method. The results obtained by the application of this equation to experimental DTA curves for the decomposition of sodium bicarbonate compared well with those from isothermal measurements, even when the DTA sample holder assembly was of the isolated cup-type instead of the block-type assembly recommended by Bae.     

Differential pulse voltammetric methods for the simultaneous estimation of uranium-iron and uranium-cadmium mixtures in solution

Differential pulse voltammetric methods have been developed for the simultaneous estimation of the constituents of uranium-iron and uranium-cadmium mixtures in solution. A mixture of 1M H₃PO₄–1M KH₂PO₄ (with a pH1.5), was found to be the most ideal supporting electrolyte for both methods, among many that were evaluated for their suitability. In uranium-iron mixtures the calibration for iron was found to be linear up to 150 g ml^–1 (r²=0.9986), while that of uranium up to 500 g ml^–1 (r²=0.999). Iron at 6.7 g ml^–1 level could be determined in the presence of 800 fold uranium (wt/wt) without significant interference. Uranium at 21 g ml^–1 level could be analyzed with 5-fold iron (wt/wt). This upper limit of iron was due to the precipitation of iron as phosphate. In the case of uranium — cadmium mixtures, cadmium calibration for cadmium was found to be linear up to 1300 g ml^–1 (r²=0.9993). Concentration levels of 4.6 g ml^–1 Cd could be determined at a 500-fold excess (wt/wt) of uranium. Uranium calibration was linear up to 500 g ml^–1 (r²=0.999) and 21 g ml^–1 uranium could tolerate up to a 1000-fold excess of cadmium (wt/wt). Both procedures could tolerate 10 g ml^–1 levels of metal ions, such as chromium, copper, manganese, molybdenum and vanadium.     

A kalman filter for calibration,evaluation of unknown samples and quality control in drifting systems: Part 2. Optimal Designs

It is well known that the optimal design for linear calibration is found at the extremes of the concentration range. For the evaluation of unknown samples, the precision is maximal at the mean value of the calibration standards. If the static model is replaced and stochastic behaviour of the parameters with extension to drift is assumed, only minor differences on these generalizations are obtained. The intelligent analyzer described here integrates on line procedures for calibration, evaluation, quality control and optimization.     

Determination of some kinetic parameters of the thermal decomposition of alkali persulfates from DTA curves

Using the DTA curves the thermal decomposition of alkali persulfates for the corresponding pyrosulfates is shown to be a second order reaction with activation energies of 72.7–75.6 kcal mol^?1 for sodium persulfate and 67.7–69.1 kcal mol^?1 for potassium persulfate.     

Computation in kinetics, IV. Regression estimation of Arrhenius equation parameters from non-isothermal kinetic data

Computer method for evaluation of Arrhenius equation parameters from non-isothermal kinetic data is offered in KILET program. Changes of temperature can be linear, hyperbolic or of any smooth function which could be expressed by a polynomial of up to 4th degree. The method is demonstrated on examples of linear and hyperbolic changes for simulated and experimental data. The advantage of non-isothermal experiment over isothermal one for evaluation of kinetic solution data is stressed.

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A simplex optimization of the experimental parameters in preparative liquid chromatography

Summary A study of the optimization of the experimental conditions for the purification or extraction of pure compounds by liquid chromatography is presented. Optimum values of the parameters of overloaded elution are derived for maximum production rate, using a Simplex algorithm and the procedure previously described for the simulation of the elution profiles of binary mixtures. The mobile phase flow velocity and the sample size have been optimized together in a first step, simulating the procedure followed in practice, when a column is available. In a second part, the influence of the column length and the average particle size of the packing material on the column performance as well as the trade-offs between the production rate and the yield are discussed.There are three major conclusions in this work. First, the optimum experimental conditions are often very different, depending whether one is primarily interested in the first or in the second eluted component of a mixture. Second, the column efficiency under analytical conditions is very important: it is traded-off for high flow rates, hence short cycle time and increased production rate. Third, the production rate depends strongly on the maximum pressure at which the equipment can be operated. Finally, the optimum production rate varies rather smoothly with the mobile phase velocity and the sample size, so a high yield (70% or more) can usually be obtained with a limited loss in production rate (30 to 60%).     

Resolution of overlapped chromatograms by means of the kalman filter : Dimensional reduction of error covariance matrices and state estimate vectors

If many samples are successively injected into a high-performance liquid chromatograph at short intervals, then overlapped chromatograms will be obtained. A simplified version of the Kalman-filter algorithm is described for rapid resolution of several such overlapped peaks on small laboratory computers; the dimensionalities of the error covariance matrix Pk and state estimate vector X?k are reduced to far less than the total number of the peaks. Computer simulation shows that at least twenty partially overlapped Gaussian peaks, used as a model of the successive injections can be successfully resolved, with the 4 × 4 matrix Pk and 4-vector X?k. This version is independent of the number of injections and is effective for the resolution of a finite number of peaks in that injection mode. The algorithm of small dimensions is applicable to a wide variety of peak resolution procedures.