Numerical methods for data- fitting problems

Self-optimising processes and real-time-optimisation of organic syntheses in a microreactor system using Nelder–Mead and design of experiments

Comparing an enhanced simplex algorithm with model-free design of experiments, this work presents a flexible platform for multi-objective, real-time optimisation.

Calculation of equilibrium constants from multiwavelength spectroscopic data--II: SPECFIT: two user-friendly programs in basic and standard FORTRAN 77

A new program (SPECFIT), written in HP BASIC or FORTRAN 77, for the calculation of stability constants from spectroscopic data, is presented. Stability constants have been successfully calculated from multiwavelength spectrophotometric and EPR data, but the program can be equally well applied to the numerical treatment of other spectroscopic measurements. The special features included in SPECFIT to improve convergence, increase numerical reliability, and minimize memory as well as computing time requirements, include (i) elimination of the linear parameters (i.e., molar absorptivities), (ii) the use of analytical instead of numerical derivatives and (iii) factor analysis. Calculation of stability constants from spectroscopic data is then as straightforward as from potentiometric titration curves and gives results of analogous reproducibility. The spectroscopic method has proved, however, to be superior in discrimination between chemical models.

Curve-fitting overlapped bands: quantification and improvement of curve-fitting robustness in the presence of errors in the model and in the data

Estimation of the band parameters of overlapped bands often relies on curve-fitting. It has been demonstrated that curve-fitting provides the maximum likelihood estimation of band parameters under a series of assumptions. One of these assumptions is that the curve-fitting model is correct and any error in the data is random. Under real conditions, we have to acknowledge the unavoidable presence of errors in the model and systematic errors in the data. Here, we derive an expression for the estimation of how these errors affect the quality of the parameters obtained from curve-fitting. In addition, we derive theoretical expressions to quantify the extent to which different methods can improve the curve-fitting robustness to these errors. The methods considered are: (i) deterministic and (ii) probabilistic constraints in the band parameters, (iii) curve-fitting band-narrowed data, and (iv) building a more accurate model. The theoretical expressions obtained are tested in the curve-fitting of a synthetic noisy spectrum with either baseline or band shape errors, and in the curve-fitting of the experimental infrared amide I band of the membrane protein bacteriorhodopsin.

Resolving factor analysis

Bilinear data matrices may be resolved into abstract factors by factor analysis. The underlying chemical processes that generated the data may be deduced from the abstract factors by hard (model fitting) or soft (model-free) analyses. We propose a novel approach that combines the advantages of both hard and soft methods, in that only a few parameters have to be fitted, but the assumptions made about the system are very general and common to a range of possible models: The true chemical factors span the same space as the abstract factors and may be mapped onto the abstract factors by a transformation matrix T, since they are a linear combination of the abstract factors. The difference between the true factors and any other linear combination of the abstract factors is that the true factors conform to known chemical constraints (for instance, nonnegativity of absorbance spectra or monomodality of chromatographic peaks). Thus, by excluding linear combinations of the abstract factors that are not physically possible (assuming a unique solution), we can find the true chemical factors. This is achieved by passing the elements of a transformation matrix to a nonlinear optimization routine, to find the best estimate of T that fits the criteria. The optimization routine usually converges to the correct minimum with random starting parameters, but more difficult problems require starting parameters based on some other method, for instance EFA. We call the new method resolving factor analysis (RFA). The use of RFA is demonstrated with computer-generated kinetic and chromatographic data and with real chromatographic (HPLC) data. RFA produces correct solutions with data sets that are refractory to other methods, for instance, data with an embedded nonconstant baseline.