Peak-fitting methods

The peak by peak fitting method is commonly used for this type of study because, since the fitting of each peak is completely independent and the calculated profiles can be adapted as best as possible, thns taking into account the possible differences in peak shapes between the various phases (microstractural effects). 

Thermal methods of analysis were used to study the polymorphism associated with pindolol [29]. The overlapping melting endotherms of the different forms were separated by peak fitting analysis and grouping of materials by the temperature of... 

Gaussian curves are fitted to the data [28], and a peak area calculation program employed to integrate the area under each peak. This method is used to determine the evolution of peak area with time. The value of the extent of reaction, a, at a time t for any given Bragg reflection (hkl) is calculated using... 

SEC calibration methods which employ a series of narrow MWD standards are based upon a peak position method and traditionally have been the most widely practiced calibration procedures. The peak position method simply correlates the peak elution volume of each standard to its nominal molecular weight or size value. A curve fitting procedure (usually a least squares regression) is used to obtain a working calibration curve. The serious limitation of polymer chemical types for which a series of narrow MWD standards covering a wide molecular weight range can be obtained led to the development of experimental approaches which could be applied to polymer chemical types other than that of the narrow MWD standards employed in calibration. 

Non-Aqueous SEC Evaluation. The SEC calibration report for a peak position method using a series of narrow MWD polystyrene standards is shown in Table I, As can be seen, a linear fit produces a high correlation to the data (r=0.9997). Figure 1 displays the molecular weight calibration plot of elution volume versus log molecular weight for the series of polystyrene standards. 

Plots of Residuals. Residuals can be plotted in many ways overall against a linear scale versus time that the observations were made versus fitted values versus any independent variable (3 ). In every case, an adequate fit provides a uniform, random scatter of points. The appearance of any stematic trend warns of error in the fitting method. Figures 4 and 5 shows a plot of area versus concentration and the associated plot of residuals. Also, the lower part of Figure 2 shows a plot of residuals (as a continuous line because of the large number of points) for the fit of the Gaussian shape to the front half of the experimental peak. In addition to these examples, plots of residuals have been used in SBC to examine shape changes in consecutive uv spectra from a diode array uv/vis spectrophotometer attached to an SBC euid the adequacy of linear calibration curve fits (1). 

Two methods used to find the area under the photometer trace are peak-height-times-half-width approximations and actual measurements with a polar planimeter. Both methods are time consuming and offer little increase in total accuracy over the peak center method. Another method involves computer fitting an assumed scattering function, usually a Gaussian or Lorentzian (though more exotic functions have also been used) to the scan data. The integrated area under the mathematical curve is then calculated. 

For bimodal MWDs none of the methods successfully resolved the two peaks for the case where a majority of the molecules were of low molecular weight (Figure 5). However, CONTIN provided the closest solution. For the case where the skewed low molecular weight peak consisted of only one quarter of the total mass, the GEX fit method gave good results. CONTIN showed three peaks, but the agreement can still be considered fair because of the difficulty in discerning two widely separated peaks of this type. As in the unimodal cases, the subdistribution method showed the poorest fits. 

Operator independence. The GEX method is virtually operator independent. The only inputs required before fitting are parameters concerning the precision of the numerical integration and exit criteria for the Marquardt algorithm. The same set of Inputs was used for all five MWDs. Currently the method also requires upper and lower limits on the GEX parameters but a simple modification of the code can eliminate this need. The CONTIN algorithm has several operator and case dependent parameters that have to be chosen before analysis. However, it is fairly stable with respect to bad choices for some of these inputs. The GEX fit method cannot fit multimodal MWDs without prior knowledge of the number of peaks in the distribution. While CONTIN does not impose this limitation it 1s recommended that the number of peaks be specified before analysis. In our experience with CONTIN if this condition is not met the algorithm tends to compute unimodal solutions for multimodal MWDs and sometimes visa versa. 

Vibrational lifetimes in supercritical fluids were obtained by fitting the data to a convolution of the instrument response and an exponential using a grid-search fit method. Vibrational peak positions were obtained by subtracting a background spectrum of the pure SCF, taken at the experimental pressure and temperature, from the solute-solvent sample spectrum. This technique removes small solvent peaks that can distort the spectrum. 

Adsorption isotherms of the bi-Langmuir-type were measured (see Tab. 2). An initial set of parameters was obtained by the perturbation method [6], Subsequently, a peak fitting approach based on the equilibrium-dispersive model was used for refinement (for details, see [4]). Fig. 4 shows a good agreement between models and experiments. 

Width (in degrees 26) of the box (window) used in the Box Car Curve Fit method for background removal. The window determines how close the background curve follows the data. An underestimated box width results in humps under the peaks, as seen in (a). An overestimated box width translates in the background that does not follow the data. A proper box width (usually established experimentally) results in the background that follows the data closely without following the peaks, as seen in (b). 

Peak Finding Method. Since the approximate location of the peaks has already been determined, the data points about each peak are fitted to a quadratic equation... 

The least squares fitting method overcomes limitations of the peak finding method, but introduces some errors of its own. It is not as sensitive to the scan rate, as long as S >> 2/P (the Nyquist frequency), and its accuracy increases with increasing number of data points. It was found empirically (8) that the error was reduced if an integral number of cycles was used in the analysis. Some error is introduced due to the fact that first and second derivatives of the raw data have to be taken. 

The following general methods are appropriate for the evaluation of model parameters Moment analysis and HETP plots, peak fitting and parameter estimation. They extract the information given by measured chromatograms to differing extents, which correspond to the reliability of the calculated values. 

---