Essential Parameters, Properties, Moments
When modeling peaks, one can accurately estimate the moments of peaks. The zeroth moment is the area, the first moment the mean or center of mass, and the second moment furnishes the standard deviation. It is generally the third moment, the skewness, and the fourth moment, the kurtosis or ‘fatness’ of the tails, which are next to impossible to estimate in raw data. Without removing the instrumental response distortion upon the true peak in some manner, these higher moments will measure the IRF as much as the actual chromatographic peak.
In PFChrom, the true moments of the peaks as eluted, as well the different deconvolutions, are accurately estimated. The higher moments of the pure chromatographic peaks, with the instrumental distortions and multiple adsorption site effects mathematically removed, can now be accurately known.
Once you have a near-zero error model fit for your chromatographic peaks, the parameters of these analytic models can be used to generate every conceivable peak property or moment, across all levels of deconvolution. PFChrom’s analytical summary offers the following items:
- Fitted Parameters – goodness of fit and fitted parameter values
- Equivalent Parameters – for generalized HVL and generalized NLC models, the equivalent diffusion and kinetic parameters
- Measured Values – measured peak characteristics; also includes analytic areas if available
- Analytic Moments – the analytic moments of the model
- Deconvolved Moments – the moments of the deconvolutions inherent in a model
- Advanced Area Analysis – additional information in breakdown of components of areas
- Parameter Statistics – peak by peak parameter statistics
- Chromatography Analysis – theoretical plates and resolutions
- Overlap Areas – matrix of overlap areas for all peaks
- Analysis of Variance – ANOVA
- Details of Fit – Convergence State, Iterations, Fit Settings
- Average Multiple Fits – Averages and Coefficients of Variations of multiple similar fits
- Confidence Statistics – 90, 95, and 99%
If a peak parameter fails significance, the parameter and the model listed for that peak will be grayed. This furnishes immediate feedback that a model may be overspecified (that a more complex model has been fitted than the data can support).
For a generalized HVL model, this feature converts the parameters from a diffusion-based HVL model to a kinetics-based NLC, furnishing the equivalent parameters had you chosen to make a generalized NLC fit.
For a generalized NLC model, this feature converts the parameters from a kinetics-based NLC model to a diffusion-based HVL, furnishing the equivalent parameters had you chosen to make a generalized HVL fit.
This interoperability between GC and LC is a unique feature of the PFChrom’s primary chromatographic models.
This summary uses minimization and root-finding algorithms to report amplitude, center (mode or apex), full width at half-maxima (FWHM), asymmetry at half-maxima, full width at 10% of maxima, and asymmetry at 10% of maxima measured values.
This summary also reports an integrated area and its percentages. Also reported are the first, second, third, and fourth moments. For convenience, the second moment is reported as an SD instead of a variance.
An analytic moment is one which has a closed form solution. PFChrom has closed form analytic moment formulas for many of its peak functions. When an analytical moment is not available, as in the case of the generalized HVL and NLC models, the analytic moments will be given for the ZDD (zero-distortion density) of the model, the moments at infinite dilution.
A given model will have its own specific deconvolutions or simplifications. If an instrument response function model is fitted, the first deconvolution will be the peak absent this IRF. There will then be a succession of up to two additional deconvolutions. For the generalized NLC, for example, the moments are for the pure NLC, the peak absent the ZDD adjustments for nonideality, and the Giddings, the peak absent all chromatographic distortion, the expected peak at infinite dilution.
Advanced Area Analysis
In most cases, the Advanced Area Analysis analysis will furnish a measure of the deviation from the non-ideality furnished by the ZDD’s higher moment parameters. This is especially useful for higher overload peaks or in instances where you would like to quantify these deviations.
This analysis also furnishes the area asymmetry relative to apex or mode of the peak, the area to the right of the apex divided by the area to the left.
The analysis also furnishes several specialty computations. You can report the area of non-overlap between the peak with and without the ZDD non-ideality. In an overload peak, this is the difference between the peak with the overload shape (minus the IRF) and the deconvolved peak that would theoretically exist if there were no overload or any other deviation from this ideality. Also reported is the area of non-overlap between the peak with and without the IRF removed.
If the model is a once or twice generalized HVL or NLC, this Advanced Area Analysis will evaluate the roots of the model and cumulatives to get the retention values at .5%, 1%, 2.5%, 5%, 25%, 50%, 75%, 95%, 97.5%, 99%, and 99.5% area.
The Parameter Statistics summary is a peak by peak display of parameter statistics, which include each parameter’s value, standard error, t-value (parameter value/std error), and confidence limits.
This analysis furnishes two computations of the theoretical plate count. The first is the rigorous procedure using moments. The second uses a Gaussian approximation. Also reported are the full width and the asymmetry at 10% of peak maximum. This analysis also contains the resolution between adjacent peaks and the retention, the k’ thermodynamic capacity factor.
The Overlap Areas analysis reports the overlap area between each peak and all other peaks.
Analysis of Variance
This analysis includes a standard ANOVA table and also reports the r² coefficient of determination, the degree of freedom adjusted r², and the standard error for the fit.
Average Multiple Fits
The Average Multiple Fits analysis is available where there are multiple fits that can be averaged. When this option is selected, and averaging is possible, there will be two separate numeric summaries at the end of summaries for the individual data sets. The first will contain the averages of the values from all of the individual summaries. The second numeric summary at the end of the individual data set summaries will contain the CV% (coefficient of variation percentages). We have found it helpful to see with effortless immediacy which parameters within an average are most tightly determined and which are more variable, and in a % format where everything is comparable. This averaging is also be useful for creating standards.