Scientific Peak Modeling

Unparalleled Fitting Accuracy

When modeling scientific peak data, the quality of the fitting id crucial for parametric accuracy. There are those who will assert that a goodness of fit <10 ppm unaccounted variance (r²>0.99999) is impossible with real world data without severe overfitting (generating over-specified fits where the significance fails on one or more of the parameters).

For many real-world spectroscopic and chromatographic data sets, PeakLab will generate fits of this exceptional accuracy, or higher, while maintaining full statistical significance on all model parameters. What is deemed impossible by certain individuals in the modeling field you will prove as not only possible but routinely achieved with your own data!

Spectroscopic Parameters with Physical Meaning

Physicists have traditionally fit only trusted theoretical models to spectroscopic peaks. Ad-hoc or empirical parameters were rare, and when these occurred, they were typically efforts to approximate the Voigt model, the convolution of a Lorentzian spectral line shape with a Gaussian broadening function, a peak which lacks a real-domain closed-form solution.

PeakLab’s predecessor always had a very high precision analytic complex Voigt function. This first edition of PeakLab adds Voigt models with IRFs, where an instrumental response function uses kinetic and/or probabilistic functions to model the asymmetry often seen with XPS, Raman, or other spectra with asymmetric peaks.

As an example, the following parameters appear in the spectroscopic model consisting of the Voigt peak convolved with the ‘ge’ IRF, the sum of a one-sided Gaussian and first order exponential. The main spectroscopic model parameters, a0-a3, are the pure Voigt peak parameters, and represent a peak that is symmetric with frequency:

a0 – the amplitude or area of the peak.

a1 – the central frequency of the peak.

a2 – the Gaussian width as a standard deviation.

a3 – the Lorentzian width or a Gaussian-Lorentzian width ratio

The auxiliary spectroscopic model parameters, a4-a6, describe the asymmetry, ideally treated as constant across all peaks in the spectrum:

a4 – the more compact instrumental distortion, here a right-sided Gaussian SD.

a5 – the less compact instrumental distortion, here a first order exponential decay time constant.

a6 – the area fraction of the narrow IRF component relative to the total instrumental distortion

In such a model, the pure Voigt peak is estimated as well as the asymmetric effects present in the peak shape. Here the asymmetry is modeled with the principal kinetic and probabilistic functions that can map such nonideality.

Chromatographic Parameters with Physical Meaning

Historically, the fitting of chromatographic peaks has yielded either ad-hoc or empirical parameters, as could be seen as true for the EMG model, or incomplete theoretical parameters, as in the case of the HVL and NLC models.

With the advent of PeakLab’s once-generalized and twice-generalized chromatographic models and the convolution fitting of true IRFs for the instrumental distortions, ALL of the fitted parameters have intrinsic physical meaning useful for characterizing every aspect of the separation, column health, and instrumental effects.

With PeakLab, you can concentrate directly on these parameters. For example, consider a single once-generalized HVL peak in a fit to analytic isocratic chromatographic data. The main chromatographic model parameters, a0-a3, are the pure HVL chromatographic peak model parameters absent instrumental distortions and theoretical non-ideality:

a0 – the fitted area of the peak.

a1 – the fitted center of mass of the pure zero-distortion peak, absent the a3 chromatographic distortion and its concentration dependency.

a2 – a statistical width as a standard deviation of the broadening of the pure zero distortion peak, absent the a3 chromatographic distortion and its concentration dependency.

a3 – the chromatographic distortion, the concentration-dependent fronting or tailing, in the pure chromatographic peak.

The a4 parameter adjusts for peak non-ideality and a5-a7 parameters are usually data-wide instrumental distortions shared by all of the peaks.

a4 – the non-ideality associated with the third moment skewness in the zero-distortion density, often treated as common across all peaks.

a5 – the narrow width instrumental distortion, in LC we believe this is mostly axial dispersion, usually fitted with a right-sided Gaussian for the delay introduced, and generally treated as common across all peaks.

a6 – the larger width instrumental distortion likely associated with first order exponential delays in the flow path and detection cell, and usually close to a constant across all peaks.

a7 – the area fraction of the narrow IRF component relative to the total instrumental distortion, also often close to constant across all analytic peaks

In this model, the ideal HVL peak is deconvolved, the deviation from the underlying Gaussian is estimated, and the instrumental distortions are quantified.

True R&D Peak Modeling Capabilities

PeakLab was developed with years of numerical research that included over one-hundred exploratory instrument response functions and an even larger count of experimental continuous probability density models.

PeakLab v1 consists of 797 built-in functions, 685 of which are peak functions. These built-in functions include:

46 built-in Spectroscopy Peak Functions
392 built-in Chromatography Peak Functions
98 built-in Statistical Peak Functions

For advanced peak modeling researchers, PeakLab also offers you up to sixty user defined peaks at any given time, and these are can be specified so that they are automatically placed at the peak locations in the data for fitting. The user-defined peaks also offer automatic IRF fitting for the most commonly used IRF models, and if needed, user-defined peaks can be created with your own custom IRFs. You can also import highly elaborate peak models from Mathematica and Maple in a single import step.