Curve-Fit Peak Functions¶
TableCurve 2D's built-in peak functions consist of a set of 74 total equations. There are 37 peak models with and without an intercept. To make peak function fitting as flexible as possible, you are first presented with a peak configuration dialog which contains custom fit settings specifically for the built-in peak functions. You can choose only the symmetric or asymmetric peaks, one or both of the intercept and zero-intercept versions as well as any one of the four Minimization Criteria available in the non-linear fitting engine. You can choose individual peak functions or you may have the AI Expert feature pick the settings for you based upon an analysis of the data profile. The non-linear fitting controls override the non-linear fit preferences in the Curve-Fit Preferences option. Your peak function customization will be automatically saved across sessions.
Also fitted with this option are all installed UDFs, and all active TCEX.DLL equations.
Symmetric Peak Functions¶
Basic three-parameter (single width) symmetric peaks include the Gaussian, Lorentzian, Logistic, Complementary Error Function, and Laplace models. Peaks with a second width term that control the decay or shape of the peak include the Modified Gaussian, SDS, SDC, Pearson VII, Gaussian-Lorentzian Cross Product, Error, and Student t peak functions. The Pearson VII, Student t, and Gaussian-Lorentzian XP each produce a symmetric peak function that can vary between a Gaussian and Lorentzian. The Pearson VII and Student t models, which are actually two different parametrizations of the same model, can actually go further, producing a distribution with wider tails than the Lorentzian. In the Modified Gaussian and Error models, the Gaussian's power of 2 is replaced by an adjustable parameter expression. Both models are capable of producing the Laplace or double-sided exponential. The SDS (symmetric double sigmoidal) and the SDC (symmetric double cumulative) functions can vary from approximating a Gaussian at one extreme to that of a square wave profile at the other. The SDS and SDC functions enable fitting profiles where there is a plateau at the peak maximum.
Asymmetric Peak Functions¶
TableCurve 2D's asymmetric peaks include the statistical distribution functions, the single width Log-Normal and Extreme-Value, the two-width term Log-Normal, Extreme Value (fronted and tailed), Gamma, Weibull, Inverted Gamma, and Chi-Square, and the three width term Beta, F-variance, and Pearson IV models. All are parametrized to have an amplitude and peak center consistent with all of TableCurve 2D's peak functions.
The asymmetric functions include two convolution models, the EMG (Exponentially Modified Gaussian), and the GMG (Half-Gaussian Modified Gaussian). Also included are the ADS and ADC, asymmetric versions of the SDS and SDC peaks. There are several forms of asymmetric logistic peaks as well as the peak formed by the derivative of the LDR (logistic dose response) equation. There are three forms of a pulse peak for fitting profiles with a very rapid rise and much slower decay. There are also two kinetics peaks, one for a first order intermediate and another for a first order equilibrium. Unlike the versions in the kinetics equation set which assume the reaction begins at time X=0, these two include a lag term.
Synthetic Functions¶
TableCurve 2D's built-in peak functions have all been constructed to be defined for all values of X, even if the native function has undefined regions or is undefined at certain parameter values. These synthetic functions will either return zero (for zero-intercept versions) or return to the intercept value (for intercept versions) when an undefined condition is present. This is especially useful with peaks such as the Weibull, Gamma, Inverted Gamma, Log Normal-4, Chi-Square, Beta, F-variance, and Pulse peaks which are undefined below and/or above certain X-values. If you are fitting data that are at zero or a constant value until a peak profile appears starting at a certain X, you need not remove these initial points. They can be fitted as a part of the actual model.
AI Expert¶
When selecting this option, TableCurve 2D references raw data measurements of baseline and asymmetry to determine which peaks are most appropriate to the data as well as whether zero-intercept, intercept, or both versions should be fitted. The AI Expert option will only be effective if there is a full peak present so that an accurate baseline can be estimated, and only if there are sufficient points near the left and right half-maxima points, since the asymmetry determination is made at this half-maxima value.
Reading and Saving Peak Fitting Configurations¶
The most recent peak configuration is saved across sessions and presented each time this option is used. If you select the AI Expert and are not satisfied with the suggested fit, you can press Reset to restore the configuration present when first opening this dialog. If you will be fitting several types of specific peak profiles, you may want to save a configuration specific to each or to a given application or problem. The Save item saves all information shown in the dialog to a binary file with an [FPK] extension. The Read item can then be used to access this configuration whenever desired.
Nomenclature¶
During fitting, a function name with an underscore suffix, such as Gaussian_, represents the zero-intercept version of the function. When the underscore is absent, the intercept version of the function is being referenced. In the Review, non-linear equations can be presented in either a full mathematical representation or in a symbolic notational format. Since some of the peak functions have rather long mathematical expressions, you may prefer to use the notational format offered in the Preferences option in the main Edit menu or in the File menu of the Curve-Fit graph.
Automated Fitting¶
Upon closing the peak configuration via the Fit button, TableCurve 2D's Automated Curve-Fit Processing begins and fits the peak functions as configured as well as any installed UDFs and TCEX.DLL functions. The non-linear fitting controls in the peak configuration also apply to such UDFs and external functions.