Log-Normal 4-Parameter Cumulative (LN4 Cumulative)

The LN4 cumulative is the integral of TableCurve's four-parameter log-normal peak (LogNorm4, the LN4). It is an asymmetric transition whose steepness on each side is set independently of its width, and its parameters are those of the peak that is its first derivative: a the height of the step, b the position of the derivative's apex (the steepest point of the transition), c the derivative's full width at half maximum, and d its asymmetry at half maximum (Asym50: the right half-width over the left). With d > 1 the transition rises steeply and approaches its final level slowly; with d < 1 the reverse; d = 1 is the limit of the Gaussian cumulative (not allowed exactly).

Direction

A negative c gives the falling transition with the same asymmetry, y = a - F(x; |c|). TableCurve uses the sign of the width for the reverse cumulatives throughout.

Support

The underlying peak is zero beyond x = b - |c| d / (d²-1): for d > 1 below that point, for d < 1 above it. There the cumulative holds its limit (0 before the rise, a after it). Exported code keeps the function defined at every x (the LOGSAFE form described in the Code Export topic).

Equations

8790 LN4Cum_ zero intercept: a, b, c, d 8791 LN4Cum with a constant: constant + a, b, c, d 8792 LN4Cum+ with a linear baseline: constant, slope + a, b, c, d 8754-8771 2- and 3-component sums, including shared-asymmetry (S) and shared-width-and-asymmetry (WS) forms, 4 parameters per component minus the shared ones

Fitting

Select LN4 Cumulative in Curve-Fit Transition Functions (Process menu), with the baseline forms wanted; tick 2-Component Models / 3-Component Models for sums of transitions. Curve-Fit All Equations fits the single forms; Curve-Fit All Equations plus Multiple Components fits all of them. Starting estimates are automatic: each transition is located on the smoothed third derivative, which also gives its width and asymmetry directly. The 2- and 3-component models are fitted only when the data's S/N (Time SNRdb in the numeric pane) is at least 35 dB and the data shows that many transitions.

When to use it

Any single or overlapping step whose rise and approach differ in steepness: titration and binding curves with an asymmetric approach, dissolution and release profiles, thermal transitions, breakthrough curves. Fit it beside the Gaussian, logistic and EMG cumulatives and compare: when the fitted d is close to 1 the simpler symmetric transitions describe the data as well with one parameter fewer.