Appendix AE. The mystery peak challenge

[FitResults,GOF]=peakfit([x y],0,0,1,3)

for a 4 exponentially broadened Gaussian model (shape 39)
[FitResults,GOF]=peakfit([x y],0,0,4,39,1)

Alternatively, it is even easier to use the interactive peak fitter ipf.m for this purpose; you can quickly change the model shape, number of peaks, starting guesses, data region to be fitted, etc., with single keystrokes and mouse clicks. Either way, the three initial fits in the figures show that the signal contains a small amount of random noise, which appears to be white (so the signal has probably not been smoothed, which is fortunate) and which has a relative standard deviation of about 0.2%, based on 1/5^th of the visual peak-to-peak value in the residual plot. But unfortunately these fits are not successful because their fitting errors (0.5 to 0.8%) are all significantly larger than the 0.2% random noise. Trying different shapes and greater numbers of peaks does not help either, resulting in either higher fitting errors, unstable fits, or zero peak heights for additional peaks (as in the four-Gaussian fit). 

Another approach to the problem of asymmetrical peaks is to use the technique of first-derivative symmetrization described previously. This applies specifically to exponential broadening, a commonly- encountered peak broadening mechanism. The idea is that if you compute the first derivative of an exponentially broadened peak, multiply it by a weighting factor equal to the time constant of the exponential, and add it to the original broadened signal, the result will be the original peak before broadening, which makes the peak overlap less severe. This works for any original peak shape. Even if you do not know the time constant beforehand, you can try different values until the baseline after the peak is as low as possible but not negative, as shown in this GIF animation. This is easily done interactively in iSignal, version 7, which has smoothing, derivatives, symmetrization, and curve fitting, or by using the symmetrize.m function. The derivative of y with respect to x, by the "derivxy.m" function, shown by the green line in the figure above, is quite noisy. As usual, we must smooth the derivatives of noisy signals to make them useful, but we must not over-smooth and distort the signals. As a rule of thumb, a smooth width equal to 1/10^th of the number of data points in the halfwidth does not distort the signal visibly. Our peak has about 530 points (

), so a smooth width near 53 will not distort the signal peak, but it does eliminate most of the noise from the derivative (red line, above right). Also, we can see that the derivative, in y/x units, is comparable in numerical magnitude to the original signal, so the weighting factor (in x units) is probably somewhere near 1.0. Next, we add the first derivative times that factor to the original signal, looking at the trailing edge as we try six different factor values near 1.0. The graph on the left shows that the optimum value is about 1.25.

 When this symmetrization is applied to the entire signal, the result, shown below on the right, has more distinct bumps. When that modified signal is used for curve fitting, it is found that a 3-Gaussian model works quite well, with a fitting error of only 0.25%. This is evidence that the signal actually consists of three exponentially broadened Gaussians. Ordinarily there is no independent way to check the accuracy of the peak parameters so measured, but - full disclosure - the signal in the case was not actually unknown but rather was generated by the file MysteryPeaks.m. The true time constant is 1.3, the peak positions are 53, 55, and 57.5 and the peak areas are each equal to 1.0. The curve-fit results after symmetrization (shown  on the right) are within 0.1% for the peak positions and within 2% of the areas. In contrast, direct fitting of three equal-alpha exponential Gaussians to the original data, which is now possible because we have determined a value for the time constant