Tips for Getting Better Curve Fits

Band Selection

In general, the selection of bands should reflect assumptions about the probable chemical state of the data and not be chosen purely on the basis of modeling the experimental data with the synthetic curve. If noise or other factors play a part in determining the artificial peaks used to model the data, it is best to prepare the data to remove these factors, either by transformations (satellite subtraction, smoothing) or by editing the data.

Also, minimizing the number of parameters used will result in a more predictable fit. Where possible, it is best to use the least complex band type.

Limits

While there are no hard and fast rules for choosing the default limits, keep in mind that the choice of limits has a great affect on curve fit optimization.

If it is difficult to come up with initial estimates that provide a good initial fit, it may be best to keep the limits loose at first and tighten them up later.

Important: The limits for each parameter adjust with the estimate when the estimate is adjusted manually, and after the curve fit routine is performed. This is why a second command to perform a curve fit may result in a different (and possible improved) fit after the curve fit routine has completed. Since the limits track the estimates, they are “loosened” after each fit. As you get closer to the desired fit, you may wish to tighten up the limits or adjust the invariance.

Tightening Limits Using Invariance and Locking

Setting the invariance to a higher number is an easy way to tighten the limits for a particular band for each fit. As stated previously, the limits track the estimate for each fit, but the invariance stays the same. Setting the invariance higher for a selected parameter causes that parameter to be adjusted later in the fit, therefore ensuring that it does not change as dramatically.

Locking a parameter for a fit is essentially the same as setting the limits equal to the estimate for that parameter. The parameter does not adjust at all during the fit. This can be especially useful for the position parameter, since it is often likely that the position of a peak is the most easily characterized parameter.

Re-fitting Data

It is often the case that the first time a curve fit is performed, there is convergence before the optimal desired fit is obtained. This is sometimes due to hitting the limit on the maximum number of iterations, but it is more often due to tight constraints. In this case, the usual remedy is to click the fit button again and observe whether the goodness of fit has improved and that the synthetic curve more closely resembles the expected outcome.

Why do you get a better fit the second time, even though you have not readjusted any parameters? This is simply due to the fact that after the first fit, the limits self adjust to track the estimates, in effect, “loosening” them between fits. This is often desirable, since the computer “nudges” the estimates towards convergence.

In some cases, however, you truly want your limits for some parameters to be fixed for the duration of all the subsequent fits you perform on that data set. In that case, you must be careful to readjust those limits in between fits. Additionally, you can lock those parameters for some fits (or increase the invariance), which has the affect of “tightening” the existing limits.

Correcting Unsatisfactory Fits

The choice of initial estimates greatly determines the direction the curve fit algorithm will take in adjusting parameters. In some cases, the least-squares curve fit algorithm may be converging to a local minimum, instead of the desired solution. One of the easiest ways to combat this is to choose new estimates on the “opposite” side of the solution. For example, if the initial estimates produce a curve that lies above the experimental data, adjust the estimates to produce a curve that lies below the data, and then re-fit.

There are times where re-fitting data multiple times results in some estimates adjusting far out of expected range. This is often the case with asymmetric bands, where the tail scale and tail length often increase out of proportion because of small peaks that haven’t been modeled by additional bands. If this is the case, you may want to readjust these estimates to expected values, and try to determine what aspect of the experimental data is not being modeled properly by the existing bands.

Correcting Low-Intensity Fits

One particular problem that often occurs is that the artificial peaks often appear lower in intensity than the experimental data. There is a specific reason for this, and there are ways to combat the problem.

The goodness-of-fit value (which is used by the curve fit algorithm to determine success of a particular fit) is weighted by the intensity. However, lower intensity portions of a curve, the “wider” area, also contribute to the goodness-of-fit value. This is desirable. Because of the gaussian curve shape, the lower intensity data covers a greater range of eV and therefore contributes more to the area. The highest portion of the peak, while greater intensity, often contributes relatively little. Since all estimates except position contribute to the area, the total area is an important factor in the correctness of the fit and obviously important in determining chemical state and correctness of the results. The algorithm models this behavior.

However, these types of results – lower intensity synthetic peaks – are often undesirable both from an objective and subjective viewpoint (they are more obvious than incorrectness at the lower portion of the peak when viewing a graph). Keep in mind that the problem is probably not with the peak itself but with the lower intensity data often associated with the background or with some lower intensity peaks around the peak base. In effect, the algorithm is trying to widen the lower portion of the curve to correct for this, and does this by “flattening” out the peaks, therefore widening the bottoms. All parameters except position contribute to this effect.

To solve this problem, begin by studying the lower intensity experimental data. Often, some of the background has been left after baseline subtraction and choosing a new baseline results in removal of this data. Also, there may be one or more lower intensity peaks that have not been modeled, and additional peaks may need to be added. Lastly, noise present in the low intensity data often causes the algorithm to flatten the bands. The data may need to be smoothed, or in some cases spurious noise can just be manually edited out of the data.

Additional Curve Fitting Topics