Abbe Criterion

contributed by Prof Matthew R. Linford (USA)

The Abbe Criterion

The Abbe criterion is a figure of merit that helps us to assess the quality of a peak fit. It is given by:

Here, R(i) and R(i + 1) refer to the residuals of the fit at the ith and (i + 1)th data points, respectively. In essence, the Abbe criterion tells a user how the residuals of a fit are distributed. Let’s
consider two possibilities.

First, imagine that the residuals of a fit consistently have the same value. This would occur if the calculated fit were always above or below the measured data points by a fixed amount. Clearly this situation is undesirable. Now, if R(i) = R(i + 1) for the N – 1 data points considered in the Equation shown here, Abbe = 0.

Next, let’s consider the possibility that R(i) = -R(i+1) for the N – 1 data points in the Abbe equation shown here. In other words, each subsequent residual has the same magnitude, but a sign opposite to the previous residual. A little math shows that this equation approaches a value of two. This situation is also undesirable.

In summary, we would not want the fit to our data to be consistently above or below it (Abbe = 0), and we should be suspicious if we find that our residuals are anticorrelated, i.e., of the same magnitude, but alternating in sign (Abbe → 2). It turns out that for random noise and statistically distributed residuals, Abbe ≈ 1.

This value of unity for the Abbe criterion is the desired value for this parameter, and significant deviations from it suggest issues with a data fit.

Original Source Publication:

Vacuum Technology & Coating, pages 25-30, Dec, 2015 (vtcmag@vtcmag.com)