F-measure as Equivalent Electric Resistance

Occasionally, the precision (P) and recall (R) scores of a classification model alone aren’t enough to give a significant measure for its fitness. Rather, a combination of such quantities can be a more realistic measure. One of these combinations is known as the F-measure, whose popular variant is the F1-score.

In short, the F-measure is a function of precision, recall, and a real number β. It is defined as follows:

Notice that, for β equals zero, the F-score equals the precision whereas it approaches the recall as β tends to infinity. A more insightful way to write the F-measure is like so:

Taking the first term as a normalization factor, the remaining is half the harmonic mean of ‘P’ and ‘R/β²’. By restricting β to take only natural values and looking at ‘P’ and ‘R’ as values of electric resistors, we can extend the interpretation of the F-measure as the parallel combination of one ‘P’ resistor with β² ‘R’ resistors. The circuit would look like this for β=2:

With this interpretation, it is much obvious how the role of recall dominates the F-measure for values of β greater than 1. A similar interpretation can be done for β within the interval (-1,1). Writing β=1/ɑ and picking natural values for ɑ instead switches the role of precision and recall in the F-measure, since F(1/ɑ,P,R)=F(ɑ,R,P). So for these values, much more weight is given to the precision.