# F-Measure

An F-Measure is a performance metric for binary classification functions that is based on the harmonic mean for the classifier's precision and recall.

## References

### 2011

• (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/F1_score
• QUOTE:In statistics, the F1 score (also F-score or F-measure) is a measure of a test's accuracy. It considers both the precision p and the recall r of the test to compute the score: p is the number of correct results divided by the number of all returned results and r is the number of correct results divided by the number of results that should have been returned. The F1 score can be interpreted as a weighted average of the precision and recall, where an F1 score reaches its best value at 1 and worst score at 0.

The traditional F-measure or balanced F-score (F1 score) is the harmonic mean of precision and recall: :$\displaystyle{ F = 2 \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{\mathrm{precision} + \mathrm{recall}} }$.

The general formula for positive real $\displaystyle{ β }$ is: :$\displaystyle{ F_\beta = (1 + \beta^2) \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{(\beta^2 \cdot \mathrm{precision}) + \mathrm{recall}} }$.

The formula in terms of Type I and type II errors: :$\displaystyle{ F_\beta = \frac {(1 + \beta^2) \cdot \mathrm{true\ positive} }{((1 + \beta^2) \cdot \mathrm{true\ positive} + \beta^2 \cdot \mathrm{false\ negative} + \mathrm{false\ positive})}\, }$.

Two other commonly used F measures are the $\displaystyle{ F_{2} }$ measure, which weights recall higher than precision, and the $\displaystyle{ F_{0.5} }$ measure, which puts more emphasis on precision than recall.

The F-measure was derived so that $\displaystyle{ F_\beta }$ "measures the effectiveness of retrieval with respect to a user who attaches $\displaystyle{ β }$ times as much importance to recall as precision" [1]. It is based on van Rijsbergen's effectiveness measure :$\displaystyle{ E = 1 - \left(\frac{\alpha}{P} + \frac{1-\alpha}{R}\right)^{-1} }$.

Their relationship is $\displaystyle{ F_\beta = 1 - E }$ where $\displaystyle{ \alpha=\frac{1}{1 + \beta^2} }$.

### 2009

1. van Rijsbergen, C. J. (1979). Information Retrieval (2nd ed.). Butterworth.