# 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.

**AKA:**Fβ-Metric.**Context:**- Input(s):
- True Positives, False Positives, and False Negatives (or recall and precision).
- Positive Real weight
*β*(a Non-Negative Real), otherwise set to One.

- Output: an F Score.
- It can be calculated as :[math]\displaystyle{ F_\beta = \frac{(1 + \beta^2)\times TP}{(1 + \beta^2)TP + \beta^2FN + 1FP} }[/math]

- Input(s):
**Example(s):****Counter-Example(s):**- a Recall Metric.
- a Precision Metric.
- an Accuracy Metric.
- a PMI Measure.
- an F-Statistic.

**See:**Tagging Task, Annotation Task.

## References

### 2011

- (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/F1_score
- QUOTE:In statistics, the
**F**(also F-score or_{1}score**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 F_{1}score can be interpreted as a weighted average of the precision and recall, where an F_{1}score reaches its best value at 1 and worst score at 0.The traditional F-measure or balanced F-score (F

_{1}score) is the harmonic mean of precision and recall: :[math]\displaystyle{ F = 2 \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{\mathrm{precision} + \mathrm{recall}} }[/math].The general formula for positive real [math]\displaystyle{ β }[/math] is: :[math]\displaystyle{ F_\beta = (1 + \beta^2) \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{(\beta^2 \cdot \mathrm{precision}) + \mathrm{recall}} }[/math].

The formula in terms of Type I and type II errors: :[math]\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})}\, }[/math].

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

The F-measure was derived so that [math]\displaystyle{ F_\beta }[/math] "measures the effectiveness of retrieval with respect to a user who attaches [math]\displaystyle{ β }[/math] times as much importance to recall as precision"

^{[1]}. It is based on van Rijsbergen's effectiveness measure :[math]\displaystyle{ E = 1 - \left(\frac{\alpha}{P} + \frac{1-\alpha}{R}\right)^{-1} }[/math].Their relationship is [math]\displaystyle{ F_\beta = 1 - E }[/math] where [math]\displaystyle{ \alpha=\frac{1}{1 + \beta^2} }[/math].

- QUOTE:In statistics, the

### 2009

- (Hu et al., 1999) ⇒ Xiaohua Hu, Xiaodan Zhang, Caimei Lu, E. K. Park, and Xiaohua Zhou. (2009). “Exploiting Wikipedia as External Knowledge for Document Clustering.” In: Proceedings of ACM SIGKDD Conference (KDD-2009). doi:10.1145/1557019.1557066
- QUOTE:Cluster quality is evaluated by three metrics, purity [14], F-score [10], and normalized mutual information (NMI) [15]. … F-score combines the information of precision and recall which is extensively applied in information retrieval. … All the three metrics range from 0 to 1, and the higher their value, the better the clustering quality is.

- ↑ van Rijsbergen, C. J. (1979).
*Information Retrieval*(2nd ed.). Butterworth.