F1 score

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:

$F_1 = 2 \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{\mathrm{precision} + \mathrm{recall}}$.

The general formula for positive real β is:

$F_\beta = (1 + \beta^2) \cdot \frac{\mathrm{precision} \cdot \mathrm{recall}}{(\beta^2 \cdot \mathrm{precision}) + \mathrm{recall}}$.

The formula in terms of Type I and type II errors:

$F_\beta = \frac {(1 + \beta^2) \cdot \mathrm{true\ positive} }{((1 + \beta^2) \cdot \mathrm{true\ positive} + \beta^2 \cdot \mathrm{false\ negative} + \mathrm{false\ positive})}\,$.

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

The F-measure was derived so that $F_\beta$ "measures the effectiveness of retrieval with respect to a user who attaches β times as much importance to recall as precision". It is based on van Rijsbergen's effectiveness measure

$E = 1 - \left(\frac{\alpha}{P} + \frac{1-\alpha}{R}\right)^{-1}$.

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

Applications

The F-score is often used in the field of information retrieval for measuring search, document classification, and query classification performance. Earlier works focused primarily on the F₁ score, but with the proliferation of large scale search engines, performance goals changed to place more emphasis on either precision or recall and so $F_\beta$ is seen in wide application.

The F-score is also used in machine learning. Note, however, that the F-measures do not take the true negative rate into account, and that measures such as the Matthews correlation coefficient, Informedness or Cohen's kappa may be preferable to assess the performance of a binary classifier.

See also

• Precision and recall
• BLEU
• NIST (metric)
• METEOR
• ROUGE (metric)
• Word Error Rate (WER)
• Receiver operating characteristic
• Matthews correlation coefficient