F-score

In

The F₁ score is the harmonic mean of the precision and recall. It thus symmetrically represents both precision and recall in one metric. The more generic ${\displaystyle F_{\beta }}$ score applies additional weights, valuing one of precision or recall more than the other.

The highest possible value of an F-score is 1.0, indicating perfect precision and recall, and the lowest possible value is 0, if either precision or recall are zero.

Etymology

The name F-measure is believed to be named after a different F function in Van Rijsbergen's book, when introduced to the Fourth Message Understanding Conference (MUC-4, 1992).^[1]

Definition

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The traditional F-measure or balanced F-score (F₁ score) is the harmonic mean of precision and recall:^[2]

${\displaystyle F_{1}={\frac {2}{\mathrm {recall} ^{-1}+\mathrm {precision} ^{-1}}}=2{\frac {\mathrm {precision} \cdot \mathrm {recall} }{\mathrm {precision} +\mathrm {recall} }}={\frac {2\mathrm {tp} }{2\mathrm {tp} +\mathrm {fp} +\mathrm {fn} }}}$.

F_β score

A more general F score, ${\displaystyle F_{\beta }}$, that uses a positive real factor ${\displaystyle \beta }$, where ${\displaystyle \beta }$ is chosen such that recall is considered ${\displaystyle \beta }$ times as important as precision, is:

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

In terms of Type I and type II errors this becomes:

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

Two commonly used values for ${\displaystyle \beta }$ are 2, which weighs recall higher than precision, and 0.5, which weighs recall lower than precision.

The F-measure was derived so that ${\displaystyle F_{\beta }}$ "measures the effectiveness of retrieval with respect to a user who attaches ${\displaystyle \beta }$ times as much importance to recall as precision".^[3] 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 ${\displaystyle \alpha ={\frac {1}{1+\beta ^{2}}}}$.

Diagnostic testing

This is related to the field of binary classification where recall is often termed "sensitivity".

Dependence of the F-score on class imbalance

Precision-recall curve, and thus the ${\displaystyle F_{\beta }}$ score, explicitly depends on the ratio ${\displaystyle r}$ of positive to negative test cases.^[13] This means that comparison of the F-score across different problems with differing class ratios is problematic. One way to address this issue (see e.g., Siblini et al, 2020^[14] ) is to use a standard class ratio ${\displaystyle r_{0}}$ when making such comparisons.

Applications

The F-score is often used in the field of

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[16] and so ${\displaystyle F_{\beta }}$ is seen in wide application.

The F-score is also used in

• The F₁-score of a classifier which always predicts the positive class converges to 1 as the probability of the positive class increases.
• The F₁-score of a classifier which always predicts the positive class is equal to 2 * proportion_of_positive_class / ( 1 + proportion_of_positive_class ), since the recall is 1, and the precision is equal to the proportion of the positive class.[21]
• If the scoring model is uninformative (cannot distinguish between the positive and negative class) then the optimal threshold is 0 so that the positive class is always predicted.
• F₁ score is concave in the true positive rate.^[22]

Criticism

David Hand and others criticize the widespread use of the F₁ score since it gives equal importance to precision and recall. In practice, different types of mis-classifications incur different costs. In other words, the relative importance of precision and recall is an aspect of the problem.^[23]

According to Davide Chicco and Giuseppe Jurman, the F₁ score is less truthful and informative than the

Another source of critique of F₁ is its lack of symmetry. It means it may change its value when dataset labeling is changed - the "positive" samples are named "negative" and vice versa. This criticism is met by the P4 metric definition, which is sometimes indicated as a symmetrical extension of F₁.^[26]

Difference from Fowlkes–Mallows index

While the F-measure is the harmonic mean of recall and precision, the Fowlkes–Mallows index is their geometric mean.^[27]

Extension to multi-class classification

The F-score is also used for evaluating classification problems with more than two classes (Multiclass classification). In this setup, the final score is obtained by micro-averaging (biased by class frequency) or macro-averaging (taking all classes as equally important). For macro-averaging, two different formulas have been used by applicants: the F-score of (arithmetic) class-wise precision and recall means or the arithmetic mean of class-wise F-scores, where the latter exhibits more desirable properties.^[28]

See also

• BLEU
• Confusion matrix
• Hypothesis tests for accuracy
• METEOR
• NIST (metric)
• Receiver operating characteristic
• ROUGE (metric)
• Uncertainty coefficient, aka Proficiency
• Word error rate
• LEPOR

References