# F1 score

Precision and recall

In statistical analysis of binary classification, 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 positive results divided by the number of all positive results returned by the classifier, and r is the number of correct positive results divided by the number of all relevant samples (all samples that should have been identified as positive). The F1 score is the harmonic mean of the precision and recall, where an F1 score reaches its best value at 1 (perfect precision and recall) and worst at 0.

## Etymology

The name F-measure is believed to be named after a different F function in Van Rijsbergen's book, when introduced to MUC-4. [1]

## Definition

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

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

The general formula for positive real 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 weighs recall higher than precision (by placing more emphasis on false negatives), and the ${\displaystyle F_{0.5}}$ measure, which weighs recall lower than precision (by attenuating the influence of false negatives).

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

The F1 score is also known as the Sørensen–Dice coefficient or Dice similarity coefficient (DSC).

## Diagnostic testing

This is related to the field of binary classification where recall is often termed as Sensitivity. There are several reasons that the F1 score can be criticized in particular circumstances.[3]

 True condition Total population Condition positive Condition negative Prevalence = Σ Condition positive/Σ Total population Accuracy (ACC) = Σ True positive + Σ True negative/Σ Total population Predictedcondition Predicted conditionpositive True positive Positive predictive value (PPV), Precision = Σ True positive/Σ Predicted condition positive False discovery rate (FDR) = Σ False positive/Σ Predicted condition positive Predicted conditionnegative True negative False omission rate (FOR) = Σ False negative/Σ Predicted condition negative Negative predictive value (NPV) = Σ True negative/Σ Predicted condition negative True positive rate (TPR), Recall, Sensitivity, probability of detection, Power = Σ True positive/Σ Condition positive False positive rate (FPR), Fall-out, probability of false alarm = Σ False positive/Σ Condition negative Positive likelihood ratio (LR+) = TPR/FPR Diagnostic odds ratio (DOR) = LR+/LR− F1 score = 2 · Precision · Recall/Precision + Recall False negative rate (FNR), Miss rate = Σ False negative/Σ Condition positive Specificity (SPC), Selectivity, True negative rate (TNR) = Σ True negative/Σ Condition negative Negative likelihood ratio (LR−) = FNR/TNR

## Applications

The F-score is often used in the field of information retrieval for measuring search, document classification, and query classification performance.[4] 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[5] and so ${\displaystyle F_{\beta }}$ is seen in wide application.

The F-score is also used in machine learning.[6] Note, however, that the F-measures do not take the true negatives 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.[3]

The F-score has been widely used in the natural language processing literature,[7] such as the evaluation of named entity recognition and word segmentation.

## 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.[8]

## Difference from G-measure

While the F-measure is the harmonic mean of recall and precision, the G-measure is the geometric mean.[3]