F-score

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]

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

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

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

		Predicted condition		Sources: [4][5][6][7][8][9][10][11][12]
	Total population = P + N	Positive (PP)	Negative (PN)	Informedness, bookmaker informedness (BM) = TPR + TNR − 1	Prevalence threshold (PT) =${\displaystyle {\mathsf {\tfrac {{\sqrt {{\text{TPR}}\times {\text{FPR}}}}-{\text{FPR}}}{{\text{TPR}}-{\text{FPR}}}}}}$
Actual condition	Positive (P)	True positive (TP), hit	False negative (FN), type II error, miss, underestimation	True positive rate (TPR), recall, sensitivity (SEN), probability of detection, hit rate, power = TP/P = 1 − FNR	False negative rate (FNR), miss rate = FN/P = 1 − TPR
	Negative (N)	False positive (FP), type I error, false alarm, overestimation	True negative (TN), correct rejection	False positive rate (FPR), probability of false alarm, fall-out = FP/N = 1 − TNR	True negative rate (TNR), specificity (SPC), selectivity = TN/N = 1 − FPR
	Prevalence = P/P + N	Positive predictive value (PPV), precision = TP/PP = 1 − FDR	False omission rate (FOR) = FN/PN = 1 − NPV	Positive likelihood ratio (LR+) = TPR/FPR	Negative likelihood ratio (LR−) = FNR/TNR
	Accuracy (ACC) = TP + TN/P + N	False discovery rate (FDR) = FP/PP = 1 − PPV	Negative predictive value (NPV) = TN/PN = 1 − FOR	Markedness (MK), deltaP (Δp) = PPV + NPV − 1	Diagnostic odds ratio (DOR) = LR+/LR−
	Balanced accuracy (BA) = TPR + TNR/2	F1 score = 2 PPV × TPR/PPV + TPR = 2 TP/2 TP + FP + FN	Fowlkes–Mallows index (FM) = ${\displaystyle \scriptstyle {\mathsf {\sqrt {{\text{PPV}}\times {\text{TPR}}}}}}$	Matthews correlation coefficient (MCC) =${\displaystyle \scriptstyle {\mathsf {\sqrt {{\text{TPR}}\times {\text{TNR}}\times {\text{PPV}}\times {\text{NPV}}}}}}$${\displaystyle \scriptstyle -{\mathsf {\sqrt {{\text{FNR}}\times {\text{FPR}}\times {\text{FOR}}\times {\text{FDR}}}}}}$	Threat score (TS), critical success index (CSI), Jaccard index = TP/TP + FN + FP

Normalised harmonic mean plot where x is precision, y is recall and the vertical axis is F₁ score, in percentage points

Precision-Recall Curve: points from different thresholds are color coded, the point with optimal F-score is highlighted in red

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.

The F-score is often used in the field of information retrieval for measuring search, document classification, and query classification performance.[15] 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 machine learning.[17] However, the F-measures do not take true negatives into account, hence measures such as the Matthews correlation coefficient, Informedness or Cohen's kappa may be preferred to assess the performance of a binary classifier.[18]

The F-score has been widely used in the natural language processing literature,[19] such as in the evaluation of named entity recognition and word segmentation.

The F₁ score is the Dice coefficient of the set of retrieved items and the set of relevant items.[20]

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.[21]

According to Davide Chicco and Giuseppe Jurman, the F₁ score is less truthful and informative than the Matthews correlation coefficient (MCC) in binary evaluation classification.[22]

David Powers has pointed out that F₁ ignores the True Negatives and thus is misleading for unbalanced classes, while kappa and correlation measures are symmetric and assess both directions of predictability - the classifier predicting the true class and the true class predicting the classifier prediction, proposing separate multiclass measures Informedness and Markedness for the two directions, noting that their geometric mean is correlation.[23]

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.[24]

While the F-measure is the harmonic mean of recall and precision, the Fowlkes–Mallows index is their geometric mean.[25]

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.[26]

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