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 average 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:

F_{1}=\left({\frac {\mathrm {recall} ^{-1}+\mathrm {precision} ^{-1}}{2}}\right)^{-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 weighs recall higher than precision (by placing more emphasis on false negatives), and the $F_{0.5}$ measure, which weighs recall lower than precision (by attenuating the influence of false negatives).

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".^[2] 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 $\alpha ={\frac {1}{1+\beta ^{2}}}$ .

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
Predicted condition	Predicted condition positive	True positive, Power	False positive, Type I error	Positive predictive value (PPV), Precision = Σ True positive/Σ Predicted condition positive	False discovery rate (FDR) = Σ False positive/Σ Predicted condition positive
	Predicted condition negative	False negative, Type II error	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 = 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 = 2/1/Recall + 1/Precision
		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 $F_{\beta }$ 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, such as the evaluation of named entity recognition and word segmentation.

Difference from G-measure

While the F-measure is the harmonic mean of Recall and Precision, the G-measure is the geometric mean.^[3]

References

↑ Sasaki, Y. (2007). "The truth of the F-measure" (PDF).
↑ Van Rijsbergen, C. J. (1979). Information Retrieval (2nd ed.). Butterworth-Heinemann.
1 2 3 Powers, David M W (2011). "Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation" (PDF). Journal of Machine Learning Technologies. 2 (1): 37–63.
↑ Beitzel., Steven M. (2006). On Understanding and Classifying Web Queries (Ph.D. thesis). IIT. CiteSeerX 10.1.1.127.634.
↑ X. Li; Y.-Y. Wang; A. Acero (July 2008). Learning query intent from regularized click graphs (PDF). Proceedings of the 31st SIGIR Conference.
↑ See, e.g., the evaluation of the .

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Sasaki, Y. (2007). "The truth of the F-measure" (PDF).

[2] Van Rijsbergen, C. J. (1979). Information Retrieval (2nd ed.). Butterworth-Heinemann.

[Powers2011-3] 1 2 3 Powers, David M W (2011). "Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation" (PDF). Journal of Machine Learning Technologies. 2 (1): 37–63.

[4] Beitzel., Steven M. (2006). On Understanding and Classifying Web Queries (Ph.D. thesis). IIT. CiteSeerX 10.1.1.127.634.

[5] X. Li; Y.-Y. Wang; A. Acero (July 2008). Learning query intent from regularized click graphs (PDF). Proceedings of the 31st SIGIR Conference.

[6] See, e.g., the evaluation of the .