Why is the F value the harmonic mean?

from F value Why is the F value the harmonic mean?

What is F value http://scikit-learn.org/stable/modules/generated/sklearn.metrics.f1_score.html

• F1 = 2 * (precision * recall) / (precision + recall)

• Max 1 Min 0

• When precision == recall, F value is the same value

• When either precision or recall is 0, F1 is also 0.

• TP: True Positive

• FP: False Positive

• FN: False Negative

• TN: True Negative

• Precision:

• $P=\frac{TP}{TP+FP}$

• Recall:

• $R=\frac{TP}{TP+FN}$

• Sum of reciprocals

• $\frac{1}{P}+\frac{1}{R}=\frac{2TP+FP+FN}{TP}$

• F1 score is the harmonic mean of P and R

• $F1=\frac{2PR}{P+R}=\frac{2\frac{TP}{TP+FP}\frac{TP}{TP+FN}}{\frac{TP}{TP+FP}+\frac{TP}{TP+FN}}=\frac{2TP}{(TP+FN)+(TP+FP)}=\frac{2TP}{2TP+FN+FP}$

• $\frac{2}{\frac{1}{P}+\frac{1}{R}}=\frac{2TP}{2TP+FP+FN}$

• Normalized Symmetric Difference http://www.dcs.gla.ac.uk/Keith/pdf/Chapter7.pdf p.128

• $E=\frac{FP+FN}{2TP+FP+FN}$

• F1 and E add 1: F1 + E = 1

• Fβ Score

• $F_{\beta }=\frac{(1+{\beta }^2)TP}{(1+{\beta }^2)TP+{\beta }^{2}FN+FP}$

• $\frac{1}{P}+\frac{{\beta }^2}{R}=\frac{(TP+FP)+{\beta }^2(TP+FN)}{TP}$

• In short, Recall with ${\beta }^2$ weighted harmonic mean

• Why ${\beta }^2$ instead of $\beta$?

• $F_{\beta }=\frac{(1+{\beta }^2)TP}{(1+{\beta }^2)TP+{\beta }^{2}FN+FP}=\frac{1+{\beta }^2}{\frac{1}{P}+\frac{{\beta }^2}{R}}$

• The coefficients are omitted and written as Z

• $\frac{\partial F_{\beta }}{\partial R}=Z\frac{{\beta }^2}{R^2},\frac{\partial F_{\beta }}{\partial P}=Z\frac{1}{P^2}$

• That is, the gradients match when [$ \beta P = R

• The fact that the gradients match means that “the effect on the score is the same when P and R are moved by the same amount.

• Does it mean that equilibrium is reached when R is β times greater than P?

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