False Negative Rate - Aetherwisp

The False Negative Rate (FNR) is a performance index that states how many positive cases are wrongly classified as negative. Formally:

f_\text{FNR}(\{ y^{i},\hat{y}^{i} \}_{i})=\frac{\sum_{i=1}^{N}\mathrm{I}(y^{i}=\text{pos or }y^{i}\neq \hat{y}^{i})}{\sum_{i=1}^{N} \mathrm{I}(y^{i}=\text{pos})}

FNR is defined between 0 and 1 (a percentage). If there are no negatives, it is undefined (divides by zero). FNR is best when its lowest. It's commonly provided alongside the False Positive Rate.

A more convenient notation is

\text{FNR}=\frac{\text{FN}}{\text{P}}

where $\text{FN}$ is the number of false negatives and $\text{P}$ is the number of positives. This assumes the presence of a real and predicted set $\{ y^{i},\hat{y}^{i} \}$ even if not written. Counting false negatives needs both $y^{i}$ and $\hat{y}^{i}$ , whereas counting positives only need $y^{i}$ .

A sibling quantity is the True Negative Rate, which is the count of correctly classified negatives. It is the complement of FNR:

\text{TNR}=1-\text{FNR}

Like FNR, TNR is defined on $[0,1]$ . You want TNR to be as close to 1 as possible. TNR is also called specificity, because it measures how precisely the model can tell you what's wrong. This term is more common in diagnostic tests.

FPR is related to the error rate and accuracy of a binary classifier by

\begin{align} \text{Err}&=\frac{\text{FP}+\text{FN}}{\text{N}+\text{P}}=\frac{\text{P}\cdot\text{FNR}+\text{N}\cdot\text{FPR}}{\text{N}+\text{P}} \\ \text{Acc}=1-\text{Err}&=\frac{\text{TP}+\text{TN}}{\text{N}+\text{P}}=\frac{\text{P}\cdot\text{TPR}+\text{N}\cdot\text{TNR}}{\text{N}+\text{P}} \end{align}