F₁-score#

The F₁-score, also known as balanced F-score or F-measure, is a metric that combines two competing metrics, precision and recall, with an equal weight. F₁-score is the harmonic mean between precision and recall, and symmetrically represents both in one metric.

Guides: Precision and Recall

Read the precision and the recall guides if you're not familiar with those metrics.

Precision and recall offer a trade-off: increasing precision often reduces recall, and vice versa. This is called the precision/recall trade-off.

Ideally, we want to maximize both precision and recall to obtain the perfect model. This is where the F₁-score comes in play. Because the F₁-score is the harmonic mean of precision and recall, maximizing the F₁-score implies simultaneously maximizing both precision and recall. Thus, the F₁-score has become a popular metric for the evaluation of many workflows, such as classification, object detection, semantic segmentation, and information retrieval.

API Reference: f1_score ↗

Implementation Details#

Using TP / FP / FN / TN, we can define precision and recall. The F₁-score is computed by taking the harmonic mean of precision and recall.

The F₁-score is defined:

\[ \begin{align} \text{F}_1 &= \frac {2} {\frac {1} {\text{Precision}} + \frac {1} {\text{Recall}}} \\[1em] &= \frac {2 \times \text{Precision} \times \text{Recall}} {\text{Precision} + \text{Recall}} \end{align} \]

It can also be calculated directly from true positive (TP) / false positive (FP) / false negative (FN) counts:

\[ \text{F}_1 = \frac {\text{TP}} {\text{TP} + \frac 1 2 \left( \text{FP} + \text{FN} \right)} \]

Examples#

Perfect inferences:

Metric	Value
TP	20
FP	0
FN	0

\[ \begin{align} \text{Precision} = \frac{20}{20 + 0} &= 1.0 \\[1em] \text{Recall} = \frac{20}{20 + 0} &= 1.0 \\[1em] \text{F}_1 = \frac{20}{20 + \frac 1 2 \left( 0 + 0 \right)} &= 1.0 \end{align} \]

Partially correct inferences, where every ground truth is recalled by an inference:

Metric	Value
TP	25
FP	75
FN	0

\[ \begin{align} \text{Precision} = \frac{25}{25 + 75} &= 0.25 \\[1em] \text{Recall} = \frac{25}{25 + 0} &= 1.0 \\[1em] \text{F}_1 = \frac{25}{25 + \frac 1 2 \left( 75 + 0 \right)} &= 0.4 \end{align} \]

Perfect inferences but some ground truths are missed:

Metric	Value
TP	25
FP	0
FN	75

\[ \begin{align} \text{Precision} = \frac{25}{25 + 0} &= 1.0 \\[1em] \text{Recall} = \frac{25}{25 + 75} &= 0.25 \\[1em] \text{F}_1 = \frac{25}{25 + \frac 1 2 \left( 0 + 75 \right)} &= 0.4 \end{align} \]

Zero correct inferences with non-zero false positive and false negative:

Metric	Value
TP	0
FP	15
FN	10

\[ \begin{align} \text{Precision} = \frac{0}{0 + 15} &= 0.0 \\[1em] \text{Recall} = \frac{0}{0 + 10} &= 0.0 \\[1em] \text{F}_1 = \frac{0}{0 + \frac 1 2 \left( 15 + 10\right)} &= 0.0 \end{align} \]

Zero correct inferences with zero false positive and false negative:

Metric	Value
TP	0
FP	0
FN	0

Undefined F₁

This example shows an edge case where both precision and recall are undefined. When either metric is undefined, F₁ is also undefind. In such cases, it's often interpreted as 0.0 instead.

\[ \begin{align} \text{Precision} &= \frac{0}{0 + 0} \\[1em] &= \text{undefined} \\[1em] \text{Recall} &= \frac{0}{0 + 0} \\[1em] &= \text{undefined} \\[1em] \text{F}_1 &= \frac{0}{0 + \frac 1 2 \left( 0 + 0\right)} \\[1em] &= \text{undefined} \\[1em] \end{align} \]

Multiple Classes#

In workflows with multiple classes, the F₁-score can be computed per class. In the TP / FP / FN / TN guide, we learned how to compute per-class metrics when there are multiple classes, using the one-vs-rest (OvR) strategy. Once you have TP, FP, and FN counts computed for each class, you can compute precision, recall, and F₁-score for each class by treating each as a single-class problem.

Aggregating Per-class Metrics#

If you are looking for a single F₁-score that summarizes model performance across all classes, there are different ways to aggregate per-class F₁-scores: macro, micro, and weighted. Read more about these methods in the Averaging Methods guide.

F\(_\beta\)-score#

The F\(_\beta\)-score is a generic form of the F₁-score with a weight parameter, \(\beta\), where recall is considered \(\beta\) times more important than precision:

\[ \text{F}_{\beta} = \frac {(1 + \beta^2) \times \text{precision} \times \text{recall}} {(\beta^2 \times \text{precision}) + \text{recall}} \]

The three most common values for the beta parameter are as follows:

F_0.5-score \(\left(\beta = 0.5\right)\), where precision is more important than recall, it focuses more on minimizing FPs than minimizing FNs
F₁-score \(\left(\beta = 1\right)\), the true harmonic mean of precision and recall
F₂-score \(\left(\beta = 2\right)\), where recall is more important than precision, it focuses more on minimizing FNs than minimizing FPs

Limitations and Biases#

While the F₁-score can be used to evaluate classification/object detection models with a single metric, this metric is not adequate to use for all applications. In some applications, such as identifying pedestrians from an autonomous vehicle, any false negatives can be life-threatening. In these scenarios, having a few more false positives as a trade-off for reducing the chance of any life-threatening events happening is preferred. Here, recall should be weighted much more than the precision as it minimizes false negatives. To address the significance of recall, \(\text{F}_\beta\) score can be used as an alternative.

Threshold-Dependence#

Precision, recall, and F₁-score are all threshold-dependent metrics. Threshold-dependent means that, before computing these metrics, a confidence score threshold must be applied to inferences to decide which should be used for metrics computation and which should be ignored.

A small change to this confidence score threshold can have a large impact on threshold-dependent metrics. To evaluate a model across all thresholds, rather than at a single-threshold, use threshold-independent metrics, like average precision.

F1-score#