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False Positive Rate (FPR)#

False positive rate (FPR) measures the proportion of negative ground truths that a model incorrectly predicts as positive, ranging from 0 to 1. A low false positive rate indicates that the model is good at avoiding false alarms, where a high false positive rate suggests that the model is incorrectly classifying a significant number of negative cases as positive.

As shown in this diagram, false positive rate is the fraction of all negative ground truths that are incorrectly predicted:

\[\text{FPR} = \frac{\text{FP}}{\text{TN} + \text{FP}}\]

In the above formula, \(\text{TN}\) is the number of true negative inferences and \(\text{FP}\) is the number of false positive inferences.

Guide: True Negative / False Positive

Read the TP / FP / FN / TN guide if you're not familiar with "TN" and "FP" terminology.

FPR Image FPR Image

FPR is often used in conjunction with TPR (recall). By measuring both FPR and TPR, a more complete picture of a model's performance can be drawn.

Implementation Details#

FPR is used to evaluate the performance of a classification model, particularly in tasks like binary classification, where the goal is to classify data into one of two possible classes. FPR is closely related to specificity. While specificity measure the model's ability to correctly identify the negative class, the FPR focuses on the negative class instances that are incorrectly classified as positive.

Here is how FPR is calculated:

\[ \text{FPR} = \frac {\text{# False Positives}} {\text{# True Negatives} + \text{# False Positives}} \]


Perfect model inferences, where every negative ground truth is recalled by an inference:

Metric Value
TN 20
FP 0
\[ \begin{align} \text{FPR} &= \frac{0}{20 + 0} \\[1em] &= 0.0 \end{align} \]

Partially correct inferences, where some negative ground truths are correctly recalled (TN) and others are missed (FP):

Metric Value
TN 85
FP 15
\[ \begin{align} \text{FPR} &= \frac{15}{85 + 15} \\[1em] &= 0.15 \end{align} \]

Zero correct inferences — no negative ground truths are recalled:

Metric Value
TN 0
FP 20
\[ \begin{align} \text{FPR} &= \frac{20}{0 + 20} \\[1em] &= 1.0 \end{align} \]

Multiple Classes#

So far, we have only looked at binary classification cases, but in multiclass or multi-label cases, FPR is computed per class. In the TP / FP / FN / TN guide, we went over multiple-class cases and how these metrics are computed. Once you have these four metrics computed per class, you can compute FPR for each class by treating each as a single-class problem.

Aggregating Per-class Metrics#

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

Limitations and Biases#

While FPR is a valuable metric for evaluating the performance of classification models, it does have limitations and potential biases that should be considered when interpreting results:

  1. Sensitivity to Class Imbalance: FPR is sensitive to class imbalance in dataset, just like specificity. If one class significantly outnumbers the other, a low FPR can be achieved simply by predicting the majority class most of the time. This can lead to a misleadingly low FPR score while neglecting the model's ability to correctly classify the minority class.

  2. Ignoring False Negatives: FPR focuses exclusively on the true negatives (correctly classified negative cases) and false positives (negative cases incorrectly classified as positive), but it doesn't account for false negatives (positive cases incorrectly classified as negative). Ignoring false negatives can be problematic in applications where missing positive cases is costly or has severe consequences.

  3. Incomplete Context: FPR alone does not provide a complete picture of a model's performance. It is often used in conjunction with other metrics like sensitivity (recall), precision, and F1-score to provide a more comprehensive assessment. Depending solely on FPR might hide issues related to other aspects of classification, such as the models' ability to identify true positives.

  4. Threshold Dependence: FPR is a binary metric that doesn't take into account the probability or confidence levels associated with predictions. Models with different probability distributions might achieve the same FPR score, but their operational characteristics can vary significantly. To address this limitation, consider using threshold-independent metrics like the area under the receiver operating characteristic curve (AUC-ROC) which can provide a more comprehensive understanding of model performance.