Computer Science & Data Science · Classification and evaluation

ROC-AUC (area under the ROC curve)

ROC-AUC summarizes binary score discrimination across decision thresholds as an area.

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Formula

AUC = \int_{0}^{1} TPR (FPR) d FPR

dimensionless, 0 to 1; 0.5 is expected for random ranking

Variables & units

Variables & units
Symbol	Meaning	Unit
$AUC$	Area under the ROC curve	dimensionless
$TPR$	True positive rate or recall	dimensionless
$FPR$	False positive rate	dimensionless

LaTeX

\mathrm{AUC}=\int_0^1 TPR(FPR)\,dFPR

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Derivation & background

The ROC curve plots TPR against FPR as the classification threshold varies. For continuous scores, AUC is the probability that a randomly selected positive case ranks above a randomly selected negative one; ties usually receive half credit. It assesses ranking, not calibration or one operating threshold. With severe class imbalance, precision-recall analysis can provide useful additional context.

Exam blueprint

Validity range

Derivation steps

ROC-AUC summarizes binary score discrimination across decision thresholds as an area.

1Read the relation from its definition: \mathrm{AUC}=\int_0^1 TPR(FPR)\,dFPR.
2Identify the input, requested output and assumptions of the relation.
3For two positives scored 0.9 and 0.7 and negatives scored 0.8 and 0.4, three of four positive-negative pairs rank correctly, so AUC=3/4=0.75.

Rearrangements

Defining relation

\mathrm{AUC}=\int_0^1 TPR(FPR)\,dFPR

The equation states the defining relation within its stated domain.

Task variant

Use the example for ROC-AUC (area under the ROC curve) to explain which inputs determine the requested output.

The example connects concrete inputs to its result.
ROC-AUC summarizes binary score discrimination across decision thresholds as an area.
The relation applies only under its stated assumptions.

Positive scores: 0.9, 0.7, 0.4. Negative scores: 0.8, 0.5, 0.3. What is ROC-AUC with no ties?

The scores rank 6 of 9 positive-negative pairs correctly: 3+2+1.
Therefore AUC=6/9≈0.667.

Common mistakes

Applying the relation to a different case without checking its stated assumptions.

Check the validity range before substituting values: The ROC curve plots TPR against FPR as the classification threshold varies. For continuous scores, AUC is the probability that a randomly selected positive case ranks above a randomly selected.

Exam context

Compare classifiers without fixing one threshold and evaluate ranking separation.

Formula cluster

Classification and evaluation

ROC-AUC summarizes binary score discrimination across decision thresholds as an area.

F₁ = 2TP / (2TP + FP + FN)

F₁ score

Recall = TP/(TP+FN)

Recall (sensitivity)

TNR = TN/(TN+FP)

Specificity (true negative rate)

Precision = TP/(TP+FP)

Precision (positive predictive value)

Worked example

For two positives scored 0.9 and 0.7 and negatives scored 0.8 and 0.4, three of four positive-negative pairs rank correctly, so AUC=3/4=0.75.

Applications

Compare classifiers without fixing one threshold and evaluate ranking separation.

How do you calculate with ROC-AUC (area under the ROC curve)?

Here is how to work through a typical ROC-AUC (area under the ROC curve) (AUC = ∫ TPR dFPR) task step by step:

1
Task
Use the example for ROC-AUC (area under the ROC curve) to explain which inputs determine the requested output.
Solution path
1. The example connects concrete inputs to its result.
2. ROC-AUC summarizes binary score discrimination across decision thresholds as an area.
3. The relation applies only under its stated assumptions.
2
Task
Positive scores: 0.9, 0.7, 0.4. Negative scores: 0.8, 0.5, 0.3. What is ROC-AUC with no ties?
Solution path
1. The scores rank 6 of 9 positive-negative pairs correctly: 3+2+1.
2. Therefore AUC=6/9≈0.667.

Frequently asked questions about ROC-AUC (area under the ROC curve)

What does ROC-AUC (area under the ROC curve) describe?+

The relation \mathrm{AUC}=\int_0^1 TPR(FPR)\,dFPR summarizes ROC-AUC summarizes binary score discrimination across decision thresholds as an area. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example For two positives scored 0.9 and 0.7 and negatives scored 0.8 and 0.4, three of four positive-negative pairs rank correctly, so AUC=3/4=0.75., symbols are connected to concrete values and the result is interpreted in context. Validity limit: The ROC curve plots TPR against FPR as the classification threshold varies. For continuous scores, AUC is the probability that a randomly selected positive case ranks above a randomly selected Source: scikit-learn API Reference, roc_auc_score.

How should I work through a calculation?+

First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example For two positives scored 0.9 and 0.7 and negatives scored 0.8 and 0.4, three of four positive-negative pairs rank correctly, so AUC=3/4=0.75. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: The ROC curve plots TPR against FPR as the classification threshold varies. For continuous scores, AUC is the probability that a randomly selected positive case ranks above a randomly selected Source: scikit-learn API Reference, roc_auc_score.

Which assumption must I keep in mind?+

The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: The ROC curve plots TPR against FPR as the classification threshold varies. For continuous scores, AUC is the probability that a randomly selected positive case ranks above a randomly selected Source: scikit-learn API Reference, roc_auc_score.

Which units and quantities are involved?+

The stated units are: dimensionless, 0 to 1; 0.5 is expected for random ranking. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: The ROC curve plots TPR against FPR as the classification threshold varies. For continuous scores, AUC is the probability that a randomly selected positive case ranks above a randomly selected Source: scikit-learn API Reference, roc_auc_score.

When is comparison with f1-score and recall-sensitivitaet useful?+

Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source scikit-learn API Reference, roc_auc_score. supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: The ROC curve plots TPR against FPR as the classification threshold varies. For continuous scores, AUC is the probability that a randomly selected positive case ranks above a randomly selected Source: scikit-learn API Reference, roc_auc_score.

Quanta exam set

Curated exam set for "ROC-AUC (area under the ROC curve)":

Question (front)

Which formula describes ROC-AUC (area under the ROC curve)?

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Question (front)

How do you rearrange ROC-AUC (area under the ROC curve) for Defining relation?

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Which common mistake happens with ROC-AUC (area under the ROC curve)?

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]scikit-learn API Reference, roc_auc_score.

More formulas: Data analysis and machine learning

H(p,q)=-\sum_{i=1}^{K}p_i\ln q_i

Cross-entropy (classification loss)

\theta_{t+1}=\theta_t-\eta\nabla J(\theta_t)

Gradient descent parameter update

\mathrm{BA}=\frac{1}{2}\left(\frac{TP}{TP+FN}+\frac{TN}{TN+FP}\right)

Balanced accuracy

\mathrm{MCC}=\frac{TP\,TN-FP\,FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}

Matthews correlation coefficient (MCC)

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Quanta Study editorial team

Symbol

Meaning

Unit

AUC

Area under the ROC curve

dimensionless

TPR

True positive rate or recall

dimensionless

FPR

False positive rate

dimensionless

Derivation & background

Exam blueprint

Validity range

Derivation steps

ROC-AUC summarizes binary score discrimination across decision thresholds as an area.

1Read the relation from its definition: \mathrm{AUC}=\int_0^1 TPR(FPR)\,dFPR.
2Identify the input, requested output and assumptions of the relation.
3For two positives scored 0.9 and 0.7 and negatives scored 0.8 and 0.4, three of four positive-negative pairs rank correctly, so AUC=3/4=0.75.

How do you calculate with ROC-AUC (area under the ROC curve)?

Here is how to work through a typical ROC-AUC (area under the ROC curve) (AUC = ∫ TPR dFPR) task step by step:

Task

Use the example for ROC-AUC (area under the ROC curve) to explain which inputs determine the requested output.

Solution path

The example connects concrete inputs to its result.
ROC-AUC summarizes binary score discrimination across decision thresholds as an area.
The relation applies only under its stated assumptions.

Task

Positive scores: 0.9, 0.7, 0.4. Negative scores: 0.8, 0.5, 0.3. What is ROC-AUC with no ties?

Solution path

The scores rank 6 of 9 positive-negative pairs correctly: 3+2+1.
Therefore AUC=6/9≈0.667.

Frequently asked questions about ROC-AUC (area under the ROC curve)

What does ROC-AUC (area under the ROC curve) describe?+

How should I work through a calculation?+

Which assumption must I keep in mind?+

Which units and quantities are involved?+

When is comparison with f1-score and recall-sensitivitaet useful?+

Retain ROC-AUC (area under the ROC curve) for exams

Create a curated FSRS exam set for AUC = ∫ TPR dFPR: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential