Explain the ROC curve and AUC.

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The ROC curve (Receiver Operating Characteristic curve) is a graphical tool used to evaluate the performance of a binary classification model by plotting the trade-off between its True Positive Rate (TPR) and False Positive Rate (FPR) at various threshold settings.

Key Components:

• True Positive Rate (TPR) / Sensitivity / Recall:
  The proportion of actual positives correctly identified.

  $$\text{TPR} = \frac{\text{True Positives}}{\text{True Positives} + \text{False Negatives}}$$

• False Positive Rate (FPR):
  The proportion of actual negatives incorrectly classified as positive.

  $$\text{FPR} = \frac{\text{False Positives}}{\text{False Positives} + \text{True Negatives}}$$

How It Works:

• The model outputs probabilities or scores.

• By varying the classification threshold from 0 to 1, you get different TPR and FPR values.

• Plotting TPR (y-axis) against FPR (x-axis) for these thresholds creates the ROC curve.

Interpretation:

• A curve closer to the top-left corner indicates better performance (high TPR, low FPR).

• The diagonal line from bottom-left to top-right represents a random classifier (no skill).

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AUC (Area Under the Curve):

• AUC measures the area under the ROC curve, ranging from 0 to 1.

• It quantifies the model’s ability to distinguish between classes.

• AUC = 1 means perfect classification.

• AUC = 0.5 means no better than random guessing.

Why Use ROC and AUC?

• They provide a threshold-independent evaluation.

• Useful when classes are imbalanced.

• Help compare different models easily.

In summary, ROC curves visualize a model’s diagnostic ability, while AUC provides a single scalar metric summarizing its overall performance.

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