Q15Machine Learning

When a Logistic Regression model predicts cancer, it does not output a binary Yes/No. It outputs a continuous mathematical probability from 0.0 to 1.0 (e.g., 0.85). The engineer must artificially select a Threshold (e.g., 0.5). If Probability > 0.5, predict Yes. Changing this threshold violently alters the Precision and Recall. The Receiver Operating Characteristic (ROC) Curve is a graphical architecture designed to evaluate the absolute quality of the model across EVERY SINGLE mathematically possible threshold simultaneously.

Constructing the ROC Curve

• Y-Axis: True Positive Rate (Sensitivity / Recall) $\Rightarrow \frac{TP}{TP+FN}$.
• X-Axis: False Positive Rate (1 - Specificity) $\Rightarrow \frac{FP}{FP+TN}$.
• Execution: The algorithm mathematically iterates the threshold from 0.0 to 1.0. At each threshold, it calculates the TPR and FPR and plots a single point on the 2D graph. A perfect model shoots violently straight up the Y-axis to (0, 1). A useless model (random guessing) plots a straight diagonal 45-degree line from (0, 0) to (1, 1).

Area Under the Curve (AUC)

Looking at a graph is not mathematically precise. The AUC (Area Under the Curve) is the exact mathematical integral (the physical 2D area) underneath the ROC curve.

• An AUC of exactly 1.0 indicates absolute mathematical perfection (100% TPR with 0% FPR).
• An AUC of 0.5 mathematically proves the model is garbage (no better than flipping a coin).
• The AUC represents the strict probability that the model will score a randomly chosen positive instance mathematically higher than a randomly chosen negative instance.