Logistic Regression, ROC-AUC & Classification Metrics
Model binary class probabilities via log-odds, Maximum Likelihood Estimation, and evaluate imbalanced datasets.
1. The Logit Function & Sigmoid Activation
Logistic Regression models the log-odds of a positive class as a linear combination of input features:
\[ \ln\left(\frac{p}{1 - p}\right) = \beta^T \mathbf{x} \implies p = \sigma(\beta^T \mathbf{x}) = \frac{1}{1 + e^{-\beta^T \mathbf{x}}} \]
2. Confusion Matrix Metrics for Imbalanced Datasets
- Precision: \(\frac{\text{TP}}{\text{TP} + \text{FP}}\) (Quality of positive predictions).
- Recall (Sensitivity): \(\frac{\text{TP}}{\text{TP} + \text{FN}}\) (Completeness of positive predictions).
- F1-Score: Harmonic mean: \(2 \times \frac{\text{Precision} \times \text{Recall}}{\text{Precision} + \text{Recall}}\).
🎯 Module Mastery Certification Quiz
+100 XPFor a medical disease detection model where missing a sick patient (False Negative) is catastrophic, which metric should be maximized?