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unit 2/03_Logistic_Regression.md

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+# Logistic Regression
+
+Logistic Regression is used for **classification** problems (predicting categories), even though it has "Regression" in its name.
+
+## Odds vs Probability
+
+### Probability
+- The chance of an event happening out of **all** possibilities.
+- **Formula**: `Probability = (Events in favour) / (Total observations)`
+- Range: 0 to 1.
+
+### Odds
+- The ratio of events **happening** to events **not happening**.
+- **Formula**: `Odds = (Events in favour) / (Events NOT in favour)`
+
+### Log of Odds (Logit)
+- We use the **Log of Odds** because odds can vary widely in magnitude.
+- Taking the log keeps the magnitude consistent across classes.
+- This is the function used in Logistic Regression.
+
+## The Sigmoid Function
+Linear regression fits a straight line, which doesn't work well for classification (where we want output between 0 and 1).
+Logistic regression uses an **S-shaped curve** called the **Sigmoid Function**.
+
+- **Formula**: `S(z) = 1 / (1 + e^-z)`
+- **Output**: Always between **0 and 1**.
+- **Usage**:
+  - If output > Threshold (e.g., 0.5) -> Classify as **Positive** (e.g., Presence of fish).
+  - If output < Threshold -> Classify as **Negative** (e.g., Absence of fish).
+
+## Assumptions of Logistic Regression
+1. **Independence of Error**: Sample outcomes are separate (no duplicates).
+2. **Linearity in the Logit**: Relationship between independent variables and log-odds is linear.
+3. **Absence of Multicollinearity**: Independent variables should not be highly correlated with each other.
+4. **No Strong Outliers**: Extreme values should not heavily influence the model.