Add/logistic regression

docs/algorithms/machine-learning/supervised/classifications/index.md

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 # Classification Algorithms 🤖 
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docs/algorithms/machine-learning/supervised/regressions/index.md

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     <p style="font-size: 12px;">📅 2025-01-19 | ⏱️ 3 mins</p>
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+<!-- logistic regression -->
+<a href="logistic-regression" style="padding: 0 2px 0 16px; background-color: rgba(39, 39, 43, 0.4); border: 1px solid rgba(76, 76, 82, 0.4); border-radius: 10px; box-shadow: 0 4px 8px rgba(0,0,0,0.1); overflow: hidden; transition: transform 0.2s; display: flex; align-items: center;">
+  <img src="https://static.javatpoint.com/tutorial/machine-learning/images/logistic-regression-in-machine-learning.png" alt="" style="width: 300px; height: 150px; object-fit: cover; border-radius: 10px;" />
+  <div style="padding: 15px;">
+    <h2 style="margin: 0; font-size: 20px;">Logistic Regression</h2>
+    <p style="font-size: 16px;">Classifying data into discrete categories.</p>
+    <p style="font-size: 12px;">📅 2025-01-19 | ⏱️ 2 mins</p>
+  </div>
+</a>
 
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docs/algorithms/machine-learning/supervised/regressions/logistic-regression.md

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+# 🧮 Logistic Regression Algorithm
+
+<div align="center"> <img src="https://static.javatpoint.com/tutorial/machine-learning/images/logistic-regression-in-machine-learning.png" alt="Logistic Regression Poster" /> </div>
+
+## 🎯 Objective 
+Logistic Regression is a supervised learning algorithm used for classification tasks. It predicts the probability of a data point belonging to a particular class, mapping the input to a value between 0 and 1 using a logistic (sigmoid) function.
+
+## 📚 Prerequisites 
+- Basic understanding of Linear Algebra and Probability.
+- Familiarity with the concept of classification.
+- Libraries: NumPy, Pandas, Matplotlib, Scikit-learn.
+
+--- 
+
+## 🧩 Inputs 
+- *Input Dataset*: A structured dataset with features (independent variables) and corresponding labels (dependent variable).
+- The dependent variable should be categorical (binary or multiclass).
+- Example: A CSV file with columns like `age`, `income`, and `purchased` (label).
+
+
+## 📤 Outputs 
+- *Predicted Class*: The output is the probability of each data point belonging to a class.
+- *Binary Classification*: Outputs 0 or 1 (e.g., Yes or No).
+- *Multiclass Classification*: Outputs probabilities for multiple categories.
+
+---
+
+## 🏛️ Algorithm Architecture 
+
+### 1. Hypothesis Function
+The hypothesis function of Logistic Regression applies the sigmoid function:
+
+\[
+h_\theta(x) = \frac{1}{1 + e^{-\theta^T x}}
+\]
+
+---
+
+### 2. Cost Function
+The cost function used in Logistic Regression is the log-loss (or binary cross-entropy):
+
+\[
+J(\theta) = -\frac{1}{m} \sum_{i=1}^m \left[ y^{(i)} \log(h_\theta(x^{(i)})) + (1 - y^{(i)}) \log(1 - h_\theta(x^{(i)})) \right]
+\]
+
+---
+
+### 3. Gradient Descent
+The parameters of the logistic regression model are updated using the gradient descent algorithm:
+
+\[
+\theta := \theta - \alpha \frac{\partial J(\theta)}{\partial \theta}
+\]
+
+---
+
+## 🏋️‍♂️ Training Process 
+- **Model**: Logistic Regression model from sklearn.
+
+- **Validation Strategy**: A separate portion of the dataset can be reserved for validation (e.g., 20%), but this is not explicitly implemented in the current code.
+
+- **Training Data**: The model is trained on the entire provided dataset.
+
+
+
+---
+
+## 📊 Evaluation Metrics 
+- Accuracy is used to evaluate the classification performance of the model.
+
+\[
+\text{Accuracy} = \frac{TP + TN}{TP + TN + FP + FN}
+\]
+
+Where:
+
+- **TP**: True Positives
+- **TN**: True Negatives
+- **FP**: False Positives
+- **FN**: False Negatives
+
+---
+
+## 💻 Code Implementation 
+
+```python
+import numpy as np
+from sklearn.linear_model import LogisticRegression
+from sklearn.metrics import accuracy_score
+
+# Generate Example Dataset
+np.random.seed(42)
+X = np.random.rand(100, 2)  # Features
+y = (X[:, 0] + X[:, 1] > 1).astype(int)  # Labels: 0 or 1 based on sum of features
+
+# Train Logistic Regression Model
+model = LogisticRegression()
+model.fit(X, y)
+
+# Predictions
+y_pred = model.predict(X)
+accuracy = accuracy_score(y, y_pred)
+
+# Output Accuracy
+print("Accuracy:", accuracy)
+```
+
+## 🔍 Scratch Code Explanation
+1. **Dataset Generation**:
+
+    - A random dataset with 100 samples and 2 features is created.
+
+    - Labels (`y`) are binary, determined by whether the sum of feature values is greater than 1.
+
+2. **Model Training**:
+    - The `LogisticRegression` model from `sklearn` is initialized and trained on the dataset using the fit method.
+
+3. **Predictions**:
+
+    - The model predicts the labels for the input data (`X`) using the `predict` method.
+
+    - The `accuracy_score` function evaluates the accuracy of the predictions.
+
+4. **Output**:
+
+    - The calculated accuracy is printed to the console.
+
+
+### 🛠️ Example Usage: Predicting Customer Retention
+
+```python
+# Example Data: Features (e.g., hours spent on platform, number of purchases)
+X = np.array([[5.0, 20.0], [2.0, 10.0], [8.0, 50.0], [1.0, 5.0]])  # Features
+y = np.array([1, 0, 1, 0])  # Labels: 1 (retained), 0 (not retained)
+
+# Train Logistic Regression Model
+model = LogisticRegression()
+model.fit(X, y)
+
+# Predict Retention for New Customers
+X_new = np.array([[3.0, 15.0], [7.0, 30.0]])
+y_pred = model.predict(X_new)
+
+print("Predicted Retention (1 = Retained, 0 = Not Retained):", y_pred)
+```
+
+- This demonstrates how Logistic Regression can be applied to predict customer retention based on behavioral data, showcasing its practicality for real-world binary classification tasks.
+
+
+
+--- 
+
+## 🌟 Advantages 
+   -  Simple and efficient for binary classification problems.
+
+    - Outputs probabilities, allowing flexibility in decision thresholds.
+
+    - Easily extendable to multiclass classification using the one-vs-rest (OvR) or multinomial approach.
+
+## ⚠️ Limitations 
+
+-  Assumes a linear relationship between features and log-odds of the target.
+
+- Not effective when features are highly correlated or when there is a non-linear relationship.
+
+## 🚀 Application 
+
+=== "Application 1"
+    **Medical Diagnosis**: Predicting the likelihood of a disease based on patient features.
+
+
+=== "Application 2"
+    **Marketing**: Determining whether a customer will purchase a product based on demographic and behavioral data.
+