Logistic regression is a fundamental building block in machine learning, yet many practitioners treat it as a black box. Let's break it down and implement it from scratch.

What is Logistic Regression?
At its core, logistic regression is a statistical model that uses a logistic function to model a binary dependent variable. Despite its name, it's used for classification rather than regression.

Implementation from Scratch:
```python
import numpy as np

class LogisticRegression:
    def __init__(self, learning_rate=0.01, iterations=1000):
        self.lr = learning_rate
        self.iterations = iterations
        self.weights = None
        self.bias = None

    def sigmoid(self, z):
        return 1 / (1 + np.exp(-z))

    def fit(self, X, y):
        n_samples, n_features = X.shape
        self.weights = np.zeros(n_features)
        self.bias = 0

        for _ in range(self.iterations):
            linear_pred = np.dot(X, self.weights) + self.bias
            predictions = self.sigmoid(linear_pred)

            # Gradient descent
            dw = (1/n_samples) * np.dot(X.T, (predictions - y))
            db = (1/n_samples) * np.sum(predictions - y)

            self.weights -= self.lr * dw
            self.bias -= self.lr * db
```

Understanding the Math:
1. The sigmoid function transforms our input to a probability between 0 and 1
2. We use gradient descent to minimize the log loss
3. The weights represent the importance of each feature

This implementation helps understand the inner workings of logistic regression and builds a foundation for more complex ML concepts.

An exercise in gaining an in-depth understanding of logistic regression.

DIY AI & ML: Logistic Regression

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Architecture Best Practices:

1. File-based Routing
```typescript
// pages/blog/[slug].tsx
export default function BlogPost({ post }) {
  return (
    <article>
      <h1>{post.title}</h1>
      <div>{post.content}</div>
    </article>
  )
}

export async function getStaticPaths() {
  // Generate paths at build time
}

export async function getStaticProps({ params }) {
  // Fetch data at build time
}
```

2. API Route Optimization
- Implement API middleware for authentication
- Use edge functions for global performance
- Cache responses when possible

3. Database Strategies
- Implement connection pooling
- Use proper indexing
- Consider read replicas for scaling

4. Performance Optimization
- Implement image optimization
- Use incremental static regeneration
- Leverage edge caching

5. State Management
- Choose appropriate solutions (Redux, Zustand, Jotai)
- Implement proper data fetching strategies
- Use SWR or React Query for data fetching

Following these practices ensures your Next.js application remains performant as it scales.

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