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This image shows the mathematical formulation of the least squares cost function with weights. The least squares problem is often used for optimization, particularly in linear regression and fitting models to data. 

The formula is:

\[
\mathcal{L}_{\text{least squares}} = \sum_{i} w_i \| d_i \|^2
\]

Where:
- \( w_i \) are the weights associated with each term.
- \( d_i \) represents the differences or residuals between the observed and predicted values.
- The double vertical bars \( \| d_i \| \) denote the magnitude or norm of the difference \( d_i \).
- The sum is over all terms \( i \), which could represent different data points or residuals.

This expression aims to minimize the weighted squared errors, where different data points are given different levels of importance based on the weights \( w_i \).

Would you like a deeper explanation of any part of this? Or would you like to know how to apply this in a specific context?

Here are some related questions:
1. How does weighting impact the least squares method?
2. What is the difference between standard least squares and weighted least squares?
3. How can the least squares cost function be applied to linear regression?
4. What is the interpretation of the residuals \( d_i \) in this formula?
5. How do you compute the least squares solution in practice?

**Tip:** In weighted least squares, larger weights give more importance to those data points in the optimization process.

What does the least squares formula with weights represent?

Explore the least squares method with weights, an important tool in optimization and linear regression. The formula involves minimizing weighted squared errors where different data points have varying levels of importance based on their associated weights.

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