Add Weighted Euclidean Distance Metric

### Describe the workflow you want to enable

The workflow I want to enable is the ability for users to easily incorporate feature importance into distance-based algorithms like clustering (e.g., KMeans) and nearest neighbors (e.g., KNeighborsClassifier). Currently, scikit-learn allows users to define custom distance metrics, but there is no built-in support for weighted distance metrics, which are essential when certain features are more important than others.

Example Workflow:
A user has a dataset where some features are more relevant than others (e.g., in customer segmentation, age and income might be more important than the number of children).

The user wants to use a clustering algorithm like KMeans or a nearest neighbors algorithm like KNeighborsClassifier but needs to account for the varying importance of features.

The user specifies a vector of weights corresponding to the importance of each feature.

The algorithm uses the weighted Euclidean distance metric to compute distances, ensuring that more important features have a greater influence on the results.

### Describe your proposed solution

I propose adding a Weighted Euclidean Distance Metric to scikit-learn as a built-in distance metric. This will allow users to specify feature weights directly, making it easier to incorporate feature importance into distance-based algorithms.
 **Key Components of the Solution:**

1. New Class: 

- Add a WeightedEuclideanDistance class to the sklearn.metrics.pairwise module.
This class will accept a vector of weights during initialization.
- It will compute the weighted Euclidean distance between two points using the formula:
**d(x, y) = sqrt( summation from i = 1 to n of [ w_i * (x_i - y_i) squared ] )**
where ​wi are the user-defined weights.



### Describe alternatives you've considered, if relevant

_No response_

### Additional context

Why This Feature is Needed:

- Feature Importance: In many real-world datasets, not all features are equally important. For example, in medical diagnosis, certain biomarkers might be more relevant than others. A weighted distance metric allows users to account for this.
- 
- Ease of Use: While scikit-learn allows users to define custom distance metrics, this requires writing additional code and can be error-prone. A built-in weighted distance metric would simplify the process and make it more accessible to users.
- 
- Alignment with scikit-learn’s Design: Scikit-learn emphasizes ease of use and consistency. Adding a weighted distance metric aligns with this philosophy by providing a simple, consistent way to incorporate feature importance into distance-based algorithms.

Existing Alternatives:

- Custom Distance Functions: Users can define custom distance functions using the metric parameter. However, this requires additional coding and lacks the convenience of a built-in solution.
- 
- Feature Weighting via Preprocessing: Users can scale features by their importance before applying distance-based algorithms. However, this approach is less intuitive and can be cumbersome.

Potential Impact:

- This feature would benefit users in domains like bioinformatics, finance, and customer analytics, where feature importance is critical.
- 
- It would also make scikit-learn more competitive with other machine learning libraries that offer weighted distance metrics.

COMMENT:
> Do you or someone else have a bit of an overview that can be shared here that illustrates how popular and/or widespread the use of weighted euclidean distance is?
> 
> For scikit-learn we try to include things that are very widely used and leave things which are only "widely used" for third-party packages or users to implement themselves. The reasoning behind this is that everything we add needs to be maintained and adds to the problem of "can't see the forest for the trees" when people look at the scikit-learn documentation.

**"Thank you for your response! Weighted Euclidean distance is widely used in various domains, and its integration into scikit-learn would be highly beneficial "**
Research Applications:

- **Healthcare Decision-Making:** A study titled "A Weighted Euclidean Distance - Statistical Variance Procedure based Approach for Improving The Healthcare Decision Making System In Yemen" discusses the application of weighted Euclidean distance in healthcare to enhance decision-making processes
SOURCE : https://www.researchgate.net/publication/331160671_A_Weighted_Euclidean_Distance_-_Statistical_Variance_Procedure_based_Approach_for_Improving_The_Healthcare_Decision_Making_System_In_Yemen

**Safety Risk Management in Railway Projects:** The paper "The Use of Weighted Euclidean Distance to Provide Assistance in the Selection of Safety Risk Prevention and Control Strategies for Major Railway Projects" illustrates how weighted Euclidean distance aids in selecting appropriate safety risk prevention strategies
SOURCE : https://www.mdpi.com/2075-5309/14/5/1270?utm_source=chatgpt.com

**Advantages of Including It in Scikit-Learn:**

- Ease of Use: A built-in solution eliminates the need for users to manually preprocess data or define custom functions.

- Consistency: Aligns with scikit-learn’s philosophy of providing intuitive tools for common ML tasks.

- Competitive Edge: Other ML libraries offer weighted distance options, and adding this feature would enhance scikit-learn’s capabilities.

Integrating weighted Euclidean distance into scikit-learn would not only simplify implementation for users but also broaden the library's applicability across various fields.

Labels: New Feature, Needs Decision - Include Feature