k-Nearest Neighbors

1. Introduction to k-Nearest Neighbors

The k-Nearest Neighbors (k-NN) algorithm is arguably the simplest machine learning method. It is a lazy learning algorithm, meaning it does not explicitly learn a model but stores the training dataset and makes predictions based on it when queried.

For classification or regression, the algorithm examines the k closest points in the training data to the query point.
The "closeness" or distance is usually measured by a distance metric like Euclidean distance.
The predicted output depends on the majority label in classification or average value in regression of the k neighbors.

2. How k-NN Works

Training phase: Simply store all the training samples (features and labels)—no explicit model building.
Prediction phase:

1. For a new input sample, compute the distance to all points in the training dataset.

2. Identify the k closest neighbors.

3. Classification: Use majority voting among these neighbors to assign a class label.

4. Regression: Average the target values of these neighbors to predict the output.

Example of 1-nearest neighbor: The prediction is the label of the single closest training point.

3. Role of k (Number of Neighbors)

The parameter k controls the smoothness of the model.
k=1: Predictions perfectly fit the training data but can be noisy and unsteady (i.e., overfitting).
k increasing: Produces smoother predictions, less sensitive to noise but may underfit (fail to capture finer patterns),.
Commonly used values are small odd numbers like 3 or 5 to avoid ties.

4. Distance Metrics

The choice of distance metric influences performance.
Euclidean distance is the default and works well in many cases.
Other metrics include Manhattan distance, Minkowski distance, or domain-specific similarity measures.
Selecting the correct distance metric depends on the problem and data characteristics.

5. Strengths and Weaknesses of k-NN

Strengths

Simple to implement and understand.
No training time since model retention is just the dataset.
Naturally handles multi-class classification.
Makes no parametric assumptions about data distribution.

Weaknesses

Computationally expensive at prediction time because distances are computed to all training samples.
Sensitive to irrelevant features and the scaling of input data.
Performance can degrade with high-dimensional data ("curse of dimensionality").
Choosing the right k and distance metric is crucial.

6. k-NN for Classification Example

In its simplest form, considering just one neighbor (k=1), the predicted class for a new sample is the class of the closest data point in the training set. When considering more neighbors, the majority vote among the neighbors' classes determines the prediction.

Visualizations (like in Figure 2-4) show how the k-NN classifier assigns labels based on proximity to known labeled points.

7. k-NN for Regression

Instead of voting for a label, k-NN regression predicts values by averaging the output values of the k nearest points. This can smooth noisy data but is still sensitive to outliers and requires careful choice of k.

8. Feature Scaling

Because distances are involved, feature scaling (standardization or normalization) is important to ensure no single feature dominates due to scale differences.
For example, differences in units like kilometers vs. meters could skew neighbor calculations.

9. Practical Recommendations

Start with k=3 or 5.
Use cross-validation to select the best k.
Scale features appropriately before applying k-NN.
Try different distance metrics if necessary.
For large datasets, consider approximate nearest neighbor methods or dimensionality reduction to speed up predictions.

10. Summary

k-NN’s simplicity makes it a good baseline model.
It directly models local relationships in data.
The choice of k controls the balance of bias and variance.
Proper data preprocessing and parameter tuning are essential for good performance.

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