Generally speaking, kd-tree is a binary tree which nodes represent vector sub-spaces. It's especially useful in high-dimensional data programming, especially laser point cloud programming. It's very important because it highly improves the prediction time complexity of knn.
Every node records a data structure, including a vector sample, a segmentation line which means we make a division along $r^{th}$ dimension with a fixed value, and two pointers pointing to its son nodes.
The basic intuition of k-dimensional tree, in my belief, is a famous algorithm which is used to detect the nearest two vectors in a vector space. It continuously divide the vector space and solve the problem using divide-and-conquer method. We try to use similar way to make the searching through the samples and finding k nearest samples process faster. 
For building k-dimensional tree, we calculate the samples' mid-value and variance for every feature. We recursively selected the feature with largest variance and divide the set to 2 sub-set according to this feature's mid-value.
For finding nearest neighbors, we just need to find the matching branch now. And we only need to do comparison along the branch path.
The picture provided is a sketch map for an easy k-dimensional tree.

k-dimensional tree

KNN can be useful in solving problems that have solutions that depend on identifying similar objects.
1. Recommender Systems
For example: Recommending products on Amazon, articles on Medium, movies on Netflix, or videos on YouTube.
2. Concept Search
This is used to search for semantically similar documents (i.e., documents containing similar topics). We could find it in many e-Discovery software packages, which are used for helping companies find all of the e-mails, contracts, etc.
3. Herta Security
They then use k-NN to identify a person by compare the face to their watchlist. 

University of Michigan - Ann Arbor

K-nearest neighbors is a non-parametric statistical method, which predicts based on the *k* nearest neighbors. k-NN classifiers are an example of what's called instance based or memory based supervised learning.

It can be applied to both classification and regression problems.
- When applied to classification problems, the prediction result would be the most common one among the *k* nearest training observations.
- When applied to regression problems, the prediction result would be the average of the *k* nearest training observations.

Some key notes about this method:
- It makes few assumptions about structure of data
- It gives potentially accurate but sometimes unstable predictions 
- It is sensitive to small changes in training data



K-nearest Neighbors (k-NN)

Statquest with Josh Starmer.(2017, June 26). Statquest: K-nearest neighbors, Clearly Explained [Video]. Youtube.  [www.youtube.com/watch?v=HVXime0nQeI](https://www.youtube.com/watch?v=HVXime0nQeI) 

Reference Video: K-Nearest Neighbors

Dhiraj, K. (2019, August 29). Difference between K-Nearest Neighbor(K-NN) and K-Means Clustering. Medium. [medium.com/@dhiraj8899/difference-between-k-nearest-neighbor-k-nn-and-k-means-clustering-d9a44859182f](https://medium.com/@dhiraj8899/difference-between-k-nearest-neighbor-k-nn-and-k-means-clustering-d9a44859182f) 

Medium: Difference between K-Means and KNN

This reference gets into mathmatical ways how KNN and K-Nearest Neighbors are different concepts

Umamaheswaran, Venkatesh. (2018, November 12). Comprehending K-means and KNN Algorithms. Medium. Retrieved From [becominghuman.ai/comprehending-k-means-and-knn-algorithms-c791be90883d](https://becominghuman.ai/comprehending-k-means-and-knn-algorithms-c791be90883d) 

Math/Python Explanation: Difference Between K-Means and KNN

A good source that talks about KNN, it's various uses, its algorithm, and examples.

Harrison, O. (2019, July 14). Machine Learning Basics with the K-Nearest Neighbors Algorithm. Retrieved March 12, 2020, from [towardsdatascience.com/machine-learning-basics-with-the-k-nearest-neighbors-algorithm-6a6e71d01761](https://towardsdatascience.com/machine-learning-basics-with-the-k-nearest-neighbors-algorithm-6a6e71d01761) 

Machine Learning Basics with KNN Algorithm

When using KNN for regression problems, the model is based off of ‘feature similarity’ which means that the output value of an observation you are trying to find is based on how closely it resembles the points in the training data set. With regression using KNN, this similarity is based on the average distance, typically using Euclidean distance, between the chosen observation and those around it.

KNN Regression

Singh, A. (2019, August 9). A Practical Introduction to K-Nearest Neighbor for Regression. Retrieved March 18, 2020, from https://www.analyticsvidhya.com/blog/2018/08/k-nearest-neighbor-introduction-regression-python/

A Practical Introduction to K-Nearest Neighbors Algorithm for Regression (Reference)

KNN in practice

The UM online course: Applied Machine Learning in Python->Week 2-> K-Nearest Neighbors: Classification and Regression. The link is: https://www.online.umich.edu/courses/applied-machine-learning-in-python/



Reference video: K-Nearest Neighbors: Classification and Regression

Scikit-learn enables the creation of classifier objects. For example, 
```python
from sklearn.neighbors import KNeighborsClassifier
knn = KNeighborsClassifier(n_neighbors = 5, weights = 'distance')
```
In this example, the classifier object (named knn) will consider 5 neighbors in its kneighbors queries.   Setting the weights parameter to 'distance' means that closer neighbors will be more influential than those farther away  (in terms of detemrining which category the "target" point belongs to).

sklearn.neighbors.KNeighborsClassifier

K-nearest neighbors can be used to classify data. Given a training set X_train with labels y_train, and given a new instance x_test to be classified:

- Find the k most similar instances (X_NN) to x_test that are in X_train.
- Get the labels y_NN for the instances in X_NN.
- Predict the label for x_test by combining the labels y_NN. The combination can be done through different methods, e.g., a simple majority vote can find the label that is the most common among the k most similar examples.

For example, if K = 1, then the first closest neighbor will be used to classify the unknown data point. If K = 14, then the 14 nearest neighbors will be used. If the data point falls between multiple clusters, then you choose the category that holds the most.

Classification Algorithm of K-Nearest Neighbors

Advantages
- It is easy and simple to implement
- Since it is non-parametric, you don’t have to make any assumptions about $f$
- A model does not need to be created
- It can be used for various statistical methods (classification, regression, etc.)

Disadvantages
- As the number of predictors increase, KNN can get very slow and at some point, not usable anymore.


K-Nearest Neighbors Advantages and Disadvantages

Used the data for classes 0, 1, and 2 plotted in the image.

What class would a KNeighborsClassifer classify the new point as for k = 1 and k = 3?

Which of the following is true for the nearest neighbor classifier?

The $$1$$-nearest neighbor algorithm is the simplest case of the $$k$$-nearest neighbor approach, where $$k=1$$. At training time, the learner memorizes the dataset, and at prediction time, it assigns a new data point the label of its single closest neighbor based on a chosen distance function $$d$$. Because it memorizes the data, the algorithm always achieves a training error of zero by perfectly interpolating the training data. Despite this, it can still generalize; under mild conditions, the $$1$$-nearest neighbor algorithm is a consistent estimator, meaning it eventually converges to the optimal predictor.

1-Nearest Neighbor Algorithm

In nonparametric methods like the $$k$$-nearest neighbor algorithm, a distance function, denoted as $$d$$, must be specified to measure the similarity between data points. Equivalently, this involves defining a vector-valued basis function $$\phi(\mathbf{x})$$ for featurizing the data. The choice of distance metric is critical because it encodes different inductive biases and represents specific assumptions about the underlying data patterns. With a finite amount of available data, different distance functions will yield different predictors depending on how compatible these assumptions are with the observed data.

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