Non-parametric classification method
In statistics, the k"-nearest neighbors algorithm (k"-NN) is a non-parametric supervised learning method. It was first developed by Evelyn Fix and Joseph Hodges in 1951, and later expanded by Thomas Cover. 
Most often, it is used for classification, as a "k"-NN classifier, the output of which is a class membership. An object is classified by a plurality vote of its neighbors, with the object being assigned to the class most common among its "k" nearest neighbors ("k" is a positive integer, typically small). If "k" = 1, then the object is simply assigned to the class of that single nearest neighbor.
The "k"-NN algorithm can also be generalized for regression. In "k-NN regression", also known as "nearest neighbor smoothing", the output is the property value for the object. This value is the average of the values of "k" nearest neighbors. If "k" = 1, then the output is simply assigned to the value of that single nearest neighbor, also known as "nearest neighbor interpolation".
For both classification and regression, a useful technique can be to assign weights to the contributions of the neighbors, so that nearer neighbors contribute more to the average than distant ones. For example, a common weighting scheme consists of giving each neighbor a weight of 1/"d", where "d" is the distance to the neighbor.
The input consists of the "k" closest training examples in a data set. 
The neighbors are taken from a set of objects for which the class (for "k"-NN classification) or the object property value (for "k"-NN regression) is known. This can be thought of as the training set for the algorithm, though no explicit training step is required.
A peculiarity (sometimes even a disadvantage) of the "k"-NN algorithm is its sensitivity to the local structure of the data.
In "k"-NN classification the function is only approximated locally and all computation is deferred until function evaluation. Since this algorithm relies on distance, if the features represent different physical units or come in vastly different scales, then feature-wise normalizing of the training data can greatly improve its accuracy.
Statistical setting.
Suppose we have pairs formula_1 taking values in formula_2, where Y is the class label of X, so that formula_3 for formula_4 (and probability distributions formula_5). Given some norm formula_6 on formula_7 and a point formula_8, let formula_9 be a reordering of the training data such that formula_10.
Algorithm.
The training examples are vectors in a multidimensional feature space, each with a class label. The training phase of the algorithm consists only of storing the feature vectors and class labels of the training samples.
In the classification phase, "k" is a user-defined constant, and an unlabeled vector (a query or test point) is classified by assigning the label which is most frequent among the "k" training samples nearest to that query point.
A commonly used distance metric for continuous variables is Euclidean distance. For discrete variables, such as for text classification, another metric can be used, such as the overlap metric (or Hamming distance). In the context of gene expression microarray data, for example, "k"-NN has been employed with correlation coefficients, such as Pearson and Spearman, as a metric. Often, the classification accuracy of "k"-NN can be improved significantly if the distance metric is learned with specialized algorithms such as Large Margin Nearest Neighbor or Neighbourhood components analysis.
A drawback of the basic "majority voting" classification occurs when the class distribution is skewed. That is, examples of a more frequent class tend to dominate the prediction of the new example, because they tend to be common among the "k" nearest neighbors due to their large number. One way to overcome this problem is to weight the classification, taking into account the distance from the test point to each of its "k" nearest neighbors. The class (or value, in regression problems) of each of the "k" nearest points is multiplied by a weight proportional to the inverse of the distance from that point to the test point. Another way to overcome skew is by abstraction in data representation. For example, in a self-organizing map (SOM), each node is a representative (a center) of a cluster of similar points, regardless of their density in the original training data. "K"-NN can then be applied to the SOM.
Parameter selection.
The best choice of "k" depends upon the data; generally, larger values of "k" reduces effect of the noise on the classification, but make boundaries between classes less distinct. A good "k" can be selected by various heuristic techniques (see hyperparameter optimization). The special case where the class is predicted to be the class of the closest training sample (i.e. when "k" = 1) is called the nearest neighbor algorithm.
The accuracy of the "k"-NN algorithm can be severely degraded by the presence of noisy or irrelevant features, or if the feature scales are not consistent with their importance. Much research effort has been put into selecting or scaling features to improve classification. A particularly popular approach is the use of evolutionary algorithms to optimize feature scaling. Another popular approach is to scale features by the mutual information of the training data with the training classes.
In binary (two class) classification problems, it is helpful to choose "k" to be an odd number as this avoids tied votes. One popular way of choosing the empirically optimal "k" in this setting is via bootstrap method.
The 1-nearest neighbor classifier.
The most intuitive nearest neighbour type classifier is the one nearest neighbour classifier that assigns a point x to the class of its closest neighbour in the feature space, that is formula_11.
As the size of training data set approaches infinity, the one nearest neighbour classifier guarantees an error rate of no worse than twice the Bayes error rate (the minimum achievable error rate given the distribution of the data).
The weighted nearest neighbour classifier.
The k-nearest neighbour classifier can be viewed as assigning the k nearest neighbours a weight formula_12 and all others 0 weight. This can be generalised to weighted nearest neighbour classifiers. That is, where the ith nearest neighbour is assigned a weight formula_13, with formula_14. An analogous result on the strong consistency of weighted nearest neighbour classifiers also holds.
Let formula_15 denote the weighted nearest classifier with weights formula_16. Subject to regularity conditions, which in asymptotic theory are conditional variables which require assumptions to differentiate among parameters with some criteria. On the class distributions the excess risk has the following asymptotic expansion
formula_17
for constants formula_18 and formula_19 where formula_20 and formula_21.
The optimal weighting scheme formula_22, that balances the two terms in the display above, is given as follows: set formula_23, 
formula_24 for formula_25 and 
formula_26 for formula_27.
With optimal weights the dominant term in the asymptotic expansion of the excess risk is formula_28. Similar results are true when using a bagged nearest neighbour classifier.
Properties.
"k"-NN is a special case of a variable-bandwidth, kernel density "balloon" estimator with a uniform kernel.
The naive version of the algorithm is easy to implement by computing the distances from the test example to all stored examples, but it is computationally intensive for large training sets. Using an approximate nearest neighbor search algorithm makes "k-"NN computationally tractable even for large data sets. Many nearest neighbor search algorithms have been proposed over the years; these generally seek to reduce the number of distance evaluations actually performed.
"k-"NN has some strong consistency results. As the amount of data approaches infinity, the two-class "k-"NN algorithm is guaranteed to yield an error rate no worse than twice the Bayes error rate (the minimum achievable error rate given the distribution of the data). Various improvements to the "k"-NN speed are possible by using proximity graphs.
For multi-class "k-"NN classification, Cover and Hart (1967) prove an upper bound error rate of
formula_29
where formula_30 is the Bayes error rate (which is the minimal error rate possible), formula_31 is the asymptotic "k-"NN error rate, and M is the number of classes in the problem. This bound is tight in the sense that both the lower and upper bounds are achievable by some distribution. For formula_32 and as the Bayesian error rate formula_30 approaches zero, this limit reduces to "not more than twice the Bayesian error rate".
Error rates.
There are many results on the error rate of the k nearest neighbour classifiers. The k-nearest neighbour classifier is strongly (that is for any joint distribution on formula_34) consistent provided formula_35 diverges and formula_36 converges to zero as formula_37.
Let formula_38 denote the k nearest neighbour classifier based on a training set of size n. Under certain regularity conditions, the excess risk yields the following asymptotic expansion
formula_39
for some constants formula_40 and formula_41.
The choice formula_42 offers a trade off between the two terms in the above display, for which the formula_43-nearest neighbour error converges to the Bayes error at the optimal (minimax) rate formula_44.
Metric learning.
The K-nearest neighbor classification performance can often be significantly improved through (supervised) metric learning. Popular algorithms are neighbourhood components analysis and large margin nearest neighbor. Supervised metric learning algorithms use the label information to learn a new metric or pseudo-metric.
Feature extraction.
When the input data to an algorithm is too large to be processed and it is suspected to be redundant (e.g. the same measurement in both feet and meters) then the input data will be transformed into a reduced representation set of features (also named features vector). Transforming the input data into the set of features is called feature extraction. If the features extracted are carefully chosen it is expected that the features set will extract the relevant information from the input data in order to perform the desired task using this reduced representation instead of the full size input. Feature extraction is performed on raw data prior to applying "k"-NN algorithm on the transformed data in feature space.
An example of a typical computer vision computation pipeline for face recognition using "k"-NN including feature extraction and dimension reduction pre-processing steps (usually implemented with OpenCV):
Dimension reduction.
For high-dimensional data (e.g., with number of dimensions more than 10) dimension reduction is usually performed prior to applying the "k"-NN algorithm in order to avoid the effects of the curse of dimensionality.
The curse of dimensionality in the "k"-NN context basically means that Euclidean distance is unhelpful in high dimensions because all vectors are almost equidistant to the search query vector (imagine multiple points lying more or less on a circle with the query point at the center; the distance from the query to all data points in the search space is almost the same).
Feature extraction and dimension reduction can be combined in one step using principal component analysis (PCA), linear discriminant analysis (LDA), or canonical correlation analysis (CCA) techniques as a pre-processing step, followed by clustering by "k"-NN on feature vectors in reduced-dimension space. This process is also called low-dimensional embedding.
For very-high-dimensional datasets (e.g. when performing a similarity search on live video streams, DNA data or high-dimensional time series) running a fast approximate "k"-NN search using locality sensitive hashing, "random projections", "sketches" or other high-dimensional similarity search techniques from the VLDB toolbox might be the only feasible option.
Decision boundary.
Nearest neighbor rules in effect implicitly compute the decision boundary. It is also possible to compute the decision boundary explicitly, and to do so efficiently, so that the computational complexity is a function of the boundary complexity.
Data reduction.
Data reduction is one of the most important problems for work with huge data sets. Usually, only some of the data points are needed for accurate classification. Those data are called the "prototypes" and can be found as follows:
Selection of class-outliers.
A training example surrounded by examples of other classes is called a class outlier. Causes of class outliers include:
Class outliers with "k"-NN produce noise. They can be detected and separated for future analysis. Given two natural numbers, "k" ;"r" ;0, a training example is called a ("k","r")NN class-outlier if its "k" nearest neighbors include more than "r" examples of other classes.
Condensed Nearest Neighbor for data reduction.
Condensed nearest neighbor (CNN, the "Hart algorithm") is an algorithm designed to reduce the data set for "k"-NN classification. It selects the set of prototypes "U" from the training data, such that 1NN with "U" can classify the examples almost as accurately as 1NN does with the whole data set.
Given a training set "X", CNN works iteratively:
Use "U" instead of "X" for classification. The examples that are not prototypes are called "absorbed" points.
It is efficient to scan the training examples in order of decreasing border ratio. The border ratio of a training example "x" is