3: K-Nearest Neighbors (KNN)
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)3.1: K nearest neighbors
Assume we are given a dataset where \(X\) is a matrix of features from an observation and \(Y\) is a class label. We will use this notation throughout this article. \(k\)-nearest neighbors then, is a method of classification that estimates the conditional distribution of \(Y\) given \(X\) and classifies an observation to the class with the highest probability. Given a positive integer \(k\), \(k\)-nearest neighbors looks at the \(k\) observations closest to a test observation \(x_0\) and estimates the conditional probability that it belongs to class \(j\) using the formula
\[Pr(Y=j|X=x_0)=\frac{1}{k}\sum_{i\in \mathcal{N}_0}I(y_i=j)\]
where \(\mathcal{N}_0\) is the set of \(k\)-nearest observations and \(I(y_i=j)\) is an indicator variable that evaluates to 1 if a given observation \((x_i,y_i)\) in \(\mathcal{N}_0\) is a member of class \(j\), and 0 if otherwise. After estimating these probabilities, \(k\)-nearest neighbors assigns the observation \(x_0\) to the class which the previous probability is the greatest. The following plot can be used to illustrate how the algorithm works:
- If we choose \(K = 3\), then we have 2 observations in Class B and one observation in Class A. So, we classify the red star to Class B.
- If we choose \(K = 6\), then we have 2 observations in Class B but four observations in Class A. So, we classify the red star to Class A.
3.2: The Calculation of Distance
Since in k-NN algorithm, we need k nearest points, thus, the first step is calculating the distance between the input data point and other points in our training data. Suppose \(x\) is a point with coordinates \((x_1,x_2,...,x_p)\) and \(y\) is a point with coordinates \((y_1,y_2,...,y_p)\), then the distance between these two points is: \[d(x,y) = \sqrt{\sum_{i=1}^{p}(x_i - y_i)^2}\]
3.3: The Effect of K
As most statistical learning model does, if \(K\) is small, then, we use a smaller region of data to learn. This may cause over-fitting.This may cause under-fitting. And if \(K\) is large, then we use a larger region of data to learn. The following plot shows the effects of \(K\). as \(K\) gets larger, the decision boundary appears linear.
