Estimation Using Nearest Neighbor - Rhea

Introduction

In this slecture, basic principles of implementing nearest neighbor rule will be covered. The error related to the nearest neighbor rule will be discussed in detail including convergence, error rate, and error bound. Since the nearest neighbor rule mostly relies on a metric function between patterns, the properties of metrics will be studied in detail. Several examples will be illustrated to help understanding throughout the lecture.

Error Rate & Bound using NN

Let's consider a testing sample x. Based on labeled training sample D$ ^n $ $ = x_{1},... ,x_{n}, $ the nearest neighbor technique will find the closest point x' to x. Then we assign the class of x' to x. This is how the classification based on the nearest neighbor rule is processed. Although this rule is very simple, it is also reasonable. The label $ \theta' $ used in the nearest neighbor is random variable which means $\theta' = w_{i}$ is same as a posterior probability $ P(w_{i}|x'). $ If sample sizes are big enough, it could be assumed that x' is sufficiently close to x that $ P(w_{i}|x') = P(w_{i}|x). $

Metrics Type

However, the closest distance between x' and x is determined by which metrics are used for feature space. A "metric" on a space S is a function $ D: S\times S\rightarrow \mathbb{R} $ which has following 4 properties:

• Non-negativity : $ D(\vec{x_{1}},\vec{x_{2}}) \geq 0, \forall \vec{x_{1}},\vec{x_{2}} \in S $
• Symmetry : $ D(\vec{x_{1}},\vec{x_{2}}) = D(\vec{x_{2}},\vec{x_{1}}), \forall \vec{x_{1}},\vec{x_{2}} \in S $
• Reflexivity : $ D(\vec{x},\vec{x}) = 0, \forall \vec{x} \in S $
• Triangle Inequality : $ D(\vec{x_{1}},\vec{x_{2}}) + D(\vec{x_{2}},\vec{x_{3}}) \geq D(\vec{x_{1}},\vec{x_{3}}), \forall \vec{x_{1}},\vec{x_{2}},\vec{x_{3}} \in S $

Real Application

References

Questions and comments

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