Nearest Neighbor Predictors

Why

We might expect similar precepts to lead to similar postcepts.

Definition

Consider a set of inputs $X$ with a metric $d: X \times X \to \R $ Let $D = (x^1, y^1), \dots , (x^n, y^n)$ a dataset in $X \times Y$ The nearest-neighbor predictor is the predictor $f: X \to Y$ which assigns to $x \in X$ the value ...

Notation

Let $D = ((a^1, b^1), \dots , (a^n, b^n))$ be a dataset in $A \times B$, where $A$ and $B$ are non-empty sets. Let $f$ be the nearest neighbor inductor. Then $\iota (D)(x)$ is Let $n$ be a natural number. Let $\Xi $ be a length $n$ paired record sequence in $\mathcal{U} \times \mathcal{V} $; so

\[ \Xi = ((u^1, v^1), \dots , (u^n, v^n)) \]

with $u^i \in \mathcal{U} $ and $v^i \in \mathcal{V} $ for $i = 1,\dots ,n$.

The nearest neighbor induction associates $\Xi $ with the function $f_{\Xi }$ such that

\[ f_{\Xi }(u) = v^j \]

where $j < n$ is the largest integer such that

\[ d(u, u^j) = \min_{i} \set{d(u, u^i)}. \]

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