Non-Parametric Decision Rule

Let ${\underline{x_{1}}, \underline{x_{2}}, \dots, \underline{x_{n}}}$ be a supervised sample.
Take a ball (of volume $V$ ) around $\underline{x_{0}}$ , the ball will embrace $k$ patterns.
Among these $k$ patterns, let $k_{i}$ be the number of patterns belonging to class $ω_{i}$ .

Then we will have that the estimated probability $p_{n} (\underline{x_{0}}, ω_{i})$ , the probability that $\underline{x_{0}}$ will belong to the class $ω_{i}$ is:

p_{n} (\underline{x_{0}}, ω_{i}) = \frac{k _{i}}{n V}

Also, the estimated probability $\hat{P} (ω_{i} ∣ \underline{x_{0}})$ is given by:

\hat{P} (ω_{i} ∣ \underline{x_{0}}) = \frac{p _{n} ( x _{0} , ω _{i} )}{\sum _{j = 1}^{c} p _{n} ( x _{0} , ω _{j} )} = \frac{\frac{k _{i}}{n V}}{\sum _{j = 1}^{c} \frac{k _{j}}{n V}} = \frac{k _{i}}{k}

Where:

$\sum_{j = 1}^{c} p_{n} (\underline{x_{0}}, ω_{j})$ : means the probability that $\underline{x_{0}}$ will belong to one of the classes $k_{j}$ , so it is equal to $\frac{k}{n V}$ .

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Non-Parametric Decision Rule

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