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Generalisation

A measure of generalisation is defined similarly, simply reversing the direction of the nearest-neighbour distance measure:

$$G_{\lambda}(\mathcal{I};p)\doteq \frac{1}{\mathcal{N}}\sum\l... ...ft(\vert\mathbf{I}_{i}-\mathbf{I}_{\mu}\vert\right)^{\lambda},$$ (6.4)

with standard error:

$$\sigma_{G}=\frac{SD_{i} \left\{\mathbf{min}_{\mu}\{\vert\math... ...athbf{I}_{\mu}\vert^\lambda\}\right\}}{\sqrt{\mathcal{N}-1}}.$$ (6.5)

That is, for each member of the training set $\mathbf{I}_{i}$, the distance to the nearest-neighbour in the sample set $\{\mathbf{I}_{\mu}\}$ is computed.

Note here that both measures can be further extended, by considering the sum of distances to $k$-nearest-neighbours, rather than just to the single nearest-neighbour. However, the choice of $k$ would require careful consideration and in what follows, experiments shown are restricted to the single nearest-neighbour case.