K-nearest neighbors algorithm: Difference between revisions

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Let <math>C^{wnn}_n</math> denote the weighted nearest classifier with weights <math>\{w_{ni}\}_{i=1}^n</math>. Subject to regularity conditions, {{explain|date=Januarywhich 2019}}in onasymptotic theory are conditional variables which require assumptions to differentiate among parameters with some criteria. On the class distributions the excess risk has the following asymptotic expansion<ref name="Samworth12" />

:<math>\mathcal{R}_\mathcal{R}(C^{wnn}_{n}) - \mathcal{R}_{\mathcal{R}}(C^{Bayes}) = \left(B_1 s_n^2 + B_2 t_n^2\right) \{1+o(1)\},</math>

for constants <math>B_1</math> and <math>B_2</math> where <math>s_n^2 = \sum_{i=1}^n w_{ni}^2</math> and <math>t_n = n^{-2/d}\sum_{i=1}^n w_{ni}\{i^{1+2/d} - (i-1)^{1+2/d}\}</math>.