Nearest-neighbor methods use those observations in the training set T closest in input space to x form Y-hat.
Specifically, the k-nearest neighbor fit for Y-hat is difined as follows: Y(x)=1/kΣyi,xi belong to Nk(x).
where Nk(x) is the neighborhood of x defined by the k closest points xi in the traing sample.
Closeness implies a metric, which for the moment we assume is Euclidean distance. So, in words, we find the k observations with xi closest to x in input space, and average their responses.