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< x < x 2 . Then the weight of x 1 , the inverse of its distance from x , is 1/( x x 1 ), and the
weight of x 2 is 1/( x 2 x ). The weighted average of the labels is
which, when we multiply numerator and denominator by ( x x 1 )( x 2 x ), simplifies to
This expression is the linear interpolation of the two nearest neighbors, as shown in
Fig. 12.23(a) . When both nearest neighbors are on the same side of the query x , the
same weights make sense, and the resulting estimate is an extrapolation . We see ex-
trapolation in Fig. 12.23(a) in the range x = 0 to x = 1. In general, when points are
unevenly spaced, we can find query points in the interior where both neighbors are on
one side.
(4) Average of Three Nearest Neighbors . We can average any number of the nearest
neighbors to estimate the label of a query point. Figure 12.23(b)
shows what happens on our example training set when the three nearest neighbors are
used.
Figure 12.22 Results of applying the first two rules in Example 12.12
Figure 12.23 Results of applying the last two rules in Example 12.12
12.4.4
Kernel Regression
A way to construct a continuous function that represents the data of a training set well is
to consider all points in the training set, but weight the points using a kernel function that
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