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(ie, on all axes in the Euclidean space containing the instances) This lies in contrast to methods such as rule and decision tree learning systems that select only a subset of the instance attributes when forming the hypothesis To see the effect of this policy, consider applying k-NEAREST NEIGHBOR a problem in which to each instance is described by 20 attributes, but where only 2 of these attributes are relevant to determining the classification for the particular target function In this case, instances that have identical values for the 2 relevant attributes may nevertheless be distant from one another in the 20-dimensional instance space As a result, the similarity metric used by k-NEAREST NEIGHBOR--depending on all 20 attributes-will be misleading The distance between neighbors will be dominated by the large number of irrelevant attributes This difficulty, which arises when many irrelevant attributes are present, is sometimes referred to as the curse of dimensionality Nearest-neighbor approaches are especially sensitive to this problem One interesting approach to overcoming this problem is to weight each attribute differently when calculating the distance between two instances This corresponds to stretching the axes in the Euclidean space, shortening the axes that correspond to less relevant attributes, and lengthening the axes that correspond to more relevant attributes The amount by which each axis should be stretched can be determined automatically using a cross-validation approach To see how, first note that we wish to stretch (multiply) the jth axis by some factor z j , where the values z l z, are chosen to minimize the true classification error of the learning algorithm Second, note that this true error can be estimated using crossvalidation Hence, one algorithm is to select a random subset of the available , data to use as training examples, then determine the values of z l z that lead to the minimum error in classifying the remaining examples By repeating this process multiple times the estimate for these weighting factors can be made more accurate This process of stretching the axes in order to optimize the performance of k-NEAREST NEIGHBOR provides a mechanism for suppressing the impact of irrelevant attributes An even more drastic alternative is to completely eliminate the least relevant attributes from the instance space This is equivalent to setting some of the zi scaling factors to zero Moore and Lee (1994) discuss efficient cross-validation methods for selecting relevant subsets of the attributes for k-NEAREST NEIGHBOR algorithms In particular, they explore methods based on leave-one-out crossvalidation, in which the set of m training instances is repeatedly divided into a training set of size m - 1 and test set of size 1, in all possible ways This leave-oneout approach is easily implemented in k-NEAREST NEIGHBOR algorithms because no additional training effort is required each time the training set is redefined Note both of the above approaches can be seen as stretching each axis by some constant factor Alternatively, we could stretch each axis by a value that varies over the instance space However, as we increase the number of degrees of freedom available to the algorithm for redefining its distance metric in such a fashion, we also increase the risk of overfitting Therefore, the approach of locally stretching the axes is much less common.

6 Save and exit Interface Builder 7 Open SwitchSliderViewController and implement a method named handleSwitch and a

Figure 323 demonstrates the area of the display used when different image shapes are letterboxed to fit it (that is, the dimensions are equalized in the largest direction to make the smaller box fit inside the larger box)24 The 133:1 row makes it clear how much smaller a letterboxed 235:1 movie is: Only 57 percent of the screen is used for the picture On the other hand, the 2:1 row makes it clear how much expensive screen space goes unused by a 4:3 video program or even a 185:1 movie The two middle rows are quite similar, so the mathematical relationship of 16:9 to 4:3 gives it the edge In summary, the only way to support multiple aspect ratios without mattes would be to use a display that can physically change shape a mighty morphin television Since this is currently impossible (or outrageously expensive), 16:9 is the most reasonable compromise This said, the designers of DVD could have improved things slightly by allowing more than one anamorphic distortion ratio If a 235 movie were stored using a 235 anamorphic squeeze, then 24 percent of the internal picture would not be wasted on the black mattes, and the player could automatically generate the mattes for either 4:3 or 16:9 displays This was not done, probably because of the extra cost and complexity it would add to the player The limited set of aspect ratios presently supported by the MPEG-2 standard (4:3, 16:9, and 221:1) also may have had something to do with it

method named handleSlider (Listings 12-3 and 12-4) Also implement a property for the UISwitch named mySwitch

One additional practical issue in applying k-NEAREST NEIGHBOR efficient is memory indexing Because this algorithm delays all processing until a new query is received, significant computation can be required to process each new query Various methods have been developed for indexing the stored training examples so that the nearest neighbors can be identified more efficiently at some additional cost in memory One such indexing method is the kd-tree (Bentley 1975; Friedman et al 1977), in which instances are stored at the leaves of a tree, with nearby instances stored at the same or nearby nodes The internal nodes of the tree sort the new query x, to the relevant leaf by testing selected attributes of x,