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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.

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,

Even the simplest applications will have a Model, View, and Controller. Let s take a look at the simplest application you have built so far in this book, the simple Hello World program from 3. If you followed along with this example, open up the project now (if you didn t create this project yourself, you can download the completed project from the book s page on http://www.apress.com). Open the project and load the NIB file (MainMenu.xib). You ll remember that you created the user interface elements a Text Field control to hold the text entered by the user, a

Much of the literature on nearest-neighbor methods and weighted local regression uses a terminology that has arisen from the field of statistical pattern recognition In reading that literature, it is useful to know the following terms:

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Regression means approximating a real-valued target function Residual is the error { ( x ) - f ( x ) in approximating the target function Kernel function is the function of distance that is used to determine the weight of each training example In other words, the kernel function is the function K such that wi = K(d(xi,x,))

Label control to hold the text produced by the program and a Button control to initiate the transformation. This collection of components, including the window itself, make up the View. You also then created some connections that determined the routes for carrying messages. These connections all involve the HelloController object. If you right-click (or Control-click) on HelloController in the Document window you will see its connections. This is the Controller. You can see that the Button invokes an action method called sayHello:, and it does that by sending a message to HelloController (in effect, saying run the sayHello method ). HelloController in turn sends messages to the Text Field (amounting to tell me what you have ) and (once it has the data back from the Text Field) to the Label (amounting to display this ). The flow of messages is shown in Figure 6 1. So given that both the messages go from HelloController to these objects, how is the program to know which way around things have to happen Well, go back to the Xcode project window and check out the sayHello method in HelloController.m (see Listing 6 1).