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to class vc Similarly, p(vc jx) is the posterior probability for class vc given that x is observed A minimum error classi er maps each vector x to an element in such that the probability of misclassi cation (ie, the probability that the classi er label is different from the true label) is minimized To achieve this minimum error rate, the optimal classi er decides x has label vi if p(vi jx) for all j = i, vi , vj [ V In practice, it is very dif cult to evaluate the posterior probability in closed form Instead, one can use an appropriate discriminant function gi (x) that satis es gi (x) gj (x) if p(vi jx) p(vj jx) for j = i, for all x Then minimum error classi cation can be achieved as: decide x has label vi if gi (x) .

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gj (x) for j = i The minimum probability of misclassi cation is also known as the Bayes error, and a minimum error classi er is also known as a Bayes classi er or a maximum posterior probability (MAP) classi er Below, we brie y discuss three classi ers that approximate the optimal Bayes classi er k-Nearest-Neighbor Classi er The k-nearest-neighbor (kNN) classi er uses all the training features as the set of prototypes {pk } During the testing phase, the distance between each test vector and every prototype is calculated, and the k prototype vectors that are closest to the test vector are identi ed The class labels of these k-nearest prototype vectors are then combined using majority vote or some other method to decide the class label of the test vector When k 1, the kNN classi er is called the nearest-neighbor classi er.

14.2.3 Web Services Standards Figure 14.4 organizes the Web services standards by function area: security, messaging, discovery, security management, and description. There are elements of a protocol stack in this diagram. However, this diagram is intended primarily to graphically separate the main functional areas and map the mostly ongoing standards work to these broad categories. The core and foundation technologies are found about the base of the diagrams, and the most volatile technologies are towards the top. Given this volatility, this chapter will only address the foundation technologies of SOAP, WSDL, and the WS-I speci cation pro les. Note that WS-I speci cations are pro les of selected speci cations presented below and hence are not themselves part of the diagram with the exception of calling out the Basic

It is well-known [6] that asymptotically (in the number of training vectors), the probability of misclassi cation of a nearest-neighbor classi er approaches twice the (optimal) Bayes error Hence, the performance of a nearest-neighbor classi er can be used as a baseline to gauge the performance of other classi ers However, as the number of prototypes increases, a kNN classi er is not very suitable for actual implementation, since it requires too much memory storage and processing power for testing Maximum-Likelihood Classi er Using the Gaussian mixture density model in this classi er, the distribution of training vectors from the same class is modeled as a mixture of Gaussian density functions That is, the likelihood function is modeled as: p(xjvi ) / Gi (xjui ) X.

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