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kernel-based methods simply modify the decision surface whenever a new training example is either misclassi ed or classi ed with an insuf cient margin In that case, the example is inserted into the kernel expansion and a suitable coef cient is chosen Often this causes signi cant problems with noisy data sets and over tting, due to the usually large number of support vectors that are quickly achieved Thus kernel removal techniques have been suggested to control the complexity of the machine In particular, Bordes and colleagues proposed LASVM [34], a kernel classi er in which examples from the current kernel expansion can be removed online It is proved that the algorithm converges to the solution of the quadratic programming problem Most of the incremental SVMs are based on the algorithm of Cauwenberghs and Poggio [35].



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The key idea of the algorithm is to add the new data point to the solution by varying the margin and at the same time preserving the Kuhn-Tucker conditions for the previous trained examples The incremental approach identi es three types of patterns: 1 The nonsupport vectors represent the patterns being out of the margins and are irrelevant to training 2 The support vectors represent patterns that are strictly on the margin 3 The patterns lying between the two margins are named the error vectors these patterns are not necessarily wrongly classi ed Only the patterns belonging to the last two classes are critical and should affect directly the incremental training process In the decremental learning process, a leave-one-out (LOO) error estimation is performed on the relevant vectors.





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Because latent class analysis is based on data in the form of a contingency table, it is important to have only a few levels for each variable Extreme sparseness of cells can lead to problems in the estimation As the number of levels within variables increases, so does the number of parameters in the model A balance between retaining information in the original variables and collapsing categories must be struck, as identi cation can be dif cult to attain when too many parameters are estimated The eight questionnaire items are observed indicators of the latent variable, depression In our example, latent class analysis allows us to arrive at a model that we believe best represents the relationship between these eight items and the latent variable depression We will estimate the proportion of individuals in a sample who are in each class of depression.

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The key idea is unlearning part of the relevant patterns by decrementing its corresponding a coef cients to zero, and computing the output of the resulting SVM [36] This auxiliary technique helps decrease the complexity of the machine Particularly useful for regression problems, an incremental method, the Kernel Recursive Least Squares (KRLS) algorithm, combines the principles of recursive least squares and the bene ts of the kernel trick, providing an ef cient and nonparametric approach for online regression [37] In this method, the size of the kernel matrix is limited by mapping the input patterns to a limited set of kernels, admitting into the kernel representation a new input pattern only if its feature space representation cannot be suf ciently well approximated by a combination of the current kernel representation.

This technique permits both reducing the order of the feature space and bounding the time and memory complexity of the algorithm It should be noted that other existing architectures like least-squares support vector machines, RBF neural networks, and TS fuzzy systems are equivalent to a KRLS machine Recently, kernel least-squares approaches have been successfully applied to different areas such as anomaly detection [38], dynamic systems [39], channel identi cation [40] As with any other kernel method, the KRLS output is a predictor of the form (624), equivalent to (17): f x.

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The Mathematical Model and Parameters Estimated The LCA mathematical model is be presented here in terms of our empirical example Suppose that there are eight categorical manifest indicators with response categories i, j, k, l, m, n, o, and p, respectively, and there are c = 1, , C latent classes Let y = {i, j, k, l, m, n, o, p} represent a response pattern, and let Y represent the entire array of response patterns Each individual can respond yes or no to each of the eight indicators of depression listed in Table 261, so there are 28 = 256 possible ways to respond to the eight questions.

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