【正文】 表（ ）
。數(shù)據(jù)如表 ，這說明了三個(gè)數(shù)據(jù)集 CRA（ leptograpsus 蟹）， RSY（里普利合成數(shù)據(jù)）， HEA（心臟疾?。┦歉鶕?jù) [273]頁的數(shù)據(jù)得到的。最后，內(nèi)核矩陣逼近誤差的進(jìn)一步證實(shí)了調(diào)查結(jié)果 [206； 226]頁。人們通常認(rèn)為 LSSVM 的分類和回歸導(dǎo)致圍繞核矩陣偏置項(xiàng)。對于大內(nèi)核矩陣的特征值分解為 （ ） 此外，正如在 [294]頁： （ ）然后可以顯示 () 其中 是取自 的分塊矩陣 。這是關(guān)系到特征函數(shù) 和特征值 的積分方程 （ ） 如下： （ ） 和 分別是 和 的積分方程 ， 并且 是表示 項(xiàng)的矩陣 。此方法是用低秩逼近給定的核矩陣尋找隨機(jī)選擇 M 行 /列的核矩陣的方法。然而，在這種情況下，仍然可以嘗試找到有意義的估計(jì)值 。 （ ） 另一方面是在非線性的情況下，此情況要復(fù)雜得多。例如在線性函數(shù)估計(jì)的情況下，有消除錯(cuò)誤變量 ，其中一個(gè)可以立即解決。事實(shí)上存在的主要問題是對大型數(shù)據(jù)設(shè)置更有利詭計(jì)的雙重問題，以及找到更適合的大維輸入空間的問題。 假設(shè)存在一個(gè)線性內(nèi)核。在固定規(guī)模尺寸方法 LSSVM 中 Nystrom 方法用于估計(jì) 本征函數(shù)。 低階近似法 Nystrom 方法 圖 線性支持向量機(jī)對偶問題既是如何解決大型數(shù)據(jù)集的問題，但是大維輸入空間問題是主要的問題。下一步，我們旨在解釋在特征空間中構(gòu)造一個(gè)合適的基本成分。這部分內(nèi)容將在下一章節(jié)進(jìn)行更詳細(xì)的介紹。在這個(gè) 固定 規(guī)模 LSSVM 方法的 最小二乘支持向量機(jī) 中，單一取代了對偶，解決了存在的主要問題?；诟咚惯^程和不完整的 Cholesky 分解，我們解釋一下 Nystrom 方法對于低跌逼近的建議。 (Middle) linear bination of LSSVMs with single neuron trained by Bayesian learning。 258] and Collebert el al. [44]. The mittee method is also related to boosting [75。 … 。 179] one considers the covariance matrix () where in practice one works with a finitesample approximation () and the N data are a representative subset of the overall training data set ( or the whole training data set itself). The mittee error equals (). An optimal choice of B follows then from (). From the Lagrangian with Lagrange multiplier X () one obtains the conditions for optimality: () with optimal solution () and corresponding mittee error () with 1v = [1。 179]. We can use this method also to depose the problem of training LSSVMs into smaller problems consisting of less training data points and train individual LSSVMs on each of the subproblems. Because the mittee is a liner bination of the submodels, the functional form of the LSSVM mittee stays the same as for each of the individual submodels, . a weighted sum of kernel functions evaluated at given data points. In and this approach is shown and illustrated on the training of a noisy sinc function with 4000 given training data points. The training data were generated in the interval [10, 10]. This data set was divided into four parts corresponding to the intervals [10, 5], [5, 0], [0, 5] and [5, 10] for the purpose of visualization. Individual LSSVMs have been trained for these intervals and the results are bined into one single mittee work model. The results of individual LSSVMs and the mittee of LSSVMs are shown in and , respectively. While for the purpose of illustration the intervals are chosen to be nonoverlapping, in practice one obtains better results by randomly selecting the data points in [10, 10] and assigning it to a training set of one of the individual LSSVM models. In the example the same RBF kernel with Q = 5 and r = 1 was taken for the individual LSSVMs. Illustration of a mittee work approach to the estimation of a noisy sinc function with 4000 given data points. (Top) In this illustration the given training data are divided into four intervals. For each of the four parts a separate LSSVM has been trained on 1000 data points. the results of the four LSSVMs are bined into an overall model by means of a mittee work. (Bottom) true sinc function (solid line), estimate of the mittee of LSSVMs (dashed line). Better results are obtained by randomly selecting training data and assigning then to the subgroups for this example. Results of the individual LSSVMs for the example of the previous Figure, trained on each of the subintervals: true sinc function (solid line), individual LSSVM estimates trained on one of the four intervals (dashed line). The mittee work that condidts of the M submodels takes the form () where 公式 , H(x) is the true function to be estimated and 公式 where () is the ith LSSVM model trained on the data 公式 with resulting support values aki, bias term Bi and kernel Ki for the ith submodel and I = 1,… ,m with m the number of LSSVM submodels. As explained in [12。 Amouar in [17] for finding the reduced set of basis vectors 公式 and has been further applied in [37]. For each given data point Xk one can consider the ratio () where 公式 is the basis constituted by a subset S of selected training data points. In this way one characterized the quality of the approximation in the feature space for the reduced set. By application of the kernel trick one obtains () where SS denotes the M * M submatrix of the kernel matrix Q of size N*N and SK denotes the kth column of the matrix SS. A related criterion has also been considered in [226]. In order to minimize the error of the approximation in the feature space over all training data N, one considers the cost function () where 公式 meaning that one lets the vectors Er correspond 同 the subset S of the given training set and selects those vectors that optimize this feature space criterion. Illustration of the selection of support vectors for a multispiral problem with 1600 training data points and a fixed size LSSVM (M = 80) with RBF kernel (O = ). The entropy criterion is used to select the support vectors (depicted by the squares). No class labels are shown for the training data in this unsupervised learning process. Expressing an estimate for the model G in terms of a reduced set of basis vectors 公式 instead of 公式 with M N. a suitable selection of the vectors Wr is made as a subset of the given training data set. Mining large data sets using a mittee work of LSSVMs where the individual LSSVMs are trained on subsets of the entire training data set. Combining submodels Committee work approach In the first chapter we discussed mittee methods for bining results form estimated models which is wellknown in the area of neural works. The models could be MLP