Kernel methods

For subsequent PLS components, the NIPALS algorithm works differently than the kernel method however, the results are identical. NIPALS requires a deflation of X and of Y and the above pseudocode is continued by... 

ScHOLKOPE, B., Burges, C.J.C., and Smola, A.J. Advances in Kernel Methods — Support Vector Learning. MIT Press, Cambridge, MA, 1999. 

Some of the more popular predictive modeling methods used in drug discovery include linear methods, tree-based methods, -nearest neighbors, and kernel methods. In this section, a brief outline of these methods is given, together with references for reading and further details. 

T. Joachims, in Advances in Kernel Methods Support Vector Learning (Eds. B. Scholkopf, C. Burges,... 

SVM s are an outgrowth of kernel methods. In such methods, the data is transformed with a kernel equation (such as a radial basis function) and it is in this mathematical space that the model is built. Care is taken in the constmction of the kernel that it has a sufficiently high dimensionality that the data become linearly separable within it. A critical subset of transformed data points, the support vectors , are then used to specify a hyperplane called a large-margin discriminator that effectively serves as a hnear model within this non-hnear space. An introductory exploration of SVM s is provided by Cristianini and Shawe-Taylor and a thorough examination of their mathematical basis is presented by Scholkopf and Smola. ... 

To incorporate an even wider variety of biologic data for predicting missing enzymes in metabolic enzyme networks, kernel methods are used. A kernel is a mathematic function that can take as input a variety of data for a specific set of entities and transform it such that the input entities can be classified as distinctly as possible. This method consists of two steps a training phase and a test phase. The training phase consists of using data for which the properties are known in advance. Then, the test phase can be used to assess the applicability of the properties to new input data sets. 

Ryufuku, El. et al.. Evaluation of beta-ray skin dose based on point kernel method (in Japanese), JAERI-M-7354, 1977. 

Joachims T. Making large-scale SVM learning practical. In Scholkopf B, Burges CJC, Smola AJ, editors, Advances in kernel methods Support vector learning. Cambridge MIT Press, 1999. p. 169-84. 

Kernel methods, which include support vector machines and Gaussian processes, transform the data into a higher dimensional space, where it is possible to construct one or more hyperplanes for separation of classes or regression. These methods are more mathematically rigorous than neural networks and have in recent years been widely used in QSAR modeling. ... 

Watanabe, A. and Stark, L. 1975. Kernel method for nonlinear analysis identification of a biological control system. Math. Biosci. 27 99. 

U. KreBel, Pairwise Classification and Support Vector Machines, in Advances in Kernel Methods-Support Vector Learning, ed. by B. SchSlkopf, C. J. C. Burges, A. J. Smola (MIT Press, Cambridge, 1999), pp. 255-268... 

Scholkopf, B., Smola, A. Muller, K.R. 1999. Kernel principal components analysis. In B. Scholkopf, C. Burges A. Smola (eds.). Advances in Kernel Methods - Support Vector Learning 327-352. Cambridge (MA) MIT Press. 

Hofmann T, Scholkopf B, Smola AJ (2008) Kernel methods in machine learning. Ann Stat 36(3) 1171-1220... 

Cristianini, N., Scholkopf, B., 2002. Support vector machines and kernel methods—the new generation of learning machines. Ai Mag. 23, 31-41. 

The method introduced here is known as the method of Green s functions or the kernel method. The function (5.96) is in fact the Green s function, or elementary solution, to the diffusion equation. The general solution to the diffusion equation for a system with an arbitrary source distribution can be constructed from these elementary solutions by applying (5.97). 

The solution to this equation may be obtained by the application of the kernel method. The general solution is given by (5.97). In the present case the source distribution S(t) is given by the right-hand term of... 

If we use the relation (5.262) as the source function for the thermal-diffusion equation, then the thermal flux may again be found by using the kernel method. Thus if (5.262) is substituted into (5.97), it can be shown that... 

The general solution to this equation was developed in Sec. 5.2d by use of the kernel method [see Eq. (5.97)]. In the present case the source S(z) from (6.125) may be substituted directly into (5.97), and the integration will yield 0ih(2). We note, however, that the system in question involves only one space variable z. Thus although the general form (5.97) is applicable here, it is by no means necessary, and, in fact, some labor can be eliminated by using the one-dimensional kernel relation [see Eq. (5.78)], namely,... 

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