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Pattern analysis 3 1.1 Patterns in data 4 1.2 Pattern analysis algorithms 12 1.3 Exploiting patterns 17 1.4 Summary 22 1.5 Further reading and advanced topics 23 2 Kernel methods: an overview 25 2.1 The overall picture 26 2.2 Linear regression in a feature space 27 2.3 Other examples 36 2.4 The modularity of kernel methods 42 2.5 Roadmap of the book 43 2.6 Summary 44 2.7 Further reading and advanced topics 45 3 Properties of kernels 47 3.1 Inner products and positive semi-definite matrices 48 3.2 Characterisation of kernels 60 3.3 The kernel matrix 68 3.4 Kernel construction 74 3.5 Summary 82 3.6 Further reading and advanced topics 82 4 Detecting stable patterns 85 4.1 Concentration inequalities 86 4.2 Capacity and regularisation: Rademacher theory 93 v vi Contents

The SVM software is available at http://www.cs. columbia.edu/ approximately bgrundy/svm.

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We introduce a method of functionally classifying genes by using gene expression data from DNA microarray hybridization experiments. The method is based on the theory of support vector machines (SVMs). SVMs are considered a supervised computer learning method because they exploit prior knowledge of gene function to identify unknown genes of similar function from expression data. SVMs avoid several problems associated with unsupervised clustering methods, such as hierarchical clustering and self-organizing maps. SVMs have many mathematical features that make them attractive for gene expression analysis, including their flexibility in choosing a similarity function, sparseness of solution when dealing with large data sets, the ability to handle large feature spaces, and the ability to identify outliers. We test several SVMs that use different similarity metrics, as well as some other supervised learning methods, and find that the SVMs best identify sets of genes with a common function using expression data. Finally, we use SVMs to predict functional roles for uncharacterized yeast ORFs based on their expression data.

Supplementary data at http://noble.gs.washington.edu/proj/sdp-svm

Comparison of whole genomes has revealed that changes in the size of gene families among organisms is quite common. However, there are as yet no models of gene family evolution that make it possible to estimate ancestral states or to infer upon which lineages gene families have contracted or expanded. In addition, large differences in family size have generally been attributed to the effects of natural selection, without a strong statistical basis for these conclusions. Here we use a model of stochastic birth and death for gene family evolution and show that it can be efficiently applied to multispecies genome comparisons. This model takes into account the lengths of branches on phylogenetic trees, as well as duplication and deletion rates, and hence provides expectations for divergence in gene family size among lineages. The model offers both the opportunity to identify large-scale patterns in genome evolution and the ability to make stronger inferences regarding the role of natural selection in gene family expansion or contraction. We apply our method to data from the genomes of five yeast species to show its applicability.

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