Personalized_Medicine_A_New_Medical_and_Social_Challenge

quadratic programming (QP).^116 In most cases, the training data are not perfectly
separable, and therefore an additional term needs to be incorporated into the
optimization problem allowing some data points to be on the wrong side of the
hyperplane. By defining the slack variables,ξi0,i¼1,...,n, we can modify the
optimization problem to the following:

min

w,b,ξ

1

2

wTwþC

Xn

i¼ 1

ξi

subject to the following:

yk



wTφðxkÞþb



 1 ξk,k¼1,...,n ð 5 Þ

whereCis a regularization parameter. It has been proven that the solution of this

problem can be computed as follows:w¼

Xn

i¼ 1

αiφðÞxi, where parameters,αi, are

solutions of the following QP problem:^117

max

α

2 αTeαTdiag yðÞKdiag yðÞα



ð 6 Þ

whereeis the unity vector.
After the model learning is finished and parameters w andbare obtained,
we can classify an unlabeled data item,xnew, as belonging to×1orþ1 class
depending on the sign of the following function:

fxðÞ¼new wTφðÞþxnew b¼

Xn

i¼ 1

αiKxðÞþi;xnew b. This function,f, is used to infer

class assignment of a new, unobserved entry,xnew.
This mathematical formalism can be further extended to combine various kernel
matrices, representing distinct data types, into a single kernel matrix that is used for
data integration. Entries in these matrices can be interpreted as similarities between
biological entities. Particularly, for a set of kernels,K¼{K 1 ,...,Km}, we can define
a parameterized combination of kernels:

K¼

Xn

i¼ 1

μiKi ð 7 Þ

Following the above approach, we can generalize a solution of an SVM classifier
to a combined kernel matrix. For a detailed procedure on solving this optimization
problem and estimation of weighting parameters,μi, see Lanckriet et al. (2004a,b).^118

(^116) Boyd and Vandenberghe ( 2004 ).
(^117) Scholkopf and Smola ( 2001 ).
(^118) Lanckriet et al. (2004a,b).
164 V. Gligorijevic ́and N. Pržulj