The Wiley Finance Series : Handbook of News Analytics in Finance

with the idea that all elements must be mapped from setXinto setYwith no more than
an-deviation. A simple linear example of such a model would be

fðxiÞ¼hw;xiiþb; w2X;b2R

The notationhw;xisignifies the dot product ofwandx. Note that the equation of a
hyperplane ishw;xiþb¼0.
The idea in SVM regression is to find theflattest wthat results in the mapping from

x!y. Thus, we minimize the Euclidean norm ofw(i.e.,jjwjj¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

n

j¼ 1 w

2

j

q

). We also

want to ensure thatjyifðxiÞj; 8 i. The objective function (quadratic program)
becomes

min^12 jjwjj^2

subject to

yihw;xiib

yiþhw;xiiþb

This is a (possibly infeasible) convex optimization problem. Feasibility is obtainable by
introducing the slack variablesð;Þ. We choose a constantCthat scales the degree of
infeasibility. The model is then modified to be as follows:

min^12 jjwjj^2 þC

Xn

i¼ 1

ðþÞ

subject to

yihw;xiibþ

yiþhw;xiiþbþ

; 0

AsCincreases, the model increases in sensitivity to infeasibility.
We may tune the objective function by introducing cost functionscð:Þ;cð:Þ. Then, the
objective function becomes

min^12 jjwjj^2 þC

Xn

i¼ 1

½cðÞþcðފ

We may replace the function½fðxÞyŠwith a ‘‘kernel’’Kðx;yÞintroducing nonlinearity
into the problem. The choice of the kernel is a matter of judgment, based on the nature
of the application being examined. SVMs allow many different estimation kernels—e.g.,
the Radial Basis function kernel minimizes the distance between inputs (x) and targets
(y) based on
fðx;y; Þ¼expð jxyj^2 Þ

where is a user-defined squashing parameter.
There are various SVM packages that are easily obtained in open source. An easy-to-
use one is SVM Light—the package is available at the following URL:http://
svmlight.joachims.org/SVM Light is an implementation of Vapnik’s Support
Vector Machine for the problem of pattern recognition. The algorithm has scalable

News analytics: Framework, techniques, and metrics 53