Background - Smartphone-Based Human Activity Recognition - page 26

In order to solve the CCQP problem, many efficient algorithms have been previ-
ously proposed (Shawe-Taylor and Sun 2011 ), such as, for example, the well-known
Sequential Minimal Optimization (SMO) (Keerthi et al. 2001 ; Platt 1998 ).
Once the ʱ i coefficients are found, new patterns can be classified by apply-

ing the SVM Feed-Forward Phase (FFP) which is given by the following general
formulation:

n

f ( x ) = y i ʱ i K ( x i , x ) + b (2.16)

i = 1

where b is the bias term and can be estimated by taking into account the KKT
conditions that were formulated to obtain the dual problem as described in (Oneto
and Greco 2010 ). They consist of the derivatives and the slackness conditions of the
Lagrangian of the primal formulation. We can obtain b from the following condition:

ʱ i [ y i ( w · x i + b ) − 1] = 0 (2.17)

and assuming that we have a data sample whose ʱ i > 0, we find that:

y i ( w · x i + b ) = 1 (2.18)

From Eq. ( 2.16 ) it is also possible to realize that only the samples which affect the
classification of new samples are those with 0. They are denominated support

vectors . Geometrically, these points usually lie close to the margin bounds between
the two classes.

ʱ i >

Fig. 2.4 Example of a
multiclass problem in a
two-dimensional space.
Three classes, represented by
black circles , white circles
and crosses , are separated by
hyperplanes which provide a
possible linear solution to the
classification problem