Background - Smartphone-Based Human Activity Recognition - page 23

Fig. 2.3 Example of a
binary classification problem
in a two-dimensional space
( circles and crosses ). The
line represents a possible
solution to the problem

2.5.3.1 The Binary HF-SVM Model

In a binary classification problem only two classes are involved and it aims to pre-
dict the class that a given new sample belongs to. As a mode of illustration, Fig. 2.3
shows an example in a two-dimensional space. Two classes represented by circles
and crosses are depicted. The solution to this classification problem is to find a way
to separate the elements in two groups in order to reduce the number of misclassifi-
cations to a minimum. In this example, a simple approach involves tracing a line to
divide the space in two regions (such as the one depicted in the figure). This approach,
however, produces some classification errors because these groups are partially over-
lapped. A non-linear approach is therefore most likely to better solve this particular
data configuration.
More formally, consider a dataset composed of n samples. Each one corresponds
to an ordered pair

d are the input vectors and

( x i , y i ) ∀ i ∈ { 1 , ..., n } , where x i ∈ R

1 are their target values representing one of the two possible classes. We want
to solve this problem by finding a linear hyperplane which separates the data in two
groups. The set of linear classifiers which are possible solutions to the problem is of
the form:

y i =±

T x

f ( x ) = w + b , (2.1)

d is a vector orthogonal to the hyperplane (commonly known as
weights), and the scalar b

w ∈ R

where

∈ R is the bias. A standard linear SVM aims to find

the and b values that allow the optimal separation of the data in order to provide

the largest margin between the classes and the lowest error rate of the available data.
This can be learned by solving Convex Constrained Quadratic Programming (CCQP)
minimization problem formulated as:

w