Review: Prediction of Respiratory Motion - Prediction and Classification of Respiratory Motion - page 18

y

F(v i , )+ ε

Observation y i

Slack variables after
loss function

F(v i , )

ξ i

ε F(v i , ) − ε

Predicted

ξ i *

x

Fig. 2.9 Parameters for support vector regression. Let define e as a user defined threshold, and
v i i ¼ 1 ; ... ; ð Þ as N training samples. The loss function is defined using the threshold e, such as
if the observation is within the threshold, the loss is zero; otherwise, the loss is the amount of the
difference between the predicted value and the threshold ð j y i Fv i ; w ð Þ j e Þ

function (truth) with d-dimensional input vector x ¼ x 1 ; ... ; x ½ , F(x, ˆ )asa
function with estimation ˆ derived from minimizing a measurement error between
G(x) and F(x, ˆ ). Using N training samples v i , i ¼ 1 ; ... ; N ; the primal objective
function with a loss function L( ) can be expressed, as follows [ 86 ]:

C X

N

kk 2 ; L ½ y i F ð v i ; w Þ þ ð 2 : 7 Þ

i ¼ 1

where, C is a control value to adjust a balance, y j is the observation of G(x) in the
presence of noise. The function L( ) is a general loss function with user defined
threshold e, as shown in Fig. 2.9 , i.e., if the observation is within the threshold
y i Fx i ; w ð j ð Þ j \e Þ , the loss is zero; otherwise, the loss is the amount of the

difference between the predicted value and the threshold e, such as

j j ð Þ [ 85 , 86 ]. Based on the loss function and the threshold, the
objective function ( 2.7 ) is calculated by solving the optimization problem as
follows:

y i Fx i ; w ð Þ

!

C X n i þ X

N N

þ 1

2 ð w t w Þ;

min
w

n i ð 2 : 8 Þ

i ¼ 1 i ¼ 1

where n i and n i * are slack variables as shown in Fig. 2.9 . A control value C is used
to adjust the balance between the error term and the weight concentration [ 86 ].
This optimization problem can be resolved by the Lagrangian relaxation using
Lagrangian multipliers [ 85 , 86 ].