Review: Prediction of Respiratory Motion - Prediction and Classification of Respiratory Motion - page 17

! jj
t 0 t 1

v ð t 0 ! t 1 Þ¼ jj X 0 X 1 : ð 2 : 5 Þ

This method provides not only a statistically quantitative analysis of motion
characteristics, but also good prediction results, i.e., average RMS error less than
1 mm. However, the study on FSM is restricted to a one dimension model. This
method was enhanced into a three dimension version with hidden Markov model
by Kalet et al. [ 73 ].

2.3.1.5 Autoregressive Moving Average Model

Autoregressive moving average (ARMA) model is a mathematical generalization
of the linear model with time series data and signal noise, and widely used to
predict motion patterns of a time series from past values [ 76 , 77 , 82 ]. ARMA
consists of two models: (1) an autoregressive (AR) model represented by a
weighted sum of the present and past positions with a polynomial order p, i.e.,
u 1 x(t - 1) ? ? u p x(t - p), and (2) a moving average (MA) model repre-
sented by a weighted sum of the present and past signal noise with a polynomial
order q, i.e., h 1 e(t - 1) ? ?h q e(t - q)[ 76 , 82 ]. The mathematical notation
ARMA (p, q) with polynomial orders of p AR and q MA is expressed as follows [ 77 ]:

x ð t Þ¼ e ð t Þþ X

p

/ i x ð t i Þþ X

q

h i e ð t i Þ; ð 2 : 6 Þ

i ¼ 1 i ¼ 1

where we define u i as the parameter of the AR model, and h i as the parameter of
MA model, respectively. The error terms e(t) are the white noise assuming to be
independent and identically distributed random variables. The order of ARMA
model was built on the combination of p and q with maximizing the Akaike
information criterion. There is no limitation with sampling data and processing
time to select the orders p and q. However, McCall et al. demonstrated that up to
ARMA (4, 4) models were preferred and the ARMA (2, 1) models achieved the
optimized mean prediction errors over all the latency investigated [ 77 ]. Ren et al.
also showed that the standard deviation of the position is below 2.6 mm with
prediction in contrast with 4.6 mm without prediction [ 76 ].

2.3.1.6 Support Vector Machine

Support vector machines (SVMs) are supervised learning methods that are widely
used for classification and regression analysis [ 56 , 61 , 85 - 87 ]. For medicine
applications, they have been used to predict lung radiation-induced pneumonitis
from patient variables and compute the future location of tumors from patient
geometry and clinical variables [ 7 , 82 , 87 ]. Let define G(x) as an unknown