Review: Prediction of Respiratory Motion - Prediction and Classification of Respiratory Motion - page 13

methods to combine more than two methods, resulting in outperforming the pre-
vious solitary method. These three approaches are described in the following
Sects. 2.3.1 , 2.3.2 , and 2.3.3 , respectively. Figure 2.4 shows the key studies, which
have more than 30 references in the last 10 years, representing the salient algo-
rithms covered.

2.3.1 Model-Based Prediction Algorithms

Generally, model-based methods include (1) linear prediction [ 73 - 75 , 82 ],
(2) Kalman filter [ 72 , 73 , 75 , 81 - 84 ], (3) sinusoidal model [ 74 , 82 ], (4) finite state
model [ 73 , 78 , 82 ], (5) autoregressive moving average model [ 76 , 77 , 82 ],
(6) support vector machine [ 7 , 56 , 61 , 82 , 85 - 87 ], and (7) hidden Markov model
[ 73 , 78 ]. Especially, linear approaches and Kalman filters are widely used for the
fundamental prediction approach of respiratory motion among a variety of

investigated methods [ 5 , 16 , 38 , 39 , 55 , 56 , 73 - 78 , 88 - 92 ].

2.3.1.1 Linear Prediction

A linear prediction is a mathematical system operation where future output values
are estimated as a linear function of previous values and predictor coefficients, as
follows [ 75 , 82 ]:

x ð t Þ¼ a 0 þ a 1 x ð t 1 Þþþ a n x ð t n Þ¼ X

n

a i x ð t i Þ; ð 2 : 1 Þ

i ¼ 0

where x(t) is the predicted value or position at time t.
The predicted value is a linear combination of previous observations
x(t - n) and predictor coefficients a n that are not changing over time, as shown in
Fig. 2.5 . In a linear prediction, it is a significant task to solve a linear equation to
find out the coefficients a n that can minimize the mean squared error between the
predicted values and previous values [ 75 ]. The linear model is widely used in the
early stage to compare the prediction performance with other models, e.g. neural
network prediction and Kalman filtering [ 73 , 75 ]. Sharp et al. revealed that the root
mean squared error (RMSE) for the prediction accuracy is around 2.2 mm with
200 ms latency [ 75 ]. The limitation of this model is that it is not robust to some
changes from one linear state to another [ 73 ]. This model can be enhanced into
nonlinear (sinusoidal) and adaptive models as shown in Fig. 2.4 [ 74 ].