Page 157 - Control Theory in Biomedical Engineering

Modeling simple and complex handwriting based on EMG signals 141

with:
x e and y e : estimated outputs according to x and y coordinates,
e 1 ,e 2 : electromyography signals, inputs of the model,
a ix ,b ix ,c ix ,d ix : parameters according to x e ,
a iy ,b iy ,c iy ,d iy : parameters according to y e .
The recursive least squares algorithm is used to estimate the handwriting
model parameters (Eqs. 3–5). Parameter estimation is based on the minimi-
zation of the sum of squares of the difference between the observed and the
computed values (Kim et al., 2015, 2016; Umberto et al., 2017).
k
^ ^ X
θ kðÞ ¼ θ k 1Þ + PkðÞ yiðÞΨ iðÞ (5)
ð
i¼n +1
T
k
ð
ð
Pk 1ÞΨ kðÞΨ ðÞPk 1Þ
PkðÞ ¼ Pk 1Þ T (6)
ð
k
1+ Ψ ðÞPk 1ð ÞΨ kðÞ
^
ð
ε kðÞ ¼ ykðÞ θ k 1ÞΨ kðÞ (7)
with:
^
θ kðÞ: estimated parameters,
P(k): adaptation matrix, also called inverse correlation matrix of the input
signal,
y(k): outputs of the system,
ψ(k): observation matrix,
ε(k): error of estimation.
Fig. 7 presents the cross-validation of the handwriting model (Eq. 2). The
blue line (dark gray line in print version) is used to present experimental data
and the dotted purple line (dotted light gray line in print version) presents
the model response. The cross-validation consists of keeping the structure
and parameters of a well-defined model and applying new input-output data
that have not been used for parameter estimation.
This principle was applied to validate the developed models both within
and between writers using either similar or different traces. Multiwriter val-
idation with the same kind of graphic trace provided the highest validation
result (Gene et al., 1979; Prabir, 1989; Sylvain and Matthieu, 2016).
It is important to note that the model structure is fixed whatever the
graphic trace, however, this approach is based on nonphysical parameters,
a ix , b ix , c ix , d ix , a iy , b iy , c iy , and d iy . Therefore, parametric adjustment is
required each time we change input-output data, even if it is the same person
and the same type of graphic trace to be estimated. This leads to the