out that these models are quite heavy to deal with, in
part because, for each, the order is unknown and the
parameters are unknown and numerous.
Our search for better RP models in the context of
drowsiness monitoring led us to examine the Geomet-
ric Brownian Motion (GBM) RP model (Jeanblanc
et al., 2009). A preliminary investigation indicated
that the GBM RP model could be very appropriate for
the signals found in the context of drowsiness moni-
toring, such as a PERCLOS signal and an LoD signal.
The main goal of the present paper is to describe
the work that we did with real subjects in several
states of sleep deprivation to establish that the GBM
RP model appears to be a good, promising choice of
RP model to describe the LoD signals produced by a
specific, validated, POG-based drowsiness quantifica-
tion instrument, at least based on the data we had.
The GBM RP model lies at the heart of this paper.
We now give a brief definition of what a GBM RP is.
As one shall see, the notion of a GBM RP is rooted in
significantly advanced mathematics (Jeanblanc et al.,
2009).
A continuous-time RP X(t) is said to be a GBM,
or GBM - i.e. to follow a GBM (RP) model - if it
satisfies the stochastic differential equation (SDE)
dX(t)
X(t)
= µdt + σdW (t), (1)
where µ is a fixed, real-valued parameter, σ a fixed,
real, positive parameter, and W (t) a Weiner (random)
process also called Brownian Motion (BM) (Jean-
blanc et al., 2009).
The left side of Equation 1 is the relative incre-
ment of X(t) in the period of time [t,t + dt], i.e.
(X(t + dt) − X(t))/X(t). The right side of this equa-
tion shows that this relative increment has a determin-
istic linear trend µdt that is disturbed by a random
noise term σdW (t). The constant µ is the so-called
“drift” (or “mean rate of return” in financial mathe-
matics), and σ is the so-called “volatility”.
Recall that the goal of this paper is to show that
GBM is a good RP model for real-life LoD signals.
2 METHODS
We used data from two laboratory-based studies, re-
ferred to here as Study A and Study B. Both stud-
ies used the same overall (experimental) protocol,
and they differed only by (1) the groups of partici-
pants/subjects who took part in each study, and (2)
the nature of the tests/tasks that each participant was
asked to submit to in each study.
2.1 Participants
We recorded experimental data from N = 30 healthy
participants aged 19-33. Study A contributed 13 sub-
jects (mean age: 23.7; 7 men, 6 women), and Study B
17 subjects (mean age: 22.7; 8 men, 9 women).
2.2 Protocol
In each of the two studies (A and B), the correspond-
ing participants were each asked to submit to three
successive, time-separated test sessions in different
sleep-deprivation conditions over two days. During
each test session, the LoD signal of each participant
was produced using a drowsiness monitoring system
designed, built, and validated by our team.
In Study A, each test session consisted in driving
in a high-fidelity driving simulator; the three succes-
sive sessions had durations of 45, 45, and 60 minutes.
In Study B, each test session consisted in performing
a Psychomotor Vigilance Test (PVT); the three suc-
cessive sessions all had durations of 10 minutes.
For ease of explanation, the overall two-day ex-
periment for each participant (for either type of test)
can be viewed as the succession of Night 1, Day 1,
Night 2, and Day 2, and as consisting of three succes-
sive test sessions. Figure 1 provides an illustration of
the overall protocol used for both studies. On Night
1, the participant slept at home and was asked to re-
port the number of hours of sleep using a sleep diary
(mean ± standard deviation for all participants is 7.57
± 0.8 h of sleep, range 6.5–9.0 h). Then, the partic-
ipant was not allowed to sleep from the time he/she
woke up on Day 1 until the end of the study (12:00
noon on Day 2). (All times are in 24 h notation.) At
8:00 on Day 1, the participant arrived at our labora-
tory and submitted to the first test session, between
8:00 and 10:00. The participant was then free to leave
the laboratory to carry out his/her normal activities
but was equipped with an actigraph (either Actiwatch
2 or Philips Respironics) in order to check that he/she
had not slept while away. The participant came back
to our laboratory at 20:30 on Day 1. On Night 2,
the participant submitted to the second test session
between 2:00 and 4:00 and, after breakfast on Day
2, he/she submitted to the third test session between
11:00 and 13:00 (and after at least 28 hours of sleep
deprivation). At the end of the study, the participant
was sent back home. From noon on Day 1 until the
end of third test session, the participant was asked not
to consume any stimulant (coffee, tea, etc.). This pro-
tocol was approved by the Ethics Committee of the
University of Li
`
ege (Franc¸ois et al., 2016).
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