Page 204 - Glucose Monitoring Devices

CGM denoising by Kalman filter 207

Formulation as online self-tunable approach
A priori model for u(t)
Calling back the notation used in Eq. (10.1), if some a priori information on u(t)is
available, an “optimal” filter (i.e., determining the best trade-off between noise
reduction and signal distortion) can be determined by embedding the filtering
problem within a Bayesian context, where the Kalman filter (KF) can be adopted
in the implementation step [22].
An a priori description of the unknown signal is however necessary. A simple but
flexible way to model a smooth signal on a uniformly spaced discrete grid is to
describe it as the realization of the multiple integrations of a white noise process.
The choice of the number of integrators, to the best of our knowledge, cannot be
easily addressed on firm theoretical basis. As a matter of fact, in the smoothing/reg-
ularization literature, the choice of m is normally left to the user or handled on
empirical bases [30]. In our case, u(t) can be reliably described as the double
integration (the so-called integrated random walk model) [29], one has

uðtÞ¼ 2uðt 1Þ uðt 2Þþ wðtÞ (10.6)
2
where w(t) is a zero-mean Gaussian noise with (unknown) variance equal to l . The
choice of m ¼ 2 integrators emerges from a simulation study using a cross-
validation strategy similar to that of [27,30,31]. Bringing Eq. (11.6) into the state
space, two state variables, that is, x 1 (t) ¼ u(t) and x 2 (t) ¼ u(t 1), are needed.
T
Then, in Eq. (11.3) the state space vector at time t becomes x(t) ¼ [x 1 (t) x 2 (t)] ,
and F is consequently given by

2 1
F ¼ (10.7)
1 0
Reminding that the CGM measurement is the only output of the system, the
measurement vector y(t)of Eq. (10.4) becomes a scalar, and H ¼ [1 0]. Updating
Eq. (10.5) for the estimation of b xðtjtÞ, P tjt becomes a 2 2 matrix (with P 0j0 ¼ I 2
T
and b xð0j0Þ¼ [y(0) y( 1)] ), K t a2 1 vector, and Q and R are
" #
2
l 0 2
Q ¼ ; R ¼ s (10.8)
0 0
2
2
To arrive at an estimate of glucose u ˆ(t), both l and s are required. In our
2
2
problem, neither l nor s are known, and we need to estimate their values in real
time from the data. An additional difficulty is that, as it appears from the two panels
of Fig. 10.1, the SNR of CGM data can significantly vary from sensor to sensor and
2 2
from individual to individual. This variability suggests that “universal” l and s
estimates cannot be used for all individuals, but their values need to be individual-
ized, calling for a real time and self-tunable parameter estimation procedure to make
KF really online applicable.
2 2
Determination of l and s
2
2
As seen previously, the l and s estimation problem (i.e., the Q/R ratio estimation)
has been already faced in the literature [24,25]. However, in these studies, parameters