Page 203 - Glucose Monitoring Devices

206 CHAPTER 10 CGM filtering and denoising techniques

Finally, another proposed denoising procedure, in our opinion, more suited to
CGM applications, resorts to Kalman filter (KF). Pioneering applications of KF to
process CGM data were presented by Knobbe et al. [23], with the aim of reconstruct-
ing blood glucose concentration by employing a model of blood-to-interstitium
glucose kinetics and blood glucose concentration references, by Palerm et al.
[24,25], with the aim of predicting the glucose profile and detecting hypoglycemia,
and by Kuure-Kinsey et al. [3], with the purpose of improving CGM calibration.
In the next section, we will provide an application of KF to the GCM denoising
problem originally proposed by Facchinetti et al. [8,9].

CGM denoising by Kalman filter
Overview of the Kalman filter
Briefly, at discrete time, the KF is implemented by first-order difference equations
that recursively estimate the unknown state vector x(t) of a dynamic system exploit-
ing vectors of noisy measurements y(t) causally related to it [22,26]. The process
update equation is given by:
xðt þ 1Þ¼ FxðtÞþ wðtÞ (10.3)
where x(t) has in general size n, w(t) is usually a zero-mean Gaussian noise vector
(size n) with (unknown) covariance matrix Q (size n n), and F is a suitable matrix
(size n n). The state vector x(t) is linked to the measurement vector y(t) (size m)by
the equation:
yðtÞ¼ HxðtÞþ vðtÞ (10.4)
where v(t) is the zero-mean Gaussian noise measurement error vector (size m) with
(unknown) covariance matrix R, and which is uncorrelated with w(t), and H is a
suitable matrix (size m n). The linear minimum variance estimate of the state
vector obtainable from the measurements y(t) collected until time t is indicated by
b xðtjtÞ and can be computed by using the following linear equations:
8
1
> T T T T
> K t ¼ FP F þ Q H H FP F þ Q H þ R
>
< t 1jt 1 t 1jt 1
b xðtjtÞ¼ Fxðt 1jt 1Þþ K t ðyðtÞ Hb xðt 1jt 1ÞÞ (10.5)
>
> T
: P ¼ðI K t HÞ FP F þ Q
>
tjt t 1jt 1
where P tjt (size n n) is the covariance matrix of the estimation error affecting
b xðtjtÞ, K t (size n m) is the Kalman gain matrix, and where P 0j0 and b xð0j0Þ are
the initial conditions. The Q and R matrices, that is, the process and the measurement
noise covariance matrices (respectively), are key parameters in determining the
performance of KF. Unfortunately, Q and R are usually unknown, or sometimes
they are known except for a scale factor. The major problem of KF is the determi-
nation of Q and R, and, more specifically, of the so-called Q/R ratio [22,26].
This problem bears a close resemblance to determining the smoothing parameter
in regularization methods [27e29].