Page 241 - Glucose Monitoring Devices

Results 245

Finally, based on the Johnson family of distributions [19], we approximated the
probability distribution of the sensor error, using a transformation of the normal
density [20].

Estimation and modeling of sensor error time dependency

Classical time-series techniques were applied to the recalibrated and synchronized
sensor signal to determine the time dependence of sensor errors: the autocorrelation
function and the partial autocorrelation functions.
The autocorrelation function (ACF) is fairly straightforward: it is computed as
the correlation of an error at time t, with the errors at time t þ h, where h ¼ nT,
n is an integer, and T is a fixed time interval (generally, T is set to the time difference
between reference measures). Under weak stationary conditions (the mean and
variance of the error do not depend on time), the ACF is only dependent on the
lag (h) and not on t, and it can be computed using Eq. (12.5):
n h
P
ðε i εÞðε iþh εÞ
n i¼1
gðhÞ¼ (12.5)
n h P n 2
ðε i εÞ
i¼1
The partial autocorrelation function (PACF) can be best described as the corre-
lation between errors at time t and t þ h, h ¼ nT, excluding information transmitted
though t þ T, t þ 2T, t þ 3T, ., t þðn 1ÞT. It is similar to the concept of the
best linear predictor and is commonly computed using the DurbineLevinson
algorithm [21]. For more details on PACF, please refer to Brockwell et al. [22].

Results

Using dataset 1, we studied the sensor response at different reference glucose levels
by estimating the probability distribution of the reference/sensor pairs (recalibrated
but not synchronized). The distribution is presented in Fig. 12.1, where blue depicts
a very low probability and red a very high probability of occurrence. We observed
that sensors tend to read low at high reference values (the reference/sensor pair tends
to fall below the diagonal when the reference is above 200 mg/dL) and high at low
reference levels (the reference/sensor pair tends to fall above the diagonal when the
reference is below 110 mg/dL). In addition, the spread of reference/sensor pairs is
positively correlated with the reference level: the distribution is flatter at high
glucose levels compared to low glucose levels.

Effect of rate of change on sensor error and delay estimation
As presented in the introduction, it is widely believed that there is a delay between
BG and IG. To verify this claim, we applied the same technique described in the