CHAPTER VI. ERROR CORRECTION FOR GLUCOSE BY REDUCING
6.3 Error Correction for Glucose Computed Using a Single Transduced
Unlike previous approach in which the measured glucose value tm is obtained from the handheld device, the approach in this section corrects the glucose value which is estimated from a single point on the transduced current curve. The illustration for this approach is shown in Fig. 6.5.
We must resolve three issues in this approach:
(1) Determine a time-point k in the current curve which is the best point for computing glucose value tm to be closed to tref.
(2) Determine a mapping function p to estimate the best glucose value tm to be closed to tref. This can be described by
tm=p(xk) (6.10) so that it minimizes error function defined by
2 1
( (
N ref
p j
E t p x
=
=∑ − k)) .
Figure 6.5 An illustration of glucose correction measured from a single point on the
(6.11)
(3) Correct the measured glucose tm by reducing effects of hematocrit.
The issues (1) and (2) must be solved simultaneously because the evaluation of results of issue (1) uses results from issue (2).
d points
Transduced current curve
x=[x1 x2 … xd]T
Estimate HCT HCTm
f(x)
Correct the
measured glucose tc
Compute glucose tm by function p: tm=p(xk) (pi is separately determined)
xk tm
tm
Find residual by g(HCTm)
correlation function between hematocrit density and residual
g
Rm tm-Rm
transduced current curve.
Figure 6.6 Plot of the primary reference glucose against current point x57. We can diagnose that there would be a linear relationship between the primary reference
glucose and current-point xk.
Firstly, we should find an appropriate form for mapping function p. One of the simplest methods to diagnose the form of function is using plots. A plot of the primary reference glucose against a current value at a time point is shown in Fig. 6.6 (in this figure, we plot for x57, the current value at time point of 57). From this plot, we can diagnose that there would be a linear relationship between the primary reference glucose and a current-point xk. However, the plots are not ordinary good enough in regression and correlation analysis.
A measure of the linear association between two random variables is the coefficient of correlation. For N samples, the correlation coefficient between the primary reference glucose and a current point xk is given by:
ref k
ref k
( )(
k
covariance of x and t ρ = standard deviation of x standard deviation of t
). (6.12)
Table 6.1 Correlation coefficients between the current points and the primary
reference glucose
Curr.
Point ρ Curr.
Point Ρ Curr.
Point ρ Curr.
Point ρ Curr.
Point ρ x1 0.9743 x13 0.9919 x25 0.9918 x37 0.9910 x49 0.9912
x2 0.9744 x14 0.9918 x26 0.9915 x38 0.9918 x50 0.9912
x3 0.9785 x15 0.9919 x27 0.9913 x39 0.9918 x51 0.9914
x4 0.9842 x16 0.9919 x28 0.9914 x40 0.9917 x52 0.9915
x5 0.9881 x17 0.9919 x29 0.9911 x41 0.9915 x53 0.9910
x6 0.9897 x18 0.9920 x30 0.9912 x42 0.9917 x54 0.9912
x7 0.9906 x19 0.9919 x31 0.9910 x43 0.9916 x55 0.9910
x8 0.9911 x20 0.9918 x32 0.9912 x44 0.9915 x56 0.9910
x9 0.9915 x21 0.9917 x33 0.9910 x45 0.9915 x57 0.9910
x10 0.9915 x22 0.9918 x34 0.9910 x46 0.9916 x58 0.9911
x11 0.9918 x23 0.9916 x35 0.9911 x47 0.9914 x59 0.9910
x12 0.9919 x24 0.9917 x36 0.9911 x48 0.9913
The correlation coefficients corresponding to 59 current points are shown in Table 6.1. We can see that the correlation coefficients corresponding to current points xk
(k≥7) are very high which are larger than 0.99. This indicates that there is strong positive linear association between the primary reference glucose tref and current points xk (k≥7). Therefore, the proposed form for mapping function p can be
tm=p(xk)=apxk+bp. (6.13)
In order to determine parameters ap and bp, N training samples , j=1, 2, … N must be used. The main goal of training process is to find parameters so that
(trefj ,xkj)
Gref=XkAp, (6.14)
where
ref =[t1ref t2ref tNref T] ,
G L
Ap=[ap bp]T and
k1
k2 k
kN
1 1 1
x x x
⎡ ⎤
⎢ ⎥
⎢ ⎥
=⎢ ⎥
⎢ ⎥
⎣ ⎦
X M M .
The least mean square error solution for (6.14) is given by
1
p k k k ref
ˆ =( T )− T
A X X X G . (6.15)
We should validate the aptness of model corresponding to the selected current point. One of the highly useful means of examining the aptness is residual analysis, which is used for checking the model assumptions including that the observations are independent, and the residuals are ( ,ℵ 0 Cε). One of methods for checking the normality of residuals is the normal probability plot of the residuals, in which the plot of residuals against their expected values under normality is used. If the plot is nearly linear then the residual distribution can be considered as normality, whereas the plot that departs substantially from linearity suggests that the residual distribution
is not normal.
In statistical theory, with the ordered residuals, the expected value of the k-th smallest observation can be approximated by
0.375 0.25
MSE z k
N
⎡ ⎛ − ⎤
⎜ ⎞
⎢ ⎝ + ⎟⎠⎥
⎣ ⎦, (6.16)
where MSE stands for the mean square error, N is the number of samples, and z(A) is the Ax100 percentile of the standard normal distribution. Besides visually assessing method for checking the normality of residuals, we can still use a formal test that is correlation coefficient between the residuals and their expected values under normality. A high value of the correlation coefficient suggests agreement with normality. Looney and Gulledge provided a table containing the critical values for the correlation coefficient between ordered residuals and expected values under normality when the distribution of error terms is normal [88, 89]. If the observed correlation coefficient is at least as large as the tabled value for a given level of significance α, the residual distribution is reasonably normal.
Table 6.2 Correlation test for normality corresponding to time points
k ap bp ρr k ap bp ρr
7 6.656192 -34.5669 0.984 34 10.35636 -45.879 0.942 8 7.035924 -37.8326 0.897 35 10.26321 -43.573 0.947 9 7.203581 -36.8866 0.9863 36 10.50545 -46.0491 0.960 10 7.280892 -35.1779 0.981 37 10.56516 -46.6063 0.947 11 7.714908 -40.9227 0.931 38 10.46334 -43.6702 0.984
12 7.887218 -41.2307 0.975 39 10.498 -43.7345 0.979
13 7.951884 -40.0867 0.860 40 10.67922 -44.3428 0.902 14 8.104156 -40.3233 0.895 41 10.83406 -46.39 0.982
15 8.224674 -39.8749 0.976 42 11.02684 -48.0048 0.932 16 8.319754 -39.6246 0.949 43 10.81894 -44.8831 0.967 17 8.53971 -41.7976 0.958 44 11.15158 -47.6565 0.895 18 8.609908 -41.0147 0.938 45 11.07802 -46.4039 0.992
19 8.693864 -40.6115 0.988 46 11.33954 -48.4715 0.963 20 8.812964 -40.914 0.944 47 11.13544 -45.0948 0.980
21 8.984871 -42.8073 0.989 48 11.29035 -46.518 0.949 22 9.111073 -42.7458 0.897 49 11.18573 -44.5573 0.994
23 9.201981 -42.5743 0.986 50 11.40952 -46.4478 0.978 24 9.368159 -44.0848 0.980 51 11.60854 -49.0483 0.917 25 9.52594 -44.4165 0.888 52 11.57473 -46.5594 0.925 26 9.515565 -43.3378 0.946 53 11.38153 -43.8261 0.963 27 9.580175 -42.908 0.993 54 11.61656 -46.5014 0.991
28 9.71734 -43.9198 0.935 55 11.78842 -47.1752 0.948 29 9.95373 -46.8065 0.985 56 11.55217 -43.7301 0.969 30 9.870936 -44.5605 0.983 57 11.96745 -47.6803 0.916 31 9.924776 -42.531 0.928 58 11.98767 -47.7005 0.974 32 10.23267 -47.2393 0.902 59 11.9461 -46.6485 0.983
33 10.10795 -43.4407 0.979
In our case study, 191 samples were collected, of which 70% (134 samples) was used for training to determine the parameters of linear model ap and bp as shown
in (6.13) and the remaining 30% (57 samples) was used to test the assumptions. The computed ap, bp and correlation coefficients corresponding to the current points xk
(k≥7) are shown in Table 6.2. Note that ρr is the correlation coefficient between the residuals and their expected values. We find from Table B.6 in [88] that at the risk of 0.05, the critical value for n=60 (>57) is 0.980, this value is smaller than many observed coefficients which are shown by bold values in Table 6.2. This supports for the conclusion that the distribution of residuals corresponding to some current points does not depart substantially from a normal distribution, which confirms the aptness of model corresponding to the selected current points. In addition, we should choose the time points as small as possible in order to speed up the measuring time of the devices.
Based on this concept and the correlation coefficients in Table 6.1 and 6.2, we propose to use the time point k=19 for the estimation of glucose tm and the proposed mapping function is given by:
tm=8.6939x19-40.6115. (6.17)
However, the glucose values calculated from a single current point on the transduced current curve is still affected by the change of hematocrit. In order to improve its accuracy, the dependence on hematocrit must be reduced. An approach for resolving this issue is identical to that which is presented in section 6.2, except that the residuals are paired-differences of glucose value tm computed with a single current point minus the primary reference glucose measurements tref.