# Overcoming Individual Discrepancies, a Learning Model for Non-Invasive Blood Glucose Measurement
## 3 Theoretical Analysis
### Earlobe Model  
The measuring principle of BGC in this paper is based on the Beer-Lambert law [24], which relates the light attenuation to the properties of the material through. The Beer-Lambert law can be described in  
\[a(\lambda)=\sum_{i=1}^{N}a_{i}=\sum_{i=1}^{N}\varepsilon_{i}(\lambda)\int_{0} ^{l}c_{i}(z)dz \tag{1}\]  
\[a(\lambda)=lg\frac{I_{0}}{I} \tag{2}\]  
where \(a(\lambda)\) is the absorbance of light at wavelength of \(\lambda\). \(\varepsilon_{i}(\lambda)\) is the molar attenuation coefficient for \(\lambda\) wavelength light of the attenuating species \(i\) in the material sample. \(c_{i}\) is the amount concentration of the attenuating species \(i\) in the material sample. \(l\) is the path length of the beam of light through the material sample. \(I_{0}\) and \(I\) represent the intensity of incident and output light.  
Taking the earlobe as a material sample, Equation (3) can be simplified as  
\[a(\lambda)=\sum_{i=1}^{N}\varepsilon_{i}(\lambda)c_{i}l^{{}^{\prime}} \tag{3}\]  
where \(l^{{}^{\prime}}\) is the equivalent thickness of the earlobe, which varies from person to person.  
There are many attenuating species in the earlobe, e.g., glucose, water, fat, hemoglobin, etc. These species can be represented by symbol \(i\) in Equation (3), and let 1 represent glucose. It is remarkable that the concentrations of these species are dissimilar from everyone and change with time. These individual discrepancies should be regarded as unknown variables. According to linearity algebra, multi unknowns can not be solved by one equation. To solve the \(c_{i}(i=1,2,...\ N)\), we need at least \(N\) equations, which means that absorbance of \(N\) different wavelengths of light have to be measured to set up a system of linear equation as  
\[\left\{\begin{aligned} a(\lambda_{1})&= \varepsilon_{1}(\lambda_{1})c_{1}l^{{}^{\prime}}+\varepsilon_{2}(\lambda_{1}) c_{2}l^{{}^{\prime}}+...+\varepsilon_{N}(\lambda_{1})c_{N}l^{{}^{\prime}}\\ a(\lambda_{2})&=\varepsilon_{1}(\lambda_{2})c_{1}l^{{ }^{\prime}}+\varepsilon_{2}(\lambda_{2})c_{2}l^{{}^{\prime}}+...+\varepsilon_ {N}(\lambda_{2})c_{N}l^{{}^{\prime}}\\ a(\lambda_{N})&=\varepsilon_{1}(\lambda_{N})c_{1}l^{{ }^{\prime}}+\varepsilon_{2}(\lambda_{N})c_{2}l^{{}^{\prime}}+...+\varepsilon_ {N}(\lambda_{N})c_{N}l^{{}^{\prime}}\end{aligned}\right. \tag{4}\]  
which can be established in the form of matrix  
\[A=\Theta\cdot C \tag{5}\]  
where \(A\) and \(C\) are absorbance vector \([a(\lambda_{i})]_{N\times 1}\) and concentrations vector \([c_{i}]_{N\times 1}\). \(\Theta\) is a coefficient matrix \([\varepsilon_{j}(\lambda_{i})l^{{}^{\prime}}]_{N\times N}\).  
In theory, as long as absorbance vector \(A\) has been measured, the blood glucose concentration \(c_{1}\) can be worked out by Cramer's Rule as  
\[c_{1}=\frac{det(\Theta_{1})}{det(\Theta)} \tag{6}\]  
where \(\Theta_{1}\) is the matrix formed by replacing the 1-th column of \(\Theta\) by the absorbance vector \(A\).  
There is still an essential question that is how to determine the coefficient matrix \(\Theta\) which varies from person to person. We will solve this question by ICA and RF in Section 5.