# Overcoming Individual Discrepancies, a Learning Model for Non-Invasive Blood Glucose Measurement
## 5 Algorithm of NGM
### Independent Feature Converter  
IFC is used to transform absorbance vector \(A\) to independent components concentration vector \(C^{*}\). According to Equation (5), any component concentration can be nearly calculated by multiplication of the inverse of coefficient matrix \(\Theta^{-1}\) with absorbance vector \(A\). The key is determining the coefficient matrix \(\Theta\) that is hard to measure and, to some extent, varies from person to person. Although \(\Theta\) cannot be determined accurately, the approximate matrix can be figured out by Independent Component Analysis (ICA) [17] based on the training data set. ICA is an effective liner-transformation algorithm in multivariate and linear algebra where a matrix is factorized into two matrices, with the property that can extract independent components from a complex data matrix [17]. These independent components make the resulting matrices easier to inspect. In this paper, the two resulting matrices represent independent feature matrix \(F\) and coefficient matrix \(\Theta\) respectively. The algorithm of ICA in this paper is called fast fixed-point algorithm [27], which uses maximum entropy to determine factorized results. Figure 4: The Schematic Diagram of the Non-invasive Glucose Measurement (NGM) algorithm consisting of the Independent Feature Converter (IFC) and Glucose Estimator (GE). Figure 3: The demonstration of using the prototype of blood glucose sensor, which is used to clamp the earlobe from the upper and lower sides to allow the incident light to penetrate the tissue and the transmitted light to be received by the light receptor. In the ICA, there is a basic premise that all the component concentrations are independent. This premise is not very strong, thus making some errors in BGC estimating, but the premise is very useful and the error can be compensated by the following GE. First, we get an absorbance data matrix \(T\) from the training dataset, as  
\[T= \begin{bmatrix}A_{1}^{T}\\ A_{2}^{T}\\ \vdots\\ A_{M}^{T}\end{bmatrix} = \begin{bmatrix}a_{1}(\lambda_{1})&a_{1}(\lambda_{2})& \cdots&a_{1}(\lambda_{N})\\ a_{2}(\lambda_{1})&a_{2}(\lambda_{2})&\cdots&a_{2}(\lambda_{N})\\ \vdots&\vdots&\ddots&\vdots\\ a_{M}(\lambda_{1})&a_{M}(\lambda_{2})&\cdots&a_{M}(\lambda_{N})\end{bmatrix} , \tag{8}\]  
where \(A_{i}^{T}\) means the transpose of the absorbance vector for the \(i\)-th record in training dataset. Based on Equation (5), the matrix \(T\) can be factorized to \(F\) and \(\Theta\)  
\[T_{M\times N}=F_{M\times N}\cdot\Theta_{N\times N}, \tag{9}\]  
where  
\[F= \begin{bmatrix}C_{1}^{T}\\ C_{2}^{T}\\ \vdots\\ C_{M}^{T}\end{bmatrix} = \begin{bmatrix}c_{1}(1)&c_{2}(1)&\cdots&c_{1}(1)\\ c_{1}(2)&c_{2}(2)&\cdots&c_{2}(2)\\ \vdots&\vdots&\ddots&\vdots\\ c_{1}(M)&c_{2}(M)&\cdots&c_{N}(M)\end{bmatrix} . \tag{10}\]  
\(C_{i}^{T}\) means the transpose of concentration vector, and \(c_{j}(i)\) is the \(j\)-th component concentration of the \(i\)-th record. \(F\) is called independent feature matrix which is immeasurable. Matrix \(\Theta\) is the same coefficient matrix in Equation (5). Since the physical significance of the decomposition of matrix \(T\) has been elucidated, let us start discussing how to decompose the matrix \(T\). The principle of decomposition is to make matrix \(F\) contain as much information as possible. Considering \(A\) and \(C\) as random variables, matrix \(F\) contains the most information when random variable \(C\) obeys a Gaussian distribution according to the information theory. But [17] has pointed out that the signal analyzed by ICA cannot be subject to a Gaussian distribution and has proposed an alternative--the sigmoid function. Let \(C\) subject to a distribution whose probability density function (PDF) is the first order derivative of sigmoid function, as  
\[f_{c}(C)=\textit{sigmoid}\left(C\right)=\frac{e^{C}}{(1+e^{C})^{2}}. \tag{11}\]  
where \(f_{c}(C)\) is the PDF of vector \(C\) and \(\textit{sigmoid}\left(C\right)\) is first order derivative of the Gaussian distribution function.