Thus, we used another multiple regression analysis, which is called MRA2, to estimate the values of $CHbO$, $CHbR$, and $a$ from the regression coefficients $\alpha HbO$, $\alpha HbR$, and $\alpha 0$ that were obtained from MRA1. In MRA2, $CHbO$, $CHbR$, and $a$ were regarded as response variables, and the three regression coefficients $\alpha HbO$, $\alpha HbR$, and $\alpha 0$ in Eq. (4) were regarded as predictor variables. Their relations are written as $CHbO=\beta HbO\xb7\alpha $, $CHbR=\beta HbR\xb7\alpha $, and $a=\beta a\xb7\alpha $, where $\alpha =[1,\alpha HbO,\alpha HbR,\alpha 0]T$, $\beta HbO=[\beta HbO,0,\beta HbO,1,\beta HbO,2,\beta HbO,3]$, $\beta HbR=[\beta HbR,0,\beta HbR,1,\beta HbR,2,\beta HbR,3]$, and $\beta a=[\beta a,0,\beta a,1,\beta a,2,\beta a,3]$. The symbol $[\u2009]T$ represents the transposition of a vector. The coefficients $\beta HbO,i$, $\beta HbR,i$, and $\beta a,i(i=0,1,2,3)$ are unknown and must be determined before analysis. We adopted MCS as the foundation to establish reliable values of $\beta HbO,i$, $\beta HbR,i$, and $\beta a,i$ The simulation model used in this part also consisted of a skull layer and cortical tissue layer with the source-and-detection fibers geometry. The absorption coefficients $\mu a$ converted from the concentrations $CHbO$ and $CHbR$ and the reduced scattering coefficient $\mu s\u2032$ specified by the coefficient $a$ were provided as inputs to the simulation, while the diffuse reflectance was produced as output. The input concentrations and coefficient $a$ and the output reflectance are helpful as the data set in specifying the values of $\beta HbO,i$, $\beta HbR,i$, and $\beta a,i$ statistically for determining the absolute concentrations and coefficient $a$. The five different values of 40,172, 60,258, 80,344, 100,430, and 120,516 were calculated by multiplying the typical value^{18} of $a$ by 0.5, 0.75, 1.0, 1.25, and 1.5, respectively, and the reduced scattering coefficients $\mu s\u2032(\lambda )$ of the cortical tissue with the five different values were derived from the relation of Eq. (3). The sum of the absorption coefficients of oxyhemoglobin and deoxyhemoglobin $\mu a,HbO+\mu a,HbR(\mu a,HbT)$ for $CHbT=0.2$, 1.0, and 5.0 vol.% were used as input to the cortical tissue in the simulation. Tissue oxygen saturation $StO2$ was determined by $\mu a,HbO/\mu a,HbT$, and values of 0%, 20%, 40%, 60%, 80%, and 100% were used for simulation. For the scattering and absorption properties of the skull, the refractive index of each layer, and the source and detection geometries, the same values and conditions as those described above were again employed. In total, 90 diffuse-reflectance spectra at $\lambda =500$, 520, 540, 560, 570, 580, 584, 600, 605, 730, 760, 790, 800, 805, 830, and 850 nm were simulated under the various combinations of $CHbO$, $CHbR$, and $a$. MRA1 for each simulated spectrum based on Eq. (4) generated the 90 sets of vector $\alpha $ and concentrations $CHbO$, $CHbO$, and $a$. The coefficient vectors $\beta HbO$, $\beta HbR$, and $\beta a$ were determined statistically by performing MRA2. Once $\beta HbO$, $\beta HbR$, and $\beta a$ were obtained, $CHbO$, $CHbR$, and $a$ were calculated from $\alpha HbO$, $\alpha HbR$, and $\alpha 0$, which were derived from MRA1 for the measured reflectance spectrum, without the MCS. Therefore, the spectrum of absorption coefficient $\mu a(\lambda )$ and that of reduced scattering coefficient $\mu s\u2032(\lambda )$ were reconstructed by Eqs. (2) and (3), respectively, from the measured reflectance spectrum.