## 3 Results
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Quercetin Derivative**} & \multicolumn{2}{c}{_Range_} & \multirow{2}{*}{**Equation**} & \multirow{2}{*}{**r2**} & \multicolumn{2}{c}{**LOD**} & \multicolumn{1}{c}{**LOQ**} \\  & \multicolumn{2}{c}{**mg L-1**} & & & \multicolumn{1}{c}{**mg L-1**} & \multicolumn{1}{c}{**mg L-1**} \\ \hline quercetin-3-glucoside & 1–200 & Y = 0.001x + 0.0055 & 0.9972 & 1.17 & 3.54 \\ quercetin-3-galactoside & 1–100 & Y = 0.001x + 0.0031 & 0.9932 & 0.22 & 0.67 \\ quercetin-3-rhamnoside & 1–200 & Y = 0.0013x + 0.00003 & 0.9985 & 0.17 & 0.52 \\ \hline \hline \end{tabular}  
Results are based on two replicate samples of each standard concentration, each measured twice (n = 4).  
\end{table} 
Table 1: Linearity, limit of detection (LOD), limit of quantification (LOQ) of Folin-Ciocalteu spectrophotometric method for the determination of quercetin derivatives. Those results were further analyzed with [PERSON], Dubinin-Radushkevich and Hill adsorption isotherm equations using non-linear, improved non-linear and linear regression. To aid in understanding the improvement compared to standard non-linear regression consider Figure 2a. The data in the adsorption diagram (\(q_{\mathrm{e}}\)_vs. c_e) for one polyphenol concentration, measured several times are visible in the diagonal line corresponding to the conservation of mass constraint as previously mentioned. The value of \(q_{\mathrm{e}}\) predicted by the model (\(q_{\mathrm{e,model}}\)) should follow that constraint and be predicted on the diagonal line. However, the standard non-linear regression does not predict \(q_{\mathrm{e,model}}\) on that diagonal. Instead it drops vertically to a point that does not satisfy the mass constraint. In doing so, the standard non-linear regression residual is always larger than the improved non-linear regression residuals, for adsorption curves that are increasing functions of \(c_{\mathrm{e}}\). Moreover, as long as the adsorption curve is strictly increasing, use of the standard least squares will produce incorrect fitted maximum adsorption capacities (\(q_{\mathrm{m}}\) and \(q_{\mathrm{s}}\)) that remain incorrect by a constant factor, even in the limit of a large number of experimental observations (this is its statistical inconsistency for the correlated error setting of adsorption curve fitting). The same remains true even for linear curve fitting, if the observations occur along the diagonals specified by the initial polyphenol concentrations. Consequently, we developed the improved non-linear regression code in the R programming language that predicts the \(q_{\mathrm{e,model}}\) value at the intersection points of the curves and the diagonal lines of the data. Motivation for developing this improved way of non-linear modeling of adsorption data is not only that it gives smaller residuals, but also that it gives, for normal error distributions, a provably statistically efficient standard error asymptotically, superior to those of other consistent procedures. Figure 1: Adsorption of quercetin derivatives onto \(\beta\)-glucan. Figure 2: (**a**) Theory of non-linear regression of adsorption data with standard non-linear regression and novel improved non-linear regression. (**b**) amplified figure of example of novel improved non-linear regression of Dubinin-Radushkevich isotherms (quercetin-3-galactosidase adsorption onto \(\beta\)-glucan). As an example of the novel non-linear regression consider Figure 2b. It shows the method in this study indeed produces the \(q_{\text{e,model}}\) values at the intersections of the curve with the experimental data lines (the special curve points). These diagonal lines also can be seen in the confidence intervals of the model fits in Figure 2b. From Figure 2b it can be seen that the difference between curves obtained with improved and standard fitting is not great. This occurs when the slope of the fitted curve is not great. For adsorption isotherm, the low slope condition occurs in the settings of large concentrations.