a linear fit. The systematic trend in AC1 (blue points) is removed and the scatter is reduced from 0.32 to 0.22 dex. This demonstrates the improvement that is possible with the application of non-linear methods. However, a systematic offset remains in the GASS sample (red points) and there is no improvement in the relative systematic error.[]{data-label="fig-alfa-gass-4-par"}](fig-03.pdf){width="8cm" height="6.5cm"}

![Top panel: $\mu_i$ vs. $g-r$ for AC1 (blue) and GASS (red); the two samples are preferentially located in different regions of parameter space. Lower panel: As for Fig. \[fig-alfa-gass-4-par\], but restricting GASS data points to the region $\mu_i>20$ and $g-r<0.6$, where the AC1 sample is mostly located.[]{data-label="fig-gr-mu"}](fig-04-1.pdf "fig:"){width="8.cm" height="6.5cm"} ![Top panel: $\mu_i$ vs. $g-r$ for AC1 (blue) and GASS (red); the two samples are preferentially located in different regions of parameter space. Lower panel: As for Fig. \[fig-alfa-gass-4-par\], but restricting GASS data points to the region $\mu_i>20$ and $g-r<0.6$, where the AC1 sample is mostly located.[]{data-label="fig-gr-mu"}](fig-04-2.pdf "fig:"){width="8cm" height="6.5cm"}

In this section, we have demonstrated that for a given set of input parameters a non-linear approach can yield a significant improvement in the estimation of  over a linear fit to the data. The matrix representation provides a very good option for determining the gas 