## 4 Discussion  
In summary, four new datasets with about 200,000 additional observations were employed to test the basic model of Equation (5) on 18 independent particle sensors (14 PMS 5003 sensors and 4 PMS 6003 sensors). The individual best-fit models performed superlatively well, with intercepts mostly within \(\pm\)0.1 \(\mathrm{\SIUnitSymbolMicro m^{3}}\) and slopes mostly close to 0.99. Even the general models were well within the guidelines of a successful model, with intercept absolute values always less than 111 \(\mathrm{\SIUnitSymbolMicro g}\)/m3 and slopes between 0.95 and 1.06. The four new general models all had estimates for the a and c parameters generally within \(\pm\)10% (Table 14). Because of the small value for the additive constant d, the percentage difference was larger, at about 10-20%. A measure of the fit of a model to observations is the Mean Absolute Error (MAE). MAE results were calculated for the individual models and for the general model fits for the eight sensors in the Santa Rosa site for the 1 January 2023 to 13 July 2023 dataset. For individual sensors, MAEs ranged from 0.17 to 0.42 \(\mathrm{\SIUnitSymbolMicro g}\)/m\({}^{3}\). For the general models, the range was only slightly larger, from 0.20 to 0.52 \(\mathrm{\SIUnitSymbolMicro g}\)/m\({}^{3}\). For the four Flex sensors, the range was from 0.21 to 0.30 for the individual models and 0.23-0.34 for the general models. For the Redwood City 2023 data, the MAEs for the six sensors 7a to 9b ranged between 0.13 and 0.34 \(\mathrm{\SIUnitSymbolMicro g}\)/m\({}^{3}\) for the individual models and between 0.21 and 0.38 \(\mathrm{\SIUnitSymbolMicro g}\)/m\({}^{3}\) for the general model, with the exception of an MAE value of 1.24 \(\mathrm{\SIUnitSymbolMicro g}\)/m\({}^{3}\) for sensor 9a, which had a constant offset of 1.75 \(\mathrm{\SIUnitSymbolMicro g}\)/m\({}^{3}\). Once this offset was subtracted from every measurement, the agreement with sensor 9b was quite good. This was the only case of a bias encountered among the 18 sensors tested. These findings provide support for the original hypothesis in [1] that the CF_1 algorithm has the form shown in Equation (5), in which the numbers of particles N1 and N2 in the two smallest datasets are combined and multiplied by a single parameter. There is also support for the hypothesis that there is an additional additive component d of order \(-\)1 \(\mathrm{\SIUnitSymbolMicro g}\)/m\({}^{3}\). When this value is used in both the individual and general models, the number of negative values in the models almost exactly matches the number of zeros reported by the CF_1 algorithm. This observation suggests a reason for the otherwise mysterious appearance of multiple zeros in CF_1 estimates when in fact the number of particles in N1 is never zero. Precision of the CF_1 algorithm was consistently worse than that for the pm2.5 alt algorithm, with 26-78% of values unable to meet the standard of 20%. For the pm2.5 alt algorithm, 90-99% of values met the standard for eight of the nine monitors, and even the ninth monitor lost only 15% of the values compared to 54% for the Cf_1 algorithm. Accuracy, as found by regressing CF_1 values on the average of the four calibrated sensors, ranged between 60 and 70% overestimates, agreeing with multiple other studies. The limit of detection for the CF_1 algorithm, although not obviously high at 1.22 \(\mathrm{\SIUnitSymbolMicro g}\)/m\({}^{3}\), was more than 10 times higher than the LOD of 0.11 for the pm2.5 alt method, resulting \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Site** & **Year** & **N obs** & **a** & **c** & **d** \\ \hline Santa Rosa 1  & 2020 & 134,800 & 0.0042 & 0.10 & \(-\)1.1 \\ \hline Redwood City & 2021–2022 & 78,900 & 0.0045 & 0.09 & \(-\)1.0 \\ \hline Santa Rosa & 2021 & 52,400 & 0.0040 & 0.11 & \(-\)1.0 \\ \hline Redwood City & 2022–2023 & 39,900 & 0.0045 & 0.09 & \(-\)1.0 \\ \hline Santa Rosa & 2023 & 24,700 & 0.0040 & 0.10 & \(-\)0.8 \\ \hline \hline \end{tabular}
\end{table}