## 1 Introduction
### Objectives of this Study  
#### 1.3.1 Objective #1  
A main objective of this study is to rigorously test the recent \"decoded\" CF_1 algorithm [26]. In that article, the CF_1 algorithm was found to be nearly perfectly matched by the following equation:  
\[\text{PM2.5 (CF\_1)}=\text{a(N1+N2)}+\text{cN3}+\text{d} \tag{5}\]  
That is, despite providing specific numbers N1 and N2 of particles in the 0.3-0.5 \(\upmu\)m and 0.5-1 \(\upmu\)m size categories, the CF_1 algorithm instead uses a single coefficient to multiply the sum of the numbers in the two smallest size categories. The best fit to observed CF_1 PM2.5 estimates required an additive component \(d\). Reference [26] used a single six-month data series of collocated PurpleAir monitors inside and outside a Santa Rosa home from 18 June 2020 to 31 December 2020 to test the model in Equation (4) against observed values of PM2.5 provided by the CF_1 algorithm. Four PA-II monitors (eight independent sensors) were used to give eight independent best-fit estimates of PM2.5 as reported using the CF_1 algorithm. The eight individual models were then averaged to give a single general model, which was then applied again to the observed data. The general model (Equation (4)) had the following values for the coefficients: a = 0.0042, c = 0.10, and d = \(-\)1.17 \(\upmu\)g/m\({}^{3}\).  
The individual best-fit models were all in excellent agreement with the CF_1 observations. The general model applied to all cases also gave good results. One interesting finding was that the additive component d was negative and on the order of \(-\)1 \(\upmu\)g/m\({}^{3}\). When the model was compared to the observations, because of this negative value for d, some model estimates of PM2.5 were negative. Interestingly, nearly all of the 18,000 negative results in the model corresponded to values of zero in the observed CF_1 data. This suggested that, indeed, the CF_1 algorithm is of the form in Equation (4), and that rather than report negative concentrations, the [PERSON] approach was to provide zero values instead. This was apparently the first explanation of the otherwise incomprehensible CF_1 estimates of zero, since N1 and N2 are never zero.  
Although the general model developed in [26] was convincingly shown to provide good agreement with the CF_1 algorithm, the agreement was tested against only one fairly old (2020) database for only one site. A reasonable question arises whether the model will hold up if tested at other sites with different PurpleAir monitors and using more recent data.  
Therefore, this study further tests the conclusions of reference [26] by using four additional databases and adding a second site in Redwood City, CA. The newer Santa Rosa data include two new PurpleAir Flex monitors employing four PMS 6003 sensors, adding to the other four monitors to provide extensive data on 12 sensors. The Redwood City databases add three collocated PurpleAir monitors (six independent PMS 5003 sensors). Both datasets include indoor and outdoor measurements. For each of the four datasets, individual models of the CF_1 algorithm for each of the independent sensors are created, and a general model is also estimated. We regress both the individual and general model predictions on the observed CF_1 values and analyze the effectiveness of the individual and general models by their intercepts, slopes, and R\({}^{2}\) correlations resulting from the regressions. We also calculate the Mean Absolute Error (MAE) to test the performance of the individual and general models.