## 5 Results
### Determination of the PAR  
Our approach is to use a linear model to calculate the PPFD from the raw output of the AS7341 channels. With the obtained data, including the PPFD of the reference, we use multiple linear regression (MLR) to find the parameters of a linear conversion function of the form  
\[y_{i}=b_{0}+\sum_{i=1}^{n}(b_{i}x_{i})+e. \tag{1}\]  
In Equation (1), each channel output of the AS7341 corresponds to a single explanatory variable \(x_{i}\). The reference value of the PAR sensor acts as the dependent variable (\(y_{i}\)). The slope coefficients \(b_{i}\) for each explanatory are then determined with the MLR. The set of 50 measurement points and usage of the eight channels lead to the derived slope coefficients summarized in Table 1. The model has an adjusted coefficient of determination of 0.991 and a root-mean-square error of \(16\,\mathrm{\SIUnitSymbolMicro mol\,m^{-1}\,s^{-2}}\). Both values indicate that the usage of the AS7341 output is suitable to calculate the PAR with a simple MLR algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(b_{0}\) & \(b_{1}\) & \(b_{2}\) & \(b_{3}\) & \(b_{4}\) \\ \hline \(-9.8853\) & \(0.0046\) & \(0.0136\) & \(0.0243\) & \(0.0459\) \\ \hline \(b_{5}\) & \(b_{6}\) & \(b_{7}\) & \(b_{8}\) & \\ \hline \(-0.0471\) & \(0.0195\) & \(0.0178\) & \(-0.0026\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Resulting slope coefficients of the multiple linear regression including the AS7341’s channels F1–F8 for the complete data set. Figure 4: Three example data sets measured with the AS7341 in the insulator box. In each set, either a narrow-bandwidth LED with violet (\(425\,\mathrm{nm}\)), green (\(530\,\mathrm{nm}\)), or red (\(629\,\mathrm{nm}\)) emission (in the figure shown from left to right) is turned on. As expected, channels corresponding to the LED spectrum show a high signal. The AS7341’s clear channel shows an output for every LED due to its broadband sensitivity. The near infrared (NIR) channel, sensitive in the near infrared, shows as expected negligible output. However, the experimental data set is not ideal to achieve a representative model built via the MLR. Low light conditions are over-represented as shown in Figure 5. When measuring outside under clear sky conditions, a PPFD in the range of \(300\,\mathrm{\SIUnitSymbolMicro m}^{-1}\,\mathrm{s}^{-2}\) to \(550\,\mathrm{\SIUnitSymbolMicro m}^{-1}\,\mathrm{s}^{-2}\) was determined. This matches the results of [PERSON] et al., who have been measuring the PPFD during a diurnal course in Basel, Switzerland, with a spectrometer in November 2018 [6]. For a calibration that is valid over the whole possible light intensity range, further calibration is needed at higher PPFD. The determined residuals are slightly lower, except for some outliers in data sets gathered outside where only sunlight is recorded (Figure 5) compared to those taken in the box. This may be due to the usage of very narrow-band LEDs in the insulator box, which we expect were not measured accurately by our PAR reference. In addition to the MLR model, using the eight channels sensitive in the PAR wavelength region, all other possible combinations, including the NIR and clear channels, are used to calculate a regression model for the PPFD. The models are then sorted by their adj. \(R^{2}\) and RMSE and a f-test is performed. This test is indicating whether the model is valid beyond our samples. All combinations showed a low p-value below 0.01, stating that there is a good linear relationship between the channel output and the PPFD. The lowest adj. \(R^{2}\) of 0.994 and RMSE of \(12\,\mathrm{\SIUnitSymbolMicro m}^{-1}\,\mathrm{s}^{-2}\) is obtained for the solely inclusion of the channels one, three, six, eight and further the NIR. This is remarkable, as the near infrared channel is, as said, only sensitive in the infrared. In an ideal case, the NIR output should be negligible, since, according to the datasheet, the reference PAR sensor is not sensitive in this region.