## 3 Proposed Methods
### Multi-Model Integration Strategies  
So far, we have proposed four models for PM\({}_{2.5}\) forecasting. Let the forecasts made by the four models be denoted by \(\hat{z}_{1},\hat{z}_{2},\hat{z}_{3}\), and \(\hat{z}_{4}\), respectively. To combine their forecasts to produce an improved one, five multi-model integration strategies and a lower bound are presented as follows. * Averaged strategy. This strategy simply determines the final forecast as the mean of the four individual forecasts as calculated by \(\hat{z}=\frac{1}{4}\sum\limits_{i=1}^{4}\hat{z}_{i}\). * Weighted strategy. The strategy considers the final forecast as the sum of weighted forecasts made by individual models. Let the prediction RMSE error estimated for the \(i\)th model during its training phase be \(e_{i}\). The normalized reciprocal error is adopted as the weight for the model, i.e., \(w_{i}=e_{i}^{-1}/\sum\limits_{j=1}^{4}e_{j}^{-1}\). The final forecast is then calculated by \(\hat{z}=\sum\limits_{i=1}^{4}w_{i}\hat{z}_{i}\). The rest of the strategies work with the classified value range of PM\({}_{2.5}\). We use the long-term training set to produce the distributions of PM\({}_{2.5}\) concentrations. The 10-percentile and the 90-percentile are used to separate the PM\({}_{2.5}\) concentrations into low, middle, and high ranges. A random forest classifier is trained with the long-term dataset with the labeled ranges. During the test process, the random forest classifier receives the inputs and estimates the most probable value range of the PM\({}_{2.5}\). Let the estimate made by the random forest classifier be denoted as \(C_{\text{rfc}}\), which is L, M, or H, if the classification result is labeled as low, middle, or high PM\({}_{2.5}\) range. We propose three advanced multi-model integration strategies as follows. * Max_Avg_Min strategy. This strategy determines the final forecast according to the classified value range of the PM\({}_{2.5}\). If the test instance is classified as in the high/low range, the maximal/minimal forecast value made by individual models is output as the final forecast. If it is classified in the middle range, the strategy outputs the same forecast value as that made by the Averaged strategy. In other words, the final prediction determined by the Max_Avg_Min strategy can be calculated as follows. \[\hat{z}=\left\{ \begin{array}{ll}\max(\hat{z}_{1},\hat{z}_{2},\hat{z}_{3},\hat {z}_{4})&\text{if }C_{\text{rfc}}=\ ^{\prime}\text{H}^{\prime}\\ \sum\limits_{i=1}^{4}\hat{z}_{i}/4&\text{if }C_{\text{rfc}}=\ ^{\prime}\text{M}^{ \prime}\\ \min(\hat{z}_{1},\hat{z}_{2},\hat{z}_{3},\hat{z}_{4})&\text{if }C_{\text{rfc}}=\ ^{\prime}\text{L}^{ \prime}\end{array} \right.\] (4)
* Max_Wgt_Min strategy. This strategy resembles the Max_Avg_Min strategy by assigning the maximal or minimal forecast value made by individual models as the final forecast if the classified range is high or low. However, if the test instance is classifiedas in the middle range, the strategy outputs the same forecast value as that made by the weighted strategy, i.e., \[\hat{z}=\left\{ \begin{array}{ll}\max(\hat{z}_{1},\hat{z}_{2},\hat{z}_{3},\hat{z} _{4})&\text{if }C_{\text{rfc}}=~{}^{\prime}\text{H}^{\prime}\\ \sum\limits_{i=1}^{4}w_{i}\hat{z}_{i}&\text{if }C_{\text{rfc}}=~{}^{\prime}\text{M}^{ \prime}\\ \min(\hat{z}_{1},\hat{z}_{2},\hat{z}_{3},\hat{z}_{4})&\text{if }C_{\text{rfc}}=~{}^{ \prime}\text{L}^{\prime}\end{array} \right.\] (5)
* Adpt_Wgt strategy. This strategy adopts the adaptive weighting scheme to produce the final forecast. In precise terms, the set of the individual forecast which falls in the classified PM\({}_{2.5}\) range is identified. The final forecast is determined by calculating the sum of the weighted forecasts which are contained in the identified set. So the Adpt_Wgt strategy will adapt to the models which are validated by the random forest. The Adpt_Wgt strategy can be realized by the following formula. \[\hat{z}=\left\{