# Integrated Predictor Based on Decomposition Mechanism for PM2.5 Long-Term Prediction
## 2 Distributed Decomposition Model
### Decomposition  
Assume that PM2.5 time-series data \(Y_{t}\) has \(N\) data, which means \(t=1,2,\ldots,N\). The relation with \(Y_{t}\) and its three independent components (i.e., trend, period, and residual), as shown in Equation (1).  
\[Y_{t}=T_{t}+S_{t}+R_{t}\quad t=1,2,\ldots,N \tag{1}\]  
Figure 2: Flowchart of the prediction framework.  
where \(T_{t}\), \(S_{t}\), and \(R_{t}\) are the trend component, period component, and residual component, respectively. Then, we have to obtain these three components \(T_{t}\), \(S_{t}\), and \(R_{t}\) from the one time-series data \(Y_{t}\), but this is mathematically an unsolvable equation.  
Each component has different frequency bands, that is, the trend and period component are in the low and middle band, respectively. We used the cycle calculation process, shown in Figure 3, to calculate the trend and period components until these two components tended to be stable and no longer changed. In order to start the loop operation, we assumed \(T_{t}\) = 0 in the first loop. The time series is fitted iteratively until the trend and period component stabilize, and at the end of the cycle calculation process, the trend component \(T_{t}\) and period component \(S_{t}\) are extracted from the data series, and the residual component is obtained by  
\[R_{t}=Y_{t}-T_{t}-S_{t},t=1,2,\ldots,N \tag{2}\]  
Figure 3 shows that the decomposition has two important steps: one is \"Remove high frequencies\", and the other is \"Remove low frequencies\". Next, we will discuss the method of these two steps.  
To remove the high frequencies, we used the locally weighted scatterplot smoothing (LOESS) smoother, which is based on fitting a weighted polynomial regression for a given time of observation, where weights decrease with distance from the nearest neighbor [30]. LOESS is a combination of the local fitting of polynomials and iteratively weighted least squares. At each point \(x\) in the dataset, a linear or quadratic polynomial is fit using the weighted least squares, giving more weight to points near point \(x\) that need to be smoothed and less weight to points further away.  
Suppose \(x\) is the point to be smoothed, \(i\) is the number of data points around \(x\) to be smoothed, and \(x^{(i)}\) is the point around \(x\) within the width \(i\). We chose a Gaussian function as the weighting factor \(\omega^{(i)}\)  
\[\omega^{(i)}=\exp(-\frac{\left(x^{(i)}-x\right)^{2}}{2\tau^{2}}) \tag{3}\]  
where \(\tau\) is a constant to be set. It is always set according to the number of the data, and for PM2.5, which is a complex nonlinear data, we suggest that it is set to \(\tau\) as within [3; 19]. The value of the regression function for the point is then obtained by evaluating the local polynomial for that data point. By minimizing the value of the following  
\[F=\sum_{i}\omega^{(i)}\left(y^{(i)}-\theta^{T}x^{(i)}\right)^{2} \tag{4}\]  
Figure 3: Flowchart of decomposition.  
to find the \(\theta\) parameter, then the smoothed value \(\hat{y}\) of point \(x\) can be obtained by  
\[\hat{y}=\theta^{T}x \tag{5}\]  
and the LOESS fit is complete after the regression function values have been computed for each of the data points.  
On the other hand, we used a high-pass filter with cutoff frequency to remove low frequency bands. Obviously, it is critical to ensure that the loop converges to the trend component and the period component. We chose the so called seasonal-trend decomposition procedure based on LOESS (STL) [31] to achieve the loop convergence.  
The STL method can adaptively adjust parameters according to each cycle calculation, so it can obtain good performance in ensuring convergence, and has many applications in data decomposition [32]. Here, we have omitted the details of the STL in order to focus our work (please refer [31] for the details of STL).