# Multi-Attention Recurrent Neural Network for Multi-Step Prediction of ChlorophyllI Concentration
## 3 Baseline Methods and Results
### Results  
**Comparison with baselines** For fairness reasons, we set the length of the input sequences of all models to 24 (i.e., time window size \(T=24\)) and gradually increase the length of the output sequences (i.e., the value of forecasting horizons \(\mathbf{\tau}\)) along the set {1, 3, 6, 12, 24}. This means that when the value of \(\mathbf{\tau}\) increases from 1 to 24, the model predicts the chlorophyll concentration within the next 30 to 720 min. We list the results of MaRNN and baseline methods based on different prediction horizons in Tables 1 and 2, with the best performance highlighted in bold. According to Tables 1 and 2, the MaRNN model is more competent than the baseline method for the multi-step prediction of chlorophyll concentration. We also observe that the predictive performance of both the MaRNN model and baseline method decrease as \(\mathbf{\tau}\) gradually increases along the set {1, 3, 6, 12, 24}. As we know, a time series is a set of random variables arranged in the order in which the data is produced, which is usually the sampling of a certain phenomenon according to a given frequency. Obviously, time series data describes a highly nonlinear time-varying process, and its uncertainty will intensify with the increase of time intervals. Therefore, the further away the observation period (time window) from the time point, the more difficult it is to predict, which should be the reason for the decrease in model performance due to the increase in predicted horizon. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{2}{c}{**Horizon = 1**} & \multicolumn{2}{c}{**Horizon = 3**} & \multicolumn{2}{c}{**Horizon = 6**} & \multicolumn{2}{c}{**Horizon = 12**} & \multicolumn{2}{c}{**Horizon = 24**} \\ \cline{2-11}  & **MAE** & **MAPE** & **MAE** & **MAPE** & **MAE** & **MAPE** & **MAE** & **MAPE** \\ \hline MTSMFF & 0.4850 & 0.2104 & 0.5702 & 0.2697 & 0.7895 & 0.3367 & 0.8522 & 0.4413 & 0.9539 & 0.5238 \\ DSTP-RNN & 0.3856 & 0.1741 & **0.4796** & 0.2148 & 0.5430 & 0.2555 & 0.5907 & **0.2647** & 0.7517 & 0.4109 \\ CGA-LSTM & 0.3701 & **0.1676** & 0.4963 & 0.2462 & 0.5758 & 0.2647 & 0.6526 & 0.3258 & 0.6642 & 0.3398 \\ DyAt-Nets & 0.4384 & 0.2055 & 0.5549 & 0.2523 & 0.5889 & 0.2790 & 0.6365 & 0.3129 & 0.8205 & 0.4609 \\ HCA-GRU & 0.3900 & 0.1675 & 0.4849 & 0.2223 & 0.5652 & 0.2717 & 0.6173 & 0.3211 & 0.6393 & 0.3443 \\ MaRNN & **0.3633** & 0.1685 & 0.4803 & **0.2202** & **0.5327** & **0.2532** & **0.5813** & 0.2761 & **0.6212** & **0.3182** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance of different methods compared in WWA_Data for different predictive horizons. The statistical performance data are from Tables 1 and 2, and the results demonstrate that the MaRNN achieves the best performance in most application cases (14 out of 20 cases). This indicates that the increase in \(\tau\) has a relatively small negative impact on the MaRNN model. In addition, we visualize the above methods in the application scenario of \(\tau=24\) to analyze and compare the performance of MaRNN and baseline methods at different prediction points, as shown in Figure 2. In Figure 2, the values on the \(x\)-axis represent different prediction points in a given predicted horizon, while the y-value represents the MAE of different methods. Taking the red curve as an example, for a point where the \(x\)-axis value is equal to 1, it represents the predictive performance of MaRNN at the first time step when the predicted horizon \(\tau=24\). Intuitively, compared with baseline methods based on deep learning (MTSMF, DSTP-RNN, DyAt-Nets, HCA-GRU, and CGA-LSTM), the MaRNN proposed in this paper has more advantages in practical applications of chlorophyll concentration prediction tasks. **Effects of different factors on prediction** In Figure 3, we show the weight distribution of exogenous factors for WWA_Data and TAB_Data. The \(x\)-axis denotes different time steps, while the \(y\)-axis represents the attention weights obtained by the factors.