# DERN: Deep Ensemble Learning Model for Short- and Long-Term Prediction of Baltic Dry Index
## 3 Method
### Data Pre-Processing and Analysis  
The BDI data plotted in Figure 1a shows a sharp increase and a dramatic decrease between 2007 and 2008. Therefore, we pre-processed the data in a way to make it more stationary. By using a decomposition technique, the BDI data is separated into three components: trend, seasonality, and noise, as depicted in Figure 1c. The trend can be observed as increasing or decreasing the trend value in the time-series data; however, the BDI data does not show any significant increasing or decreasing trend, but rather a peak in 2008 and two long tails. The seasonality repeats the short-term cycle in the BDI, and the noise corresponds to random variation in the series.  
Due to the complexity of the bulk shipping market and the non-linear nature of freight rates series [31], in this study, some data transformation techniques, including difference transform, power transform, log transform, standardization and normalization, were employed. As indicated in Table 1, for each transformation technique, we conducted a Dickey-Fuller stationary test to ensure its effectiveness.  
Normalization is a rescaling of data so that all values are within a certain range. As shown in Equations (1), (8) and (11), the value range of the \(tanh\) function is between \(-1\) and \(1\); therefore, we shrunk the BDI to this range. Different from normalization, the standardization technique rescales the dataset according to the distribution of values so that the mean of the observed values is 0 and the standard deviation is 1. Further, simple transformation techniques such as power transform and log transform are performed. The 1 st difference transform applied to a time series \(x\) creates a new series \(z\) whose value at time \(t\) is the difference between \(x(t+1)\) and \(x(t)\). This method works very well in removing trends and cycles. As shown in Table 1, '1 st difference transform' results in the  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Data** & **Test Statistics** & \(p\)**-Value** & **Critical Value 1\%** & **Critical Value 5\%** & **Critical Value 10\%** \\ \hline Original data & \(-2.190442\) & \(0.209733\) & \(-3.437455\) & \(-2.864676\) & \(-2.568440\) \\ Normalization & \(-2.190442\) & \(0.209733\) & \(-3.437455\) & \(-2.864676\) & \(-2.568440\) \\ Standardization & \(-2.190442\) & \(0.209733\) & \(-3.437455\) & \(-2.864676\) & \(-2.568440\) \\ Log transform & \(-2.434086\) & \(0.132346\) & \(-3.437348\) & \(-2.864630\) & \(-2.568415\) \\ Power transform & \(-2.546063\) & \(0.104665\) & \(-3.437348\) & \(-2.864630\) & \(-2.568415\) \\
1 st difference transform & \(-7.690264\) & \(1.42\times 10^{-11}\) & \(-3.430000\) & \(-2.864676\) & \(-2.568440\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dickey-Fuller test of pre-processing data.  
Figure 1: **(a) BDI index data; (b) Difference transform of BDI data; (c) Decomposition of BDI data.**  
smallest \(p\)-value, which indicates that it generates more stationary time-series data than the original data. Therefore, in this research, we transformed the BDI data into a more stationary form by using '1 st difference transform', before feeding it into the model. The 1 st-difference-transformed data is plotted in Figure 1b.