# Dissolved Oxygen Forecasting in Aquaculture: A Hybrid Model Approach
## 3 Methodology
### Proposed Model Design
#### 3.1.1 Ensemble Empirical Mode Decomposition  
The EMD method [28] is a widely applied non-linear signal adaptive decomposition method. The EMD algorithm has demonstrated a great potential in decomposing a non-stationary and non-linear time series data into IMFs and a residual through an iterative process with individual intrinsic time scale properties [17]. Ensemble EMD (EEMD) is an improved version of the EMD methods, developed by [PERSON] et al. [29], which was aimed at overcoming the problem of intrinsic drawbacks of mode mixing that is associated with the conventional EMD algorithm [17]. EEMD is a noise-aided time series data analysis method. In EEMD method of time series data analysis, white noise is added to enable the separation of contrasting time series scales, which in turn, leads to the improved decomposition efficiency of the EMD method. The introduced white-noise is comprised of components of disparate scale which would systematically fill the entire time-frequency space. The disparate scale components of the signal are spontaneously projected onto proper scales of reference initiated by the Gaussian white-noise, as the systematically distributed white-noise is introduced to the signal. Since all the decomposed components of the introduced Gaussian white-noise consist of both the signal and the introduced white noise, all the individual trials usually end up with noisy results. However, the white-noise can be almost completely cancelled out with the aid of ensemble mean of whole trials, because the white-noise in each of the trials are unique in different trials [30]. Consequently, the actual underlying components of the water quality time series data can be represented by the ensemble mean. In other words, EEMD method sums up the components and adopts the average as the true decomposition results. Finally, the result of decomposition solves the mode mixing drawbacks associated with conventional EMD method. It is a useful method for extracting underlying and crucial components from the water quality time series data. For the DO time series data \(x(t)\), the EEMD method follows a certain procedure, which can be described as follows. Stage 1: Initialize an ensemble number \(M\) and the amplitude of the introduced Gaussian white-noise. Stage 2: Perform the \(m^{th}\) trial for introducing disparate white-noise \(W_{m}(t)\) to \(x(t)\), in order to generate the noise-augmented time series data \(x_{m}(t)\), where  
\[x_{m}(t)=x(t)+W_{m}(t) \tag{1}\]  
Stage 3: Determine all the local minima and maxima of \(x_{m}(t)\) and use them to generate both lower and upper envelopes, with the help of cubic spline interpolation functions. Stage 4: Compute the mean \(m_{1}(t)\) of both lower and upper envelopes. Stage 5: Calculate the difference \(h_{1}(t)\) that exists between the mean computed in stage 4 and the signal \(x_{m}(t)\), using,  
\[h_{1}(t)=x_{m}(t)-m_{1}(t) \tag{2}\]  
Stage 6: If the properties of IMF are satisfied by the \(h_{1}(t)\), that is, from the signal \(x_{m}(t)\), \(C_{1}(t)=h_{1}(t)\) becomes the first IMF component. Otherwise, replace \(x_{m}(t)\) with \(h_{1}(t)\) and return to Stage 3. The two properties of IMF are described as follows: (i) the number of the zero crossing and extrema must either equal or differ at most by 1 over the entire data \(x(t)\), and (ii) at any given point, the mean value \(h_{1}(t)\) of the generated envelopes given by both local minimum and local maximum must be zero. Stage 7: Separate the residue \(R_{1}(t)\) from the rest of the dataset using,  
\[R_{1}(t)=x_{m}(t)-C_{1}(t) \tag{3}\]  
Let the residue \(R_{1}(t)\) be a new signal and sift out the remaining IMFs by repeating Stage 3 through Stage 7 \(n\) times, until the stopping criterion is satisfied. The applied stopping criterion can be either of the following: (i) when the residue \(R_{n}(t)\) is reduced to a monotonic function such that no more IMF can be extracted from it; (ii) when the residue \(R_{n}(t)\) or IMF component \(C_{1}(t)\) becomes smaller than the predetermined value.