## 3 Proposed Approach
### Multivariate Empirical Mode Decomposition  
EMD, as a data-driven and adaptive signal decomposition technique, is well suited to handle non-linear and non-stationary timeseries, with which a number of oscillatory modes (i.e., IMFs) are decomposed [21]. Comparing with the original system response, the degree of non-stationarity is reduced greatly in such oscillatory modes, hence, they are comparatively easy to identify. In this sense, EMD can be regarded as a tool for feature extraction before the process modeling, and the IFGD process modeling problem reduces to identifying each extracted subprocess. The process decomposition puts forward higher requirements for EMD, since both input and output signals are required for processing. The BEMD, which has the ability to handle cross-channel property in multivariate data, is chosen to decompose the multivariate signal that is made up of the process input and output. Assuming that the reader is sufficiently familiar with the univariable EMD algorithm, its discussions may be found in [22]. In univariable EMD, the upper (lower) envelope is acquired by interpolating between all maxima (minima), then two envelopes are averaged to obtain the local mean. Since the local maxima (minima) cannot be defined in a direct  
Figure 2: Flowchart of BEMD-TCN approach. way to overcome this problem, the BEMD algorithm proposed in [23] is introduced. In [23], the multivariate input signal is projected along different directions, then envelopes can be calculated according to the projected signal, and the local mean is approximated by averaging all directional envelopes in a multidimensional space. As the uniformity of direction vectors plays a significant role in approximation accuracy, it is central to choosing a suitable direction vector set. Considering a direction vector in n-dimensional space corresponds to a point on a unit (\(n-1\)) sphere, to ensure the uniformity of direction vectors, a uniform sampling scheme should be adopted. In this work, low-discrepancy sampling approach suggested in [24] is adopted, by which more uniformly distributed points can be obtained. A multivariate IMF should ensure that the mean of the envelope must be equal to zero all the time; the difference with the univariate case is that the equality of extrema and zero crossings is not required [25]. Denoting the \(n\)-variate signal by \(\{v(t)\}_{t=1}^{T}=\{v_{1}(t),v_{2}(t),\cdots,v_{n}(t)\}\), and \(x^{\theta_{k}}=\left\{x_{1}^{k},x_{2}^{k},\cdots,x_{n}^{k}\right\}\) represents a direction vector which is characterized by angle \(\theta^{k}=\left\{\theta_{1}^{k},\theta_{2}^{k},\cdots,\theta_{n-1}^{k}\right\}\) on an (\(n-1\)) unit sphere. Then, the multivariate EMD can be formulated as,  
\[v(t)=\sum_{i=1}^{n}f_{i}(t)+R_{n}(t) \tag{4}\]  
where, \(f_{i}(t):\Re\rightarrow\Re^{n}\) denotes a decomposed IMF, and \(R_{n}(t)\) is the residue signal. The EMD algorithm for an \(n\)-variate timeseries is given below (see Algorithm 1). ```
1. Sample \(K\) points on a (\(n-1\)) unit sphere using the low-discrepancy sampling approach. 2. For an input signal \(\{v(t)\}_{t=1}^{T}\), given a direction vector \(x^{\theta_{k}}\), calculate the corresponding projected signal \(\left\{p^{\theta_{k}}(t)\right\}_{t=1}^{T}\). Then, for \(K\) directions, the projection signal set \(\left\{p^{\theta_{k}}(t)\right\}_{k=1}^{K}\) (\(t=1,2,\ldots,T\)) can be obtained. 3. Search for time instants \(\left\{t^{\theta_{k}}\right\}\) where maxima of projected signal \(\left\{p^{\theta_{k}}(t)\right\}_{k=1}^{K}\) is achieved. 4. Interpolate \([t_{i}^{\theta_{k}},v(t^{\theta_{k}})]\) such that multivariate direction envelope curves \(\left\{e^{\theta_{k}}(t)\right\}_{k=1}^{K}\) are obtained. 5. Calculate the mean \(m(t)\) based upon \(K\) direction vectors as, \[m(t)=\frac{1}{K}\sum_{k=1}^{K}e^{\theta_{k}}(t)\] (5)
6. The 'detail' \(d(t)\) is extracted as \(d(t)=v(t)-m(t)\). If \(d(t)\) is an IMF, then set \(f(t)=d(t)\) and replace \(v(t)\) with the residue \(R(t)=v(t)-f(t)\). Otherwise, use \(d(t)\) in place of \(v(t)\) and repeat step 2-5.