IEEE Computational Intelligence Magazine - November 2020 - 39
(non-stationary feature) for training. Similarly, the stationary
modes are used to train another set of RNNs (Figure 5).
VMD is an adaptive decomposition algorithm that aims to
decompose a non-linear non-stationary signal into its constructive modes [26]. These modes, which are known as Intrinsic Mode Functions (IMF), are frequency and/or amplitude
modulated signals. The sum of which constructs the original
signal (minus some noise, depending on settings).
VMD is an adaptive algorithm which solves a variational
optimisation problem for a given signal S(t) on k IMFs
{u k} = {u 1, u 2, f, u k}. It is assumed that each IMF is narrowband and, therefore, has a center frequency {~ i} where
i ! {1, 2, f, k}. The aforementioned variational optimisation
problem follows,
min
/
}
{u k}&{~ k
k
2t c d^ t h +
1) The number of modes (k) into which the signal is chosen
to be decomposed.
2) The weight of the quadratic penalty term a, which is a
denoising factor, a larger value of which admits less noise
into the decomposition process. Note that in this paper,
a is set to a relatively small value of 10 since denoising is
not a concern [27].
TABLE II KPSS test results run on the signals corresponding
to Maharashtra. Note that ST stands for stationary.
SIGNAL
2
j
) u k ^ t hm e -j~k t (5)
rt
where in the above equation, ) is the convolution operator.
The proposers of VMD argue that the solution to the minimization problem of (5) is the saddle point of the augmented
Lagrangian in a sequence of iterative sub-optimizations called
alternate direction method of multipliers (ADMM) [26]. The
readers are referred to the original paper for further details.
There are some critical parameters that need to be determined when using VMD for signal decomposition:
LAG
P-VALUE
H
ST
CC
7, 9, 11
0.02, 0.03, 0.04
1, 1, 1
✕
TR
7, 9, 11
0.01, 0.01, 0.02
1, 1, 1
✕
T
7, 9, 11
0.10, 0.10, 0.10
0, 0, 0
H
7, 9, 11
0.07 0.10 0.10
0, 0, 0
TABLE III KPSS test results run on the signals corresponding
to Tamil Nadu.
SIGNAL
LAG
P-VALUE
H
ST
CC
7, 9, 11
0.01, 0.01, 0.02
1, 1, 1
✕
TR
7, 9, 11
0.01, 0.02, 0.03
1, 1, 1
✕
T
7, 9, 11
0.10, 0.10, 0.10
0, 0, 0
H
7, 9, 11
0.05, 0.03, 0.02
1, 1, 1
✕
Stationary Features
VMD
Stationary IMF
Stationary IMF
+
VMD
Non-Stationary IMF
Stationary IMF
Non-Stationary Signal
Training
Result
RNN
RNN
Non-Stationary IMF
Final Result
Training
Result
VMD
Non-Stationary
Features
FIGURE 5 Flowchart of the proposed methodology.
NOVEMBER 2020 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE
39
IEEE Computational Intelligence Magazine - November 2020
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