## 3 Simulation and Results
### Discussion  
Based on the simulations of the algorithms namely; aIMF, WT, EKF and PF, we have found that the aIMF performed the best, following in the order to WT, EKF and PF. Theoretically, the successful application of EMD resides on the fact that the noise is not biased. Therefore, there is not so much of a restrictive constraint, comparing to the scenario of encountering with non-zero mean noise. As mentioned, the characteristics of IMF after several iterations move towards the normal distribution; see Figure 11. Thus, subtracting the averaged IMF with the original signals, given aIMF, can reduce the noise inevitably. However, the proposed aIMF algorithm using cubic spline interpolation does not intend to preserve edges of the datasets/signals. This is because of our target to reduce noise that may associate with the upper and lower boundaries of the curves, which are in time series domain. In this particular case, preserving the edges/curves can unavoidably keep the noise mixing within the signals. Unlike the images whose distribution is random walk, the noise reduction can be achieved while  
Figure 10: Plots of the original datasets, original datasets \(+\) noise, and the Particle Filter (PF) algorithm.  
the edges are preserved [27]. Referring to Section 3.1, we manually added a variety of noises into the datasets with separate simulations; and later proved that the noises have been removed significantly, displayed in Figure 7 and Tables 1-6. To continue to prove the aIMF algorithm's performance, we simulated the aIMF algorithm with the original datasets--without adding extra noise. The results measured by MSE, MAE, MAPE, R2 and Accuracy count were 8.20211E-05, 0.00719, 0.57085, 0.9980 and 99.95%, respectively. It is noted that the original datasets, EUR-USD exchange rates, contained a certain level of noise, not pure signal only. In the real application, data of exchange rates are normally fluctuated before closing hours of trading by retailers and speculators who want to manipulate the price. The manipulations are always executed with low volumes of trade, but enable the price changes at the end of trading hours. In the financial community, we deem these trades as noise. Finally, the figures from the last loss estimators shown were similar to the results in Tables 1-6. Hence, it is safe to assume that the proposed aIMF algorithm can remove the unwanted signals.  
Figure 11: Graphs (**a**)–(**c**) show plots of IMF1, IMF5 and IMF7, of which their local extrema of higher order IMF moved toward the normal distribution.  
On another front, the simulation results from the WT algorithm seemed to be unacceptable under the rationale that Fourier spectral analysis and its derivatives such as WT encountered with a limited time window width by sliding a predetermined window(s) along the time axis [28]. Moreover, there is a trade-off between the time consumed in the window width and the frequency resolution, and this phenomenon has been considered by the uncertainty principle Heisenberg [21]. In this particular case, the WT's window width must be preselected and it is known as the primary or mother wavelet which is a prototype that provides less flexibility when handling datasets were the mean and variances are highly volatile. In case of EKF simulation, the advantage of SIS is that it does not guarantee to fail as time increases and it becomes more and more skewed, especially when sampling high-dimensional spaces [26]. For the PF algorithm, we have found that the number of particles (data-points) was not adequate for Monte Carlo simulation. However, the drawback of the aIMF algorithm is that it requires long time to spline the local extrema.