January  2021, 17(1): 51-66. doi: 10.3934/jimo.2019098

Orthogonal intrinsic mode functions via optimization approach

Faculty of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, China

* Corresponding author: Bingo Wing-Kuen Ling

Received  January 2019 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by NSF grant xx-xxxx

This paper proposes an optimization approach to find a set of orthogonal intrinsic mode functions (IMFs). In particular, an optimization problem is formulated in such a way that the total energy of the difference between the original IMFs and the corresponding obtained IMFs is minimized subject to both the orthogonal condition and the IMF conditions. This formulated optimization problem consists of an exclusive or constraint. This exclusive or constraint is further reformulated to an inequality constraint. Using the Lagrange multiplier approach, it is required to solve a linear matrix equation, a quadratic matrix equation and a highly nonlinear matrix equation only dependent on the orthogonal IMFs as well as a nonlinear matrix equation dependent on both the orthogonal IMFs and the Lagrange multipliers. To solve these matrix equations, the first three equations are considered. First, a new optimization problem is formulated in such a way that the error energy of the highly nonlinear matrix equation is minimized subject to the linear matrix equation and the quadratic matrix equation. By finding the nearly global optimal solution of this newly formulated optimization problem and checking whether the objective functional value evaluated at the obtained solution is close to zero or not, the orthogonal IMFs are found. Finally, by substituting the obtained orthogonal IMFs to the last matrix equation, this last matrix equation reduced to a linear matrix equation which is only dependent on the Lagrange multipliers. Therefore, the Lagrange multipliers can be found. Consequently, the solution of the original optimization problem is found. By repeating these procedures with different initial conditions, a nearly global optimal solution is obtained.

Citation: Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098
References:
[1]

Q. ChenN. HuangS. Riemenschneider and Y. Xu, A B-spline approach for empirical mode decompositions, Advanced on Comput. Math., 24 (2006), 171-195.  doi: 10.1007/s10444-004-7614-3.  Google Scholar

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P. Flandrin and P. Gonçalvès, Empirical mode decompositions as data-driven wavelet-like expansions, Int. J. of Wavelets, Multires. and Info. Processing, 2 (2004), 477-496.  doi: 10.1142/S0219691304000561.  Google Scholar

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P. FlandrinG. Rilling and P. Gonçalvès, Empirical mode decomposition as a filter bank, IEEE Signal Processing Letters, 11 (2004), 112-114.   Google Scholar

[5]

C. Y. HoB. W. LingY. Q. LiuP. K. Tam and K. L. Teo, Optimal design of magnitude responses of rational infinite impulse response filters, IEEE Transactions on Signal Processing, 54 (2006), 4039-4046.  doi: 10.1109/TSP.2006.880317.  Google Scholar

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N. E. HuangZ. ShenS. LongM. WuH. ShihQ. ZhengN. YenC. Tung and H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society London A: Math., Phys. and Engineering Sciences, 454 (1998), 903-955.  doi: 10.1098/rspa.1998.0193.  Google Scholar

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Y. Kopsinis and S. McLaughlin, Development of EMD-based denoising methods inspired by wavelet thresholding, IEEE Transactions on Signal Processing, 57 (2009), 1351-1362.  doi: 10.1109/TSP.2009.2013885.  Google Scholar

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Y. Kopsinis and S. McLaughlin, Investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach, IEEE Transactions on Signal Processing, 56 (2008), 1-13.  doi: 10.1109/TSP.2007.901155.  Google Scholar

[10]

B. W. LingN. L. TianC. Y. HoW. C. SiuK. L. Teo and Q. Y. Dai, Maximally decimated paraunitary linear phase FIR filter bank design via iterative SVD approach, IEEE Transactions on Signal Processing, 63 (2015), 466-481.  doi: 10.1109/TSP.2014.2371779.  Google Scholar

[11]

P. Q. MuoiD. N. HáoP. Maass and M. Pidcock, Descent gradient methods for nonsmooth minimization problems in ill-posed problems, J. of Comput. and Appl. Math., 298 (2016), 105-122.  doi: 10.1016/j.cam.2015.11.039.  Google Scholar

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F. RahpeymaiiM. Kimiaei and A. Bagheri, A limited memory quasi-Newton trust-region method for box constrained optimization, J. of Comput. and Appl. Math., 303 (2016), 105-118.  doi: 10.1016/j.cam.2016.02.026.  Google Scholar

[13]

Q. S. RenQ. Yi and M. Y. Fang, Fast implementation of orthogonal empirical mode decomposition and its application into singular signal detection, IEEE International Conference on Signal Processing and Communications, 6 (2007), 1215-1218.  doi: 10.1109/ICSPC.2007.4728544.  Google Scholar

[14]

Z. J. YangB. W. Ling and C. Bingham, Joint empirical mode decomposition and sparse binary programming for underlying trend extraction, IEEE Transactions on Instrumentation and Measurement, 62 (2013), 2673-2682.  doi: 10.1109/TIM.2013.2265451.  Google Scholar

[15]

Z. J. YangB. W. Ling and C. Bingham, Trend extraction based on separations of consecutive empirical mode decomposition components in Hilbert marginal spectrum, Measurement, 46 (2013), 2481-2491.  doi: 10.1016/j.measurement.2013.04.071.  Google Scholar

show all references

References:
[1]

Q. ChenN. HuangS. Riemenschneider and Y. Xu, A B-spline approach for empirical mode decompositions, Advanced on Comput. Math., 24 (2006), 171-195.  doi: 10.1007/s10444-004-7614-3.  Google Scholar

[2]

L. E. Cheriyan and P. Janardhanan, Adaptive orthogonal signal decomposition based on empirical mode decomposition and empirical wavelet transform, 1st IRF International Conference, 9 (2014), 67-70.   Google Scholar

[3]

P. Flandrin and P. Gonçalvès, Empirical mode decompositions as data-driven wavelet-like expansions, Int. J. of Wavelets, Multires. and Info. Processing, 2 (2004), 477-496.  doi: 10.1142/S0219691304000561.  Google Scholar

[4]

P. FlandrinG. Rilling and P. Gonçalvès, Empirical mode decomposition as a filter bank, IEEE Signal Processing Letters, 11 (2004), 112-114.   Google Scholar

[5]

C. Y. HoB. W. LingY. Q. LiuP. K. Tam and K. L. Teo, Optimal design of magnitude responses of rational infinite impulse response filters, IEEE Transactions on Signal Processing, 54 (2006), 4039-4046.  doi: 10.1109/TSP.2006.880317.  Google Scholar

[6]

N. E. HuangZ. ShenS. LongM. WuH. ShihQ. ZhengN. YenC. Tung and H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proceedings of the Royal Society London A: Math., Phys. and Engineering Sciences, 454 (1998), 903-955.  doi: 10.1098/rspa.1998.0193.  Google Scholar

[7]

T. L. HuangW. X. Ren and M. L. Lou, The orthogonal Hilbert-Huang transform and its application in earthquake motion recordings analysis, The 14th World Conference on Earthquake Engineering, 10 (2008), 17-18.   Google Scholar

[8]

Y. Kopsinis and S. McLaughlin, Development of EMD-based denoising methods inspired by wavelet thresholding, IEEE Transactions on Signal Processing, 57 (2009), 1351-1362.  doi: 10.1109/TSP.2009.2013885.  Google Scholar

[9]

Y. Kopsinis and S. McLaughlin, Investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach, IEEE Transactions on Signal Processing, 56 (2008), 1-13.  doi: 10.1109/TSP.2007.901155.  Google Scholar

[10]

B. W. LingN. L. TianC. Y. HoW. C. SiuK. L. Teo and Q. Y. Dai, Maximally decimated paraunitary linear phase FIR filter bank design via iterative SVD approach, IEEE Transactions on Signal Processing, 63 (2015), 466-481.  doi: 10.1109/TSP.2014.2371779.  Google Scholar

[11]

P. Q. MuoiD. N. HáoP. Maass and M. Pidcock, Descent gradient methods for nonsmooth minimization problems in ill-posed problems, J. of Comput. and Appl. Math., 298 (2016), 105-122.  doi: 10.1016/j.cam.2015.11.039.  Google Scholar

[12]

F. RahpeymaiiM. Kimiaei and A. Bagheri, A limited memory quasi-Newton trust-region method for box constrained optimization, J. of Comput. and Appl. Math., 303 (2016), 105-118.  doi: 10.1016/j.cam.2016.02.026.  Google Scholar

[13]

Q. S. RenQ. Yi and M. Y. Fang, Fast implementation of orthogonal empirical mode decomposition and its application into singular signal detection, IEEE International Conference on Signal Processing and Communications, 6 (2007), 1215-1218.  doi: 10.1109/ICSPC.2007.4728544.  Google Scholar

[14]

Z. J. YangB. W. Ling and C. Bingham, Joint empirical mode decomposition and sparse binary programming for underlying trend extraction, IEEE Transactions on Instrumentation and Measurement, 62 (2013), 2673-2682.  doi: 10.1109/TIM.2013.2265451.  Google Scholar

[15]

Z. J. YangB. W. Ling and C. Bingham, Trend extraction based on separations of consecutive empirical mode decomposition components in Hilbert marginal spectrum, Measurement, 46 (2013), 2481-2491.  doi: 10.1016/j.measurement.2013.04.071.  Google Scholar

Figure 1.  Original signal
Figure 2.  Signal components
Figure 3.  Normalized IMFs
Figure 4.  Components obtained by the Gram Schmidt orthogonalization method
Figure 5.  Components obtained by our proposed method
Table 1.  Orthogonal errors of all the IMFs $ err_{orth} $
SignalOur proposed methodGram Schmidt orthogonalization method [8]Normalization method
$x(t)=\sin (2 \pi t) e^{-\frac{t^{2}}{2}}$
$+e^{-t} \cos (2 \pi t)$
001.3545
SignalOur proposed methodGram Schmidt orthogonalization method [8]Normalization method
$x(t)=\sin (2 \pi t) e^{-\frac{t^{2}}{2}}$
$+e^{-t} \cos (2 \pi t)$
001.3545
Table 2.  Errors between the total number of the extrema and the total number of the zero crossing points of all the IMFs $ err_{ex_-zx} $
SignalOur proposed methodGram Schmidt orthogonalization method [8]Normalization method
$x(t)=\sin (2 \pi t) e^{-\frac{t^{2}}{2}}$
$+e^{-t} \cos (2 \pi t)$
0250
SignalOur proposed methodGram Schmidt orthogonalization method [8]Normalization method
$x(t)=\sin (2 \pi t) e^{-\frac{t^{2}}{2}}$
$+e^{-t} \cos (2 \pi t)$
0250
Table 3.  Errors between the total number of the extrema and the total number of the zero crossing points of all the IMFs $ err_{ex_-zx} $
SignalOur proposed method Gram Schmidt orthogonalization method [8]Normalization method
$x(t)=\sin (2 \pi t) e^{-\frac{t^{2}}{2}}$
$+e^{-t} \cos (2 \pi t)$
$0.9139\times10^{-4}$$0.9801\times10^{-4}$$0.6368\times10^{-4}$
SignalOur proposed method Gram Schmidt orthogonalization method [8]Normalization method
$x(t)=\sin (2 \pi t) e^{-\frac{t^{2}}{2}}$
$+e^{-t} \cos (2 \pi t)$
$0.9139\times10^{-4}$$0.9801\times10^{-4}$$0.6368\times10^{-4}$
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