1. A Conjecture & a Theorem
The effects of sifting number

2. A Conjecture
Wu et al (2012)

3. Sifting number : Stoppage criteria
This is in fact the influence of stoppage criteria.
High sift number gives more detailed separation, but less physical meaning.
Should we adopt a high sifting number or low? Where is an optimal choice?

4. The Effect of Sifting Number

5. The Effect of Sifting Numbers: 10, 1,000, 100,000

6. The Effect of Sifting Number
A signal (bold black line) composed of two unit amplitude sinusoidal waves with periods of 30 (thin blue line) and 34 (thin green line), respectively.
The decompositions of the signal with different sifting numbers are displayed in the middle (the first IMF) and lower (the remainder) panels.
The red, green, and magenta lines in the middle and lower panels are the components corresponding to sifting numbers of 10, 1,000, 100,000, respectively.

7. The Effect of Sifting Number
The blue lines in the middle and lower panels correspond to the shorter period (30) waves and longer period (34) waves in the upper panel, respectively.
With low number of sifting, the amplitude modulation signal is regarded as an IMF; with high number of sifting, two separate sinusoidal waves are the answer.

8. The Effect of Sifting Number Blue: 10 times; Red : 2500 times IMF normalized

9. The Effect of Sifting Number,
The scaling factor:

10. The Effect of Sifting Number

11. The Effect of Sifting Number : White Noise

12. The Effect of Sifting Number : White Noise

13. The Effect of Sifting Number : Summary

14. The Effect of Sifting Number : Summary

15. Conclusions
The higher the number of sifting, the smaller the amplitude modulation.
The higher the number of sifting, the more IMF components.
To maintain the dyadic nature of EMD, the sifting number should be around 10.

16. A Theorem
Wang et al (2010)

17. Definition of the Intrinsic Mode Function (IMF)

18. Wang et al, 2010 Adv. Adaptive data Analysis. 3,
There is a conflict between the definition of IMF and the method proposed to implement it: The envelopes will be straight lines.
Wang et al, Adv. Adaptive data Analysis. 3,

19. Given the envelopes

20. Proof of the Theorem
All these functions are piecewise cubic spline curves. If the c(t) satisfies the definition of IMF strictly, we should have
Without loss of generality, we also assume that the extrema of IMF are spares distributed with respect to the data points, i.e., there are at lest two data points between a maximum and its neighboring minimum.

21. Mathematically

22. Proof of the Theorem
Since there are at least two extrema in [t2 , t3], equation u1(t)+l0(t)=0 has at least four roots in t∈[t2 , t3].
Notice that u1(t) and l0(t) are cubic functions, they must have the same coefficients in their expression, i.e., Ai=-Ci , i=1, 2, 3, 4.
Similarly, over the interval t∈[t3 , t4] we should have Bi=-Ci , i=1, 2, 3, 4.
>> u(t) = - l(t).

23. Proof of the Theorem
Thus the two segmental cubic curves, u1(t) and u2(t), are actually two parts of the same cubic function.
This conclusion is same with all the other segmental functions in other intervals.
Therefore, the upper (lower) envelope of an IMF is in fact a unitary cubic function.

24. The Theorem
as u(t) is a cubic natural spline as suggested by Huang et al. [1998], we should have the zero curvature at the ends
u”(a)=u”(b)=0 >> A1 & A2 =0
Therefore, u =A3 t + A4
Unless envelopes are monotonic, they have to be constant.

25. Conclusion This theorem is true for other higher order spline too.
This theorem agrees with the conjecture.
Therefore, the definition of IMF could only be an approximation.