1. Spline Interpretation ABC Introduction and outline Based mostly on Wikipedia

2. Spline Interpretation Spline is a flat flexible strip of thin narrow piece of wood, metal, or plastic used in drawing curved. Mathematically, spline means curve fitting mostly using certain piecewise low degree polynomials. The main reason to use low degree polynomial is to avoid Runge’s phenomenon.

3. Runge’s Phenomenon

4. Runge’s Phenomenon Red: data; blue: 5 th order; green: 9 th order

5. Runge’s Phenomenon

6. Definition of Spline

7. Linear Spline

8. Quadratic Spline

10. Cubic Spline

13. Hermite Spline

15. Hermite spline

18. In general, cubic spline does not guarantee monotonicity even for monotonic data. Hermite monotonic spline, however, guarantees monotonicity between data points. The advantage is the elimination of overshoots. The prices to pay are these: – we no longer have the smoothness of continuity in curvature and –we have to inject another subjective idea of monotonic and/or variations.

19. Comparisons between Cubic Natural and Hermite Splines An Example

20. Data

21. Envelopes of Maxima

22. Envelopes of Minima

23. Mean of Max and Min Envelopes

24. Difference between Mean of Max and Min Envelopes

25. First Proto-IMF

26. Observations Hermite monotonic spline seems to have better performance with limited variations. The end effect is especially severe for natural spline, a problem we have to resolve. In order to be fully adaptive with minimum subjective intervention, we selected cubic natural spline for EMD with only one assumption: smoothness.

27. B-Spline B-spline is Basis-spline. Something used extensively by Computer Aided Design (CAD), but we have found limited use of it.

28. Bézier Curve

30. Cubic Bezier Bases

31. Bézier Curve

32. An example

33. Rational Bézier Curve

34. B-Spline A B-spline is a generalization of the Bézier Curve; A B-spline with no internal knots is a Bézier Curve. It is generated through Cox-de Boor recursion formula. A further generalization is NURBS: NonUniform Rational B-Spline. See for example, David F. Rogers: An Introduction to NURBS. Morgan Kaufmann, San Francisco, 2001

35. B-Spline

36. Properties of B-Spline A B-spline could have degree n; control points, p; and knots, m, each independent to others, but m ≥ p. It is much more flexible. For n=3, p=10, and m=14, we have B-spline at left and Bezier curve at right.

37. Properties of B-Spline In general, the lower the degree, the closer the curve follows the control points. For the same numbers of control points and knots, but different degrees, the spline results are very different: left, n=7; center, n=5; right, n=3.

38. Properties of B-Spline B-spline is a piecewise curve. We must have m=n+p+1. Clamped spline has to pass the beginning and end points. Strong convex hull property Variation diminishing property Affine invariance But B-Spline does not pass through the control points!

39. References Qiuhui Chen, Norden Huang, Sherman Riemenschneider and Yuesheng Xu, 2006: A B-spline approach for empirical mode decompositions. Advances in Computational Mathematics 24: 171–195

40. Variation Diminishing Property: no straight line intersects a B-spline curve more times than it intersects the curve's control polygon.

41. Summary The differences between different splines seems to be huge. Based on principles of least interferences and maximum smoothness, we selected natural spline as the base for EMD. Different splines (except B-spline) should not change the scales of the IMFs, but it would change the shape and energy levels. EMD is unique with respect to specific spline fitting and parameters selected.