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On the Trend, Detrend and the Variability of Nonlinear and Nonstationary Time Series

On the Trend, Detrend and the Variability of Nonlinear and Nonstationary Time Series. A new application of HHT. Satellite Altimeter Data : Greenland. Two Sets of Data.

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On the Trend, Detrend and the Variability of Nonlinear and Nonstationary Time Series

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  1. On the Trend, Detrend and the Variability of Nonlinear and Nonstationary Time Series A new application of HHT

  2. Satellite Altimeter Data : Greenland

  3. Two Sets of Data

  4. The State-of-the-Arts“One economist’s trend is another economist’s cycle” Engle, R. F. and Granger, C. W. J. 1991 Long-run Economic Relationships. Cambridge University Press. • Simple trend – straight line • Stochastic trend – straight line for each quarter

  5. PhilosophicalProblem 名不正則言不順 言不順則事不成 ——孔夫子

  6. OnDefinition Without a proper definition, logic discourse would be impossible.Without logic discourse, nothing can be accomplished.Confucius

  7. Definition of the Trend Within the given data span, the trend is an intrinsically fitted monotonic function, or a function in which there can be at most one extremum. The trend should be determined by the same mechanisms that generate the data; it should be an intrinsic and local property. Being intrinsic, the method for defining the trend has to be adaptive. The results should be intrinsic (objective); all traditional trend determination methods give extrinsic (subjective) results. Being local, it has to associate with a local length scale, and be valid only within that length span as a part of a full wave cycle.

  8. Definition of Detrend and Variability Within the given data span, detrend is an operation to remove the trend. Within the given data span, the Variability is the residue of the data after the removal of the trend. As the trend should be intrinsic and local properties of the data; Detrend and Variability are also local properties. All traditional trend determination methods are extrinsic and/or subjective.

  9. The Need for HHT HHT is an adaptive (local, intrinsic, and objective) method to find the intrinsic local properties of the given data set, therefore, it is ideal for defining the trend and variability.

  10. History of HHT 1998: The Empirical Mode Decomposition Method and the Hilbert Spectrum for Non-stationary Time Series Analysis, Proc. Roy. Soc. London, A454, 903-995.The invention of the basic method of EMD, and Hilbert transform for determining the Instantaneous Frequency and energy. 1999: A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457. Introduction of the intermittence in EMD. 2003: A confidence Limit for the Empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A459, 2317-2345. Establishment of a confidence limit without the ergodic assumption. 2004: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proc. Roy. Soc. London, (in press) Defined statistical significance and predictability of IMFs. 2004: On the Instantaneous Frequency, Proc. Roy. Soc. London, (Under review) Removal of the limitations posted by Bedrosian and Nuttall theorems for instantaneous Frequency computations.

  11. Two Sets of Data

  12. Global Temperature Anomaly Annual Data from 1856 to 2003

  13. Global Temperature Anomaly 1856 to 2003

  14. IMF Mean of 10 Sifts : CC(1000, I)

  15. Mean IMF

  16. STD IMF

  17. Statistical Significance Test

  18. Data and Trend C6

  19. Data and Overall Trends : EMD and Linear

  20. Rate of Change Overall Trends : EMD and Linear

  21. Variability with Respect to Overall trend

  22. Data and Trend C5:6

  23. Data and Trends: C5:6

  24. Rate of Change Trend C5:6

  25. Trend Period C5

  26. Variability with Respect to 65-Year trend

  27. Data and Trend C4:6

  28. Data and Trend C4:6

  29. Rate of Change Trend C4:6

  30. Trend Period C4

  31. Variability with Respect to 20-Year trend

  32. Data and Trend C3:6

  33. Trend Period C3

  34. Histogram of Trend Period C3

  35. Variability with Respect to 10-Year trend

  36. Hilbert Spectrum Global Temperature Anomaly

  37. Marginal Hilbert Spectrum

  38. Morlet Wavelet Spectrum

  39. Hilbert and Morlet Wavelet Spectra

  40. Financial Data : NasDaqSC October 11, 1984 – December 29, 2000 October 12, 2004

  41. NasDaq Data

  42. NasDaq IMF

  43. NasDaq IMF Reconstruction : A

  44. NasDaq IMF Reconstruction : B

  45. NasDaq Various Overall Trends

  46. NasDaq various Overall Detrends Mean : L = 0 Exp = 73.1187 EMD = 0.3588 STD : L = 559.09 Exp = 426.66 EMD = 238.10

  47. NasDaq Trend IMF (C8-C9)

  48. NasDaq Local Period for Trend IMF (C8-C9)mean = 796.6

  49. NasDaq Trend IMF (C7-C9)

  50. NasDaq Local Period for Trend IMF (C7-C9)Mean = 425.7

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