1. On the Trend, Detrend and the Variability of Nonlinear and Nonstationary Time Series Norden E. Huang Research Center for Adaptive Data Analysis National Central University, Taiwan

2. Satellite Altimeter Data : Greenland

3. Two Sets of Data

4. IPCC Global Mean Temperature Trend

5. 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

6. Philosophical Problem 名不正則言不順 言不順則事不成 —— 孔夫子

7. On Definition Without a proper definition, logic discourse would be impossible. Without logic discourse, nothing can be accomplished. Confucius

8. Definition of the Trend Within the given data span, the trend is an intrinsically determined 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.

9. 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.

10. 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.

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. How are GSTA data derived? Noise Reduction Using Global Surface Temperature Anomaly data 1856 to 2003

28. Jones (2003) Monthly GSTA Data

29. Jones (2003) 12 Monthly GSTA Data

31. Jones (2003) GSTA Data Seasonal Variation

32. Jones (2003) GSTA Data Seasonal Variance

33. Jones Monthly GSTA Data : Fourier Spectrum

34. Observations Annual data is actually the mean of 12:1 down sample set of the original monthly data. In spite of the removal of climatologic mean, there still is a seasonal peak (1 cycle / year). Seasonal Variation and Variance are somewhat irregular. Data contain no information beyond yearly frequency, for higher frequency part of the Fourier spectrum is essentially flat. Decide to filtered the Data with HHT before down sample.

35. Need a Filter to Remove Alias Traditional Fourier filter is inadequate: –Removal of Harmonics will distort the fundaments –Noise spikes are local in time; signals local in time have broad spectral band HHT is an adaptive filter working in time space rather than frequency space.

36. EMD as filters

37. Jones Monthly GSTA Data : IMF

38. Jones Monthly GSTA Data : IMF Smoothed

39. Jones Monthly GSTA Data & HHT Smoothed

40. Jones Monthly GSTA Data : Fourier Spectrum Data & Smoothed

41. 12 Monthly GSTA Data HHT Smoothed

42. Jones (2003) 12 Monthly GSTA Data

43. GSTA : Annual Data Jones and HHT Smoothed For the Difference : Mean = - 0.082; STD = 0.01974

44. GSTA : Annual Variance Jones and HHT Smoothed Mean HHT = 0.0750; Jones = 0.1158

45. GSTA : HHT Smoothed Seasonal Variation

46. GSTA : HHT Smoothed Seasonal Variance

47. Summary Global Surface Temperature Anomaly should not be derived from simple annual average, because there are noises in the data. Noise with period shorter than one year could have caused alias in down sampling. Smoothing the data by removing any data with a period shorter than 8 months should improved the annual mean.

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

49. NasDaq Data

50. NasDaq IMF

51. NasDaq IMF Reconstruction : A

52. NasDaq IMF Reconstruction : B

53. NasDaq Various Overall Trends

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

55. NasDaq Trend IMF (C8-C9)

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

57. NasDaq Trend IMF (C7-C9)

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

59. NasDaq Trend IMF (C6-C9)

60. NasDaq Local Period for Trend IMF (C6-C9) Mean = 196.5

61. NasDaq Traditional Moving Mean Trends: Details

62. NasDaq Trends: Moving Mean and EMD : Details

63. NasDaq Period of EMD Trend (C4) Mean = 35.56

64. NasDaq Distribution of Period for EMD Trend (C4)

65. NasDaq Detrended Data (C4-C9)

66. NasDaq Detrended Data (C4-C9) : Details

67. NasDaq Histogram Detrended Data (C1-C3)

68. Various Definitions of Variability Variability defined by percentage Gain is the absolute value of the Gain. Variability defined by daily high-low is the percentage of absolute value of High-Low. Variability defined by Empirical Mode Decomposition is the percentage of the absolute value of the sum from selected IMFs. Financial data do not look like ARIMA.

69. NasDaq Variability defined by EMD : C1

70. NasDaq Variability defined by Gain

71. NasDaq Variability defined by Daily High-Low

72. NasDaq Period of Variability defined by EMD : C1 Mean = 8.38

73. NasDaq Histogram Period of EMD Variability : C1

74. NASDAQ Price gradient vs. Gain Variability

75. NASDAQ Price gradient vs. High-Low Variability

76. NASDAQ Price gradient vs. EMD Variability

77. Relationship between Variability: Gain vs. EMD

78. Relationship between Variability: Gain vs. High- Low

79. Relationship between Variability: EMD vs. High- Low

80. Statistical Significance Test Only the statistical Significant IMF components are signal above noise; therefore, they might be predictable.

81. Statistical Significance Test : Gain

82. Statistical Significance Test : High-Low

83. Statistical Significance Test : EMD

84. Statistical Significance Test : All Variability Definitions

85. The Sum of all the Statistical Significance IMFs

86. Relationship among Trends: Gain vs. EMD

87. Relationship among Trends: Gain vs. High-Low

88. Relationship among Trends: EMD vs. High-Low

89. Summary A working definition for the trend is established; it is a function of the local time scale. Need adaptive method to analysis nonstationary and nonlinear data for trend and variability. Various definitions for variability should be compared in details to determine their significance. Predictions should be made based on processes driven models, not on data.

90. Conclusion Trend is a local property of the data; it should associate with a length scale. Trend should be determined adaptively; therefore, we should not pre-select the functional form of the trend. Variability should have a reference; the trend is a good reference.