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

2. Wu, Z. , N. E. Huang, S. R. Long and C. K
Wu, Z., N. E. Huang, S. R. Long and C. K. Peng: On the Trend, Detrend and the Variability of Nonlinear and Nonstationary Time Series. Proc. Natl Acad. Sci. 140, 14,889-14,894, 2007.

3. Satellite Altimeter Data : Greenland

4. Two Sets of Data

5. 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.

6. Traditional detrending
Differencing, or differentiating
Regression
Filtering

7. The Residue from EMD The residue is the overall trend.
The trend is derived through removal of all the oscillatory modes, not through averaging or regression, which is ad hoc and arbitrary.
Trend of other time scales could be defined.

8. An Example of EMD Application
Global Warming

9. IPCC Global Mean Temperature Trend

10. How are GSTA data derived? Noise Reduction
Using Global Surface Temperature Anomaly data
1856 to 2008

11. Jones (2003) Monthly GSTA Data

12. Jones (2003) 12 Monthly GSTA Data

13. Jones (2003) 12 Monthly GSTA Data

14. Jones (2003) GSTA Data Seasonal Variation

15. Jones (2003) GSTA Data Seasonal Variance

16. Jones Monthly GSTA Data : Fourier Spectrum

17. 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.

18. 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.

19. Jones Monthly GSTA Data : IMF

20. Jones Monthly GSTA Data : IMF Smoothed

21. Jones Monthly GSTA Data & HHT Smoothed

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

23. 12 Monthly GSTA Data HHT Smoothed

24. Jones (2003) 12 Monthly GSTA Data

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

26. GSTA : Annual Variance Jones and HHT Smoothed Mean HHT = 0
GSTA : Annual Variance Jones and HHT Smoothed Mean HHT = ; Jones =

27. GSTA : HHT Smoothed Seasonal Variation

28. GSTA : HHT Smoothed Seasonal Variance

29. 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.

30. Global Climate Changes

31. Oxygen and Carbon records
Deep sea foraminifera isotope records :Zachos et al., 2001，Science
Note the Carbon concentration is not the highest during EECO!

32. Land Mass Distribution
Geological time scale changes

33. J. Zachos, et al., Science, 292,

34. Earth Orbital Parameters
Milankovitch time scales

35. 102 Years Our life time scale Instrument measured data,
The base of IPCC AAR4 report

36. GSTA

37. IPCC Global Mean Temperature Trend

38. IPCC 4th Assessment Report 2007
“Note that for shorter recent periods, the slope is greater, indicating accelerated warming.”
IPCC 4th Assessment Report 2007

39. Slope computation

40. Regression method is arbitrary and ad hoc.
The State-of-the arts: Trend “One economist’s trend is another economist’s cycle” Engle, R. F. and Granger, C. W. J Long-run Economic Relationships. Cambridge University Press.
Regression method is arbitrary and ad hoc.

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

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

43. Definition of the Trend Huang et al, Proc. Roy. Soc. Lond
Definition of the Trend Huang et al, Proc. Roy. Soc. Lond., 1998 Wu et al. PNAS 2007
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 an intrinsic and local property of the data; it is determined by the same mechanisms that generate the data.
Being local, it has to associate with a local length scale, and be valid only within that length span, and be part of a full wave length.
The method determining the trend should be intrinsic. Being intrinsic, the method for defining the trend has to be adaptive.
All traditional trend determination methods are extrinsic.

44. Let us use EMD to extract the trend and examine some relevant data
Trend should not be determined by regressions (parametric or non-parametric), but should be determined by successively removal of oscillations.
Let us use EMD to extract the trend and examine some relevant data

45. GSTA

46. AMO

47. Atlantic Multi-decadal Oscillation : AMO

48. Nature Article 1994

49. IMFs of GSTA

50. Significance Test of GSTA

51. IMFs of AMO

52. Significance Test of AMO

53. Mean Instantaneous Periods of IMF4 of GSTA

54. Mean Instantaneous Periods of IMF4 of AMO

55. Cross-Correlation between IMFs 4 of AMO and GSTA
Blue line: correlation of annual mean of GSTA and AMO
Red line: mean of correlation of each downsample of GSTA and AMO
Gray line: correlation of each downsample of GSTA and AMO

56. Global Surface Regressions

57. Detailed Comparisons between GSTA and AMO
Even on the noise level

58. Detrended GSTA

59. Detrended AMO

60. Fourier Spectra of Residues

61. Autocorr : Residues AMO

62. Autocorr : Residues GSTA

63. Cross-Correlation between IMFs 1-3 of AMO and GSTA (noise part)
Blue line: correlation of annual mean of GSTA and AMO
Red line: mean of correlation of each downsample of GSTA and AMO
Gray line: correlation of each downsample of GSTA and AMO

64. Global Ocean Surface Regressions

65. The true trend with all the cycles removed
The Warming Trend
The true trend with all the cycles removed

66. Analysis of trend, rate, and acceleration of global warming
Blue line is downsampling-mean of non-linear trend, i.e., last IMF.
Shadow area is the STD of the non-linear trend of all downsamples.

67. IPCC Global Mean Temperature Trend

68. Comparison between non-linear rate with multi-rate of IPCC
Blue shadow and blue line are the warming rate of non-linear trend.
Magenta shadow and magenta line are the rate of combination of non-linear trend and AMO-like components.
Dashed lines are IPCC rates.

69. Observations
There indeed is a cycle (MDV) and a trend (ST) co-existing.
The trend with ST+MDV is the same as IPCC. The true trend, ST, is not accelerating recently; the true rate is only half of what IPCC claimed.
The peak of the warming wave (~ 2005) seems to be over; the temperature should decrease over the next couple of decades gradually.

70. The rate of warming (oC/decade) over different temporal span.
Last 150 years
Last 100 years
Last 50 years
Last 25 years
AR4
0.04
0.07
0.13
0.18
ST and MDV
0.05
0.09
0.12
0.15 ST
0.08

71. GSAT Data and Various Trends

72. Annual Temperature Ranking : 2008
GISS
NCDC
CRU
Rank
2005
1998 1 2
2002
2003 3
2007 4
2006
2004 5 6
2001 7 8
2008
1997 9 19

73. 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.

74. 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.

75. Global Climate Change GCC is a scientific problem.
GCC is a political problem.
GCC is an economic problem.
GCC is a societal problem.
Let us work hard to understand it before it becomes a religious problem.

76. Observations
The most recent 150 years climate changes could only caused partially by CO2, and partially by natural fluctuations.
The recent global temperature is warming up, but the rate is only half of the alarming rate posted by IPCC in AR4.
Oceans seem to play a dominate and control role for climate change with years periods.
Meanwhile, we should do our best to increase energy efficiency and reduce carbon consumption for economy and national development.