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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. Norden E. Huang Research Center for Adaptive Data Analysis National Central University, Taiwan. Satellite Altimeter Data : Greenland. Two Sets of Data. IPCC Global Mean Temperature Trend.

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

  7. OnDefinition 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

  30. 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 SmoothedFor the Difference : Mean = - 0.082; STD = 0.01974

  44. GSTA : Annual Variance Jones and HHT SmoothedMean 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

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