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
On Definition Without a proper definition, logic discourse would be impossible. Without logic discourse, nothing can be accomplished. Confucius
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.
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.
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.
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.
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.
Methodology Based on observations from Monte Carlo numerical experiments on 1 million white noise data points. All IMF generated by 10 siftings. Fourier spectra based on 200 realizations of 4,000 data points sections. Probability density based on 50,000 data points data sections.
IMF Period Statistics IMF 123456789 number of peaks 34704216817683456416322087710471529026581348 Mean period2.8815.94611.9824.0247.9095.50189.0376.2741.8 period in year0.2400.4960.9982.0003.9927.95815.7531.3561.75
Statistical Significance for SOI IMFs 1 mon1 yr10 yr100 yr IMF 4, 5, 6 and 7 are 99% statistical significance signals.
Summary Not all IMF have the same statistical significance. Based on the white noise study, we have established a method to determine the statistical significant components. References: Wu, Zhaohua and N. E. Huang, 2003: A Study of the Characteristics of White Noise Using the Empirical Mode Decomposition Method, Proceedings of the Royal Society of London (in press) Flandrin, P., G. Rilling, and P. Gonçalvès, 2003: Empirical Mode Decomposition as a Filterbank, IEEE Signal Processing, (in press).
Statistical Significance Test Only the statistical Significant IMF components are signal above noise; therefore, they might be predictable.
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.