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

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.

Satellite Altimeter Data : Greenland

Two Sets of Data

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.

Traditional detrending Differencing, or differentiating Regression Filtering

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.

An Example of EMD Application Global Warming

IPCC Global Mean Temperature Trend

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

Jones (2003) Monthly GSTA Data

Jones (2003) 12 Monthly GSTA Data

Jones (2003) 12 Monthly GSTA Data

Jones (2003) GSTA Data Seasonal Variation

Jones (2003) GSTA Data Seasonal Variance

Jones Monthly GSTA Data : Fourier Spectrum

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.

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.

Jones Monthly GSTA Data : IMF

Jones Monthly GSTA Data : IMF Smoothed

Jones Monthly GSTA Data & HHT Smoothed

Jones Monthly GSTA Data : Fourier Spectrum Data & Smoothed

12 Monthly GSTA Data HHT Smoothed

Jones (2003) 12 Monthly GSTA Data

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

GSTA : Annual Variance Jones and HHT Smoothed Mean HHT = 0 GSTA : Annual Variance Jones and HHT Smoothed Mean HHT = 0.0750; Jones = 0.1158

GSTA : HHT Smoothed Seasonal Variation

GSTA : HHT Smoothed Seasonal Variance

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.

Global Climate Changes

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

Land Mass Distribution Geological time scale changes

J. Zachos, et al., 2001 Science, 292, 686-693.

Earth Orbital Parameters Milankovitch time scales

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

GSTA

IPCC Global Mean Temperature Trend

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

Slope computation

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. 1991 Long-run Economic Relationships. Cambridge University Press. Regression method is arbitrary and ad hoc.

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

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

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.

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

GSTA

AMO

Atlantic Multi-decadal Oscillation : AMO

Nature Article 1994

IMFs of GSTA

Significance Test of GSTA

IMFs of AMO

Significance Test of AMO

Mean Instantaneous Periods of IMF4 of GSTA

Mean Instantaneous Periods of IMF4 of AMO

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

Global Surface Regressions

Detailed Comparisons between GSTA and AMO Even on the noise level

Detrended GSTA

Detrended AMO

Fourier Spectra of Residues

Autocorr : Residues AMO

Autocorr : Residues GSTA

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

Global Ocean Surface Regressions

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

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.

IPCC Global Mean Temperature Trend

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.

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.

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

GSAT Data and Various Trends

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

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.

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.

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.

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 10-103 years periods. Meanwhile, we should do our best to increase energy efficiency and reduce carbon consumption for economy and national development.