Empirical Mode Decomposition Analysis using Rational Splines

Transcription

1 Empirical Mode Decomposition Analysis using Rational Splines Geoff Pegram Pegram, GGS, MC Peel & TA McMahon, (28). Empirical Mode Decomposition using rational splines: an application to rainfall time series. Proc. R. Soc. A (28) 464,

2 Outline What is EMD? The EMD algorithm with Cubic Splines Rational Splines 2D EMD on a radar field 2D EMD on the sphere

3 What is EMD? Huang et al. (1998) developed the Hilbert- Huang transform as a new spectral analysis technique. EMD of a time series. Hilbert transform of the EMD results for spectral analysis. Only looking at EMD here.

4 Why EMD / Spectral Analysis? Try to separate physical signal from random noise in a time series Understanding of physical processes Sources of variability in a process Relationship(s) with other known processes Degree of randomness Predictability Model validation

5 Prior to EMD Fourier Assumes that a time series can be decomposed into a set of linear components. Works for linear and stationary time series. However, produces many physically meaningless harmonics as the degree of non-linearity and non-stationarity increases. Most real world processes are non-linear and nonstationary (geosciences, biological, socio-economic, etc).

6 EMD Basics A time series can be decomposed into a set of Intrinsic Mode Function(s) (IMFs) and a residual The EMD algorithm extracts the IMFs and the residual from a time series Locally adaptive Robust with respect to non-linear / non-stationary data No data pre-processing (like removal of trend ) What is an IMF? IMF Rules (Huang et al., 1998) # Maxima + # Minima = # Zero Crossings ±1 Average of cubic splines fitting extrema should = at all points

7 Melbourne Monthly Precipitation (the Algorithm) Precipitation (mm) Observed Maxima Max Spline Minima Min Spline -2 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4

8 Melbourne Precipitation (Average of first fitted Spline) Precipitation (mm) Max Spline Min Spline Average -2 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4

9 Melbourne Precipitation (Observed and Average Spline) Precipitation (mm) Observed Average -2 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4

10 Melbourne Precipitation: 1 st Estimate of IMF1 (subtract average from data) 8 6 1st Estimate of IMF1 Precipitation (mm) Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4 Is it an IMF? Max. (39) + Min. (39) Zero Crossings (75) ± 1

11 EMD Basics (Recursive Sifting) Precipitation (mm) st Estimate of IMF1 Max Spline Min Spline Average -8 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4

12 Melbourne Precipitation (2 nd Estimate of IMF1) 8 6 2nd Estimate of IMF1 Precipitation (mm) Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4 Is it an IMF? Max. (38) + Min. (38) = Zero Crossings (77) ± 1

13 Melbourne Precipitation (1 st IMF) 8 6 IMF1 Precipitation (mm) After 8 sifts IMF1 is extracted Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4

14 Melbourne Precipitation (Extracting IMF2) Precipitation (mm) Observed IMF1 Residual -8 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4 Subtract IMF1 from the original time series and sift the residual.

15 EMD Basics Extracting IMFs As each IMF is identified by the sifting process, subtract all the IMFs from the original series to produce a new residual The final residual is reached when there are <= 3 extrema; it may be a trend and or an unresolved IMF There are 3 IMFs in Melbourne precipitation example

16 Melbourne Precipitation (EMD results) 3 IMFs & Residual plotted with data 16 8 Precipitation (mm) Observed Residual Precipitation (mm) 4-4 IMF2 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4 8 IMF1 IMF3 Precipitation (mm) 4-4 Precipitation (mm) Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4-8 Dec-94 Dec-95 Dec-96 Dec-97 Dec-98 Dec-99 Dec- Dec-1 Dec-2 Dec-3 Dec-4

17 Melbourne Precipitation (IMF Features) Average Period (months) Varianc e (mm 2 ) Observed 748 Variance as a % of Observed (ΣIMFs + Residual) IMF (59) IMF (13) IMF (1) Residual (18) ΣIMFs + Residual (1)

18 Issue #2: Spline end conditions How to extrapolate cubic splines to the ends? Pad the time series? Set some rules? Mirror Average Slope = (SZero) Precipitation (mm) st Estimate of IMF2 Max Spline Min Spline Dec-94 Dec-95 Dec-96 Dec-97

19 End Condition rules comparison Run EMD with the three end condition rules on the same dataset to check for differences 8135 annual precipitation stations from around the world (GHCN version 2) Sample size range from years Mean annual precipitation Maximum = 6,336mm (Hawaii) Minimum =.5mm (Egypt)

20 Results How efficient was each end condition rule? More IMFs indicates the rule may be introducing fluctuations into the analysis End Condition Rule Total Number of IMFs IMFs / Station Mirror 29, Average 31, Slope = 26,19 3.2

21 How to control the ends and variance? Meran Downs data and First fit of splines to Extrema; p = Each segment is a cubic polynomial

22 1.9 Basis functions of cubic spline over unit interval: s(x) = A.u + B.t + C.u^3 + D.t^3 t = (x-xk)/(xk-xk+1) u=1-t t^3 u^3 basic cubic

23 Rational monomial 1/(1+pu) with p = 5 (pole at 1.2) Rational Spline segment formula s k (x) = A k u + B k t + C k u 3 /(1 + pt) + D k t 3 [1/(1 + pu)]

24 1.9 Scaled sum of elemental Basis Functions of Rational Spline Segment p = 5 t u=1-t t^3/(1+pu) u^3/(1+pt) rational (scaled)

25 Meran Downs data and First fit of splines to Extrema; p =

26 Meran Downs data and First fit of splines to Extrema; p =

27 EMD of Meran Downs: p = Obs IMF 1 IMF 2 IMF 3 Resid

28 EMD of Meran Downs: p = Obs IMF 1 IMF 2 IMF 3 Resid

29 Interim Conclusion We want to look at the effect of the tautness of the spline (controlled by p =, 1, 2, 4, 8) and the EMDings (s = 1 or ) on Overshoots Variance reduction Number of sifts Number of IMFs Computational efficiency we ve supercharged the code Now for some fun

30 2D EMD of a radar rainfield scan Sinclair S & Pegram GGS (25). Empirical Mode Decomposition in 2-D space and time: a tool for space-time rainfall analysis and nowcasting Hydrology and Earth System Sciences, 9,

31 Decomposition into IM Surfaces

32 The 2D radial spectrum of IMS1 & Resid1