New Directions in Oceanographic Time Series AnalysisThe main point The tools we use to look at data...
Transcript of New Directions in Oceanographic Time Series AnalysisThe main point The tools we use to look at data...
New Directions in Oceanographic Time Series Analysis
J. M. Lilly1,S. C. Olhede2, A. M. Sykulski2, S. Elipot3, S. N. Waterman4
1NorthWest Research Associates, 2University College London,3University of Miami, 4University of British Columbia
February 26, 2014
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
The main point
The tools we use to look at data matter.
Download this talk, and Matlab toolbox JLAB, fromwww.jmlilly.net.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
A surface drifter trajectory
The box shows a region where the drifter is trapped in an eddy.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
The first tool: your eyes
Oscillatory motions due to an eddy are seen in the record center.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Rotary spectra using the periodogram
The periodogram is the squared Fourier transform. Red line is f .The rotary spectrum is the spectrum of z(t) ≡ u(t) + iv(t).
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Rotary spectral using the multitaper method
Low-frequency peak of the cyclonic eddy is clearly apparent.Mysterious super-inertial peak at negative frequencies.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Rotary wavelet transform
Strong frequency-shifting of inertial oscillations is revealed.Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Separation of eddy currents from residual
Wavelet transform is used as a basis for extracting the eddycomponent. This local best fit is called wavelet ridge analysis.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Historical dataset of eddy-resolving subsurface floats
1471 different instruments, 700,000 data points.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
A preliminary eddy atlas
Apparent eddies in Rossby number band Ro = 1/64 to Ro = 1.Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Rotary spectra using the periodogram
Moral: Our data is limited by the limitations of our methods.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
What is time series analysis for?
The goal of time series analysis is (i) to extract as much usefulinformation as possible from the data, while at the same time, (ii)avoiding mis-interpretation of artifacts and spurious features.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Trajectories from a model
From a simulation by E. Danioux.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Rotary spectra using the periodogram
The periodogram is the squared Fourier transform. Red line is f .
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Rotary spectra using the adaptive multitaper method
The periodogram (green line) is dominated by variance andblurring. Periodogram slope and apparent peaks are false.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Schematic of aliasing for rotary spectra
For rotary spectra, aliasing is not folding. It’s wrapping.Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Summary of fundamentals
Do not neglect the fundamentals. We need to know what can gowrong — or we will waste time by interpreting spurious features.
Aliasing (wrapping)
Spectral blurring (broadband bias)
Variance (false peaks)
Important!
Any ”spectrum” created from data is not the true spectrum. It isan estimate. That estimate is a joint function of the data and theestimation method.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Summary of fundamentals
“More lives have been lost looking at the raw periodogram than byany other action involving time series!” —J. W. Tukey
Recommendations
Use the multitaper method of Thomson (1982).
Read: Park et al. (1987), ”Multitaper spectral analysis ofhigh-frequency seismograms”, Journal of Geophysical Research.
Use routines mspec and sleptap from JLAB, available atwww.jmlilly.net
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Wavelets
An exceedingly brief summary of some useful recent results.
What is a wavelet? It is an intermediate basis betweendelta-functions in time and sinusoids. It lets you describe your dataas being composed of localized oscillations. But...
There has been a lot of confusion about how to choose the rightwavelet: Cauchy / Klauder, Derivative of Gaussian, Shannon,Bessel, Morlet, ... which we have sorted out.
All of these are special cases of a much broader family, thegeneralized Morse wavelets, Lilly and Olhede (2009,2013).
Recommendations
Generally speaking, you only need one wavelet, the Airy waveletΨ(ω) = ωβe−ω
3, see Lilly and Olhede (2013).
Use routines wavetrans and morsewave from JLAB, available atwww.jmlilly.net.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
A foundation for time-varying analysis
Seismogram and float trajectory:
Definitely not sinusoidal
Inhabit multiple dimensions
Contaminated by ’noise’
=⇒ Modulated multivariate oscillations
Eddies, internal wave packets,seismic waves, ENSO, ...
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Time-varying analysis
There exists an extremely powerful method for analyzing andinterpreting time-varying properties of quasi-periodic orquasi-oscillatory signals.
“The theory of instantaneous moments” –or–“The analytic signal method”
Foundation papers: Gabor (1946), Boashash (1992), Cohen(1995), Vakman and Vainshtein (1977), Picinbono (1997)
Extension to 2D (e.g. horizontal velocity currents): Lilly andOlhede (2009a)
Extension to 3D (e.g. internal wave packets): Lilly (2010)
Extension to N-D (e.g. climate fluctuations): Lilly and Olhede(2012)
Relationship to physical quantities (e.g. angular momentum,Kelvin circulation): Lilly (2012)
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Instantaneous frequency and bandwidth
Represent a signal x(t) as an oscillation with changing amplitudeand phase:
x(t) = a(t) cosφ(t)
For a given x(t), a(t) and φ(t) are uniquely defined via theanalytic signal
x+(t) ≡ x(t) + iHx(t) ≡ a(t)e iφ(t)
where x(t) = <x+(t). This defines the canonical pair a(t), φ(t).
If you wish to recover ao and ωot for x(t) = ao cos(ωot), there isno other alternative involving a linear filter (Vakman).
References
Gabor (1946), Vakman and Vainshtein (1977), Picinbono (1997)
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Instantaneous frequency and bandwidth
For a univariate signal
x(t) = a(t) cosφ(t)
the instantaneous frequency and instantaneous bandwidth are
ω(t) ≡ d
dtφ(t)
υ(t) ≡ 1
a(t)
d
dta(t).
These fundamental quantities decompose the first two moments ofthe spectrum of x+(t) across time, relating time variation tofrequency-domain structure.
References
Gabor (1946), Boashash (1992), Cohen (1995),Gabor (1946)
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Geometry of modulated oscillations in two dimensions
How can you depart from purely oscillatory (sinusoidal) behavior?
Amplitude modulation, distortion, and precession.These three signals have identical average spectra (!)Also by changing frequency with fixed geometry.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Geometry of modulated oscillations in three dimensions
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Wavelet Ridge Analysis
A wavelet is not a foundation. It is a lens. The foundation is theanalytic signal, and wavelets give us a way to extract an estimateof an analytic signal from a noisy time series.
Wavelet ridge analysis is a powerful, flexible, and rigorous methodfor extracting quasi-periodic signals of unknown frequency fromindividual time series or arrays of time series.
Potential applications: eddies, internal wave packets, ENSOsignals, seismic waves...
Start here
Lilly, J. M., and S. C. Olhede (2009). Wavelet ridge estimation ofjointly modulated multivariate oscillations.
Use routines wavetrans and ridgewalk from JLAB, available atwww.jmlilly.net.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Rotary wavelet transform
Strong frequency-shifting of inertial oscillations is revealed.Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Separation of eddy currents from residual
We also understand how to quantify the errors involved in thisquasi-periodic signal extraction process.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Deep connection to vortex dynamics
Red = one-point estimate | Lilly, Olhede, and Early (2014), in prep.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Covariance ellipse geometry and eddy forcing
Eddy vorticity flux convergence Q is related to covariance ellipses.The are only four ways that covariance ellipses can generate eddyvorticity flux divergence. Joint work with S. Waterman.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Covariance ellipse geometry and eddy forcing
For more information...
This evening’s Poster Session 058: Mesoscale ocean processes andtheir representation in earth system models
Waterman and Lilly: Geometric ingredients of eddy-mean flowfeedbacks, and a time-varying extension
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Stochastic modeling
Trajectories from a QG model. How can these be described as astochastic process? Joint work with A. Sykulski, S. Olhede,E. Danioux, and J. Early.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Stochastic modeling
For more information...
See Sykulski et al. (2013a,b), submitted manuscripts, athttp://www.ucl.ac.uk/statistics/people/adamsykulski.
Friday’s Session 071: Frontiers of oceanographic data and methods
09:30 Sykulski, Lilly, Olhede, Danioux, and Early: Stochasticmodels for Lagrangian data
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Covariance analysis using the analytic signal
SVD analysis of vector wind + SST using the analytic signal leadsto SST amplitude and phase, wind ellipses, and wind phase.=⇒ Eastward propagation of wind signals.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
Covariance analysis using the analytic signal
For more information...
Elipot (2013), “On Singular Value Decomposition of analyticcovariance matrices of univariate and bivariate variables, withapplication to El Nino Southern Oscillation”, submitted.
Talk to Shane.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
The methods paradox
The tools we use to look at data matter... but there is a challenge.
The methods paradox
The need for powerful and trustworthy data analysis methodsincreases as datasets become larger and more complex, but
There are intrinsic barriers that prevent the appropriatemethods from reaching those who need them.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
The methods paradox
What are these barriers?
Because of the increasing size of fields, those working withdata (e.g. oceanographers) and those working on methods fortreating data (e.g. statisticians and time series analysts) arefrequently unable to find each other.
Therefore, those on the data side very frequently are forced to’reinvent the wheel’, while meanwhile,...
Those on the methods side do not have enough information todevelop the right methods for the most pressing problems.
Both sides create their own jargon, and cite primarily withintheir own literature, reinforcing the barriers.
Based on my experience in three different areas: time seriesanalysis, mapping methods, and covariance analysis.
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis
The methods paradox
Proposed solutions to the methods paradox:
We need long-term, sustained interactions between theoceanographic community and methods communities such astime series and statistics.
We need to embrace a spirit of collaboration for time seriesmethods and other data analysis expertise—rather than only’do it yourself’ or ’black box’ approaches.
We need to prioritize the creation of accessible tutorialliterature in modern data analysis methods that is tailored tothe needs of our community.
Just as a class of specialists in numerical modeling is integralto oceanographic research, we also need to train a new classof specialists in data analysis theory and methods.
The end. Thanks!
Lilly, Olhede, Sykulski, Elipot, & Waterman New Directions in Oceanographic Time Series Analysis