SOES Global Climate Cycles SOES 6047 Global Climate Cycles L16: Time Series Analysis Evolutionary and Wavelet methods Dr. Heiko Pälike Ext , Rm. 164/34

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 2 Last “spectra” lecture: ๏ Time series analysis – Frequency Domain Methods ๏ Spectral & time series analysis merely a tool ๏ Need to understand that we calculate a spectral ESTIMATE, not the true spectrum ๏ Need to understand effects of trade-off during spectral estimation ๏ after practical on Wednesday, should be able to perform your own (cross-)spectral analysis, ๏ including phase and uncertainty estimates ๏ Robust spectral estimation requires the application of a variety of methods to assess whether features obtained are meaningful or not ๏ Time series analysis – Frequency Domain Methods ๏ Spectral & time series analysis merely a tool ๏ Need to understand that we calculate a spectral ESTIMATE, not the true spectrum ๏ Need to understand effects of trade-off during spectral estimation ๏ after practical on Wednesday, should be able to perform your own (cross-)spectral analysis, ๏ including phase and uncertainty estimates ๏ Robust spectral estimation requires the application of a variety of methods to assess whether features obtained are meaningful or not

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 3 Objectives & learning outcomes ๏ Learn why one might want to use spectral methods that give time as well as frequency resolution ๏ Distinguish between evolutionary and wavelet analysis ๏ Learn which tools to use to quickly create wavelet plots ๏ Learn why one might want to use spectral methods that give time as well as frequency resolution ๏ Distinguish between evolutionary and wavelet analysis ๏ Learn which tools to use to quickly create wavelet plots

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 4 Percival, D. B. & Walden, A. T. Wavelet Methods for Time Series Analysis (Cambridge University Press, 2000a). Torrence, C. & Compo, G. P. (1998), ‘A practical guide to wavelet analysis’, B. Am. Meteorol. Soc. 79(1), Weedon, G. P., Time-Series Analysis and Cyclostratigraphy (Cambridge University Press, 2003). Yiou, P., Baert, E., & Loutre, M. F. (1996), ‘Spectral analysis of climate data’, Surveys in Geophysics 17, 619–663. Some references

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 5 Why use time-frequency methods? ๏ many examples of data sets that are not stationary ๏ such data sets cannot be analysed with “traditional” frequency-domain only methods, at least not without giving significantly misleading results! ๏ instead, use methods that allow resolution in time- domain as well as frequency domain ๏ This is called “joint time-frequency analysis” ๏ many examples of data sets that are not stationary ๏ such data sets cannot be analysed with “traditional” frequency-domain only methods, at least not without giving significantly misleading results! ๏ instead, use methods that allow resolution in time- domain as well as frequency domain ๏ This is called “joint time-frequency analysis” Redrawn based on: Weedon, G., (2003) Time-series analysis and cyclostratigraphy. 247p

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 6 Example No temporal localization Graph produced by Heiko Palike, University of Southampton

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 7 Example (II) Graph produced by Heiko Palike, University of Southampton

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 8 Problems of Fourier methods ๏ no information about how frequencies evolve over time ๏ not suitable for impulse signals ๏ low frequency resolution ๏ no information about how frequencies evolve over time ๏ not suitable for impulse signals ๏ low frequency resolution Heiko Palike, University of Southampton

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 9 Solution: time-frequency methods ๏ There are two main types of frequency analysis that also give us information about the time evolution of a signal: ๏ “Evolutive” methods: same as frequency domain methods covered in last lecture ๏ simply chop up original signal into a number of “windows” (can be overlapping or separate), and perform frequency analysis as before ๏ disadvantage: lose frequency resolution because fewer data points per window than total data set ๏ “Wavelet” methods: new development, adapt the window length according to the frequency (best of both worlds) ๏ for low frequencies, which change slower, use smaller number of windows ๏ for higher frequencies, which change faster in time, use larger number of windows ๏ There are two main types of frequency analysis that also give us information about the time evolution of a signal: ๏ “Evolutive” methods: same as frequency domain methods covered in last lecture ๏ simply chop up original signal into a number of “windows” (can be overlapping or separate), and perform frequency analysis as before ๏ disadvantage: lose frequency resolution because fewer data points per window than total data set ๏ “Wavelet” methods: new development, adapt the window length according to the frequency (best of both worlds) ๏ for low frequencies, which change slower, use smaller number of windows ๏ for higher frequencies, which change faster in time, use larger number of windows

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 10 Examples for sliding windows: Figure produced by L. Hinnov

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 11 Examples for sliding windows: Figure produced by L. Hinnov

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 12 More useful examples ๏ example here: sliding window (evolutive) multitaper-method spectral analysis of astronomical data ๏ note effect of the Earth’s precession slow-down ๏ example here: sliding window (evolutive) multitaper-method spectral analysis of astronomical data ๏ note effect of the Earth’s precession slow-down Heiko Palike, University of Southampton

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 13 ๏ A time-varying frequency component would yield identical Fourier spectra, but can be resolved by joint time-frequency analysis Example for time resolution Graphs produced by Heiko Palike, University of Southampton

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 14 Time Fourier Wavelets Frequency Windowed (evolutive) Fourier Time-Frequency Plane: Tilings Heiko Palike, University of Southampton

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 15 ๏ Wavelets can be interactively calculated at ๏ note “cone of influence” ๏ Wavelets can be interactively calculated at ๏ note “cone of influence” Wavelet spectrum example Graphs created by Heiko Palike, University of Southampton, using ResearchSystems software

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 16 ๏ wavelets are usually computed in power-of-two frequency steps ๏ average of wavelet spectrum gives “Fourier type” global spectrum ๏ wavelets are usually computed in power-of-two frequency steps ๏ average of wavelet spectrum gives “Fourier type” global spectrum Sunspot data example Graphs created by Heiko Palike, University of Southampton, using ResearchSystems software

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 17 Detecting signal changes ๏ Example from ODP Leg 199 (Init. Repts)... detect changes in downhole logs... Courtesy of IODP: ODP Leg 199 (Init. Repts) Shipboard Scientific Party, Chapter 12, Site College Station, TX (Ocean Drilling Program).

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 18 Analysis of data in depth-domain ๏ use wavelet analysis of data in depth domain (pre-agemodel) ๏ superimpose predicted Milankovitch pattern using existing age model ๏ check if predicted bands co-incide with what is actually contained within the data ๏ use wavelet analysis of data in depth domain (pre-agemodel) ๏ superimpose predicted Milankovitch pattern using existing age model ๏ check if predicted bands co-incide with what is actually contained within the data From: Palike, H., Norris, R.D., Herrle,J.O., Wilson, P.A., Coxall, H.K., Lear, C.H.,Palike, H., Norris, R.D., Herrle,J.O., Wilson, P.A., Coxall, H.K., Lear, C.H., Shackleton, N.J., Tripati, A.K., Wade, B.S (2006), Supporting Online Material for The Heartbeat of the Oligocene Climate System. Science, v. 314, p Reprinted with permission from AAAS. This figure may be used for non-commercial, classroom purposes only. Any other uses requires the prior written permission from AAAS.

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 19 The aim of evolutive analysis ๏ to detect and quantify periodically re-occurring components in data, that change their frequency characteristics with time ๏ to give an exploratory view of data; where is it worth analysing ๏ to determine changing sedimentation rates ๏ to make colourful plots... ๏ to detect and quantify periodically re-occurring components in data, that change their frequency characteristics with time ๏ to give an exploratory view of data; where is it worth analysing ๏ to determine changing sedimentation rates ๏ to make colourful plots...

L16 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 20 Resources: Evolutive Spectral Analysis ๏ Matlab (built-in toolkits for evolutive and wavelet analysis) ๏ Interactive (Web) Wavelet analysis ๏ Software used for interactive website (Fortran programmes) ๏ Matlab (built-in toolkits for evolutive and wavelet analysis) ๏ Interactive (Web) Wavelet analysis ๏ Software used for interactive website (Fortran programmes)

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 21 Key point summary ๏ joint time-frequency analysis a useful tool to detect signals that vary with time in any way, in either frequency or amplitude ๏ two types of joint time-frequency methods: ๏ evolutive (sliding window) methods ๏ wavelet methods ๏ a bit more complicated to use that simple frequency domain methods, but built into tools such as Matlab ๏ joint time-frequency analysis a useful tool to detect signals that vary with time in any way, in either frequency or amplitude ๏ two types of joint time-frequency methods: ๏ evolutive (sliding window) methods ๏ wavelet methods ๏ a bit more complicated to use that simple frequency domain methods, but built into tools such as Matlab

L16 Time Series Analysis: Evolutionary Methods SOES Global Climate Cycles 22 ๏ This resource was created by the University of Southampton and released as an open educational resource through the 'C-change in GEES' project exploring the open licensing of climate change and sustainability resources in the Geography, Earth and Environmental Sciences. The C-change in GEES project was funded by HEFCE as part of the JISC/HE Academy UKOER programme and coordinated by the GEES Subject Centre. ๏ This resource is licensed under the terms of the Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales license ( ๏ However the resource, where specified below, contains other 3rd party materials under their own licenses. The licenses and attributions are outlined below: ๏ The University of Southampton and the National Oceanography Centre, Southampton and its logos are registered trade marks of the University. The University reserves all rights to these items beyond their inclusion in these CC resources. ๏ The JISC logo, the C-change logo and the logo of the Higher Education Academy Subject Centre for the Geography, Earth and Environmental Sciences are licensed under the terms of the Creative Commons Attribution -non-commercial-No Derivative Works 2.0 UK England & Wales license. All reproductions must comply with the terms of that license. ๏ All content reproduced from the American Association for the Advancement of Science (AAAS) may be reproduced for non commercial classroom purposes only, any other uses requires the prior written permission from AAAS. Copyright statement