L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 1 SOES 6047 Global Climate Cycles L14: Time Series Analysis Frequency Domain Methods ๏ Dr. Heiko Pälike ๏ ๏ Ext , Rm. 164/34 ๏ Dr. Heiko Pälike ๏ ๏ Ext , Rm. 164/34

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 2 Last “spectra” lecture: ๏ Time series analysis – Why? ๏ Concepts and terminology of time series analysis ๏ Overview of spectral analysis rationale and methods ๏ The concept of “windows” ๏ Resources: where to find software & tools ๏ Time series analysis – Why? ๏ Concepts and terminology of time series analysis ๏ Overview of spectral analysis rationale and methods ๏ The concept of “windows” ๏ Resources: where to find software & tools

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 3 Objectives & learning outcomes ๏ 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 ๏ 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

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 4 Lecture outline ๏ What is the output of spectral analysis when dealing with geological (climatic) data ๏ some necessary maths background ๏ The principle of resolution statistical confidence trade- off ๏ Examples ๏ Resources: where to find software & tools ๏ What is the output of spectral analysis when dealing with geological (climatic) data ๏ some necessary maths background ๏ The principle of resolution statistical confidence trade- off ๏ Examples ๏ Resources: where to find software & tools

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 5 Note: this is probably one of the few lectures where textbooks are the appropriate source for information! Bloomfield, P., Fourier analysis of time series: An introduction (John Wiley & Sons, 1976). Ghil, M., et al. (2002), ‘Advanced specctral methods for climatic time series’, Reviews of Geophysics 40, Hinnov, L. A. (2000), ‘New perspectives on orbitally forced stratigraphy’, Annual Review of Earth and Planetary Science 28, 419–475. Jenkins, G. M. & Watts, D. G., Spectral Analysis and its applications (Holden-Day, Oakland, CA, 1968). Percival, D. & Walden, A. T., Wavelet methods for Time Series Analysis (Cambridge University Press, 2002). Press, W. H., et al., Numerical Recipes: The art of scientific computing (Cambridge University Press, 1992). Schulz, M. & Mudelsee, M. (2002), ‘REDFIT: estimating red-noise spectra directly from unevenly spaced paleoclimatic time series’, Computers & Geosciences 28, 421–426. Schulz, M. & Stattegger, K. (1997), ‘Spectrum: Spectral Analysis of unevenly spaced paleoclimatic time series’, Computers & Geoscience 23, 929–945. Thomson, D. J. (1982), ‘Spectrum estimation and harmonic analysis’, Proceedings of the IEEE 70, 1055–1096. Thomson, D. J. (1990), ‘Quadratic inverse spectrum estimates; applications to palaeoclimatology’, Philosophical Transactions of the Royal Society of London Series A - Mathematical Physical and Engineering Sciences 332, 539–597. Vautard, R., G. M. (1989), ‘Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series’, Physica D35, 395–424. 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

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 6 The aim of spectral analysis ๏ to detect and quantify periodically re-occurring components in data ๏ to statistically assess whether periodicities are potentially part of the noise background, or significant ๏ by way of cross-spectral analysis, to assess periodic signals that are common between two (time) series ๏ to assess phase relationships between two (time) series as a function of frequency (leads & lags) ๏ to assess the statistical significance of phase relationships ๏ to compute the coherence between two (time) series, i.e., what percentage of signal in one time series co-varies with that of another ๏ to determine periodicities that can be filtered ๏ to determine the noise background spectrum ๏ to detect and quantify periodically re-occurring components in data ๏ to statistically assess whether periodicities are potentially part of the noise background, or significant ๏ by way of cross-spectral analysis, to assess periodic signals that are common between two (time) series ๏ to assess phase relationships between two (time) series as a function of frequency (leads & lags) ๏ to assess the statistical significance of phase relationships ๏ to compute the coherence between two (time) series, i.e., what percentage of signal in one time series co-varies with that of another ๏ to determine periodicities that can be filtered ๏ to determine the noise background spectrum

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 7 History of Fourier series ๏ 19th/20th century: two paths for Fourier analysis: Continuous & Discrete. ๏ CONTINUOUS ๏ Fourier extends the analysis to arbitrary function (Fourier Transform). ๏ Dirichlet, Poisson, Riemann, Lebesgue address F. eries convergence. ๏ Other FT variants born from varied needs (ex.: Short Time FT - speech analysis). ๏ DISCRETE: Fast calculation methods (FFT) ๏ Gauss, first usage of FFT (manuscript in Latin went unnoticed!!! Published 1866). ๏ IBM’s Cooley & Tukey “rediscover” FFT algorithm (“An algorithm for the machine calculation of complex Fourier series”). ๏ Other DFT variants for different applications (ex.: Warped DFT - filter design & signal compression). ๏ FFT algorithm refined & modified for most computer platforms. ๏ 19th/20th century: two paths for Fourier analysis: Continuous & Discrete. ๏ CONTINUOUS ๏ Fourier extends the analysis to arbitrary function (Fourier Transform). ๏ Dirichlet, Poisson, Riemann, Lebesgue address F. eries convergence. ๏ Other FT variants born from varied needs (ex.: Short Time FT - speech analysis). ๏ DISCRETE: Fast calculation methods (FFT) ๏ Gauss, first usage of FFT (manuscript in Latin went unnoticed!!! Published 1866). ๏ IBM’s Cooley & Tukey “rediscover” FFT algorithm (“An algorithm for the machine calculation of complex Fourier series”). ๏ Other DFT variants for different applications (ex.: Warped DFT - filter design & signal compression). ๏ FFT algorithm refined & modified for most computer platforms.

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 8 ๏ Aim of this lecture: ๏ for you to understand, ๏ and to produce a figure as shown here ๏ Need to understand principles in order to evaluate what these data show us... ๏ Aim of this lecture: ๏ for you to understand, ๏ and to produce a figure as shown here ๏ Need to understand principles in order to evaluate what these data show us... What do these figures mean? Reproduced by permission of American Geophysical Union: Wade, B., Palike, H., Oligocene climate dynamics. Paleoceanography, v19 PA December Copyright [2004]. American Grophysical Union.

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 9 ๏ Aim of this lecture: ๏ for you to understand, ๏ and to produce a figure as shown here ๏ Need to understand principles in order to evaluate what these data show us... ๏ Aim of this lecture: ๏ for you to understand, ๏ and to produce a figure as shown here ๏ Need to understand principles in order to evaluate what these data show us... What do these figures mean? Reproduced by permission of American Geophysical Union: Wade, B., Palike, H., Oligocene climate dynamics. Paleoceanography, v19 PA December Copyright [2004]. American Grophysical Union.

L12 Time Series Analysis: Introduction SOES Global Climate Cycles 10 Time vs. Frequency domain (recap)! ๏ Given certain conditions, signals can be represented ๏ and manipulated in the TIME or the FREQUENCY domain. ๏ For infinitely long & stationary signals with a fixed period, ๏ the spectrum is an infinitely narrow peak at the central frequency ๏ For signals of finite length, the spectrum is smeared out, ๏ and the spectrum turns into a spectral estimate ๏ next slides: how to express this mathematically ๏ Given certain conditions, signals can be represented ๏ and manipulated in the TIME or the FREQUENCY domain. ๏ For infinitely long & stationary signals with a fixed period, ๏ the spectrum is an infinitely narrow peak at the central frequency ๏ For signals of finite length, the spectrum is smeared out, ๏ and the spectrum turns into a spectral estimate ๏ next slides: how to express this mathematically Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 11 time,t frequency,f F s(t)S(f) = F [s(t)] Frequency analysis: why?  Fast & efficient insight on signal’s building blocks.  Simplifies original problem - ex.: filtering  Powerful & complementary to time domain analysis techniques. analysis synthesis s(t), S(f) : Transform Pair General Transform as problem-solving tool Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 12 ๏ In essence, what Fourier analysis does, is to represent a given signal as a set of amplitude and phase information pairs for a number of frequency bins ๏ So, in principle, we could re-construct a signal (or at least approximate it), by adding a number of different sine and cosine terms together ๏ have to get down to a little maths to give it justice..... brace !! ๏ In essence, what Fourier analysis does, is to represent a given signal as a set of amplitude and phase information pairs for a number of frequency bins ๏ So, in principle, we could re-construct a signal (or at least approximate it), by adding a number of different sine and cosine terms together ๏ have to get down to a little maths to give it justice..... brace !! Fourier de-composition J. Laskar, P. Robutel, F. Joutel, M. Gastineau, A. C. M. Correia, and B. Levrard. A long-term numerical solution for the insolation quantities of the Earth. Astronomy and Astrophysics v. 428, p 261–285. (2004) Reproduced with permission © ESO.

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 13 Fourier Series (FS) * see next slide A periodic function s(t) satisfying Dirichlet’s conditions * can be expressed as a Fourier series, with harmonically related sine/cosine terms. a 0, a k, b k : Fourier coefficients. k: harmonic number, T: period,  = 2  /T For all t but discontinuities Note: {cos(k ω t), sin(k ω t) } k form “orthogonal” base (signal average over a period, i.e. average term & zero-frequency component.) analysis synthesis Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 14 Fourier Series convergence s(t) piecewise-continuous; s(t) piecewise-monotonic; s(t) absolutely integrable, (a) (b) (c) Dirichlet conditions In any period: Example: square wave (a)(b)(c) Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 15 Fourier Series analysis (I) * Even & Odd functions Odd : s(-x) = -s(x) Even : s(-x) = s(x) FS of odd * function: square wave. (zero average) (odd function) Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 16 Fourier Series analysis (II) f k =k  /2  r K = amplitude,  K = phase v k = r k cos (  k t +  k ) Polar Fourier spectrum representations v k = a k cos(  k t) - b k sin(  k t) Fourier spectrum of square-wave. Rectangular Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 17 Fourier Series time shifting FS of even function:  /2-advanced square-wave  /2-advanced square-wave (even function) (zero average) phase amplitude BUT Note:amplitudes unchanged BUT phases advance by k   /2. Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 18 Fourier Series synthesis Square wave reconstruction from spectral terms Convergence may be slow (~1/k) - ideally need infinite terms. Practically, series truncated when remainder below computer tolerance Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 19 Fourier Integral (FI) Fourier analysis tools for aperiodic signals. Any aperiodic signal s(t) can be expressed as a Fourier integral if s(t) piecewise smooth (1) in any finite interval (- L,L) and absolute integrable (2). Fourier Integral Theorem (3) (2) s(t) continuous, s’(t) monotonic (1) (3) Fourier Transform (Pair) - FT analysisanalysis synthesissynthesis Complex form Real-to-complex link Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 20 Fourier analysis - tools Input Time Signal Frequency spectrum Discrete DFS Periodic (period T) Continuous DTFT Aperiodic Discrete DFT Periodic (period T) Discrete ContinuousFT Aperiodic FS Continuous Note: j =√-1,  = 2  /T, s[n]=s(t n ), N = No. of samples Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 21 Spectrum estimates ๏ For continuous time series of finite duration, spectral analysis gives an estimate of the true spectrum, i.e., it treats the data series as a sample of the total (infinitely long) series ๏ Thus, any spectral estimate will have some artefacts ๏ leakage of spectral power ๏ possibly, aliasing of spectral power, if frequency content of data is such that Nyquist frequency is exceeded ๏ statistical significance of peaks detected is a function of assumed noise model -- a partially subjective evaluation ๏ Spectral analysis is all about trade-off: ๏ Resolution versus Statistical significance ๏ Leakage versus side-lobe suppression ๏ For continuous time series of finite duration, spectral analysis gives an estimate of the true spectrum, i.e., it treats the data series as a sample of the total (infinitely long) series ๏ Thus, any spectral estimate will have some artefacts ๏ leakage of spectral power ๏ possibly, aliasing of spectral power, if frequency content of data is such that Nyquist frequency is exceeded ๏ statistical significance of peaks detected is a function of assumed noise model -- a partially subjective evaluation ๏ Spectral analysis is all about trade-off: ๏ Resolution versus Statistical significance ๏ Leakage versus side-lobe suppression

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 22 Periodogramme problems The spectral analysis of discrete samples results in discrete frequency bins of the estimated spectrum For the simple periodogramme (one frequency,phase pair per data point), the true spectral peak might not fall exactly into a frequency bin In this case, the spectral power at this frequency is smeared out to neighbouring bins, AND BEYOND! as number of samples is increased, the number of frequency bins is increased, BUT the standard deviation error remains the same and HIGH! Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 23 A few simple examples... ๏ Example: create an artificial periodic signal, with two frequency components, exactly with 40kyr and 200kyr periods ๏ The TRUE spectrum should be two single lines with known amplitudes ๏ Example: create an artificial periodic signal, with two frequency components, exactly with 40kyr and 200kyr periods ๏ The TRUE spectrum should be two single lines with known amplitudes Graph produced by Heiko Palike, University of Southampton, using excel for mac

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 24 ๏ The expected spectral peaks in the frequency domain should be at 1/(0.040kyr) and 1/(0.200kyr), i.e., at 25 and 5 cycles /Myr. The total series has 401 data points ๏ We use AnalySeries (Practical on Wednesday) to compute a Blackman-Tukey Spectral estimate, with a medium trade-off between frequency resolution (bandwidth) and significance ๏ The expected spectral peaks in the frequency domain should be at 1/(0.040kyr) and 1/(0.200kyr), i.e., at 25 and 5 cycles /Myr. The total series has 401 data points ๏ We use AnalySeries (Practical on Wednesday) to compute a Blackman-Tukey Spectral estimate, with a medium trade-off between frequency resolution (bandwidth) and significance A few simple examples... Graphs produced by Heiko Palike, University of Southampton using AnalySEries software AnalySeries software settings scree nshot settings are outlined by Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 25 Bandwidth: a practical example ๏ using AnalySeries, we first create a time series of the Earth’s eccentricity over the past 2000 kyr ๏ we now want to assess the effects of the trade-off between bandwidth and confidence by performing a “Blackman-Tukey” spectral analysis: ๏ using AnalySeries, we first create a time series of the Earth’s eccentricity over the past 2000 kyr ๏ we now want to assess the effects of the trade-off between bandwidth and confidence by performing a “Blackman-Tukey” spectral analysis: Graphs produced by Heiko Palike, University of Southampton using AnalySeries software

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 26 ๏ choosing two different settings for the number of lags in the Blackman-Tukey window, we get the following results: ๏ Note the difference in the error bounds (different Y-scales!) ๏ choosing two different settings for the number of lags in the Blackman-Tukey window, we get the following results: ๏ Note the difference in the error bounds (different Y-scales!) Bandwidth: a practical example Graphs produced by Heiko Palike, University of Southampton using AnalySeries software AnalySeries software settings screenshot settings are outlined by Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 27 Phase estimation ๏ Now an example to estimate phases... ๏ start with two periodic samples, with a longer and shorter frequency component ๏ plotting age from the right to the left, the red signal has a small phase lag with respect to the blue one, for the shorter period only. ๏ Now an example to estimate phases... ๏ start with two periodic samples, with a longer and shorter frequency component ๏ plotting age from the right to the left, the red signal has a small phase lag with respect to the blue one, for the shorter period only. Graphs produced by Heiko Palike, University of Southampton using AnalySeries software

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 28 Phase estimation ๏ our aim now is to show that there is a phase difference, and to quantify what it is (it should be 1/5th of the shorter period 25 kyr, i.e. 5 kyr, or in angular terms, (5/25)*360°, or in radians, (5/25)*2π ๏ This is an example of cross-spectral analysis... ๏ our aim now is to show that there is a phase difference, and to quantify what it is (it should be 1/5th of the shorter period 25 kyr, i.e. 5 kyr, or in angular terms, (5/25)*360°, or in radians, (5/25)*2π ๏ This is an example of cross-spectral analysis... AnalySeries software settings screenshot settings are outlined by Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 29 ๏ First, the two spectra from the two curves should be identical... we have only changed the phase, not the amplitude..... ๏ indeed, we find two spectral peaks for each, at around 10 cycles/ Myr, and 40 cycles/Myr ๏ First, the two spectra from the two curves should be identical... we have only changed the phase, not the amplitude..... ๏ indeed, we find two spectral peaks for each, at around 10 cycles/ Myr, and 40 cycles/Myr Phase estimation Graphs produced by Heiko Palike, University of Southampton using AnalySeries software

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 30 ๏ Now we investigate the phase plots, together with their uncertainty estimates... ๏ Cross-spectral analysis computes phase estimates for all frequencies... only those where we actually have a signal are meaningful, and this occurs where the (blue and green) uncertainty estimates become narrow. ๏ we find zero phase lag for the longer period signal, and ~1.25 radians for the shorter period one. Is this expected? ๏ Now we investigate the phase plots, together with their uncertainty estimates... ๏ Cross-spectral analysis computes phase estimates for all frequencies... only those where we actually have a signal are meaningful, and this occurs where the (blue and green) uncertainty estimates become narrow. ๏ we find zero phase lag for the longer period signal, and ~1.25 radians for the shorter period one. Is this expected? Phase estimation Graphs produced by Heiko Palike, University of Southampton using AnalySeries software

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 31 Noise background ๏ starting with a data set of benthic d13C, our aim is to estimate noise backgrounds ๏ it is visible from the plot of the data below, that there appears to be a periodic oscillations every ~400 kyr for these data ๏ starting with a data set of benthic d13C, our aim is to estimate noise backgrounds ๏ it is visible from the plot of the data below, that there appears to be a periodic oscillations every ~400 kyr for these data Graphs produced by Heiko Palike, University of Southampton using AnalySeries software

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 32 Noise background ๏ We use the software packet “SSA-MTM Toolkit” to do the noise- background calculation. ๏ After loading the data, we perform a Multitaper-method spectral analysis, again noticing the resolution significance trade-off ๏ We use the software packet “SSA-MTM Toolkit” to do the noise- background calculation. ๏ After loading the data, we perform a Multitaper-method spectral analysis, again noticing the resolution significance trade-off SSA-MTM Toolkit software package settings scree nshot settings are outlined by Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 33 ๏ We use the software packet “SSA-MTM Toolkit” to do the noise- background calculation. ๏ After loading the data, we perform a Multitaper-method spectral analysis, again noticing the resolution significance trade-off ๏ Note the overall fit of a median noise background, and the offset 90%, 95% and 99% significance curves ๏ Remember that these were obtained with certain assumptions.... ๏ We use the software packet “SSA-MTM Toolkit” to do the noise- background calculation. ๏ After loading the data, we perform a Multitaper-method spectral analysis, again noticing the resolution significance trade-off ๏ Note the overall fit of a median noise background, and the offset 90%, 95% and 99% significance curves ๏ Remember that these were obtained with certain assumptions.... Noise background Screenshot of graph produced by Heiko Palike, University of Southampton using the SSA-MTM Toolkit software package

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 34 Software tools: get your hands dirty Heiko Palike, University of SouthamptonAnalySeries software information screen shot by Heiko Palike, University of Southampton SSA-MTM Toolkit, version 4.2 software package information screen shot by Heiko Palike, University of Southampton

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 35 Resources: Spectral Analysis ๏ AnalySeries ( ๏ SSA-MTM Toolkit ( ๏ Multi-taper spectral techniques [Fortran] ( ๏ Multi-taper spectral analysis [MATLAB] ( ๏ SPECTRUM [works with unevenly spaced data!] bremen.de/~mschulz/index.html#software ๏ REDFIT, ENVELOPE (same address as SPECTRUM) computes the red-noise envelope, and amplitude spectrum ๏ ARAND (old Mac software which was developed for Specmap) ( ๏ Interactive (Web) Wavelet analysis ๏ AnalySeries ( ๏ SSA-MTM Toolkit ( ๏ Multi-taper spectral techniques [Fortran] ( ๏ Multi-taper spectral analysis [MATLAB] ( ๏ SPECTRUM [works with unevenly spaced data!] bremen.de/~mschulz/index.html#software ๏ REDFIT, ENVELOPE (same address as SPECTRUM) computes the red-noise envelope, and amplitude spectrum ๏ ARAND (old Mac software which was developed for Specmap) ( ๏ Interactive (Web) Wavelet analysis

L14 Time Series Analysis: Frequency Domain Methods SOES Global Climate Cycles 36 Key point summary ๏ 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 ๏ 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

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