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2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise

Nick Watkins
Nick Watkins

This document summarizes a talk given by Nick Watkins on distinguishing between the Hurst effect, 1/f spectra, and long range dependence (LRD) often conflated in time series analysis. It discusses how the Hurst effect refers to anomalous growth of the range over time, 1/f refers to a singularity in the Fourier spectrum, and LRD describes long memory seen in stationary processes with 1/f spectra. Mandelbrot originally showed the Hurst effect could be explained by fractional Gaussian noise exhibiting LRD. The document cautions against assuming stationarity and reviews alternative non-stationary models proposed by Mandelbrot to explain 1/f spectra without LRD.

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Balancing Fact and Formula in the Science of Complex Systems: The Example of 1/f Spectra Nick Watkins NickWatkins@mykolab.com
際際滷s 1-24 were given as an invited talk on 19th December 2014 at the Fall AGU in San Francisco, beginning the Session on Intermittency and Dynamical Complexity in Space Plasmas from the Sun to Interplanetary and Planetary Environments. I have also included the spares prepared in case of questions and being developed for future talks.
Thanks Tom, Giuseppe and Marius for inviting me Holger Kantz for hosting me in Dresden since September 2013 Co-authors Tim Graves, Christian Franzke, Bobby Gramacy, Scott Osprey and Paulo Davini Discussions with all of the above and Sandra Chapman, Eli Barkai, Igor Sokolov, Rainer Klages and Aleksei Chechkin among others
Theme Hurst effect Will today distinguish three things often taken as same Observed growth of range in time series: Hurst effect
Theme 1/f Hurst effect Will today distinguish three things often taken as same Observed growth of range in time series: Hurst effect Observation of a singularity at zero in Fourier spectra: 1/f
Theme (S)LRD 1/f Hurst effect Will today distinguish three things often taken as same Observed growth of range in time series: Hurst effect Observation of a singularity at zero in Fourier spectra: 1/f The long range dependence seen in stationary 1/f case: (S)LRD. Using 1/f as a diagnostic of LRD assumes stationarity
Fact: Anomalous growth of range Hurst Effect Hurst, Nature, 1957 I heard about the Nile in '64, ... the variance doesn't draw like time span as you take bigger and bigger integration intervals; it goes like time to a certain power different from one. Hurst was getting results that were incomprehensible. Mandelbrot, 1998
Formula: Long Range Dependence (S)LRD Hurst Effect Mandelbrot, van Ness, and Wallis, 1965-69 First [history in Graves et al, arXiv, 2014a] demonstration that Hurst effect could be explained by stationary long range dependent process Model, fractional Gaussian noise [see also Kolmogorovs Wiener Spiral], had singular spectral density at lowest frequencies. '( ) ~ S f f
The 1/f paradox If spectral density '( ) then i) it is singular as and ii) if we define an autocorrelation function via ( ) ( ) ( ) and use Wiener-Khinchine theorem to get from Fourier transform of ~ 0 緒 S f f f x t x t S falls off as power law, and '( ) then summed lags "blow up" its ( ) ワ f
Ionosphere Magnetosphere 1 1 2 2 , 22fBm: ( ) ( ) ( )~ ( ) 件 H H R H X t t s s dL s Infinite range memory kernel Gaussian
Fractional motions and noises 0 100 200 300 400 500 600 700 800 900 1000 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 Fractional Brownian motion, H=0.7 1 1 2 2 , 22fBm: ( ) ( ) ( )~ ( ) 件 H H R H X t t s s dL s Build a nonstationary, self similar walk (used wfbm in Matlab) fractional motion 2 1 H
Fractional motions and noises 0 100 200 300 400 500 600 700 800 900 1000 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 Fractional Brownian motion, H=0.7 1 1 2 2 , 22fBm: ( ) ( ) ( )~ ( ) 件 H H R H X t t s s dL s Build a nonstationary, self similar walk (used wfbm in Matlab) 0 100 200 300 400 500 600 700 800 900 1000 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fractional Gaussian noise fractional motion Then differentiate to give a stationary LRD noise fractional noise 2 1 H 2 1 H
Ionosphere Magnetosphere [], if infinite dependence is necessary it does not mean that IBM's details of ten years ago influence IBM today, because there's no mechanism within IBM for this dependence. However, IBM is not alone. The River Nile is [not] alone. They're just one- dimensional corners of immensely big systems. The behaviour of IBM stock ten years ago does not influence its stock today through IBM, but IBM the enormous corporation has changed the environment very strongly. The way its price varied, went up or went up and fluctuated, had discontinuities, had effects upon all kinds of other quantities, and they in turn affect us. Mandelbrot, interviewed in 1998 by B. Sapoval for Web of Stories One resolution of 1/f paradox In modern fractional Langevin models fGn is noise term e.g. reviews of Metzler et al, PCCP, 2014; Watkins GRL, 2013.
Ionosphere Magnetosphere [], if infinite dependence is necessary it does not mean that IBM's details of ten years ago influence IBM today, because there's no mechanism within IBM for this dependence. However, IBM is not alone. The River Nile is [not] alone. They're just one- dimensional corners of immensely big systems. The behaviour of IBM stock ten years ago does not influence its stock today through IBM, but IBM the enormous corporation has changed the environment very strongly. The way its price varied, went up or went up and fluctuated, had discontinuities, had effects upon all kinds of other quantities, and they in turn affect us. Mandelbrot, interviewed in 1998 by B. Sapoval for Web of Stories One resolution of 1/f paradox In modern fractional Langevin models fGn is noise term e.g. reviews of Metzler et al, PCCP, 2014; Watkins GRL, 2013. Resolution of apparent paradox is that world as a whole is Markovian, the memory is a consequence of looking at a piece of it. Generalises the Mori-Zwanzig approach.
Often see 1/f spectra and heavy tails Ionosphere Magnetosphere Ground-based magnetometers sense ionospheric currents 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 5 0 500 1000 1500 2000 2500 3000 1978 AE data: threshold percentile=99 Time, t [minutes] AE[nanoTesla] AE data threshold Slow 1/f region in AE power spectrum Tsurutani et al, GRL, 1991 See also Hush et al, Poster SH53A-4209 this afternoon Moscone South
Joint inference on LRD and heavy tails. Tim Graves PhD developed Bayesian method: tested on 留-stable ARFIMA(0,d,0) where heavy tails & LRD co-exist, see Graves et al, arXiv, submitted CSDA 2014b; & my talk from AGU, 2012 on 際際滷share. 1.5 0.15d Test on Synthetic ARFIMA data
Other models for 1/f Ionosphere Magnetosphere Selecta H Selecta N
1/f without (S)LRD (S)LRD 1/f Hurst effect Before (S)LRD models, Mandelbrot [1963-67] had proposed other 1/f models which were not stationary LRD in same sense as fGn. Solved 1/f paradox by a different route. Still little known in the geosciences [but see Klemes, WRR, 1974].
Ionosphere Magnetosphere Abrupt state changes Fat distributions of switching times: Levy (E[t^2] = ) case.
The conditional spectrum: Magnetosphere Mandelbrot 1967 reviewed in N2, Selecta, 1999
The conditional spectrum: Magnetosphere Mandelbrot 1967 reviewed in N2, Selecta, 1999 Numerical 1/f spectrum need not estimate Wiener- Khinchine spectrum. Instead depends on conditioning length T. Unlike stationary LRD model, singularity is an artefact.
Hurst effect from state changes Ionosphere Magnetosphere Franzke et al, in review, Sci. Rep., 2014 Interestingly, the classic Lorenz 63 model (columns 1 and 2) can generate Hurst effect in some measures, such as DFA, even without long tailed waiting times between regime shifts. Confirms that Hurst effect is easier to generate than 1/f or full blown S(LRD).
Formula versus fact Like the ear, the eye is very sensitive to features that the spectrum does not reflect. Seen side by side, different 1/f noises, Gaussian [i.e. fGn], dustborne [i.e. fractional renewal] and multifractal, obviously differ from one another- Mandelbrot, Selecta N, 1999. Nothing can be more fatal to progress than a too confident reliance on mathematical symbols; for the student is only too apt to consider the formula and not the fact as the physical reality. Thomson (Kelvin) & Tait, 1890 edition.
Summary The 1/f paradox: The "1/f" spectral shape seen throughout physics, & why it seems paradoxical. Many faces of 1/f: Mandelbrot promoted two resolutions of paradox in the mid 60s: stationary long range dependence (S)LRD , or what he dubbed conditionally stationary renewal models. History is in Graves et al, arXiv, 2014a. One formula consistent with different facts. Ways forward ? Mandelbrots Selecta (Vols. N &H) urged use of our eyes as well as formalism, and Ive advertised two new results in this spirit: Bayesian method for choosing between (S)LRD models, Graves et al, arXiv, 2014b, and work on a dynamical origin for Hurst effect in the Lorenz model (Franzke et al, Scientific Reports, in review, 2014 ).
SPARE SLIDES
THE ROAD TO MULTIFRACTALS
So what were the other models ? Additive models extending fGn like fractional hyperbolic model of Mandelbrot & Wallis [1969]. Multiplicative, multifractal models exhibiting volatility bunching as well as 1/f spectra and fat tails-1972 (turbulence), 1990s (finance). And the class he referred to as dustborne: the least known of his papers, from 1965-67, though closely related to the Alternating Fractional Renewal Process, the CTRW and modern work on weak ergodicity breaking.
Multiplicative multifractal cascades Many systems have aggregation, but not by an additive route. Classic example is turbulence. One indicator: lognormal or stretched exponential pdf Selecta Volume N 1999
Multifractals and volatility clustering another: correlations between the absolute value of the time series- or here, in ionospheric data, the first differences. 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 4 -600 -400 -200 0 200 400 600 increments,r First differences of AE index January-June 1979 -100 -80 -60 -40 -20 0 20 40 60 80 100 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 lag acf AE data: acf of returns -100 -80 -60 -40 -20 0 20 40 60 80 100 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 lag acf AE data: acf of squared returns
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise
CONDITIONAL STATIONARITY, NON- ERGODICTY ETC
The infrared catastrophe as mirage: Ionosphere Magnetosphere Reviewed in N2, Selecta, 1999
The infrared catastrophe as mirage: Ionosphere Magnetosphere Rather than representing a true singularity at lowest frequencies, as seen in stationary LRD, in this case he described the infrared catastrophe as a ``mirage arising from fact that here measured square Fourier moduli do not estimate a Wiener-Khinchine density. Same formula changes meaning because of different facts Reviewed in N2, Selecta, 1999
Ionosphere Magnetosphere (Fifth Berkeley Symposium on Probability, 1965)
Non-ergodicity: Ionosphere Magnetosphere M 1967 illustrated this first in case of single jump, infinite interval
Non-ergodicity: Ionosphere Magnetosphere Mandelbrot 1967 then discussed case of many state changes with power law waiting times
FRACTIONAL STABLE MODELS
Distinct from fBm and fGN: Ionosphere Magnetosphere Mandelbrot 1967 was prepared in the same period as Mandelbrot and van Ness on fBm and fGn, which it cites as ``to be published". In it contrast is clearly drawn between the conditionally stationary, non-Gaussian renewal process as a 1/f model and his stationary, Gaussian (fGn) model:
LRD and its controversies Ionosphere Magnetosphere Mandelbrot [1965]; M and Van Ness[1968] proposed use of fractional Brownian motion. Non stationary self similar model which generalises Wiener process, has spectral index between -1 and -3. and its derivative, fractional Gaussian noise, which is stationary, and long range dependent. Unlike the stable amplitude distribution we just saw, the power spectra of fBm and fGn are singular at zero frequency. In Bm (and fBm) this is a result of its nonstationarity [Selecta, N2, 1999]
Ionosphere MagnetosphereMandelbrot & Wallis [1969] first attempt to unify long range memory kernel of fGn with heavy tailed amplitude fluctuations - called it fractional hyperbolic model because of its power law tails. Anticipated the versatile linear fractional stable noises, but it didnt satisfy him completely for problems he was looking at. Additive fractional stable class
HISTORY
A neglected paper ? Ionosphere Magnetosphere Mandelbrot 1967 received far less attention than either papers on heavy tails in finance in early 1960s or the series with Van Ness and Wallis in 1968-69 on stationary fractional Gaussian models for LRD. Only about 20 citations in its first 20 years ! Was apparently unknown to Vit Klemes [Water Resources Research, 1974], who essentially reinvented it to criticise fBm. Still seems a relatively little known paper. Not cited by Beran et al [2013], and while listed in the citations of Beran [1994] I cant find in the text. Some exceptions, e.g. Grigolini et al, Physica A, 2009. Although he revisited the paper with new commentary in Selecta Volume N [1999] dealing with multifractals and 1/f noise, Mandelbrot neglected to mention it explicitly in his popular and historical accounts of genesis of LRD such as Mandelbrot and Hudson [2008]. Why ? Because it wasnt as popular as fBm/fGn ? Or because it wasnt as beautiful ?
Whose time has come ? Ionosphere Magnetosphere Should we pay more attention to this class of models ? As the AFRP [Lowen and Teichs book], and as models for weak ergodicity breaking [c.f. Niemann et al, 2013], we already are ! Also links to the CTRW. Particular value of looking back nearly 50 years to how Mandelbrot saw these models is to see how they fit into the panorama of grid-bound self-affine variability as he later put it [Selecta, 1999, N1]. Helps link maths and physics, the formula & the fact. And inform future work.
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise
Ionosphere Magnetosphere And so my argument has always [sic] been that each of these causal chains is totally incomprehensible in detail, probably exponentially decaying. There are so many of them that a very strong dependence may be perfectly compatible. Now I would like to mention that this is precisely the reason why infinite dependence exists, for example, in physics. In a magnet- because two parts far away have very minor dependence along any path of actual dependence. There are so many different paths that they all combine to create a global structure. In other words, there is no global structure in one dimension, but there's one in two and three dimensions etc. for magnets -the basis of Onsager's work and the whole theory. And in economics there is nothing comparable to these calculations, but the intuition of what they represent is the same BBM, op cit A critical phenomenon ?
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise
2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise

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2014 AGU Balancing fact and formula in complex sytems: the example of 1/f noise

  • 1. Balancing Fact and Formula in the Science of Complex Systems: The Example of 1/f Spectra Nick Watkins NickWatkins@mykolab.com
  • 2. 際際滷s 1-24 were given as an invited talk on 19th December 2014 at the Fall AGU in San Francisco, beginning the Session on Intermittency and Dynamical Complexity in Space Plasmas from the Sun to Interplanetary and Planetary Environments. I have also included the spares prepared in case of questions and being developed for future talks.
  • 3. Thanks Tom, Giuseppe and Marius for inviting me Holger Kantz for hosting me in Dresden since September 2013 Co-authors Tim Graves, Christian Franzke, Bobby Gramacy, Scott Osprey and Paulo Davini Discussions with all of the above and Sandra Chapman, Eli Barkai, Igor Sokolov, Rainer Klages and Aleksei Chechkin among others
  • 4. Theme Hurst effect Will today distinguish three things often taken as same Observed growth of range in time series: Hurst effect
  • 5. Theme 1/f Hurst effect Will today distinguish three things often taken as same Observed growth of range in time series: Hurst effect Observation of a singularity at zero in Fourier spectra: 1/f
  • 6. Theme (S)LRD 1/f Hurst effect Will today distinguish three things often taken as same Observed growth of range in time series: Hurst effect Observation of a singularity at zero in Fourier spectra: 1/f The long range dependence seen in stationary 1/f case: (S)LRD. Using 1/f as a diagnostic of LRD assumes stationarity
  • 7. Fact: Anomalous growth of range Hurst Effect Hurst, Nature, 1957 I heard about the Nile in '64, ... the variance doesn't draw like time span as you take bigger and bigger integration intervals; it goes like time to a certain power different from one. Hurst was getting results that were incomprehensible. Mandelbrot, 1998
  • 8. Formula: Long Range Dependence (S)LRD Hurst Effect Mandelbrot, van Ness, and Wallis, 1965-69 First [history in Graves et al, arXiv, 2014a] demonstration that Hurst effect could be explained by stationary long range dependent process Model, fractional Gaussian noise [see also Kolmogorovs Wiener Spiral], had singular spectral density at lowest frequencies. '( ) ~ S f f
  • 9. The 1/f paradox If spectral density '( ) then i) it is singular as and ii) if we define an autocorrelation function via ( ) ( ) ( ) and use Wiener-Khinchine theorem to get from Fourier transform of ~ 0 緒 S f f f x t x t S falls off as power law, and '( ) then summed lags "blow up" its ( ) ワ f
  • 10. Ionosphere Magnetosphere 1 1 2 2 , 22fBm: ( ) ( ) ( )~ ( ) 件 H H R H X t t s s dL s Infinite range memory kernel Gaussian
  • 11. Fractional motions and noises 0 100 200 300 400 500 600 700 800 900 1000 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 Fractional Brownian motion, H=0.7 1 1 2 2 , 22fBm: ( ) ( ) ( )~ ( ) 件 H H R H X t t s s dL s Build a nonstationary, self similar walk (used wfbm in Matlab) fractional motion 2 1 H
  • 12. Fractional motions and noises 0 100 200 300 400 500 600 700 800 900 1000 -18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 Fractional Brownian motion, H=0.7 1 1 2 2 , 22fBm: ( ) ( ) ( )~ ( ) 件 H H R H X t t s s dL s Build a nonstationary, self similar walk (used wfbm in Matlab) 0 100 200 300 400 500 600 700 800 900 1000 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Fractional Gaussian noise fractional motion Then differentiate to give a stationary LRD noise fractional noise 2 1 H 2 1 H
  • 13. Ionosphere Magnetosphere [], if infinite dependence is necessary it does not mean that IBM's details of ten years ago influence IBM today, because there's no mechanism within IBM for this dependence. However, IBM is not alone. The River Nile is [not] alone. They're just one- dimensional corners of immensely big systems. The behaviour of IBM stock ten years ago does not influence its stock today through IBM, but IBM the enormous corporation has changed the environment very strongly. The way its price varied, went up or went up and fluctuated, had discontinuities, had effects upon all kinds of other quantities, and they in turn affect us. Mandelbrot, interviewed in 1998 by B. Sapoval for Web of Stories One resolution of 1/f paradox In modern fractional Langevin models fGn is noise term e.g. reviews of Metzler et al, PCCP, 2014; Watkins GRL, 2013.
  • 14. Ionosphere Magnetosphere [], if infinite dependence is necessary it does not mean that IBM's details of ten years ago influence IBM today, because there's no mechanism within IBM for this dependence. However, IBM is not alone. The River Nile is [not] alone. They're just one- dimensional corners of immensely big systems. The behaviour of IBM stock ten years ago does not influence its stock today through IBM, but IBM the enormous corporation has changed the environment very strongly. The way its price varied, went up or went up and fluctuated, had discontinuities, had effects upon all kinds of other quantities, and they in turn affect us. Mandelbrot, interviewed in 1998 by B. Sapoval for Web of Stories One resolution of 1/f paradox In modern fractional Langevin models fGn is noise term e.g. reviews of Metzler et al, PCCP, 2014; Watkins GRL, 2013. Resolution of apparent paradox is that world as a whole is Markovian, the memory is a consequence of looking at a piece of it. Generalises the Mori-Zwanzig approach.
  • 15. Often see 1/f spectra and heavy tails Ionosphere Magnetosphere Ground-based magnetometers sense ionospheric currents 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10 5 0 500 1000 1500 2000 2500 3000 1978 AE data: threshold percentile=99 Time, t [minutes] AE[nanoTesla] AE data threshold Slow 1/f region in AE power spectrum Tsurutani et al, GRL, 1991 See also Hush et al, Poster SH53A-4209 this afternoon Moscone South
  • 16. Joint inference on LRD and heavy tails. Tim Graves PhD developed Bayesian method: tested on 留-stable ARFIMA(0,d,0) where heavy tails & LRD co-exist, see Graves et al, arXiv, submitted CSDA 2014b; & my talk from AGU, 2012 on 際際滷share. 1.5 0.15d Test on Synthetic ARFIMA data
  • 17. Other models for 1/f Ionosphere Magnetosphere Selecta H Selecta N
  • 18. 1/f without (S)LRD (S)LRD 1/f Hurst effect Before (S)LRD models, Mandelbrot [1963-67] had proposed other 1/f models which were not stationary LRD in same sense as fGn. Solved 1/f paradox by a different route. Still little known in the geosciences [but see Klemes, WRR, 1974].
  • 19. Ionosphere Magnetosphere Abrupt state changes Fat distributions of switching times: Levy (E[t^2] = ) case.
  • 20. The conditional spectrum: Magnetosphere Mandelbrot 1967 reviewed in N2, Selecta, 1999
  • 21. The conditional spectrum: Magnetosphere Mandelbrot 1967 reviewed in N2, Selecta, 1999 Numerical 1/f spectrum need not estimate Wiener- Khinchine spectrum. Instead depends on conditioning length T. Unlike stationary LRD model, singularity is an artefact.
  • 22. Hurst effect from state changes Ionosphere Magnetosphere Franzke et al, in review, Sci. Rep., 2014 Interestingly, the classic Lorenz 63 model (columns 1 and 2) can generate Hurst effect in some measures, such as DFA, even without long tailed waiting times between regime shifts. Confirms that Hurst effect is easier to generate than 1/f or full blown S(LRD).
  • 23. Formula versus fact Like the ear, the eye is very sensitive to features that the spectrum does not reflect. Seen side by side, different 1/f noises, Gaussian [i.e. fGn], dustborne [i.e. fractional renewal] and multifractal, obviously differ from one another- Mandelbrot, Selecta N, 1999. Nothing can be more fatal to progress than a too confident reliance on mathematical symbols; for the student is only too apt to consider the formula and not the fact as the physical reality. Thomson (Kelvin) & Tait, 1890 edition.
  • 24. Summary The 1/f paradox: The "1/f" spectral shape seen throughout physics, & why it seems paradoxical. Many faces of 1/f: Mandelbrot promoted two resolutions of paradox in the mid 60s: stationary long range dependence (S)LRD , or what he dubbed conditionally stationary renewal models. History is in Graves et al, arXiv, 2014a. One formula consistent with different facts. Ways forward ? Mandelbrots Selecta (Vols. N &H) urged use of our eyes as well as formalism, and Ive advertised two new results in this spirit: Bayesian method for choosing between (S)LRD models, Graves et al, arXiv, 2014b, and work on a dynamical origin for Hurst effect in the Lorenz model (Franzke et al, Scientific Reports, in review, 2014 ).
  • 26. THE ROAD TO MULTIFRACTALS
  • 27. So what were the other models ? Additive models extending fGn like fractional hyperbolic model of Mandelbrot & Wallis [1969]. Multiplicative, multifractal models exhibiting volatility bunching as well as 1/f spectra and fat tails-1972 (turbulence), 1990s (finance). And the class he referred to as dustborne: the least known of his papers, from 1965-67, though closely related to the Alternating Fractional Renewal Process, the CTRW and modern work on weak ergodicity breaking.
  • 28. Multiplicative multifractal cascades Many systems have aggregation, but not by an additive route. Classic example is turbulence. One indicator: lognormal or stretched exponential pdf Selecta Volume N 1999
  • 29. Multifractals and volatility clustering another: correlations between the absolute value of the time series- or here, in ionospheric data, the first differences. 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 4 -600 -400 -200 0 200 400 600 increments,r First differences of AE index January-June 1979 -100 -80 -60 -40 -20 0 20 40 60 80 100 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 lag acf AE data: acf of returns -100 -80 -60 -40 -20 0 20 40 60 80 100 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 lag acf AE data: acf of squared returns
  • 33. The infrared catastrophe as mirage: Ionosphere Magnetosphere Reviewed in N2, Selecta, 1999
  • 34. The infrared catastrophe as mirage: Ionosphere Magnetosphere Rather than representing a true singularity at lowest frequencies, as seen in stationary LRD, in this case he described the infrared catastrophe as a ``mirage arising from fact that here measured square Fourier moduli do not estimate a Wiener-Khinchine density. Same formula changes meaning because of different facts Reviewed in N2, Selecta, 1999
  • 36. Non-ergodicity: Ionosphere Magnetosphere M 1967 illustrated this first in case of single jump, infinite interval
  • 37. Non-ergodicity: Ionosphere Magnetosphere Mandelbrot 1967 then discussed case of many state changes with power law waiting times
  • 39. Distinct from fBm and fGN: Ionosphere Magnetosphere Mandelbrot 1967 was prepared in the same period as Mandelbrot and van Ness on fBm and fGn, which it cites as ``to be published". In it contrast is clearly drawn between the conditionally stationary, non-Gaussian renewal process as a 1/f model and his stationary, Gaussian (fGn) model:
  • 40. LRD and its controversies Ionosphere Magnetosphere Mandelbrot [1965]; M and Van Ness[1968] proposed use of fractional Brownian motion. Non stationary self similar model which generalises Wiener process, has spectral index between -1 and -3. and its derivative, fractional Gaussian noise, which is stationary, and long range dependent. Unlike the stable amplitude distribution we just saw, the power spectra of fBm and fGn are singular at zero frequency. In Bm (and fBm) this is a result of its nonstationarity [Selecta, N2, 1999]
  • 41. Ionosphere MagnetosphereMandelbrot & Wallis [1969] first attempt to unify long range memory kernel of fGn with heavy tailed amplitude fluctuations - called it fractional hyperbolic model because of its power law tails. Anticipated the versatile linear fractional stable noises, but it didnt satisfy him completely for problems he was looking at. Additive fractional stable class
  • 43. A neglected paper ? Ionosphere Magnetosphere Mandelbrot 1967 received far less attention than either papers on heavy tails in finance in early 1960s or the series with Van Ness and Wallis in 1968-69 on stationary fractional Gaussian models for LRD. Only about 20 citations in its first 20 years ! Was apparently unknown to Vit Klemes [Water Resources Research, 1974], who essentially reinvented it to criticise fBm. Still seems a relatively little known paper. Not cited by Beran et al [2013], and while listed in the citations of Beran [1994] I cant find in the text. Some exceptions, e.g. Grigolini et al, Physica A, 2009. Although he revisited the paper with new commentary in Selecta Volume N [1999] dealing with multifractals and 1/f noise, Mandelbrot neglected to mention it explicitly in his popular and historical accounts of genesis of LRD such as Mandelbrot and Hudson [2008]. Why ? Because it wasnt as popular as fBm/fGn ? Or because it wasnt as beautiful ?
  • 44. Whose time has come ? Ionosphere Magnetosphere Should we pay more attention to this class of models ? As the AFRP [Lowen and Teichs book], and as models for weak ergodicity breaking [c.f. Niemann et al, 2013], we already are ! Also links to the CTRW. Particular value of looking back nearly 50 years to how Mandelbrot saw these models is to see how they fit into the panorama of grid-bound self-affine variability as he later put it [Selecta, 1999, N1]. Helps link maths and physics, the formula & the fact. And inform future work.
  • 47. Ionosphere Magnetosphere And so my argument has always [sic] been that each of these causal chains is totally incomprehensible in detail, probably exponentially decaying. There are so many of them that a very strong dependence may be perfectly compatible. Now I would like to mention that this is precisely the reason why infinite dependence exists, for example, in physics. In a magnet- because two parts far away have very minor dependence along any path of actual dependence. There are so many different paths that they all combine to create a global structure. In other words, there is no global structure in one dimension, but there's one in two and three dimensions etc. for magnets -the basis of Onsager's work and the whole theory. And in economics there is nothing comparable to these calculations, but the intuition of what they represent is the same BBM, op cit A critical phenomenon ?