jitter dynamics, outlier’s data right and fractional calculus

MESA LAB

Jitter Dynamics, Outlier’s Data Right and Fractional Calculus

YangQuan Chen, Ph.D., Director, MESA (Mechatronics, Embedded Systems and Automation)LAB

ME/EECS/SNRI/UCSolar, School of Engineering,University of California, Merced

E: [email protected]; or, [email protected]: (209)228-4672; O: SE1-254; Lab: Castle #22 (T: 228-4398)

June 2, 2014. Monday 9:00AM-5:00PMENGCAS 820, MESA LAB @ UC Merced Castle Research Facility

Fractional Calculus Day @ UC Merced 2014 Edition

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Outline• Inverse Power Law and Heavy-tailedness• Jitter Dynamics• Outlier’s Data Right • Fractional Calculus Based Models

06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition

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What is “Fractional Calculus”?• Calculus: integration and differentiation.• “Fractional Calculus”: integration and

differentiation of non-integer orders.– Orders can be real numbers (and even complex

numbers!)– Orders are not constrained to be “integers” or even

“fractionals”How this is possible? Why should I care?

Any (good) consequences (to me)?06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition

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Slide credit: Igor Podlubny

06/02/2014

The animation shows the derivative operator oscillating between the antiderivative (α=-1) and the derivative (α=1) of the simple power function y=x continuously.http://en.wikipedia.org/wiki/Fractional_calculus

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We are here

Fractional Calculus: a response to more advanced characterization of our more complex world at smaller scale


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Slide credit: Igor Podlubny06/02/2014

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Rule of thumb for “Fractional Order Thinking”

• Self-similar• Scale-free/Scale-

invariant• Power law• Fat/Heavy tail• Long range

dependence (LRD)• 1/f a noise

• Porous media• Particulate• Granular• Lossy• Anomaly• Disorder• Soil, tissue, electrodes,

bio, nano, network, transport, diffusion, soft matters (biox) …

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Power Law

• “Scaling laws in cognitive sciences” by CT Kello, GDA Brown, R Ferrer-i-Cancho, JG Holden, K Linkenkaer-Hansen, T. Trends in Cognitive Sciences 14 (5), 223-232, 2010


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When k is negative: Inverse power law

Scale invariance

Scale-free

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In Different Contexts• Scale-free networks (degree distributions)• Pink noise (power spectrum)• Probability density function (PDF)• Autocorelation function (ACF)• Allometry (Y=a Xb )• Anomalous relaxation (evolving over time)• Anomalous diffusion (MSD versus time)• Self-similar


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MESA LABOther connectedness? (hidden)

• Fractal, irregular, anomalous, rough, Hurst– Multifractal, multi-scale, scale-rich

• Renormalization (?), Universality• Extreme events– spikiness, bursty, intermittence• Fluctuation in fluctuations; Variability, • Emergence, Surprise• Nonlocality, Long term memory• Complex (behavior, processes, network, fluid,

dynamics, systems …)06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition

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MESA LABInteger Order Calculus

Exponential Law


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btextx

xxtbxtx

0

0

)(

)0(),()(

MESA LABFractional Order Calculus

Power Law


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!)1()1(

)1()(

)1(/)(

,)(

)(

)(

kknote

tkT

kt

cttx

Rctx

kk

MESA LABFractional Order Calculus

Power Law


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)()(

)1,0()0(),()(

,1

0

0)1()(

btEtxtx

xxtbxtx

Mittag-Leffler function in two parameters:

E1,1(x)=ex)()( 1, xExE

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G. M. Mittag-Leffler(1846-1927)

Professor Donald E. Knuth, creator of TEX:

“As far as the spacing in mathematics is concerned...I took Acta Mathematica, from 1910 approximately; this was a journal in Sweden ... Mittag-Leffler was the editor, and his wife was very rich, and they had the highest budget for making quality mathematics printing. So the typography was especially good in Acta Mathematica.”(Questions and Answers with Prof. Donald E. Knuth, Charles University, Prague, March 1996)


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MESA LABThe Mittag-Leffler

function


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Root of long (algebraic) tail, or inverse power law

Tail MATTERS!

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Heavy tail, fat tail


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Brownian motion Levy flights

Long jumps, intermittence

MESA LABSpikiness/Burstiness

MESA LABFractional noises / fluctuations

fBm

fGn


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anti-persistent persistent

Normal Gaussian noise

http://www.frontiersin.org/Fractal_Physiology/10.3389/fphys.2012.00208/full

Brownian motion, orWeiner process

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A.-L. Barabási. The origin of bursts and heavy tails in human dynamics. Nature 435 207–211 (2005).

Poisson PDF

http://seeingcomplexity.wordpress.com/2011/03/07/global-android-activations-and-the-power-law/

Inverse power law PDF

Spikiness/Burstiness

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Stable Distributions

http://academic2.american.edu/~jpnolan/stable/stable.html

=2: Gaussian1, b=0: Cauchy1.5,b=1: Levy

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Connection to FC via PDF• “Fractional Calculus and Stable Probability

Distributions” (1998) by by Rudolf Gorenflo , Francesco Mainardi http://arxiv.org/pdf/0704.0320.pdf


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MESA LAB100+ years of LRD/HT research

• Distribution of wealth, income of individuals• City sizes vs. ranks - given the population, what is the city rank?• Graphs of gene regulatory & protein-protein networks are scale free• Long neuron inter-spike intervals in depressed mice• Internet and WWW - scale free network (graph): fault tolerant, hubs

are both the strength and Achilles’ heels• Scene lengths in VBR and MPEG video are heavy-tailed• Computer files, Web documents, frequency of access are heavy-tailed • Stock price fluctuations and company sizes• Inter occurrence of catastrophic events, earthquakes - applications to

reinsurance• Frequency of words in natural languages (often called Zipf’s law)


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Ubiquity of Power Laws, Jankovic, 2007

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MESA LABJitter in IP networks: a Cauchy approach

• http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5403630• Communications Letters, IEEE (Volume:14 , Issue: 2 ) Jitter is

recognized as an important phenomenon that degrades the communication performance. Particularly, in real time services such as voice and video over the Internet, there is evidence that jitter departs from already proposed Laplacian models and that it has a heavy tail behavior. In this paper, we show that an Alpha-Stable jitter model is adequate, and that in some cases the Cauchy distribution provides a satisfactory approximation. Furthermore, this work shows how the jitter dispersion increases with the number of hops in the path, following a power law with scaling exponent dependent on the index of stability 𝛼. This allows us to predict the expected QoS in terms of the number of nodes and traffic parameters.


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http://en.wikipedia.org/wiki/Cauchy_distribution

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5403630

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=5403630

http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234

http://ieeexplore.ieee.org/xpl/tocresult.jsp?isnumber=5403597

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NCS – delay is random, time-varying


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… and spiky


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MESA LABPROBLEM? running variance estimate

is not convergent! LS methods fail


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MESA LAB“Outlier modeling” – A New Fractional

Order Statistic Point of View• Paradigm shift• “How do you know outlier is not part of the

dynamic system’s behavior?” – YangQuan Chen• Data has “equal rights”

• Outliers are of “spiky nature”• “Event of low probability can still happen often”

– Hint of “heavy-tailedness” of PDF06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition

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“Spikiness”

Long-range

dependenceSelf-similar

Fractional Brownian motion (FBm)

Fractional Gaussian

noise (FGn)

-stable distributio

ns

Hurst parameter

“Heavy tails”

Jitters as outliers?NETWORK TRAFFIC

ARFIMA

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Key message• Outliers could and should be modelled well

using fractional calculus!


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• Self-similarity => Hurst parameter

• “Spikiness”.

?????

Jitter dynamics

MESA LABSpikiness/Burstiness

MESA LABThank you for your attention!

Q & A


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06/02/2014

Integer-Order Calculus Fractional-Order Calculus

Slide credit: Richard L. Magin, ICCC12

jitter dynamics, outlier’s data right and fractional calculus

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