Jitter Dynamics, Outlier’s Data Right and Fractional Calculus

1. 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: yqchen@ieee.org; or, yangquan.chen@ucmerced.edu T: (209)228-4672; O: SE1-254; Lab: Castle #22 (T: 228-4398) June 2, 2014. Monday 9:00AM-5:00PM ENGCAS 820, MESA LAB @ UC Merced Castle Research Facility Fractional Calculus Day @ UC Merced 2014 Edition

2. Outline • Inverse Power Law and Heavy-tailedness • Jitter Dynamics • Outlier’s Data Right • Fractional Calculus Based Models Fractional Calculus Day @ UC Merced 2014 Edition

3. 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)? Fractional Calculus Day @ UC Merced 2014 Edition

4. 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. Slide credit: Igor Podlubny http://en.wikipedia.org/wiki/Fractional_calculus Fractional Calculus Day @ UC Merced 2014 Edition

5. We are here Fractional Calculus: a response to more advanced characterization of our more complex world at smaller scale Slide credit: Igor Podlubny Fractional Calculus Day @ UC Merced 2014 Edition

6. Rule of thumb for “Fractional Order Thinking” • Porous media • Particulate • Granular • Lossy • Anomaly • Disorder • Soil, tissue, electrodes, bio, nano, network, transport, diffusion, soft matters (biox) … • Self-similar • Scale-free/Scale-invariant • Power law • Fat/Heavy tail • Long range dependence (LRD) • 1/f a noise Fractional Calculus Day @ UC Merced 2014 Edition

7. 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 When k is negative: Inverse power law Scale-free Scale invariance Fractional Calculus Day @ UC Merced 2014 Edition

8. 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 Fractional Calculus Day @ UC Merced 2014 Edition

9. Other 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 …) Fractional Calculus Day @ UC Merced 2014 Edition

10. Integer Order CalculusExponential Law Fractional Calculus Day @ UC Merced 2014 Edition

11. Fractional Order CalculusPower Law Fractional Calculus Day @ UC Merced 2014 Edition

12. Fractional Order CalculusPower Law Mittag-Leffler function in two parameters: E1,1(x)=ex Fractional Calculus Day @ UC Merced 2014 Edition

13. G. M. Mittag-Leffler(1846-1927) Professor Donald E. Knuth, creator of TEX: “As far as the spacing in mathematics is concerned...I took ActaMathematica, 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 ActaMathematica.” (Questions and Answers with Prof. Donald E. Knuth, Charles University, Prague, March 1996) Fractional Calculus Day @ UC Merced 2014 Edition

14. The Mittag-Leffler function Fractional Calculus Day @ UC Merced 2014 Edition

15. Root of long (algebraic) tail, or inverse power law Tail MATTERS! Fractional Calculus Day @ UC Merced 2014 Edition

16. Heavy tail, fat tail Fractional Calculus Day @ UC Merced 2014 Edition

17. Long jumps, intermittence Levy flights Brownian motion Fractional Calculus Day @ UC Merced 2014 Edition

18. Spikiness/Burstiness

19. Fractional noises / fluctuations Brownian motion, or Weiner process anti-persistent persistent fBm fGn Normal Gaussian noise http://www.frontiersin.org/Fractal_Physiology/10.3389/fphys.2012.00208/full Fractional Calculus Day @ UC Merced 2014 Edition

20. Spikiness/Burstiness Poisson PDF Inverse power law PDF A.-L. Barabási. The origin of bursts and heavy tails in human dynamics. Nature 435207–211 (2005). http://seeingcomplexity.wordpress.com/2011/03/07/global-android-activations-and-the-power-law/ Fractional Calculus Day @ UC Merced 2014 Edition

21. Stable Distributions a=2: Gaussian a=1, b=0: Cauchy a=1.5,b=1: Levy http://academic2.american.edu/~jpnolan/stable/stable.html Fractional Calculus Day @ UC Merced 2014 Edition

22. Connection to FC via PDF • “Fractional Calculus and Stable Probability Distributions” (1998) by byRudolf Gorenflo , Francesco Mainardi http://arxiv.org/pdf/0704.0320.pdf Fractional Calculus Day @ UC Merced 2014 Edition

23. 100+ 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) Ubiquity of Power Laws, Jankovic, 2007 Fractional Calculus Day @ UC Merced 2014 Edition

24. Outline • Inverse Power Law and Heavy-tailedness • Jitter Dynamics • Outlier’s Data Right • Fractional Calculus Based Models Fractional Calculus Day @ UC Merced 2014 Edition

25. Jitter 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. http://en.wikipedia.org/wiki/Cauchy_distribution Fractional Calculus Day @ UC Merced 2014 Edition

26. NCS – delay is random, time-varying Fractional Calculus Day @ UC Merced 2014 Edition

27. … and spiky Fractional Calculus Day @ UC Merced 2014 Edition

28. PROBLEM? running variance estimate is not convergent! LS methods fail Fractional Calculus Day @ UC Merced 2014 Edition

29. Outline • Inverse Power Law and Heavy-tailedness • Jitter Dynamics • Outlier’s Data Right • Fractional Calculus Based Models Fractional Calculus Day @ UC Merced 2014 Edition

30. “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 PDF Fractional Calculus Day @ UC Merced 2014 Edition

31. Long-range • dependence • Hurst • parameter • Self-similar • ARFIMA • -stable • distributions • Jitters as outliers? • NETWORK • TRAFFIC • Fractional Gaussian • noise (FGn) • “Spikiness” • “Heavy tails” • Fractional Brownian motion (FBm)

32. Outline • Inverse Power Law and Heavy-tailedness • Jitter Dynamics • Outlier’s Data Right • Fractional Calculus Based Models Fractional Calculus Day @ UC Merced 2014 Edition

33. Key message • Outliers could and should be modelled well using fractional calculus! Fractional Calculus Day @ UC Merced 2014 Edition

34. Jitter dynamics ????? • Self-similarity => Hurstparameter • “Spikiness”.

35. Spikiness/Burstiness

36. Thank you for your attention!Q & A Integer-Order Calculus Fractional-Order Calculus Slide credit: Richard L. Magin, ICCC12 Fractional Calculus Day @ UC Merced 2014 Edition