MESA LAB Jitter Dynamics Outliers Data Right and
MESA LAB Jitter Dynamics, Outlier’s Data Right and Fractional Calculus Yang. Quan 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: SE 1 -254; Lab: Castle #22 (T: 228 -4398) June 2, 2014. Monday 9: 00 AM-5: 00 PM ENGCAS 820, MESA LAB @ UC Merced Castle Research Facility Fractional Calculus Day @ UC Merced 2014 Edition
Slide-2/1024 MESA LAB 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
Slide-3/1024 MESA LAB 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” 06/02/2014 How this is possible? Why should I care? Any (good) consequences (to me)? Fractional Calculus Day @ UC Merced 2014 Edition
MESA LAB Slide-4 of 1024 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 06/02/2014 Slide credit: Igor Podlubny Fractional Calculus Day @ UC Merced 2014 Edition
MESA LAB Slide-5/1024 Fractional Calculus: a response to more advanced characterization of our more complex world at smaller scale We are here 06/02/2014 Slide credit: Igor Podlubny Fractional Calculus Day @ UC Merced 2014 Edition
Slide-6 of 1024 Rule of thumb for “Fractional Order Thinking” • Self-similar • Scale-free/Scaleinvariant • Power law • Fat/Heavy tail • Long range dependence (LRD) • 1/f a noise 06/02/2014 • • MESA LAB Porous media Particulate Granular Lossy Anomaly Disorder Soil, tissue, electrodes, bio, nano, network, transport, diffusion, soft matters (biox) … Fractional Calculus Day @ UC Merced 2014 Edition
MESA LAB Slide-7/1024 Power Law When k is negative: Inverse power law Scale-free Scale invariance • 06/02/2014 “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 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-8/1024 MESA LAB 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 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-9/1024 MESA LAB 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 …) 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-10/1024 MESA LAB Integer Order Calculus Exponential Law 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-11/1024 MESA LAB Fractional Order Calculus Power Law 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
MESA LAB Slide-12/1024 Fractional Order Calculus Power Law Mittag-Leffler function in two parameters: 06/02/2014 E 1, 1(x)=ex Fractional Calculus Day @ UC Merced 2014 Edition
MESA LAB Slide-13/1024 Professor Donald E. Knuth, creator of TEX: G. M. Mittag-Leffler (1846 -1927) 06/02/2014 “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) Fractional Calculus Day @ UC Merced 2014 Edition
Slide-14/1024 MESA LAB The Mittag-Leffler function 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
MESA LAB Slide-15/1024 Root of long (algebraic) tail, or inverse power law 06/02/2014 Tail MATTERS! Fractional Calculus Day @ UC Merced 2014 Edition
Slide-16/1024 Heavy tail, fat tail 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition MESA LAB
MESA LAB Slide-17/1024 Long jumps, intermittence Brownian motion 06/02/2014 Levy flights Fractional Calculus Day @ UC Merced 2014 Edition
MESA LAB Spikiness/Burstiness
MESA LAB Slide-19/1024 Fractional noises / fluctuations anti-persistent Brownian motion, or Weiner process persistent Normal Gaussian noise f. Bm f. Gn http: //www. frontiersin. org/Fractal_Physiology/10. 3389/ fphys. 2012. 00208/full 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-20/1024 Spikiness/Burstiness MESA LAB Poisson PDF Inverse power law PDF A. -L. Barabási. The origin of bursts and heavy tails in human dynamics. Nature 435 207– 211 (2005). http: //seeingcomplexity. wordpress. com/2011/03/07/global-android-activations-and-the-power-law/ 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-21/1024 MESA LAB Stable Distributions =2: Gaussian =1, b=0: Cauchy =1. 5, b=1: Levy http: //academic 2. american. edu/~jpnolan/stable. html 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-22/1024 MESA LAB 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 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-23 of 1024 • • • MESA LAB 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 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-24/1024 MESA LAB 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
Slide-25/1024 MESA LAB Jitter in IP networks: a Cauchy approach • http: //ieeexplore. ieee. org/xpl/article. Details. 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 ��. This allows us to predict the expected Qo. S in terms of the number of nodes and traffic parameters. http: //en. wikipedia. org/wiki/Cauchy_distribution 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-26 of 1024 MESA LAB NCS – delay is random, time-varying 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-27 of 1024 … and spiky 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition MESA LAB
Slide-28 of 1024 MESA LAB PROBLEM? running variance estimate is not convergent! LS methods fail 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Slide-29/1024 MESA LAB 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
Slide-30 of 1024 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? ” – Yang. Quan Chen • Data has “equal rights” • Outliers are of “spiky nature” • “Event of low probability can still happen often” – Hint of “heavy-tailedness” of PDF 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
Long-range dependence Hurst parameter Self-similar -stable distributions Jitters as outliers? NETWORK TRAFFIC “Spikiness” MESA LAB ARFIMA Fractional Gaussian noise (FGn) “Heavy tails” Fractional Brownian motion (FBm)
Slide-32/1024 MESA LAB 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
Slide-33/1024 MESA LAB Key message • Outliers could and should be modelled well using fractional calculus! 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
MESA LAB Jitter dynamics ? ? ? • • Self-similarity => Hurst parameter “Spikiness”.
MESA LAB Spikiness/Burstiness
MESA LAB Slide-36/1024 Thank you for your attention! Q&A Integer-Order Calculus Fractional-Order Calculus Slide credit: Richard L. Magin, ICCC 12 06/02/2014 Fractional Calculus Day @ UC Merced 2014 Edition
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