Infonet Group INF NET University of Namur www.infonet.fundp.ac.be 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes Steve UHLIG suh@infonet.fundp.ac.be Infonet Group 15 th ITC Specialist Seminar on Internet Traffic Engineering and Traffic Management Wurzburg, Germany 22-24 July 2002 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.1/17
Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Schedule Scaling basics Wavelet basics Scaling detection Conclusion 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.2/17
✓ ✘ ✒ ✎ ✝ ✞ ☎ ✠ ✡ ☛ ✞ ✎ ✄ ✔ ✕ ✖ ✗ ✌ ✙ ✗ ✘ ✂ ☎ ✂ ✝ ✘ ✜ ✛ ✑ ✙ ✘ � ☎ ✘ ✙ ✁ ✑ ✂ ✝ ✚ ☎ ✠ ✡ ☛ ✄ ✂ ✘ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Scaling basics Scaling no particularly important timescale Two examples: 1. Self-similarity (SS): ☞✍✌ - formally: ✄✆☎ ✝✟✞ ✁✏✎ affinity between original and zoom (all timescales behave in the same way) - property: ✄✆☎ non-stationarity dependence of moments on time Note: H is “self-similarity” parameter 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.3/17
✬ ✄ ✣ ✘ ✫ ✖ ✢ ★ ✘ ✣ ✘ ✣ ✭ ✣ ✄ ✢ ✝ ✮ ✔ ✱ ✛ ✛ ✘ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Scaling basics (contd) 2. Long-range dependence (LRD): ✑✧✩ ★✥✪ - definition: ✝✥✤ ✎✧✦ power-law in auto-correlation ( ) - persistence in correlations but requires 2 nd -order stationarity Relationships between self-similarity and LRD: : self-similarity LRD ✕✰✯ LRD self-similarity 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.4/17
✞ ✷ ✩ ✷ ✶ ✴ ✾ ✷ ✂ ✺ ✣ ✳ ✌ ✚ ✞ ✿ ✞ ✯ ✯ ✯ ✸ ✩ ✂ ❁ ✱ ✔ ✮ ✜ ✭ ✲ ✌ ✬ ✶ ✴ ✂ ✂ ✑ ✶ ✄ ✣ ✝ ✌ ✷ ❁ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Scaling basics (contd bis) Refining SS: strict SS statistical SS: statistical SS (discrete-time processes) often encountered in the literature (see Park and Willinger (2000)) : 1. build aggregated sequence ✸✹✴✻✺✽✼ ✳✵✴ ☞❀✌ ✳✵✴ 2. then as statistical SS LRD for ✕✰✯ ! statistical SS requires 2 nd -order stationarity ! 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.5/17
Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Scaling examples: LRD vs. SS LRD through large and persistent correlations vs. SS through fBm (gaussian noise increments) LRD and self-similarity LRD and self-similarity (10-aggregated series) 11000 11000 LRD LRD self-similarity self-similarity 10500 10500 10000 10000 9500 9500 9000 9000 0 20000 40000 60000 80000 100000 0 20000 40000 60000 80000 100000 Time Time LRD and self-similarity (100-aggregated series) LRD and self-similarity (1000-aggregated series) 11000 11000 LRD LRD self-similarity self-similarity 10500 10500 10000 10000 9500 9500 9000 9000 0 20000 40000 60000 80000 100000 0 20000 40000 60000 80000 100000 Time Time 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.6/17
❍ ❃ ❋ ❊ ❇ ❅ ❍ ❋ ❋ ❊ ▲ ❈ ❍ ❇ ▲ ❅ ❂ ❈ ❉ ❇ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be A Wavelet Primer Wavelet : bandpass oscillating function where is ❂❄❃❆❅ timescale and is time ❊●❋ form an orthonormal basis of ■❑❏ internal product < ❍◆▼ > matches irregularities in ❂❖❃ ❊●❋ at time and timescale A look at the beast: 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.7/17
❘ ✞ ❵ ❱ ❳ ✸ ❛ P ✄ P ✣ ❴ ✝ ❜ ❱ ❨ ✸ ❛ ✝ ❬ ❭❪ ❱ ❫ ✌ ✸ ✝ ✣ ✄ ❝ ✂ ✂ ✝ ✝ ✌ ✣ ❭❪ ✄ ✞ ✞ ✣ ✝ ❘ ✄ ✸ ❬ ❫ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be A Wavelet Primer (contd) dyadic grid signal decomposition: ✄✆☎ ✄✆☎ ❯❲❱ ❘❚❙ ✎◗P ❳❩❨ signal approximation + details at each timescale time-frequency plane tiling: Fourier vs. wavelets study instead of process increments 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.8/17
❇ ♣ ❤ ❉ ❍ ✐ ❤ q ❥ ❋ ▼ ❊ ❃ ❋ ❍ ♠ ❋ ♥ ♦ ▼ ❉ ❞ ♦ ❋ r ♣ t ❈ ✐ ❡ ✈ r ❂ ❃ ❅ ❇ ✉ s ❊ ❅ ❇ q ❍ ❞ ✐ ❃ ❣ ❏ ❂ ▼♣ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be A Wavelet Primer (contd bis) “Design” constraints on the wavelet function : built-in scaling : ❊●❋ ❂❄❃ ❡✆❢ ❡✆❢ scaling in process automatically captured vanishing moments : ❂❧❦ polynomial non-stationarity automatically removed Another (very) nice property: almost decorrelation : under , LRD in increments become SRD in coefficients “almost” independence among timescales 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.9/17
Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Toy example 3 different “periods” in this synthetic process, but what’s in there ? 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.10/17
✣ ✸ ✌ ✘ ✝ ★ ✞ ❘ ✄ P ❛ ✘ ✷ ✼ ② ④ ① ② ① ✷ ✌ ❱ ✇ ✄ ✇ ❱ ✝ ❘ ❘ ❱ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Scaling detection Logscale diagram: ★③✞ 1. Compute 2. then plot against (+ confidence intervals) ⑤⑦⑥⑧ Scaling detection: straight line in slope of the LD for some range of alignment in confidence intervals ! LD is a second-order statistic (variance of wavelet coefficients), hence we know nothing about local scaling properties. 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.11/17
❻ ❺ ⑩ ❹❺ ⑩ ❸ ❷ ❷ ❶ ⑩ ⑨ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be LD of toy example Logscale diagram for the ‘‘toy’’ simulation 40 35 LD Slope Scaling property 30 uncorrelated 25 y(j) 20 correlation or LRD 15 self-similarity 10 5 0 0 2 4 6 8 10 12 14 16 18 Octave j Guessing scaling of the signal with LD: uncorrelated for [1,4], self- similar for [6,15] ? 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.12/17
Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Back to scaling detection Principle for 3D-LD: 1. break the process into constant-size time intervals 2. compute the LD over each interval 3. plot in 3D the time-evolution of the LD Scaling detection: same principle as for LD but additional time dimension... now checking for stationarity in scaling. 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.13/17
Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Back to scaling detection (contd) Statistical nature of the process? (3D-LD version) Estimator LD time-dependency Type of stationarity none strict stationarity 3D-LD change in LD level quantitative non-stationarity no change in LD slope qualitative stationarity slope of LD changes with time qualitative non-stationarity 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.14/17
Infonet Group INF NET University of Namur www.infonet.fundp.ac.be 3D-LD of toy example 3D-LD for the ‘‘toy’’ simulation y(j) 30 25 20 15 10 5 0 10 12 14 -5 5 8 10 6 Octave j 15 4 20 2 0 25 Time A better guess for scaling ? 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.15/17
Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Conclusions LD considers each timescale as a homogeneous (stationary) process 3D-LD allows for identifying changes in scaling with time 3D-LD is more than just a binary stationarity check : it also tells what type of (non-)stationarity 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.16/17
✤ Infonet Group INF NET University of Namur www.infonet.fundp.ac.be Case study: TCP flow arrivals See paper for details...also longer tech-report available at http://www.info.fundp.ac.be/ suh/3D-LD/tech-report.ps.gz A scaling model for flow arrivals: 1. “time of the day” and “day of the week” seasonalities for timescales longer than hours, 2. self-similar process for timescales between minutes and hours, 3. non-stationary correlations for timescales between seconds and minutes, 4. non-stationary Poisson process following the first two components. 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.17/17
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