3D-LD: a graphical Wavelet-based method for Analyzing Scaling - - PowerPoint PPT Presentation

Infonet Group University of Namur

www.infonet.fundp.ac.be

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3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes

Steve UHLIG

suh@infonet.fundp.ac.be

Infonet Group 15th 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 University of Namur

www.infonet.fundp.ac.be

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Schedule

Scaling basics Wavelet basics Scaling detection Conclusion

3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.2/17

Infonet Group University of Namur

www.infonet.fundp.ac.be

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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:

dependence of moments on time

non-stationarity Note: H is “self-similarity” parameter

3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.3/17

Infonet Group University of Namur

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Scaling basics (contd)

• 2. Long-range dependence (LRD):
• definition:

power-law in auto-correlation (

)

• persistence in correlations but requires 2nd-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 University of Namur

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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 2nd-order stationarity !

3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.5/17

Infonet Group University of Namur

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Scaling examples: LRD vs. SS

LRD through large and persistent correlations vs. SS through fBm (gaussian noise increments)

9000 9500 10000 10500 11000 20000 40000 60000 80000 100000 Time LRD and self-similarity LRD self-similarity 9000 9500 10000 10500 11000 20000 40000 60000 80000 100000 Time LRD and self-similarity (10-aggregated series) LRD self-similarity 9000 9500 10000 10500 11000 20000 40000 60000 80000 100000 Time LRD and self-similarity (100-aggregated series) LRD self-similarity 9000 9500 10000 10500 11000 20000 40000 60000 80000 100000 Time LRD and self-similarity (1000-aggregated series) LRD self-similarity 3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.6/17

Infonet Group University of Namur

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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

Infonet Group University of Namur

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A Wavelet Primer (contd)

dyadic grid signal decomposition:

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

Infonet Group University of Namur

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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 University of Namur

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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

Infonet Group University of Namur

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Scaling detection

Logscale diagram:

• 1. Compute

• 2. then plot

against

(+ confidence intervals) 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

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LD of toy example

5 10 15 20 25 30 35 40 2 4 6 8 10 12 14 16 18 y(j)

Octave j Logscale diagram for the ‘‘toy’’ simulation

LD Slope Scaling property

uncorrelated

correlation or LRD

self-similarity

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

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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 University of Namur

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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 University of Namur

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3D-LD of toy example

3D-LD for the ‘‘toy’’ simulation 5 10 15 20 25 Time 2 4 6 8 10 12 14 Octave j

• 5

5 10 15 20 25 30 y(j)

A better guess for scaling ?

3D-LD: a graphical Wavelet-based method for Analyzing Scaling Processes – p.15/17

Infonet Group University of Namur

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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 University of Namur

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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