Long Range Dependence

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Long Range Dependence

• Processes With Long Memory

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Introduction

• ARMA models have exponentially decaying ACF
• This is not always true sometimes it has
  important implications
• E.g. Artificial traffic generation

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Nile River Minima
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- Slope 0.27 ?
- Slope 0.73 1 - ?
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Ethernet, Bytes
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AR(1)
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Properties of LRD sequences
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Spectral Properties (i.e. in Fourier domain)
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Draw Y(t) on the board
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Nile Time Series, Fractionally Differenced
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Fractionally Integrated White Noise
H0.5
H0.99
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Fractional Gaussian Noise
H0.5
H0.99
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Aggregation Properties
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Quiz a
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Quiz b
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Quiz c
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LRD comes from Heavy Tail
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Nile
Ethernet Byte
Ethernet Packet
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All you need to know about Wavelets
Original time series (Sprint)

• Wavelet localized Fourier analysis
• Sj Approximation at octave j
• D_j detail at octave j
• Original time series is the sum of all details

D1 Detail at octave 1
S1 Coarse approximation at octave 1
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Wavelet decomposition
Wavelet coefficient
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Nile
Ethernet Bytes
Ethernet Packets
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ARIMA (p0,d1,q0)
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ARIMA (p3,d1,q3)
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Sprint data, raw
Differenced at lags 1 and 16
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Wavelet Analysis of Non Stationary TS, nb
vanishing moments 1
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Wavelet Analysis of Non Stationary TS, nb
vanishing moments 2
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Application to Sprint Data wavelet analysis does
not find LRD
Sprint data, raw
Sprint data, differenced
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Nile Data, FARIMA forecast
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Nile Data, ARIMA forecast
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Case Studies
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Impact on Confidence Intervals

• Example mobility model
• Mobility models may exhibit some aspects of long
  range dependence
• See Augustin Chaintreau, Pan Hui, Jon Crowcroft,
  Christophe Diot, Richard Gass, and James Scott.
  "Impact of Human Mobility on the Design of
  Opportunistic Forwarding Algorithms".
• The random trip model supports LRD

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Long Range Dependent Random Waypoint

• Consider the random waypoint without pause, like
  before, but change the distribution of speed

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LRD means high variability
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Practical Implications
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Average Over Independent Runs
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Compare to Single Long Run