A short introduction to Long memory The model Estimation Bibliography Estimation of the long Memory parameter using an Infinite Source Poisson model applied to transmission rate measurements Fran¸ cois Roueff Ecole Nat. Sup. des T´ el´ ecommunications 46 rue Barrault, 75634 Paris cedex 13, France http://www.tsi.enst.fr/~roueff/ August 13, 2005 Joint work with Gilles Fa¨ y (UST de Lille) and Philippe Soulier (Univ. Paris X) Fran¸ cois Roueff Estimation of the long memory parameter, p. 1
A short introduction to Long memory The model Estimation Bibliography Outline A short introduction to Long memory The model Infinite Source Poisson model Heavy Tails and Long Memory Asymptotic properties Some sample paths Estimation Observation schemes Heavy tails VS long memory Whittle wavelet estimator Simulations Bibliography Fran¸ cois Roueff Estimation of the long memory parameter, p. 2
A short introduction to Long memory The model Estimation Bibliography Second order long memory A second-order stationary process X ( t ) has long memory parameter (or Hurst index) H ∈ (1 / 2 , 1) if cov ( X (0) , X ( t )) = ℓ ( t ) t 2 H − 2 for some ℓ slowly varying at infinity. A possible extension to H ≥ 1 ( X is non-stationary), is �� t � = L ( t ) t 2 H var X ( s ) d s 0 for some L slowly varying at infinity. Fran¸ cois Roueff Estimation of the long memory parameter, p. 3
A short introduction to Long memory The model Estimation Bibliography Examples 1. Linear processes : 1.1 FGN (Gaussian, [Mandelbrot and Van Ness(1968)]), 1.2 ARFIMA ([Granger and Joyeux(1980)]); 2. Non-linear processes : 2.1 Shot noise ([Giraitis et al.(1993)]), 2.2 Renewal-reward ([Taqqu and Levy(1986)]), 2.3 On-Off sources ([Taqqu et al.(1997)]), 2.4 Infinite Source Poisson ([Mikosch et al.(2002)], [Maulik et al.(2002)]). Fran¸ cois Roueff Estimation of the long memory parameter, p. 4
A short introduction to Long memory The model Estimation Bibliography Linear VS Non-linear models Remark 1 Non-linear Long memory models are often derived from Point processes : appropriate for traffic models. Fran¸ cois Roueff Estimation of the long memory parameter, p. 5
A short introduction to Long memory The model Estimation Bibliography Linear VS Non-linear models Remark 1 Non-linear Long memory models are often derived from Point processes : appropriate for traffic models. Remark 2 Little is known about the estimation of H in the non-linear case. Here we investigate the non-linear Poisson case. Fran¸ cois Roueff Estimation of the long memory parameter, p. 6
A short introduction to Long memory The model Estimation Bibliography An example : traffic measurements Here is a 10ms aggregated traffic (obtained form packet counts through some point of the Internet network) 450 1 0.9 400 0.8 350 empirical auto−corr for 10ms aggregated trafic 0.7 300 nb. of paquets per 10 ms 0.6 250 0.5 200 0.4 150 0.3 100 0.2 50 0.1 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 50 100 150 200 250 300 350 400 450 s x 10 4 lag Fran¸ cois Roueff Estimation of the long memory parameter, p. 7
A short introduction to Long memory The model Estimation Bibliography Same example (aggregated again) The same 100ms aggregated traffic (obtained form packet counts through some point of the Internet network) 2600 1.2 2400 1 2200 empirical auto−corr for 100ms aggregated trafic 0.8 2000 nb. of paquets per 100 ms 1800 0.6 1600 0.4 1400 1200 0.2 1000 0 800 600 −0.2 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 50 100 150 200 250 300 350 400 450 s lag Fran¸ cois Roueff Estimation of the long memory parameter, p. 8
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Infinite Source Poisson model 1. { S n } n ∈ N : points of a unit rate homogeneous Poisson process; 2. { ( U n , η n ) } : i.i.d. independent of { S n } ; U n are referred to as transmission rates, η n are the flows durations, U n and η n may be dependent. The n th flow starts at time S n , has rate U n and is transmitting for a duration η n . We observe the cumulative rate � X ( t ) := U n 1 [ S n , S n + η n ) ( t ) . for t ∈ [0 , T ] n ∈ N If E η < ∞ , this process has a stationary version defined by � X S ( t ) := U n 1 [ S n , S n + η n ) ( t ) . n ∈ Z Fran¸ cois Roueff Estimation of the long memory parameter, p. 9
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Second order properties If E [ U 2 ] < ∞ , then, for all s ≥ 0, E[ X 2 ( s )] < ∞ and for s ≤ t , cov ( X ( s ) , X ( t )) = E [ U 2 { s − ( t − η ) + } + ] . The stationary version is weakly stationary if E[ U 2 η ] < ∞ . Then cov ( X S (0) , X S ( t )) = E [ U 2 ( η − t ) + ] . Fran¸ cois Roueff Estimation of the long memory parameter, p. 10
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Long memory is a consequence of heavy tails Assume E[ U 2 ] < ∞ and define U 2 1 { η> t } � � H ( t ) = E . Assumption H is regularly varying with index α ∈ (0 , 2): H ( t ) = ℓ ( t ) t − α , where ℓ is slowly varying. In words, durations are heavy tailed with non-necessarily independent rates. Fran¸ cois Roueff Estimation of the long memory parameter, p. 11
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Examples Example 1 U and η are independent and η is heavy tailed, e.g. the M/G/ ∞ queue, when U = 1. Fran¸ cois Roueff Estimation of the long memory parameter, p. 12
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Examples Example 1 U and η are independent and η is heavy tailed, e.g. the M/G/ ∞ queue, when U = 1. Example 2 In a more general Internet traffic framework : the workload U × η of the flow is heavy tailed and independent of the transmission rate U (and the latter stays away from zero). Fran¸ cois Roueff Estimation of the long memory parameter, p. 13
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Consequence on the weak dependence of X Non-stationary case, α ∈ (0 , 2) �� T � ∼ c α ℓ ( T ) T 2 H as T → ∞ X ( s ) d s var 0 Stationary case, E[ η ] < ∞ 1 1 α − 1 ℓ ( t ) t 1 − α = 2 − 2 H ℓ ( t ) t 2 H − 2 . cov ( X S (0) , X S ( t )) ∼ Fran¸ cois Roueff Estimation of the long memory parameter, p. 14
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Consequence on the dependence structure of X The processes X and X S have Hurst index H = (3 − α ) / 2 Remark H < 1 ⇔ α ∈ (1 , 2) ; H ≥ 1 ⇔ α ∈ (0 , 1] . Remark α ∈ (1 , 2) ⇒ E [ η ] < ∞ ⇒ α ∈ [1 , 2) Fran¸ cois Roueff Estimation of the long memory parameter, p. 15
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Asymptotic properties The asymptotic behavior of � t Y ( t ) := ( X ( s ) − E X ( s )) d s 0 conveniently renormalized has been studied in various situations by [Mikosch et al.(2002)], [Maulik et al.(2002)] or [Mikosch and Resnick(2004)]. Fran¸ cois Roueff Estimation of the long memory parameter, p. 16
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Two very different cases Let T → ∞ . Stable case : 1 < α < 2, i.e. 1 / 2 < H < 1 T − 1 /α Y ( Tt ) converges weakly to an α -stable Levy process. Fran¸ cois Roueff Estimation of the long memory parameter, p. 17
A short introduction to Long memory Infinite Source Poisson model The model Heavy Tails and Long Memory Estimation Asymptotic properties Bibliography Some sample paths Two very different cases Let T → ∞ . Stable case : 1 < α < 2, i.e. 1 / 2 < H < 1 T − 1 /α Y ( Tt ) converges weakly to an α -stable Levy process. Unstable case : 0 < α < 1, i.e. 1 < H < 3 / 2 H − 1 / 2 ( T ) T − H Y ( Tt ) converges weakly to the Gaussian process W with auto-covariance function � t � s 1 { ( u ∨ v ) 1 − α − | u − v | 1 − α } d u d v . cov ( W ( s ) , W ( t )) = 1 − α 0 0 Fran¸ cois Roueff Estimation of the long memory parameter, p. 18
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