L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to its intuitive appeal and a lack of suitable alternatives, self a#nity and the linear relationship (1) have been believed to be warranted by a large number of real world data sets. The mingling of fractals, long memory dependence, and self a#nity has drawn cautious criticism. Kaplan and Kuo [21] note that fractional Brownian motion and fractional Gaussian noise are limited in the sense that their behavior at small scales is completely determined by the Hurst parameter. Davies and Hall [9, p. 7] point out that It is sometimes argued that the assumption of self a#nity is necessary for ....

L. M. Kaplan and C.-C. J. Kuo, Extending self-similarity for fractional Brownian motion, IEEE Transactions on Signal Processing, 42 (1994), pp. 3526--3530.

.... that affect queuing have been discovered [14 16] Wavelet based multiscale models provide generalizations of fBm with more flexible correlation structures [17 20] Using efficient multiscale tree structures, these models provide fast O(N) synthesis algorithms to synthesize N point data sets [21, 22]. Due to their additive nature, these models are inherently Gaussian, and so we will term them wavelet domain independent Gaussian (WIG) models. As a consequence of its Gaussian nature, unfortunately, a WIG model can produce unrealistic synthetic traffic traces in certain situations. First, ....

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

.... that affect queuing have been discovered [14 16] Wavelet based multiscale models provide generalizations of fBm with more flexible correlation structures [17 20] Using efficient multiscale tree structures, these models provide fast O(N) synthesis algorithms to synthesize N point data sets [21, 22]. Due to their additive nature, these models are inherently Gaussian, and so we will term them wavelet domain independent Gaussian (WIG) models. As a consequence of its Gaussian nature, unfortunately, a WIG model can produce unrealistic synthetic traffic traces in certain situations. First, ....

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....wavelet coefficients with variance decaying appropriately with scale form the building blocks for modeling both the long and short term correlations of a target data set. Efficient O(N) algorithms based on the tree structure of wavelet coefficients are available to synthesize N point data sets [22,23]. We will collect such models under the term wavelet domain independent Gaussian (WIG) models. As a consequence of their Gaussian nature, the fBm fGn WIG models can produce unrealistic synthetic traffic traces in certain situations. In many networking applications, for instance, we are nowhere ....

....[k] 2 U n;k ; k = 0; 1; 2 Gamma 1: 12) We will focus on modeling C [k] in this paper. 2. 3 Wavelet domain Independent Gaussian (WIG) model Wavelets serve as an approximate Karhunen Lo eve or decorrelating transform for fBm [18] fGn, and more general LRD signals [23]. Hence, the difficult task of modeling these highly correlated signals in the time domain reduces to a simple one of modeling them approximately by an uncorrelated process in the wavelet domain. The WIG model synthesizes a Gaussian LRD process by generating the parent node U 0;0 of the scaling ....

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....wavelet coefficients with variance decaying appropriately with scale form the building blocks for modeling both the long and short term correlations of a target data set. Efficient O(N) algorithms based on the tree structure of wavelet coefficients are available to synthesize N point data sets [22, 23]. We will term all such models waveletdomain independent Gaussian (WIG) models. As a consequence of their Gaussian nature, the fBm fGn WIG models can produce unrealistic synthetic traffic traces in certain situations. In many networking applications, for instance, we are nowhere near the Gaussian ....

....We will focus on modeling the finest scale scaling coefficients: k] 2 n=2 U n;k ; k = 0; 1; 2 1: 11) C. Wavelet domain independent Gaussian (WIG) model Wavelets serve as an approximate Karhunen Loeve or decorrelating transform for fBm [18] fGn, and more general LRD signals [23]. Hence, the difficult task of modeling these highly correlated signals in the time domain reduces to a simple one of modeling them approximately by an uncorrelated process in the wavelet domain. The WIG model synthesizes a Gaussian process capturing both the long and short term correlations, by ....

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....coefficients can be recursively computed via [35] u j Gamma1;k = 2 (u j;2k u j;2k 1 ) w j Gamma1;k = 2 (u j;2k Gamma u j;2k 1 ) 9) 3. 2 Modeling LRD data Wavelets serve as an approximate Karhunen Lo eve or decorrelating transform for fBm [6] fGn, and more general LRD signals [36]. Hence, modeling and processing of these signals in the wavelet domain is often more efficient and powerful than in the time domain. j 2 j 2 j j j y f (t) j,k (t) j,k j 2,4k 3 j 2,4k 2 j 2,4k 1 j 1,2k 1 U U U U j,k U j 1,2k U j 2,4k U j,k W j,k (a) b) c) Figure 2: ....

.... and identically distributed within scale according to W j;k N(0; oe j ) The variance of the wavelet coefficients of continuous time fBm decays with scale according to a power law in H [6] For fGn, an exact power law in H also holds for decay of the Haar wavelet coefficient variances [36]. This power law decay, along with the decorrelation property of wavelets, has led to fast, robust algorithms for estimation [36, 37] Gaussian LRD processes can be approximately synthesized by generating wavelet coefficients as independent zero mean Gaussian random variables, identically ....

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L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....can be recursively computed via [35] u j Gamma1;k = 2 Gamma1=2 (u j;2k u j;2k 1 ) w j Gamma1;k = 2 Gamma1=2 (u j;2k Gamma u j;2k 1 ) 9) 3. 2 Modeling LRD data Wavelets serve as an approximate Karhunen Lo eve or decorrelating transform for fBm [6] fGn, and more general LRD signals [36]. Hence, modeling and processing of these signals in the wavelet domain is often more efficient and powerful than in the time domain. The variance of the wavelet coefficients of continuous time fBm decays with scale according to a power law in H [6] For fGn, an exact power law in H also holds for ....

....in the wavelet domain is often more efficient and powerful than in the time domain. The variance of the wavelet coefficients of continuous time fBm decays with scale according to a power law in H [6] For fGn, an exact power law in H also holds for decay of the Haar wavelet coefficient variances [36]. This power law decay, along with the decorrelation property of wavelets, has led to fast, robust algorithms for estimation [36, 37] Gaussian LRD processes can be approximately synthesized by generating wavelet coefficients as independent zero mean Gaussian random variables, identically ....

[Article contains additional citation context not shown here]

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....a global scale invariance, initially assumed to hold uniformly over all scales. The approaches proposed have been based on either a phenomenological modeling, where the variance progression from scale to scale can be controlled by a function that does not necessarily identify to a strict powerlaw (Kaplan and Kuo (1994)) or on deeper ideas of cascade analysis, in which probability distribution functions of detail coecients at di erent scales are related to each other (Arn eodo, Muzy and Roux (1997) Such extensions can possibly be applied for example to long range dependent processes, for which scale ....

Kaplan, L. M. & Kuo, C. C. J. (1994), `Extending self-similarity for fractional Brownian motion', IEEE Transactions on Signal Processing 42(12), 3526-3530.

....respectively. The scaling coefficients may be viewed as providing a coarse approximation of the signal, with the wavelet coefficients providing higherfrequency detail information. Wavelets serve as an approximate Karhunen Lo eve transform for fBm [3] fGn, and more general LRD signals [9]. Thus, highly correlated, LRD signals become nearly uncorrelated in the wavelet domain. In addition, the energy of the wavelet coefficients of continuous time fBm decays with scale according to a power law [3] While for sampled fBm the power law decay is not exact [3] the Haar wavelet transform ....

....uncorrelated in the wavelet domain. In addition, the energy of the wavelet coefficients of continuous time fBm decays with scale according to a power law [3] While for sampled fBm the power law decay is not exact [3] the Haar wavelet transform of fGn exhibits power law scaling of the form 1 [9] var(W j;k ) oe 2 2 (2H Gamma1) j Gamma1) 2 Gamma 2 2H Gamma1 ) 7) where oe 2 is the variance of the fGn process. 4. THE MWM The basic idea behind the MWM is simple. To model nonnegativity, we use the Haar wavelet transform with special wavelet domain constraints. To capture ....

[Article contains additional citation context not shown here]

L. M. Kaplan and C.-C. J. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. on Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....wavelet coefficients with variance decaying appropriately with scale form the building blocks for modeling both the long and short term correlations of a target data set. Efficient O(N) algorithms based on the tree structure of wavelet coefficients are available to synthesize N point data sets [22, 23]. We will term all such models waveletdomain independent Gaussian (WIG) models. As a consequence of their Gaussian nature, the fBm fGn WIG models can produce unrealistic synthetic traffic traces in certain situations. In many networking applications, for instance, we are nowhere near the Gaussian ....

....modeling the finest scale scaling coefficients: C (n) k] 2 Gamman=2 U n;k ; k = 0; 1; 2 n Gamma 1: 11) C. Wavelet domain independent Gaussian (WIG) model Wavelets serve as an approximate Karhunen Loeve or decorrelating transform for fBm [18] fGn, and more general LRD signals [23]. Hence, the difficult task of modeling these highly correlated signals in the time domain reduces to a simple one of modeling them approximately by an uncorrelated process in the wavelet domain. The WIG model synthesizes a Gaussian process capturing both the long and short term correlations, by ....

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....processes do correspond to a type of nonGaussian 1=f noise. However, considering the difficulties associated with the FFT power spectrum (e.g. it is not truly 1=f ) the terminology ESS provides a more precise characterization. For more general processes, we can define the structure function g B [25, 26] such that Var [B(t s) Gamma B(t) g B (s)oe 2 X (20) and for T = 1 (now in a discrete time setting) Var h X (m) n] i = g B [m] m 2 Var[X[n] 21) When g B (s) oe 2 X jsj 2H , we have the self similarity condition for fBm and fGn. The structure function directly ....

....and short term covariances will play a major role in queuing behavior. We investigate two parametric models that overcome this limitation and prove that the FFT synthesis is exact for these two models. 4. 1 Asymptotic discrete fractional Gaussian noise The first class, proposed by Kaplan and Kuo [25], results from the following covariance r X [n] oe 2 X ; n = 0 oe 2 X 2 Theta (A Gamma 1) 1 Gamma jpj)jpj n Gamma1 A Gamma jn 1j 2H jn Gamma 1j 2H Gamma 2jnj 2H Delta ; jnj 1 (36) with A = 2H p(2 Gamma2H) 2H Gammap(2 Gamma2H) They call this process ....

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....with matching across bin boundaries) to introduce the desired level of SRD. Better still would be a method of synthesizing self similar sample paths that consistently integrates the presence of LRD and SRD. One such method, based on the Haar wavelet transform, is discussed by Kaplan and Kuo in [KK94]. At the moment their method is somewhat limited due to difficulties in parameter estimation, but still appears promising. In general, we believe wavelet methods hold great promise for characterizing and synthesizing self similar traffic, due to the natural match between the notion of scaling in ....

L. Kaplan and C.-C. Kuo, "Extending SelfSimilarity for Fractional Brownian Motion," IEEE Transactions on Signal Processing, 42(12), pp. 3526-3530, December 1994.

....decay across scale. We choose the pdfs for the A (j) s to control the wavelet coefficients scaling behavior via (24) The fact that this scaling behavior allows us to model correlations can be explained as follows. Consider the Karhunen Lo eve properties of the wavelet transform. Previous work [11,12,47] has demonstrated that the wavelet transform approximately decorrelates or whitens a general class of LRD signals, including 1=f processes. If the decorrelation were exact, then specifying the correct variances of the wavelet coefficients would fully capture the correlation structure of the ....

L. M. Kaplan and C.-C. J. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. on Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....decay across scale. We choose the pdfs for the A (j) s to control the wavelet coefficients scaling behavior via (26) The fact that this scaling behavior allows us to model correlations can be explained as follows. Consider the Karhunen Lo eve properties of the wavelet transform. Previous work [11,12,47] has demonstrated that the wavelet transform approximately decorrelates or whitens a general class of LRD signals, including 1=f processes. If the decorrelation were exact, then specifying the correct variances of the wavelet coefficients would fully capture the correlation structure of the ....

L. M. Kaplan and C.-C. J. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

....with matching across bin boundaries) to introduce the desired level of SRD. Better still would be a method of synthesizing self similar sample paths that consistently integrates the presence of LRD andSRD. One such method, based on the Haar wavelet transform, is discussed by Kaplan and Kuo in [KK94]. At the moment their method is somewhat limited due to difficulties in parameter estimation, but still appears promising. In general, we believe wavelet methods hold great promise for characterizing and synthesizing self similar traffic, due to the natural match between the notion of scaling in ....

L. Kaplan and C.-C. Kuo, "Extending SelfSimilarity for Fractional Brownian Motion," IEEE Transactions on Signal Processing, 42(12), pp. 3526-3530, December 1994.

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L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

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L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

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L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

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L. M. Kaplan and C.-C. J. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. on Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

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L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

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L. M. Kaplan and C.-C. J. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

No context found.

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

No context found.

L. Kaplan and C.-C. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

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L. M. Kaplan and C.-C. J. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

No context found.

L. M. Kaplan and C.-C. J. Kuo, "Extending self-similarity for fractional Brownian motion," IEEE Trans. Signal Proc., vol. 42, pp. 3526--3530, Dec. 1994.

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