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In [[computer networks]], [[self-similarity]] is a feature of network data transfer dynamics. When modeling network data dynamics the traditional time series models, such as an [[autoregressive moving average model]] (ARMA(p, q)), are not appropriate. This is because these models only provide a finite number of parameters in the model and thus interaction in a finite time window, but the network data usually have a [[Long-range dependency|long-range dependent]] temporal structure. A self-similar process is one way of modeling network data dynamics with such a long range correlation. This article defines and describes network data transfer dynamics in the context of a self-similar process. Properties of the process are shown and methods are given for  [[Conceptual graph|graphing]] and estimating parameters modeling the self-similarity of network data.


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== Definition ==


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Suppose <math>X</math> be a [[Stationary process#Weaker forms of stationarity|weakly stationary (2nd-order stationary) process]]
with mean <math>\mu</math>, variance <math>\sigma^2</math>, and [[autocorrelation]] function <math>\gamma(t)</math>.
Assume that the autocorrelation function <math>\gamma(t)</math> has the form
<math>\gamma(t)\rightarrow t^{-\beta}L(t)</math> as <math>t\to\infty</math>, where <math>0<\beta<1</math>
and <math>L(t)</math> is a [[slowly varying function]] at [[infinity]], that is <math>\lim_{t\to\infty}\frac{L(tx)}{L(t)}=1</math> for all <math>x>0</math>.
For example, <math>L(t)=const</math> and <math>L(t)=\log (t)</math> are slowly varying functions.<br>
Let <math>X_k^{(m)}=\frac{1}{m^H}(X_{km-m+1}+\cdot\cdot\cdot+X_{km})</math>,
where <math>k=1,2,3,\ldots</math>, denote an aggregated point series over non-overlapping blocks of size <math>m</math>, for each <math>m</math> is a [[positive integer]].


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=== Exactly self-similar process ===


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* <math>X</math> is called an exactly self-similar process if there exists a self-similar parameter <math>H</math> such that <math>X_k^{(m)}</math> has the same distribution as <math>X</math>. An example of exactly self-similar process with <math>H</math> is [[Fractional Brownian motion|Fractional Gaussian Noise]] (FGN) with <math>\frac{1}{2}<H<1</math>.
'''Definition:Fractional Gaussian Noise (FGN)''' {{-}}
<math>X(t)=B_H(t+1)-B_H(t),~\forall t\geq 1</math> is called the Fractional Gaussian Noise, where <math>B_H(\cdot)</math> is a [[Fractional Brownian motion]].<ref>W. E. Leland, W. Willinger, M. S. Taqqu, D. V. Wilson, "On the self-similar nature of Ethernet traffic",ACM SIGCOMM Computer Communication Review 25,202-213,1995.</ref>


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=== exactly second order self-similar process ===


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* <math>X</math> is called an exactly second order self-similar process if there exists a self-similar parameter <math>H</math> such that <math>X_k^{(m)}</math> has the same variance and autocorrelation as <math>X</math>.
 
=== asymptotic second order self-similar process ===
 
* <math>X</math> is called an [[asymptotic]] second order self-similar process with self-similar parameter <math>H</math> if <math>\gamma^{(m)}(t)\to \frac{1}{2}[(t+1)^{2H}-2t^{2H}+(t-1)^{2H}]</math> as <math>m\to\infty</math>, <math>~\forall t=1,2,3,\ldots</math>
 
== Some relative situations of Self-Similar Processes ==
 
=== Long-Range-Dependence(LRD) ===
Suppose <math>X(t)</math> be a weakly stationary (2nd-order stationary) process with mean <math>\mu</math> and variance <math>\sigma^2</math>. The Autocorrelation Function (ACF) of lag <math>t</math> is given by <math>\gamma(t)={ \mathrm{cov}(X(h),X(h+t))\over \sigma^2} ={ E[(X(h)-\mu)(X(h+t)-\mu)]\over \sigma^2}</math>{{-}}
'''Definition:'''{{-}}
A weakly stationary process is said to be "Long-Range-Dependence" if <math> \sum_{t=0}^\infty |\gamma(t)|=\infty</math>
 
A process which satisfies <math>\gamma (t) \rightarrow t^{-\beta}L(t)</math> as <math>t\to \infty</math> is said to have long-range dependence. The [[spectral density]] function of long-range dependence follows a [[power law]] near the origin. Equivalently to <math>\gamma (t) \rightarrow t^{-\beta}L(t)</math>, <math>X</math> has long-range dependence if the spectral density function of autocorrelation function, <math>f_t(w)=\sum_{t=0}^\infty \gamma(t)e^{iwt}</math>, has the form of <math>w^{-\gamma}L(w)</math> as <math>w\to 0</math> where <math>0<\gamma<1</math>, <math>L</math> is slowly varying at 0.
 
[[Long-range dependency|also see]]
 
=== Slowly decaying variances ===
 
<math>X^{(m)}=\frac{1}{m}(X_1+\cdot\cdot\cdot+X_m)</math><br>
When an autocorrelation function of a self-similar process satisfies <math>\gamma (t) \rightarrow t^{-\beta}L(t)</math> as <math>t\to \infty</math>, that means it also satisfies <math>Var(X^{(m)})\to am^{-\beta}</math> as <math>m\to \infty</math>, where <math>a</math> is a finite positive constant independent of m, and 0<β<1.
 
== Estimating the self-similarity parameter "H" ==
 
=== R/S analysis ===
Assume that the underlying process <math>X</math> is Fractional Gaussian Noise. Consider the series <math>X_{(1)},\ldots,X_{(n)}</math>, and let <math>Y_{(n)}= \sum_{i=1}^n X_{(i)}</math>.<br>
The sample variance of <math>X_{(i)}</math> is <math>S^2(n)=\frac{1}{n}\sum_{i=1}^n X^2_{(i)}-(\frac{1}{n})^2Y^2_n</math>{{-}}
'''Definition:R/S statistic'''
 
<math>\frac{R}{S}(n)=\frac{1}{S(n)}
[\max_{0\leq t\leq n}(Y_t -\frac{t}{n}Y_n)-\min_{0\leq t\leq n}(Y_t -\frac{t}{n}Y_n)]</math><br>{{-}}
If <math>X_{(i)}</math> is FGN, then <math>E(\frac{R}{S}(n))\to C_H \times n^H</math><br>
Consider fitting a regression model :
<math>log\frac{R}{S}(n)=log(C_H)+Hlog(n)+\epsilon_n</math>, where
<math>\epsilon_n \thicksim N(0,\sigma^2)</math><br>
In particular for a time series of length <math>N</math> divide the time series data into <math>k</math> groups each of size <math>\frac{N}{k}</math>, compute <math>\frac{R}{S}(n)</math> for each group.<br>
Thus for each n we have <math>k</math> pairs of data (<math>log(n),log\frac{R}{S}(n)</math>).There are <math>k</math> points for each <math>n</math>, so we can fit a [[regression model]] to estimate <math>H</math> more accurately. If the solpe of the [[regression line]] is between 0.5~1, it is a self-similar process.
 
=== Variance-time plot ===
Variance of the sample mean is given by <math>Var(\bar{X}_n)\to cn^{2H-2},~\forall c>0</math>.<br>
For estimating H, calculate [[sample means]] <math>\bar{X}_1,\bar{X}_2,\cdots,\bar{X}_{m_k}</math> for <math>m_k</math> sub-series of length <math>k</math> .<br>
Overall mean can be given by <math>\bar{X}(k)=\frac{1}{k} \sum_{i=1}^{m_k} \bar{X}_i(k)</math>, sample variance <math>S^2(k)=\frac{1}{m_k-1}\sum_{i=1}^{m_k} (\bar{X}_i(k)-\bar{X}(k))^2</math>.<br>
The variance-time plots are obtained by plotting <math>\log S^2(k)</math> against
<math>\log k</math> and we can fit a simple least square line through the resulting points in the plane ignoring the small values of k.
 
For large values of <math>k</math>, the points in the plot are expected to be scattered around a straight line with a negative slope <math>2H-2</math>.For short-range dependence or independence among the observations, the slope of the straight line is equal to -1.<br>
Self-similarity can be inferred from the values of the estimated slope which is asymptotically between –1 and 0, and an estimate for the degree of self-similarity is given by <math>\hat{H}=1+\frac{1}{2}(slope).</math>
 
=== Periodogram-based analysis ===
 
Whittle's approximate maximum likelihood estimator ([[Maximum likelihood|MLE]]) is applied to solve the Hurst's parameter via the [[spectral density]] of <math>X</math>. It is not only a tool for visualizing the Hurst's parameter, but also a method to do some statistical inference about the parameters via the asymptotic properties of the MLE. In particular, <math>X</math> follows a [[Gaussian process]]. Let the spectral density of <math>X</math>,
<math>f_x(w;\theta)=\sigma_\epsilon^2 f_x(w;(1,\eta))</math>, where
<math>\theta=(\sigma_\epsilon^2,\eta)=(\sigma_\epsilon^2,H,\theta_3,\ldots,\theta_k),H=\frac{\gamma+1}{2}</math>, and <math>\theta_3,\ldots,\theta_k</math> construct a short-range time series autoregression (AR) model, that is <math>X_j=\sum_{i=1}^k \alpha_i X_{j-i}+\epsilon_j</math>,
with <math>Var(\epsilon_j)=\sigma_\epsilon^2</math>.
 
Thus, the Whittle's estimator <math>\hat{\eta}</math> of <math>\eta</math> minimizes
the function <math>Q(\eta)=\int_{-\pi}^{\pi} \frac{I(w)}{f(w;(1,\eta))}\, dw</math>
, where I(w) denotes the periodogram of X as <math>(2\pi n)^{-1}|\sum_{j=1}^n X_je^{iwj}|^2</math> and <math>\hat{\sigma}^2=\int_{-\pi}^{\pi} \frac{I(w)}{f(w;(1,\hat{\eta}))}\, dw</math>. These integrations can be assessed by Riemann sum.<br>
{{-}}
Then <math>n^{1/2}(\hat{\theta}-\theta)</math> asymptotically follows a normal distribution if <math>X_j</math> can be expressed as a form of an infinite moving average model.{{-}}
{{-}}
To estimate <math>H</math>, first, one has to calculate this periodogram. Since
<math>I_n(w)</math> is an estimator of the spectral density, a series with long-range dependence should have a periodogram, which is proportional to <math>|\lambda|^{1-2H}</math> close to the origin. The periodogram plot is obtained by ploting
<math>\log(I_n(w))</math> against <math>\log(w)</math>.<br>
Then fitting a regression model of the <math>\log(I_n(w))</math> on the <math>\log(w)</math> should give a slope of <math>\hat{\beta}</math>. The slope of the fitted straight line is also the estimation of <math>1-2H</math>. Thus, the estimation <math>\hat{H}</math> is obtained.
 
Note:<br>
There are two common problems when we apply the periodogram method. First, if the data does not follow a Gaussian distribution, transformation of the data can solve this kind of problems. Second, the sample spectrum which deviates from the assumed spectral density is another one. An aggregation method is suggested to solve this problem. If <math>X</math> is a Gaussian process and the spectral density function of <math>X</math> satisfies <math>w^{-\gamma}L(w)</math> as <math>w\to \infty</math>, the function,
<math>m^{-H}L^{-\frac{1}{2}}(m)\sum_{i=(j-1)m+1}^mk (X_i-E(|X_i|)),~j=1,2,\ldots,[\tfrac{n}{m}]</math>, converges in distribution to FGN as <math>m\to \infty</math>.
 
== References ==
* P. Whittle, "Estimation and information in stationary time series", Art. Mat. 2, 423-434, 1953.
* K. PARK,W. WILLINGER, Self-Similar Network Traffic and Performance Evaluation, WILEY,2000.
* W. E. Leland, W. Willinger, M. S. Taqqu, D. V. Wilson, "On the self-similar nature of Ethernet traffic",ACM SIGCOMM Computer Communication Review 25,202-213,1995.
* W. Willinger, M. S. Taqqu, W. E. Leland, D. V. Wilson, "Self-Similarity in High-Speed Packet Traffic: Analysis and Modeling of Ethernet Traffic Measurements",Statistical Science 10,67-85,1995.
 
{{Reflist}}
 
{{DEFAULTSORT:Self-Similarity Of Network Data Analysis}}
[[Category:Computer network analysis]]

Revision as of 01:23, 9 November 2013

Template:Multiple issues

In computer networks, self-similarity is a feature of network data transfer dynamics. When modeling network data dynamics the traditional time series models, such as an autoregressive moving average model (ARMA(p, q)), are not appropriate. This is because these models only provide a finite number of parameters in the model and thus interaction in a finite time window, but the network data usually have a long-range dependent temporal structure. A self-similar process is one way of modeling network data dynamics with such a long range correlation. This article defines and describes network data transfer dynamics in the context of a self-similar process. Properties of the process are shown and methods are given for graphing and estimating parameters modeling the self-similarity of network data.

Definition

Suppose be a weakly stationary (2nd-order stationary) process with mean , variance , and autocorrelation function . Assume that the autocorrelation function has the form as , where and is a slowly varying function at infinity, that is for all . For example, and are slowly varying functions.
Let , where , denote an aggregated point series over non-overlapping blocks of size , for each is a positive integer.

Exactly self-similar process

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exactly second order self-similar process

asymptotic second order self-similar process

Some relative situations of Self-Similar Processes

Long-Range-Dependence(LRD)

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Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.

The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.

As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com A weakly stationary process is said to be "Long-Range-Dependence" if

A process which satisfies as is said to have long-range dependence. The spectral density function of long-range dependence follows a power law near the origin. Equivalently to , has long-range dependence if the spectral density function of autocorrelation function, , has the form of as where , is slowly varying at 0.

also see

Slowly decaying variances


When an autocorrelation function of a self-similar process satisfies as , that means it also satisfies as , where is a finite positive constant independent of m, and 0<β<1.

Estimating the self-similarity parameter "H"

R/S analysis

Assume that the underlying process is Fractional Gaussian Noise. Consider the series , and let .
The sample variance of is Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE

To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.

Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.

The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.

As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com Definition:R/S statistic


Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE

To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.

Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.

The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.

As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com If is FGN, then
Consider fitting a regression model : , where
In particular for a time series of length divide the time series data into groups each of size , compute for each group.
Thus for each n we have pairs of data ().There are points for each , so we can fit a regression model to estimate more accurately. If the solpe of the regression line is between 0.5~1, it is a self-similar process.

Variance-time plot

Variance of the sample mean is given by .
For estimating H, calculate sample means for sub-series of length .
Overall mean can be given by , sample variance .
The variance-time plots are obtained by plotting against and we can fit a simple least square line through the resulting points in the plane ignoring the small values of k.

For large values of , the points in the plot are expected to be scattered around a straight line with a negative slope .For short-range dependence or independence among the observations, the slope of the straight line is equal to -1.
Self-similarity can be inferred from the values of the estimated slope which is asymptotically between –1 and 0, and an estimate for the degree of self-similarity is given by

Periodogram-based analysis

Whittle's approximate maximum likelihood estimator (MLE) is applied to solve the Hurst's parameter via the spectral density of . It is not only a tool for visualizing the Hurst's parameter, but also a method to do some statistical inference about the parameters via the asymptotic properties of the MLE. In particular, follows a Gaussian process. Let the spectral density of , , where , and construct a short-range time series autoregression (AR) model, that is , with .

Thus, the Whittle's estimator of minimizes the function , where I(w) denotes the periodogram of X as and . These integrations can be assessed by Riemann sum.
Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE

To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.

Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.

The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.

As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com Then asymptotically follows a normal distribution if can be expressed as a form of an infinite moving average model.Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE

To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.

Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.

The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.

As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com Benefits of Residing in a Apartment or Landed property in Singapore Property New Launches & Project Showcase In Singapore Many residential Singapore property sales involve buying property in Singapore at new launches. These are often homes underneath building, being sold new by developers. New Launch Singapore Property, 28 Imperial Residences Coming To Geylang Lorong 26 The property market is slowing down, based on personal property transactions in May Cell Apps FREE Sign Up Log in Property Brokers Feedback

Individuals all wish to be seen having the identical foresight as the experts in property investment or the massive names in their own fields. Thus the discharge of these tales works to encourage different buyers to observe suit. Bartley Ridge is the most popular new launch in district 13. Irresistible pricing from $1,1xx psf. Bartley Ridge is a ninety nine-12 months leasehold new condominium at Mount Vernon road, good next to Bartley MRT station (CC12). If you want to get more Rehda Johor chairman Koh Moo Hing said potential property consumers in the two areas Http://Modern.Dowatch.Net/Profile/Mic31K/Created/Topics are now adopting a wait-and-see attitude. How can I get the ebrochure and flooring plans of the new launch projects ? The Existing Mortgage on your HDB District 13, Freehold condominium District 11, Freehold Cluster landed house Sea Horizon EC @ Pasir Ris

FindSgNewLaunch is the main Singapore Property web site - one of the best place to begin your actual estate search whether you might be an investor, shopping for for own use, or searching for a spot to lease. With detailed details about each property, together with maps and pictures. We deliver you probably the most complete choice out there. No. For brand spanking new Singapore property gross sales, you possibly can withdraw at any time earlier than booking the unit, without penalty. On the preview, the agent will let you recognize the exact worth for you to resolve whether or not to proceed or not. Solely when you resolve to proceed will the agent book the unit for you. Pending for Sale Licence Approval All Pending for Sale Licence Approval New launch FREEHOLD condominium @ Braddell New launch condominium combined growth at Yishun PROJECT TITLE

To not worry, we'll hold you in our VIP Precedence list for future new launch VIP Preview. We'll contact you to establish your wants and advocate related tasks, both new launch or resale properties that probably match your standards. In case you're looking for resale property, such as these few years old, or just got Short-term Occupation Permit (PRIME), you might click on here right here for fast search and submit your shortlisted listings to us, we'll check and call you for viewing.

Oceanfront Suites, irresistible pricing for a 946 leasehold property with magnificent sea view. Dreaming of basking and feeling the warmth of pure sunlight is now just a click on away. Oceanfront Suites - Seaside residing no longer needs to remain an unattainable This Cambodia new launch, a mega development has also 762 residential models. Additionally located within this Oxley abroad property is a mega shopping center with 627 outlets and also up to 963 available workplace spaces and is surrounded by quite a few Embassy, resorts, Casinos and many vacationer relax space. Belysa EC @ Pasir Ris Esparina EC @ Sengkang Dell Launches World's first Gender-GEDI Female Entrepreneurship Index on 06/04/thirteen by Istanbul, Turkey. Paris Ris EC @ Paris Ris in search of indication of curiosity.

The developer should open a Venture Account with a financial institution or monetary establishment for every housing venture he undertakes, before he's issued with a Sale License (license to sell models in his development). All payments from buyers before completion of the challenge, and construction loans, go into the mission account. New launch rental LA FIESTA, an thrilling new condominium located along Sengkang Square / Compassvale Highway is a brief stroll to the bustling Sengkang City Centre the place the bus interchange, Sengkang MRT and LRT stations are located. Glorious location,Premium rental with Bayfront resort lifestyle theme and views ofwaterscape. Close to EC pricing - Worth for cash! Apr 02, 2013 Sengkang New Rental Launch, La Fiesta- Sengkang MRTstation at your gate.

As The Hillford property launch at Jalan Jurong Kechil may be very close to to beauty world mrt , the environment for the plot of land which belongs to World Class Land remains very upbeat as it is rather close to to Holland Village. Review now by visiting the brand new apartment pages on our website, each displaying complete particulars and the latest information of each new launch. You can even contact us directly to obtain quick & correct answers to all of your questions with high of the road service. An inevitable conclusion is that costs within the property market have just set new highs. The apparent connotation for potential buyers is to take motion now before prices bounce again. tract and points to his property line, marked by a big maple in a sea of Search SG Developersale.com To estimate , first, one has to calculate this periodogram. Since is an estimator of the spectral density, a series with long-range dependence should have a periodogram, which is proportional to close to the origin. The periodogram plot is obtained by ploting against .
Then fitting a regression model of the on the should give a slope of . The slope of the fitted straight line is also the estimation of . Thus, the estimation is obtained.

Note:
There are two common problems when we apply the periodogram method. First, if the data does not follow a Gaussian distribution, transformation of the data can solve this kind of problems. Second, the sample spectrum which deviates from the assumed spectral density is another one. An aggregation method is suggested to solve this problem. If is a Gaussian process and the spectral density function of satisfies as , the function, , converges in distribution to FGN as .

References

  • P. Whittle, "Estimation and information in stationary time series", Art. Mat. 2, 423-434, 1953.
  • K. PARK,W. WILLINGER, Self-Similar Network Traffic and Performance Evaluation, WILEY,2000.
  • W. E. Leland, W. Willinger, M. S. Taqqu, D. V. Wilson, "On the self-similar nature of Ethernet traffic",ACM SIGCOMM Computer Communication Review 25,202-213,1995.
  • W. Willinger, M. S. Taqqu, W. E. Leland, D. V. Wilson, "Self-Similarity in High-Speed Packet Traffic: Analysis and Modeling of Ethernet Traffic Measurements",Statistical Science 10,67-85,1995.

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  1. ↑ W. E. Leland, W. Willinger, M. S. Taqqu, D. V. Wilson, "On the self-similar nature of Ethernet traffic",ACM SIGCOMM Computer Communication Review 25,202-213,1995.
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