- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 302786, 13 pages

http://dx.doi.org/10.1155/2012/302786

## Visiting Power Laws in Cyber-Physical Networking Systems

^{1}School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road, Shanghai 200241, China^{2}Department of Computer and Information Science, University of Macau, Avenue Padre Tomas Pereira, Taipa 1356, Macau SAR, China

Received 23 February 2011; Accepted 23 March 2011

Academic Editor: Carlo Cattani

Copyright © 2012 Ming Li and Wei Zhao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Cyber-physical networking systems (CPNSs) are made up of various physical systems that are heterogeneous in nature. Therefore, exploring universalities in CPNSs for either data or systems is desired in its fundamental theory. This paper is in the aspect of data, aiming at addressing that power laws may yet be a universality of data in CPNSs. The contributions of this paper are in triple folds. First, we provide a short tutorial about power laws. Then, we address the power laws related to some physical systems. Finally, we discuss that power-law-type data may be governed by stochastically differential equations of fractional order. As a side product, we present the point of view that the upper bound of data flow at large-time scaling and the small one also follows power laws.

#### 1. Introduction

Cyber-physical networking systems (CPNSs) consist of computational and physical elements integrated towards specific tasks [1–3]. Generally, both data and systems in CPNSs are heterogeneous. For instance, teletraffic data are different from transportation traffic, letting along other data in CPNS, such as those in physiology. Therefore, one of the fundamental questions is what possible general laws are to meet CPNS in theory. The answer to that question should be in two folds. One is data. The other is systems that transmit data from sources to destinations within a predetermined restrict period of time according to a given quality of service (QoS).

In general, both data and systems in CPNS are multidimensional. For instance, data from sources to be transmitted may be from a set of sensors distributed in a certain area. Destinations receiving data may be a set of actuators, for example, a set of cars distributed in a certain area. Systems to transmit data are generally distributed.

Denote by the -dimensional Euclidean space. Denote data at sources and destinations, respectively, by which is supposed to be -dimensional and which is supposed to be -dimensional. They are given by
A stochastic equation describing an abstract relationship between and may be expressed by
where implies the transposition, is a servie matrix of order of a system, and , which is a vector with the same dimension as that of , may represent uncertainty for the operation of . The operations *⊗* and *⊕* are to be studied from a view of systems, and they are out of the scope of this paper.

Note that is usually a random field, see for example the work of Chilés and Delfiner in [4] in geosciences, the work of Uhlig in [5] in telecommunications, the work of Messina et al. in [6] in power systems, the work of Muniandy and Stanslas in [7] in medical images, the work of Mason et al. in [8] in wind engineering, and of simply citing a few. The statistics of is obviously crucial for the performance analysis of physical systems in CPNS. It is noted that the physical meaning of is diverse. For example, it may represent a two-dimensional aeromagnetic data (Spector and Grant [9]), a medical image (Fortin et al. [10]), vegetation data (Myrhaug et al. [11]), surface crack in material science (Tanaka et al. [12]), and data in physiology (Werner [13], West [14]), DNA (Cattani [15]), data in stock markets (Rosenow et al. [16]), just mentioning a few. Therefore, seeking for possible universalities of in CPNS is desired.

Without lose of generality, we rewrite (1.1) by where . The norm of is given by The autocovariance function (ACF) of is given, over the hyperrectangle for (Adler [17]), by where is the mean operator, , and The ACF measures how correlates to .

From the point of view of applications of CPNS, we are interested in two asymptotic expressions of . One is for . The other is for . The former characterizes the small scaling phenomenon of . The latter measures the large scaling one. It is quite natural for us to investigate two types of scaling phenomena. As a matter of fact, one may be interested in small scaling in some applications, for example, admission control in computer communication or monitoring sudden disaster in geoscience. On the other side, one may be interested in large scaling in applications, for example, long-term performance analysis of systems. Exact expression of is certainly useful, but it may usually be application dependent. Consequently, we study possible generalities of for and instead of its exactly full expressions. The aim of this paper is to explain that both the small scaling described by for and the large scaling described by for , in some fields related to CPNS, ranging from geoscience to computer communications, follow power laws.

The rest of paper is organized as follows. Short tutorial about power laws is explained in Section 2. Some cases of power laws relating to computational and physical systems in CPNS are described in Section 3. Stochastically differential equations to govern power-law-type data are discussed in Section 4, which is followed by our conclusions.

#### 2. Brief on Power Laws

Denote by the probability space. Then, is said to be a stochastic process when the random variable represents the value of the outcome of an experiment for every time , where represents the sample space, is the event space or sigma algebra, and is the probability measure.

As usual, is simplified to be written as . That is, the event space is usually omitted. Denote by the probability function of . Then, one can define the general th order, time varying, joint distribution function for the random variables . The joint probability density function (pdf) is written by For simplicity, we write and . Then, the probability is given by The mean and the ACF of based on pdf are written by (2.3) and (2.4), respectively, Let be the variance of . Then,

The above expressions imply that the integrals in (2.3) and (2.5) are convergent in the domain of ordinary functions if is light tailed, for example, exponentially decayed (Li et al. [18]). Light-tailed pdfs are not our interests. We are interested in heavy-tailed pdfs. By heavy tail we mean that decays so slowly that (2.3) and (2.5) may be divergent. In the following subsections, we will describe power laws in probability space, ACF, and power spectrum density (PSD) function, respectively.

##### 2.1. Power Law in pdf

A typical heavy-tailed case is the Pareto distribution. Denote by the pdf of the Pareto distribution. Then, where and are parameters and . The mean and variance of that follows are given by (2.7) and (2.8), respectively, It is easily seen that and do not exist if . Note that implies a global property of while represents a local property of . Therefore, heavy-tailed pdfs imply that is in wild randomness due to infinite or very large variance, see the work of Mandelbrot in [19] for the meaning of wild randomness.

*Note 1. *The Pareto distribution is an instance of power-law-type pdf.

##### 2.2. Power Law in ACF

A consequence of a heavy-tailed random variable in ACF is that is slowly decayed. By slowly decayed we mean that decays hyperbolically in the power law given by (Adler et al. [20]) The Taqqu theorem describes the relationship between a heavy-tailed pdf and hyperbolically decayed ACF (Abry et al. [21]).

##### 2.3. Power Law in PSD

Denote by the PSD of . Then, According to the theory of generalized functions (Kanwal [22]), one has Therefore, power law in PSD, which is usually termed noise, see the work of Wornell in [23], the work of Keshner in [24], the work of Ninness in [25], the work of Corsini and Saletti in [26], and the work of Li in [27].

##### 2.4. Power Laws in Describing Scaling Phenomena

We now turn to scaling descriptions. Small scaling phenomenon may be investigated by for and large scaling for , respectively (Li and Zhao [28]).

On the one side, following Davies and Hall [29], if is sufficiently smooth on and if where is a constant and is the fractal index of , then the fractal dimension, denoted by , of is expressed by

*Note 2. *Fractal dimension is a parameter to characterize small scaling phenomenon (Mandelbrot [30], Gneiting and Schlather [31], Li [32]).

On the other side, if then the parameter is used to measure the statistical dependence of . If , is integrable, and accordingly is short-range dependent (SRD). If is nonintegrable and is long-range dependent (LRD), see the work of Beran in [33]. Representing by the Hurst parameter yields

*Note 3. *Statistical dependence, either SRD or LRD, is a property for large scaling phenomenon.

#### 3. Cases of Power Laws in CPNS

We address some application cases of power laws in CPNS in this section.

##### 3.1. Power Laws in the Internet

Let be the teletraffic time series. It may represent the packet size of teletraffic at time . Denote the ACF of by Then, we have (Li and Lim [34]) From (3.2), the fractal dimension and the Hurst parameter of teletraffic are, respectively, given by The above exhibits that both the small scaling and the large one follow power laws.

It is worth noting that the upper bounds of teletraffic also follow power laws. In fact, the amount of teletraffic accumulated in the interval is upper bounded by where and are constants and (Cruz [35]). Following Li and Zhao [28], we have the bounds of both the small-time scaling and the large one, respectively expressed by where is a small-scale factor and is a large-scale factor. Therefore, we have the following theorem.

Theorem 3.1. *Both the small-scale factor and the large one of teletraffic obey power law, that is, and .*

*Proof. *Two scaling factors follow and , respectively. Thus, they obey power laws. This completes the proof.

In addition to teletraffic, others with respect to the Internet also follow power laws. Some are listed below.

*Note 4. *Barabasi and Albert [36] studied several large databases in the World Wide Web (WWW), where they defined vertices by HyperText Markup Language (HTML) documents. They inferred that the probability that a vertex in the network interacts with other vertices decays hyperbolically as for , hence, power law.

*Note 5. *Let and be the probabilities of a document to have outgoing and incoming links, respectively. Then, and obey power laws (Albert [37]).

*Note 6. *The probability of web pages among sites is of power law (Huberman and Adamic [38]).

##### 3.2. Power Laws in Geosciences

Let be a spatial point. The physical meaning of a random function may be diverse in the field. For instance, it may represent prospected gold amount at in a gold mine, or a value of pollution index for pollution alert at in a city.

For simplicity, denote a vector by . Let Then, one may be interested in the covariance function of . Denote by the covariance function of . Then, One of the commonly used models of covariance functions in geosciences is given by The above constant power is the case of the standard Cauchy process (Webster and Oliver [39]). It fits with some cases in geosciences, see, for example, the work of Wackernagel in [40]. We list some in the following notes.

*Note 7. *Let be the covariance function between yield densities at any two points in a region, where represents the distance difference between two points. Then,
see the work of Whittle in [41].

*Note 8. *Sea-level fluctuations, river flow, and flood height follow power laws (Li et al. [42], Lefebvre [43], Lawrance and Kottegoda [44]).

*Note 9. *Urban growth obeys power laws (Makse et al. [45]).

##### 3.3. Power Laws in Wind Engineering

Wind engineering is an important field relating to wind power generation and disaster preventions from a view of CPNS. In this field, studying fluctuations of wind speed is essential.

The PSD introduced by von Kármán [46], known as the von Kármán spectra (VKS), is widely used in the diverse fields, ranging from turbulence to acoustic wave propagation in random media, see for example, the work of Goedecke et al. in [47] and the work of Hui et al. in [48]. For the VKS expressed in (3.10), we use the term VKSW for short,
where is frequency (Hz), is turbulence integral scale, is mean speed, is friction velocity (ms^{−1}), and is friction velocity coefficient such that the variance of wind speed . Equation (3.10) implies that VKSW obeys power law for .

Another famous PSD in wind engineering is the one introduced by Davenport [49], which is expressed by
where is the normalized frequency ( (10 m)), (10 m) is the mean wind speed (ms^{−1}) measured at height 10 m, is the mean wind speed (ms^{−1}) measured at height . Davenport’s PSD exhibits a power law of wind speed. Other forms of the PSDs of wind speed, such as those discussed by Kaimal [50], Antoniou et al. [51], and Hiriart et al. [52], all follow power laws, referring [50–52] for details.

#### 4. Possible Equations for Power-Law-Type Data

The cases of power laws mentioned in the previous section are a few that people may be interested in from a view of CPNS. There are others that are essential in the field of CPNS, such as power laws in earthquake, see for example, the work of Pisarenko and Rodkin in [53]. Now, we turn to the discussions about the generality about the equations that may govern data of power law type.

Conventionally, a stationary random function may be taken as a solution of a differential equation of integer order, which is driven by white noise . This equation may be written by where and are integers.

Let and be piecewise continuous on ( and integrable on any finite subinterval of . For , denote by the Riemann-Liouville integral operator of order [54–57]. Then, where is the Gamma function. For simplicity, we write by below.

Let be a strictly decreasing sequence of nonnegative numbers. Then, for the constants , we have The above is a stochastically fractional differential equation with constant coefficients of order . This class of equations yield random functions with power laws (Li [27]). In the case of random fields, (4.3) is extended to be a partial differential equation of fractional order given by where both and are multidimensional and is an operator of partial differentiation.

Another class of stochastically differential equations of fractional order is given by (Lim and Muniandy [58])

Note that (4.3), (4.4), or (4.5) should not be taken as a simple extension of conventional equation (4.1) from integer order to fractional one. As a matter of fact, there are challenging issues with respect to differential equations of fractional order. Since data of power-law-type may be with infinite variance (Samorodnitsky and Taqqu [59]), variance analysis which is a powerful tool in the analysis of conventional random functions fails to describe random data with infinite variance. Power-law type data may be LRD, which makes the stationarity test of data a tough issue, see for example, the work of Mandelbrot in [60], the work of Abry and Veitch in [61], the work of Li et al. in [62]. Owing to power laws, stability of systems that produce such a type of data becomes a critical issue in theory, see the work of Li et al. in [63] and the references therein. In addition, the prediction of data with power laws considerably differs from that of conventional data (M. Li and J. Y. Li [64], Hall and Yao [65]). Topics in power laws are paid attention to, see for example, the work of Kamoun in [66], the work of Ng et al. in [67], the work of Song et al. in [68], the work of Cattani et al. in [69–71], and in [72–79].

#### 5. Conclusions

We have discussed the elements of power laws from both a mathematical point of view and with respect to applications to a number of fields in CPNS. The purpose of this paper is to exhibit that power laws may yet serve as a universality of data in CPNS. We believe that this point of view may be useful for data modeling and analysis in CPNS.

#### Acknowledgments

This work was supported partly by the National Natural Science Foundation of China under the project Grant nos 60873264, 61070214, and the 973 plan under the Project no. 2011CB302801/2011CB302802.

#### References

- E. A. Lee, “Cyber physical systems: design challenges,” Tech. Rep. UCB/EECS-2008-8, University of California, Berkeley, Calif, USA, 2008. View at Google Scholar
- J. A. Stankovic, I. Lee, A. Mok, and R. Rajkumar, “Opportunities and obligations for physical computing systems,”
*Computer*, vol. 38, no. 11, pp. 23–31, 2005. View at Publisher · View at Google Scholar · View at Scopus - R. Alur, D. Thao, J. Esposito et al., “Hierarchical modeling and analysis of embedded system,”
*Proceedings of the IEEE*, vol. 91, no. 1, pp. 11–28, 2003. View at Publisher · View at Google Scholar · View at Scopus - J.-P. Chilès and P. Delfiner,
*Geostatistics, Modeling Spatial Uncertainty*, Wiley Series in Probability and Statistics: Applied Probability and Statistics, John Wiley & Sons, New York, NY, USA, 1999. View at Zentralblatt MATH - S. Uhlig, “On the complexity of Internet traffic dynamics on its topology,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 167–180, 2010. View at Publisher · View at Google Scholar · View at Scopus - A. R. Messina, P. Esquivel, and F. Lezama, “Time-dependent statistical analysis of wide-area time-synchronized data,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 751659, 17 pages, 2010. View at Publisher · View at Google Scholar - S. V. Muniandy and J. Stanslas, “Modelling of chromatin morphologies in breast cancer cells undergoing apoptosis using generalized Cauchy field,”
*Computerized Medical Imaging and Graphics*, vol. 32, no. 7, pp. 631–637, 2008. View at Publisher · View at Google Scholar · View at Scopus - M. S. Mason, D. F. Fletcher, and G. S. Wood, “Numerical simulation of idealised three-dimensional downburst wind fields,”
*Engineering Structures*, vol. 32, no. 11, pp. 3558–3570, 2010. View at Google Scholar - A. Spector and F. S. Grant, “Statistical methods for interpreting aeromagnetic data,”
*Geophysics*, vol. 35, no. 2, pp. 293–302, 1970. View at Google Scholar - C. Fortin, R. Kumaresan, W. Ohley, and S. Hoefer, “Fractal dimension in the analysis of medical images,”
*IEEE Engineering in Medicine and Biology Magazine*, vol. 11, no. 2, pp. 65–71, 1992. View at Publisher · View at Google Scholar · View at Scopus - D. Myrhaug, L. E. Holmedal, and M. C. Ong, “Nonlinear random wave-induced drag force on a vegetation field,”
*Coastal Engineering*, vol. 56, no. 3, pp. 371–376, 2009. View at Publisher · View at Google Scholar · View at Scopus - M. Tanaka, R. Kato, Y. Kimura, and A. Kayama, “Automated image processing and analysis of fracture surface patterns formed during creep crack growth in austenitic heat-resisting steels with different microstructures,”
*ISIJ International*, vol. 42, no. 12, pp. 1412–1418, 2002. View at Google Scholar · View at Scopus - G. Werner, “Fractals in the nervous system: conceptual implications for theoretical neuroscience,”
*Frontiers in Fractal Physiology*, vol. 1, article 15, 28 pages, 2010. View at Publisher · View at Google Scholar - B. J. West, “Fractal physiology and the fractional calculus: a perspective,”
*Frontiers in Fractal Physiology*, vol. 1, article 12, 2010. View at Publisher · View at Google Scholar - C. Cattani, “Fractals and hidden symmetries in DNA,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 507056, 31 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - B. Rosenow, P. Gopikrishnan, V. Plerou, and H. E. Stanley, “Dynamics of cross-correlations in the stock market,”
*Physica A*, vol. 324, no. 1-2, pp. 241–246, 2003. View at Publisher · View at Google Scholar · View at Scopus - R. J. Adler,
*The Geometry of Random Fields*, Wiley Series in Probability and Mathematical Statistic, John Wiley & Sons, Chichester, UK, 1981. - M. Li, W. Zhao, and S.-Y. Chen, “mBm-based scalings of traffic propagated in Internet,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 389803, 21 pages, 2011. View at Publisher · View at Google Scholar - B. B. Mandelbrot,
*Multifractals and 1/f Noise*, Springer, New York, NY, USA, 1998. - R. J. Adler, R. E. Feldman, and M. S. Taqqu, Eds.,
*A Practical Guide to Heavy Tails: Statistical Techniques and Applications*, Birkhäuser, Boston, Mass, USA, 1998. - P. Abry, P. Borgnat, F. Ricciato, A. Scherrer, and D. Veitch, “Revisiting an old friend: on the observability of the relation between long range dependence and heavy tail,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 147–165, 2010. View at Publisher · View at Google Scholar · View at Scopus - R. P. Kanwal,
*Generalized Functions: Theory and Applications*, Birkhäuser, Boston, Mass, USA, 3rd edition, 2004. - G. W. Wornell, “Wavelet-based representations for the 1/f family of fractal processes,”
*Proceedings of the IEEE*, vol. 81, no. 10, pp. 1428–1450, 1993. View at Publisher · View at Google Scholar · View at Scopus - M. S. Keshner, “1/
*f*noise,”*Proceedings of the IEEE*, vol. 70, no. 3, pp. 212–218, 1982. View at Google Scholar · View at Scopus - B. Ninness, “Estimation of 1/
*f*noise,”*IEEE Transactions on Information Theory*, vol. 44, no. 1, pp. 32–46, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. Corsini and R. Saletti, “$1/{f}^{\gamma}$ power spectrum noise sequence generator,”
*IEEE Transactions on Instrumentation and Measurement*, vol. 37, no. 4, pp. 615–619, 1988. View at Publisher · View at Google Scholar · View at Scopus - M. Li, “Fractal time series—a tutorial review,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 157264, 26 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - M. Li and W. Zhao, “Representation of a stochastic traffic bound,”
*IEEE Transactions on Parallel and Distributed Systems*, vol. 21, no. 9, pp. 1368–1372, 2010. View at Publisher · View at Google Scholar · View at Scopus - S. Davies and P. Hall, “Fractal analysis of surface roughness by using spatial data,”
*Journal of the Royal Statistical Society. Series B*, vol. 61, no. 1, pp. 3–37, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. B. Mandelbrot,
*The Fractal Geometry of Nature*, Schriftenreihe für den Referenten, W. H. Freeman and Co., San Francisco, Calif, USA, 1982. - T. Gneiting and M. Schlather, “Stochastic models that separate fractal dimension and the Hurst effect,”
*SIAM Review*, vol. 46, no. 2, pp. 269–282, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Li, “A class of negatively fractal dimensional Gaussian random functions,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 291028, 18 pages, 2011. View at Publisher · View at Google Scholar - J. Beran,
*Statistics for Long-Memory Processes*, vol. 61 of*Monographs on Statistics and Applied Probability*, Chapman and Hall, New York, NY, USA, 1994. - M. Li and S. C. Lim, “Modeling network traffic using generalized Cauchy process,”
*Physica A*, vol. 387, no. 11, pp. 2584–2594, 2008. View at Publisher · View at Google Scholar · View at Scopus - R. L. Cruz, “A calculus for network delay—part I: network elements in isolation—part II: network analysis,”
*IEEE Transactions on Information Theory*, vol. 37, no. 1, pp. 114–141, 1991. View at Google Scholar - A.-L. Barabási and R. Albert, “Emergence of scaling in random networks,”
*Science*, vol. 286, no. 5439, pp. 509–512, 1999. View at Publisher · View at Google Scholar - R. Albert, H. Jeong, and A. L. Barabási, “Internet: diameter of the world-wide web,”
*Nature*, vol. 401, no. 6749, pp. 130–131, 1999. View at Publisher · View at Google Scholar · View at Scopus - B. A. Huberman and L. A. Adamic, “Internet: growth dynamics of the world-wide web,”
*Nature*, vol. 401, no. 6749, p. 131, 1999. View at Google Scholar · View at Scopus - R. Webster and M. A. Oliver,
*Geostatistics for Environmental Scientists*, John Wiley & Sons, 2007. - H. Wackernagel,
*Multivariate Geostatistics: An Introduction with Applications*, Springer, 2005. - P. Whittle, “On the variation of yield variance with plot size,”
*Biometrika*, vol. 43, no. 3-4, pp. 337–343, 1962. View at Google Scholar · View at Zentralblatt MATH - M. Li, C. Cattani, and S.-Y. Chen, “Viewing sea level by a one-dimensional random function with long memory,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 654284, 13 pages, 2011. View at Publisher · View at Google Scholar - M. Lefebvre, “A one- and two-dimensional generalized Pareto model for a river flow,”
*Applied Mathematical Modelling*, vol. 30, no. 2, pp. 155–163, 2006. View at Publisher · View at Google Scholar · View at Scopus - A. J. Lawrance and N. T. Kottegoda, “Stochastic modelling of riverflow time series,”
*Journal of the Royal Statistical Society. Series A*, vol. 140, no. 1, pp. 1–47, 1977. View at Google Scholar · View at Scopus - H. A. Makse, S. Havlin, and H. E. Stanley, “Modelling urban growth patterns,”
*Nature*, vol. 377, no. 6550, pp. 608–612, 1995. View at Google Scholar · View at Scopus - T. von Kármán, “Progress in the statistical theory of turbulence,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 34, no. 11, pp. 530–539, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - G. H. Goedecke, V. E. Ostashev, D. K. Wilson, and H. J. Auvermann, “Quasi-wavelet model of von Kármán spectrum of turbulent velocity fluctuations,”
*Boundary-Layer Meteorology*, vol. 112, no. 1, pp. 33–56, 2004. View at Publisher · View at Google Scholar · View at Scopus - M. C. H. Hui, A. Larsen, and H. F. Xiang, “Wind turbulence characteristics study at the Stonecutters Bridge site—part II: wind power spectra, integral length scales and coherences,”
*Journal of Wind Engineering and Industrial Aerodynamics*, vol. 97, no. 1, pp. 48–59, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. G. Davenport, “The spectrum of horizontal gustiness near the ground in high winds,”
*Quarterly Journal of the Royal Meteorological Society*, vol. 87, no. 372, pp. 194–211, 1961. View at Google Scholar - J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Coté, “Spectral characteristics of surface-layer turbulence,”
*Quarterly Journal of the Royal Meteorological Society*, vol. 98, no. 417, pp. 563–589, 1972. View at Google Scholar - I. Antoniou, D. Asimakopoulos, A. Fragoulis, A. Kotronaros, D. P. Lalas, and I. Panourgias, “Turbulence measurements on top of a steep hill,”
*Journal of Wind Engineering and Industrial Aerodynamics*, vol. 39, no. 1–3, pp. 343–355, 1992. View at Google Scholar · View at Scopus - D. Hiriart, J. L. Ochoa, and B. García, “Wind power spectrum measured at the San Pedro Mártir Sierra,”
*Revista Mexicana de Astronomia y Astrofisica*, vol. 37, no. 2, pp. 213–220, 2001. View at Google Scholar · View at Scopus - V. Pisarenko and M. Rodkin,
*Heavy-Tailed Distributions in Disaster Analysis*, vol. 30, Springer, 2010. - C. A. Monje, Y.-Q. Chen, B. M. Vinagre, D. Xue, and V. Feliu,
*Fractional Order Systems and Controls—Fundamentals and Applications*, Springer, 2010. - M. D. Ortigueira, “An introduction to the fractional continuous-time linear systems: the 21st century systems,”
*IEEE Circuits and Systems Magazine*, vol. 8, no. 3, pp. 19–26, 2008. View at Publisher · View at Google Scholar · View at Scopus - Y. Q. Chen and K. L. Moore, “Discretization schemes for fractional-order differentiators and integrators,”
*IEEE Transactions on Circuits and Systems. I*, vol. 49, no. 3, pp. 363–367, 2002. View at Publisher · View at Google Scholar - B. M. Vinagre, Y. Q. Chen, and I. Petráš, “Two direct Tustin discretization methods for fractional-order differentiator/integrator,”
*Journal of the Franklin Institute*, vol. 340, no. 5, pp. 349–362, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. C. Lim and S. V. Muniandy, “Self-similar Gaussian processes for modeling anomalous diffusion,”
*Physical Review E*, vol. 66, no. 2, Article ID 021114, 14 pages, 2002. View at Publisher · View at Google Scholar · View at Scopus - G. Samorodnitsky and M. S. Taqqu,
*Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance*, Chapman & Hall, New York, NY, USA, 1994. - B. B. Mandelbrot, “Note on the definition and the stationarity of fractional Gaussian noise,”
*Journal of Hydrology*, vol. 30, no. 4, pp. 407–409, 1976. View at Google Scholar · View at Scopus - P. Abry and D. Veitch, “Wavelet analysis of long-range-dependent traffic,”
*IEEE Transactions on Information Theory*, vol. 44, no. 1, pp. 2–15, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Li, W. S. Chen, and L. Han, “Correlation matching method for the weak stationarity test of LRD traffic,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 181–195, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Li, S. C. Lim, and S. Y. Chen, “Exact solution of impulse response to a class of fractional oscillators and its stability,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 657839, 9 pages, 2011. View at Publisher · View at Google Scholar - M. Li and J.-Y. Li, “On the predictability of long-range dependent series,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 397454, 9 pages, 2010. View at Publisher · View at Google Scholar - P. Hall and Q. Yao, “Inference in ARCH and GARCH models with heavy-tailed errors,”
*Econometrica*, vol. 71, no. 1, pp. 285–317, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - F. Kamoun, “Performance analysis of a discrete-time queuing system with a correlated train arrival process,”
*Performance Evaluation*, vol. 63, no. 4-5, pp. 315–340, 2006. View at Publisher · View at Google Scholar · View at Scopus - J. K.-Y. Ng, S. Song, and W. Zhao, “Statistical delay analysis on an ATM switch with self-similar input traffic,”
*Information Processing Letters*, vol. 74, no. 3-4, pp. 163–173, 2000. View at Publisher · View at Google Scholar - Z. Song, Y.-Q. Chen, C. R. Sastry, and N. C. Tas,
*Optimal Observation for Cyber-Physical Systems*, Springer, 2009. - C. Cattani, “Harmonic wavelet approximation of random, fractal and high frequency signals,”
*Telecommunication Systems*, vol. 43, no. 3-4, pp. 207–217, 2010. View at Publisher · View at Google Scholar · View at Scopus - C. Cattani, “Fractals and hidden symmetries in DNA,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 507056, 31 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - G. Mattioli, M. Scalia, and C. Cattani, “Analysis of large-amplitude pulses in short time intervals: application to neuron interactions,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 895785, 15 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - S. Y. Chen, Y. F. Li, and J. Zhang, “Vision processing for realtime 3-D data acquisition based on coded structured light,”
*IEEE Transactions on Image Processing*, vol. 17, no. 2, pp. 167–176, 2008. View at Publisher · View at Google Scholar - S. Y. Chen and Y. F. Li, “Vision sensor planning for 3-D model acquisition,”
*IEEE Transactions on Systems, Man, and Cybernetics, Part B*, vol. 35, no. 5, pp. 894–904, 2005. View at Publisher · View at Google Scholar · View at Scopus - W. B. Mikhael and T. Yang, “A gradient-based optimum block adaptation ICA technique for interference suppression in highly dynamic communication channels,”
*EURASIP Journal on Applied Signal Processing*, vol. 2006, Article ID 84057, 2006. View at Publisher · View at Google Scholar · View at Scopus - E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due to coherence function and time series events,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 428903, 2010. View at Publisher · View at Google Scholar · View at Scopus - E. G. Bakhoum and C. Toma, “Mathematical transform of traveling-wave equations and phase aspects of quantum interaction,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 695208, 15 pages, 2010. View at Google Scholar · View at Zentralblatt MATH - Z. Liao, S. Hu, and W. Chen, “Determining neighborhoods of image pixels automatically for adaptive image denoising using nonlinear time series analysis,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 914564, 2010. View at Publisher · View at Google Scholar · View at Scopus - Z.-W. Liao, S.-X. Hu, D. Sun, and W. F. Chen, “Enclosed Laplacian operator of nonlinear anisotropic diffusion to preserve singularities and delete isolated points in image smoothing,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 749456, 15 pages, 2011. View at Publisher · View at Google Scholar - J.-W. Yang, Z.-R. Chen, W.-S. Chen, and Y.-J. Chen, “Robust affine invariant descriptors,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 185303, 15 pages, 2011. View at Publisher · View at Google Scholar