# Power-Type Functions of Prediction Error of Sea Level Time Series

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- We established the quantitative relationship between the past sample size, used for the predicting 40 points ahead of sea level time series as a case, and the prediction error characterized in the form of mean square at measurement stations of sea level time series we investigated.
- We obtained the analytic expression of that relationship in the form of power function, providing a guideline for the quality control in sea level prediction.

## 2. Research Background

#### 2.1. Data

#### 2.2. Problem Statement

_{s}, one may obtain the linear combination of the n past records expressed by

^{2}the mean square error (MSE) given by

^{2}(n, m) does not increase as n increase (Kolmogorov [72]), which yields a consequence we express as Corollary 1.

**Corollary 1.**For a given m, the following holds

^{2}(n, m). For a given time series x(i), however, it does not tell us, quantitatively, how large e

^{2}(n, m) is with the past sample size n and the x(m) to be predicted. For that reason, we consider the solution to the problem described below, aiming at establishing a quantitative relationship between the past sample size n and the given error provided that the step number of prediction m is predetermined.

**Problem 1.**Let x(i) be a sea level time series. For a fixed m and given error e

_{1}, what is the past sample size, denoted by N

_{1}, required such that

**Note 1.**The solution to the above problem is to find the function denoted by e

_{1}(N

_{1}, m). An alternative solution is N

_{1}(e

_{1}, m), which is in fact the inverse of e

_{1}(N

_{1}, m) with respect to N

_{1}. □

#### 2.3. Research Thoughts

- First, we find the e
_{1}(n, m) expressed only by an empirical curve based on processing real data of sea level. - Then, by fitting the data, we may find the analytic expression of e
_{1}(n, m) as well as e_{1}(N_{1}, m).

#### 2.4. BP-ANN Predictor

_{max}and x

_{min}are, respectively, the maximum and minimum values of the sample data x(t). The final forecasting results are obtained by the following inverse normalized formula:

## 3. Experimental Methods

## 4. Results

#### 4.1. Prediction Results

**Note 2.**FromFigure 18, as well as Figure 20, we see that MSE decreases rapidly when the past sample size increases from 100 to 300. Then, prediction errors appear to decay slowly when the past sample size increases. Table 3 summarizes the results.

#### 4.2. Prediction Error Curve Fitting

**Note 2.**The prediction error f(n) provides us with a guideline to control the prediction error in terms of a given value of past sample size.

**Note 3.**Our present result about f(n) may serve as a specific case for the quantitative description of prediction error for that f(n) → 0 for n → ∞, which was qualitatively described by Kolmogorov, see Corollary 1.

#### 4.3. Discussions

_{yy}(τ) = E[y(t)y(t + τ)]. By LRD (Beran [94]), we mean

_{yy}(τ), which satisfies the above, is a decayed power function expressed as

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Sampaio, L.E.B.; Rezende, A.L.T.; Nieckele, A.O. The challenging case of the turbulent flow around a thin plate wind deflector, and its numerical prediction by LES and RANS models. J. Wind Eng. Ind. Aerodyn.
**2014**, 133, 52–64. [Google Scholar] - Sanayei, M.; Moore, J.A.; Brett, C.R. Measurement and prediction of train-induced vibrations in a full-scale building. Eng. Struct.
**2014**, 77, 119–128. [Google Scholar] - Tsai, J.S.-H.; Hsu, W.-T.; Wei, C.-L.; Guo, S.-M.; Shieh, L.-S. Universal prediction-based adaptive fault estimator applied to secure communication. Appl. Math. Model.
**2014**, 38, 4717–4732. [Google Scholar] - Millán, M.M. Extreme hydrometeorological events and climate change predictions in Europe. J. Hydrol.
**2014**, 518, 206–224. [Google Scholar] - Schijve, J. The significance of fatigue crack initiation for predictions of the fatigue limit of specimens and structures. Int. J. Fatigue.
**2014**, 61, 39–45. [Google Scholar] - Zhao, X.; Dryer, M. Current status of CME/shock arrival time prediction. Space Weather.
**2014**, 12, 448–469. [Google Scholar] - Stockdon, H.; Thompson, D.; Plant, N.; Long, J. Evaluation of wave runup predictions from numerical and parametric models. Coasta. Eng.
**2014**, 92, 1–11. [Google Scholar] - Dimberg, P.H.; Bryhn, A.C. Predicted effects from abatement action against eutrophication in two small bays of the Baltic Sea. Environ. Earth Sci.
**2014**, 72, 1191–1199. [Google Scholar] - Hackert, E.; Busalacchi, A.J.; Ballabrera-Poy, J. Impact of Aquarius sea surface salinity observations on coupled forecasts for the tropical Indo-Pacific Ocean. J. Geophys. Res. Ocean.
**2014**, 119, 4045–4067. [Google Scholar] - Barik, A.K.; Dash, S.K.; Guha, A. New correlation for prediction of air entrainment into an Infrared Suppression (IRS) device. Appl. Ocean Res.
**2014**, 47, 303–312. [Google Scholar] - Luhar, M.; Sharma, A.; McKeon, B. On the structure and origin of pressure fluctuations in wall turbulence: Predictions based on the resolvent analysis. J. Fluid Mech.
**2014**, 751, 38–70. [Google Scholar] - Bagheripour, P.; Asoodeh, M. Poisson’s ratio prediction through dual stimulated fuzzy logic by ACE and GA-PS. J. Appl. Geophys.
**2014**, 107, 55–59. [Google Scholar] - Camus, P.; Méndez, F.J.; Losada, I.J.; Menéndez, M.; Espejo, A.; Pérez, J.; Rueda, A.; Guanche, Y. A method for finding the optimal predictor indices for local wave climate conditions. Ocean Dyn.
**2014**, 64, 1025–1038. [Google Scholar] - Hüffer, T.; Endo, S.; Metzelder, F.; Schroth, S.; Schmidt, T.C. Prediction of sorption of aromatic and aliphatic organic compounds by carbon nanotubes using poly-parameter linear free-energy relationships. Water Res.
**2014**, 59, 295–303. [Google Scholar] - Frousios, K.; Iliopoulos, C.S.; Schlitt, T.; Simpson, M.A. Predicting the functional consequences of non-synonymous DNA sequence variants—evaluation of bioinformatics tools and development of a consensus strategy. Genomics
**2013**, 102, 223–228. [Google Scholar] - Pike, A.; Danner, E.; Boughton, D.; Melton, F.; Nemani, R.; Rajagopalan, B.; Lindley, S. Forecasting river temperatures in real time using a stochastic dynamics approach. Water Resour. Res.
**2013**, 49, 5168–5182. [Google Scholar] - Duan, W.-Q.; Stanley, H.E. Cross-correlation and the predictability of financial return series. Physica A
**2011**, 390, 290–296. [Google Scholar] - Duan, W.-Q.; Stanley, H.E. Volatility, irregularity, and predictable degree of accumulative return series. Phys. Rev. E
**2010**, 81, 066116. [Google Scholar] - Kumar, P.; Franzese, G.; Stanley, H.E. Predictions of dynamic behavior under pressure for two scenarios to explain water anomalies. Phys. Rev. Lett.
**2008**, 100, 105701. [Google Scholar] - Starr, F.W.; Angell, C.A.; Stanley, H.E. Prediction of entropy and dynamic properties of water below the homogeneous nucleation temperature. Physica A
**2003**, 323, 51–66. [Google Scholar] - Niedzielski, T.; Miziński, B. Automated system for near-real time modelling and prediction of altimeter-derived sea level anomalies. Comput. Geosci.
**2013**, 58, 29–39. [Google Scholar] - Nitsure, S.; Londhe, S.; Khare, K. Prediction of sea water levels using wind information and soft computing techniques. Appl. Ocean Res.
**2014**, 47, 344–351. [Google Scholar] - Santoro, P.; Fernández, M.; Fossati, M.; Cazes, G.; Terra, R.; Piedra-Cueva, I. Pre-operational forecasting of sea level height for the Río de la Plata. Appl. Math. Model.
**2011**, 35, 2462–2478. [Google Scholar] - Gardner, A.S.; Moholdt, G.; Cogley, J.G.; Wouters, B.; Arendt, A.A.; Wahr, J.; Berthier, E.; Hock, R.; Pfeffer, W.T.; Kaser, G. A reconciled estimate of glacier contributions to sea level rise: 2003 to 2009. Science
**2013**, 340, 852–857. [Google Scholar] - Graham, S.; Barnett, J.; Fincher, R.; Hurlimann, A.; Mortreux, C.; Waters, E. The social values at risk from sea-level rise. Environ. Impact Assess. Rev.
**2013**, 41, 45–52. [Google Scholar] - Kamb, B. Glacier Geophysics Dynamic response of glaciers to changing climate may shed light on processes in the earth’s interior. Science
**1964**, 146, 353–365. [Google Scholar] - Kim, Y.C. Handbook of Coastal and Ocean Engineering; World Scientific: Singapore, Singapore, 2010. [Google Scholar]
- Massel, S.R. Ocean Surface Waves: Their Physics and Prediction; World Scientific: Singapore, Singapore, 1996. [Google Scholar]
- Mudersbach, C.; Wahl, T.; Haigh, I.D.; Jensen, J. Trends in high sea levels of German North Sea gauges compared to regional mean sea level changes. Cont. Shelf Res.
**2013**, 65, 111–120. [Google Scholar] - Oliver, E.; Thompson, K. Madden-Julian Oscillation and sea level: Local and remote forcing. J. Geophys. Res. Oceans.
**2010**, 115. [Google Scholar] [CrossRef] - Parker, A. Impacts of sea level rise on coastal planning in Norway. Ocean Eng.
**2014**, 78, 124–130. [Google Scholar] - Atal, B.S. The history of linear prediction. IEEE Signal Process. Mag.
**2006**, 23, 154–161. [Google Scholar] - Carotti, E.S.; De Martin, J.C.; Merletti, R.; Farina, D. Compression of multidimensional biomedical signals with spatial and temporal codebook-excited linear prediction. IEEE Trans. Biomed. Eng.
**2009**, 56, 2604–2610. [Google Scholar] - Kugiumtzis, D.; Lingjærde, O.; Christophersen, N. Regularized local linear prediction of chaotic time series. Physica D
**1998**, 112, 344–360. [Google Scholar] - Lawrence, M.; Goodwin, P.; O’Connor, M.; Önkal, D. Judgmental forecasting: A review of progress over the last 25years. Int. J. Forecast.
**2006**, 22, 493–518. [Google Scholar] - Lyman, R.J.; Edmonson, W.W. Linear prediction of bandlimited processes with flat spectral densities. IEEE Trans. Signal Process.
**2001**, 49, 1564–1569. [Google Scholar] - Man, K.S. Linear prediction of temporal aggregates under model misspecification. Int. J. Forecast.
**2004**, 20, 659–670. [Google Scholar] - Mugler, D.H. Computationally efficient linear prediction from past samples of a band-limited signal and its derivative. IEEE Trans. Inf. Theory.
**1990**, 36, 589–596. [Google Scholar] - Navarro-Moreno, J.; Moreno-Kaiser, J.; Fernández-Alcalá, R.M.; Ruiz-Molina, J.C. Widely linear prediction for transfer function models based on the infinite past. Comput. Stat. Data Anal.
**2013**, 58, 139–146. [Google Scholar] - Shin, H.; Hsing, T. Linear prediction in functional data analysis. Stoch. Process. Appl.
**2012**, 122, 3680–3700. [Google Scholar] - Tugnait, J.K.; Li, T.-T. A multistep linear prediction approach to blind asynchronous CDMA channel estimation and equalization. IEEE J. Select. Areas Commun.
**2001**, 19, 1090–1102. [Google Scholar] - Dhanya, C.; Nagesh Kumar, D. Nonlinear ensemble prediction of chaotic daily rainfall. Adv. Water Res.
**2010**, 33, 327–347. [Google Scholar] - Laio, F.; Porporato, A.; Revelli, R.; Ridolfi, L. A comparison of nonlinear flood forecasting methods. Water Resour. Res.
**2003**, 39. [Google Scholar] [CrossRef] - Sugihara, G. Nonlinear forecasting for the classification of natural time series. Phil. Trans. R. Soc. Lond. A
**1994**, 348, 477–495. [Google Scholar] - Yao, Q.; Tong, H. Quantifying the influence of initial values on non-linear prediction. J. R. Stat. Soc. B
**2009**, 56, 701–725. [Google Scholar] - Yilmaz, Y.; Kozat, S.S. Competitive randomized nonlinear prediction under additive noise. IEEE Trans. Signal Process. Lett.
**2010**, 17, 335–339. [Google Scholar] - Aladag, C.H.; Yolcu, U.; Egrioglu, E.; Bas, E. Fuzzy lagged variable selection in fuzzy time series with genetic algorithms. Appl. Soft Comput.
**2014**, 22, 465–473. [Google Scholar] - Egrioglu, E. PSO-based high order time invariant fuzzy time series method: Application to stock exchange data. Econ. Model.
**2014**, 38, 633–639. [Google Scholar] - Egrioglu, E.; Aladag, C.H.; Yolcu, U.; Uslu, V.R.; Basaran, M.A. Finding an optimal interval length in high order fuzzy time series. Expert Syst. Appl.
**2010**, 37, 5052–5055. [Google Scholar] - Egrioglu, E.; Yolcu, U.; Aladag, C.H.; Kocak, C. An ARMA type fuzzy time series forecasting method based on particle swarm optimization. Math. Probl. Eng.
**2013**, 2013. [Google Scholar] [CrossRef] - Khoshnevisan, B.; Rafiee, S.; Omid, M.; Mousazadeh, H. Prediction of potato yield based on energy inputs using multi-layer adaptive neuro-fuzzy inference system. Measurement
**2014**, 47, 521–530. [Google Scholar] - Grivas, G.; Chaloulakou, A. Artificial neural network models for prediction of PM 10 hourly concentrations, in the Greater Area of Athens, Greece. Atmos. Environ.
**2006**, 40, 1216–1229. [Google Scholar] - Nastos, P.; Paliatsos, A.; Koukouletsos, K.; Larissi, I.; Moustris, K. Artificial neural networks modeling for forecasting the maximum daily total precipitation at Athens, Greece. Atmos. Res.
**2014**, 144, 141–150. [Google Scholar] - Momeni, E.; Nazir, R.; Armaghani, D.J.; Maizir, H. Prediction of pile bearing capacity using a hybrid genetic algorithm-based ANN. Measurement
**2014**, 57, 122–131. [Google Scholar] - Nastos, P.; Moustris, K.; Larissi, I.; Paliatsos, A. Rain intensity forecast using artificial neural networks in Athens, Greece. Atmos. Res.
**2013**, 119, 153–160. [Google Scholar] - Venkata Rao, K.; Murthy, B.; Mohan Rao, N. Prediction of cutting tool wear, surface roughness and vibration of work piece in boring of AISI 316 steel with artificial neural network. Measurement
**2014**, 51, 63–70. [Google Scholar] - Zanuttigh, B.; Formentin, S.M.; Briganti, R. A neural network for the prediction of wave reflection from coastal and harbor structures. Coast. Eng.
**2013**, 80, 49–67. [Google Scholar] - Yin, J.-C.; Zou, Z.-J.; Xu, F. On-line prediction of ship roll motion during maneuvering using sequential learning RBF neuralnetworks. Ocean Eng.
**2013**, 61, 139–147. [Google Scholar] - Kömürcü, M.İ.; Kömür, M.A.; Akpınar, A.; Özölçer, İ.H.; Yüksek, Ö. Prediction of offshore bar-shape parameters resulted by cross-shore sediment transport using neural network. Appl. Ocean Res.
**2013**, 40, 74–82. [Google Scholar] - Karakış, R.; Tez, M.; Kılıç, Y.A.; Kuru, Y.; Güler, İ. A genetic algorithm model based on artificial neural network for prediction of the axillary lymph node status in breastcancer. Eng. Appl. Artif. Intell.
**2013**, 26, 945–950. [Google Scholar] - Singh, U.K.; Tiwari, R.; Singh, S. Neural network modeling and prediction of resistivity structures using VES Schlumberger data over a geothermal area. Comput. Geosci.
**2013**, 52, 246–257. [Google Scholar] - Yang, W.; Xia, X. Prediction of mining subsidence under thin bedrocks and thick unconsolidated layers based on field measurement and artificial neural networks. Comput. Geosci.
**2013**, 52, 199–203. [Google Scholar] - Ceryan, N.; Okkan, U.; Kesimal, A. Prediction of unconfined compressive strength of carbonate rocks using artificial neural networks. Environ. Earth Sci.
**2013**, 68, 807–819. [Google Scholar] - Erzin, Y.; Cetin, T. The prediction of the critical factor of safety of homogeneous finite slopes using neural networks and multiple regressions. Comput. Geosci.
**2013**, 51, 305–313. [Google Scholar] - Wei, Y.; Chen, M.-C. Forecasting the short-term metro passenger flow with empirical mode decomposition and neural networks. Transp. Res. C
**2012**, 21, 148–162. [Google Scholar] - Liao, D.; Wang, Q.; Zhou, Y.; Liao, X.; Huang, C. Long-term prediction of the Earth Orientation Parameters by the artificial neural network technique. J. Geodyn.
**2012**, 62, 87–92. [Google Scholar] - Bowden, G.J.; Maier, H.R.; Dandy, G.C. Real-time deployment of artificial neural network forecasting models: Understanding the range of applicability. Water Resour. Res.
**2012**, 48. [Google Scholar] [CrossRef] - Blanco, A.M.; Sotto, A.; Castellanos, A. Prediction of the amount of wood using neural networks. J. Math. Model. Algorithms.
**2012**, 11, 295–307. [Google Scholar] - Khashei, M.; Bijari, M. Hybridization of the probabilistic neural networks with feed-forward neural networks for forecasting. Eng. Appl. Artif. Intell.
**2012**, 25, 1277–1288. [Google Scholar] - Panella, M. Advances in biological time series prediction by neural networks. Biomed. Signal Process. Control
**2011**, 6, 112–120. [Google Scholar] - Gilhousen, D.B. A field evaluation of NDBC moored buoy winds. J. Atmos. Ocean. Technol.
**1987**, 4, 94–104. [Google Scholar] - Kolmogorov, A.N. On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition. Am. Math. Soc. Transl.
**1963**, 28, 55–59. [Google Scholar] - Hecht-Nielsen, R. Theory of the backpropagation neural network, Proceedings of International Joint Conference on Neural Networks, Washington, DC, USA, 18–22 June 1989; pp. 593–605.
- Zhang, Y.; Wu, L. Stock market prediction of S&P 500 via combination of improved BCO approach and BP neural network. Expert Syst. Appl.
**2009**, 36, 8849–8854. [Google Scholar] - Ciampi, A.; Zhang, F. A new approach to training back-propagation artificial neural networks: empirical evaluation on ten data sets from clinical studies. Stat. Med.
**2002**, 21, 1309–1330. [Google Scholar] - Guodong, L.; Jing, D. Discussion on problems of BP neural networks applied to hydrological prediction. J. Hydraul. Eng.
**1999**, 1, 66–71, In Chinese. [Google Scholar] - Guo, Z.-H.; Wu, J.; Lu, H.-Y.; Wang, J.-Z. A case study on a hybrid wind speed forecasting method using BP neural network. Knowl. -Based Syst.
**2011**, 24, 1048–1056. [Google Scholar] - Huang, S.-C.; Huang, Y.-F. Bounds on the number of hidden neurons in multilayer perceptrons. IEEE Trans. Neural Netw.
**1991**, 2, 47–55. [Google Scholar] - Sheela, K.G.; Deepa, S. Review on methods to fix number of hidden neurons in neural networks. Math. Probl. Eng.
**2013**, 2013. [Google Scholar] [CrossRef] - Hecht-Nielsen, R. Kolmogorov’s mapping neural network existence theorem. Proceedings of the First IEEE International Conference On Neural Networks, San Diego, CA, USA; 1987; pp. 11–14. [Google Scholar]
- He, Y.-G.; Tan, Y.-H.; Sun, Y. A neural network approach for fault diagnosis of large-scale analogue circuits. IEEE Int. Symp. Circuits Syst.
**2002**, 1, 153–156. [Google Scholar] - Liu, X.; Chen, X.; Wu, W.; Peng, G. A neural network for predicting moisture content of grain drying process using genetic algorithm. Food Control
**2007**, 18, 928–933. [Google Scholar] - Jwo, D.-J.; Chin, K.-P. Applying back-propagation neural networks to GDOP approximation. J. Navig.
**2002**, 55, 97–108. [Google Scholar] - Sola, J.; Sevilla, J. Importance of input data normalization for the application of neural networks to complex industrial problems. IEEE Trans. Nucl. Sci.
**1997**, 44, 1464–1468. [Google Scholar] - Demuth, H.B.; Beale, M.H. Neural Network Toolbox for Use with MATLAB: Computation, Visualization, Programming-User’s Guide; MathWorks: Natick, MA, USA, 2000. [Google Scholar]
- Chen, W.; Ma, R.X.; Hao, Y.H. BP artificial neural network design based on MATLAB. Comput. Study.
**2005**, 2, 30–31, In Chinese. [Google Scholar] - Han, C.; Lv, Y.; Yang, D.; Hao, Y. An intrusion detection system based on neural network, Proceedings of 2011 International Conference on Mechatronic Science, Electric Engineering and Computer, Jilin, China, 19–22 August 2011; pp. 2018–2021.
- Affandi, A.K.; Watanabe, K.; Tirtomihardjo, H. Application of an artificial neural network to estimate groundwater level fluctuation. J. Spat. Hydrol.
**2007**, 7, 23–46. [Google Scholar] - Keyhani, R.; Deriche, M.; Palmer, E. A high impedance fault detector using a neural network and subband decomposition, Proceedings of the Sixth International Symposium on Signal Processing and its Applications, Kuala Lumpur, Malaysia, 13–16 August 2001; pp. 458–461.
- Ercan, A.; Kavvas, M.L.; Abbasov, R.K. Long-Range Dependence and Sea Level Forecasting; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Barbosa, S.; Fernandes, M.; Silva, M. Long-range dependence in North Atlantic sea level. Physica A
**2006**, 371, 725–731. [Google Scholar] - Beretta, A.; Roman, H.E.; Raicich, F.; Crisciani, F. Long-time correlations of sea-level and local atmospheric pressure fluctuations at Trieste. Physica A
**2005**, 347, 695–703. [Google Scholar] - Li, M.; Cattani, C.; Chen, S.Y. Viewing sea level by a one-dimensional random function with long memory. Math. Probl. Eng.
**2011**, 2011. [Google Scholar] [CrossRef] - Beran, J. Statistical methods for data with long-range dependence. Stat. Sci.
**1992**, 7, 404–416. [Google Scholar] - Jiao, Z.; Chen, Y.-Q.; Podlubny, I. Distributed-Order Dynamic Systems; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Sheng, H.; Chen, Y.-Q.; Qiu, T.-S. Fractional Processes and Fractional Order Signal Processing; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Sun, H.G.; Chen, Y.-Q.; Chen, W. Random-order fractional differential equation models. Signal Process.
**2011**, 91, 525–530. [Google Scholar] - Muniandy, S.V.; Chew, W.X.; Wong, C.S. Fractional dynamics in the light scattering intensity fluctuation in dusty plasma. Phys. Plasmas.
**2011**, 18, 013701. [Google Scholar] - Asgari, H.; Muniandy, S.V.; Wong, C.S. Stochastic dynamics of charge fluctuations in dusty plasma: A non-Markovian approach. Phys. Plasmas.
**2011**, 18, 083709. [Google Scholar] - Eab, C.H.; Lim, S.C. Accelerating and retarding anomalous diffusion. J. Phys. A
**2012**, 45, 145001. [Google Scholar] - Yang, X.-J.; Srivastava, H.M.; He, J.-H.; Baleanu, D. Cantor-type cylindrical-coordinate method for differential equations with local fractional derivatives. Phys. Lett. A
**2013**, 377, 1696–1700. [Google Scholar] - Yang, X.-J.; Baleanu, D. Fractal heat conduction problem solved by local fractional variation iteration method. Therm. Sci.
**2013**, 17, 625–628. [Google Scholar] - Cattani, C.; Ciancio, A. Separable transition density in the hybrid model for tumor-immune system competition. Comput. Math. Methods Med.
**2012**, 2012, 610124. [Google Scholar] - Cattani, C.; Ciancio, A.; Lods, B. On a mathematical model of immune competition. Appl. Math. Lett.
**2006**, 19, 678–683. [Google Scholar] - Bakhoum, E.G.; Toma, C. Mathematical transform of traveling-wave equations and phase aspects of quantum interaction. Math. Probl. Eng.
**2010**, 2010. [Google Scholar] [CrossRef] - Toma, C. Advanced signal processing and command synthesis for memory-limited complex systems. Math. Probl. Eng.
**2012**, 2012. [Google Scholar] [CrossRef] - Beran, J.; Shernan, R.; Taqqu, M.S.; Willinger, W. Long-range dependence in variable bit-rate video traffic. IEEE Trans. Commun.
**1995**, 43, 1566–1579. [Google Scholar] - Gu, C.G.; Coomans, C.P.; Hu, K.; Scheer, F.A.J.L.; Stanley, H.E.; Meijer, J.H. Lack of exercise leads to significant and reversible loss of scale invariance in both aged and young mice. Proc. Natl. Acad. Sci. USA
**2015**, 112, 2320–2324. [Google Scholar] - Lévy Véhel, J. Beyond multifractional Brownian motion: New stochastic models for geophysical modeling. Nonlinear Process. Geophys.
**2013**, 20, 643–655. [Google Scholar] - Pinchas, M. Symbol error rate for non-blind adaptive equalizers applicable for the SIMO and FGn case. Math. Probl. Eng.
**2014**, 2014. [Google Scholar] [CrossRef] - Arzano, M.; Calcagni, G. Black-hole entropy and minimal diffusion. Phys. Rev. D
**2013**, 88, 084017. [Google Scholar] - Werner, G. Fractals in the nervous system: conceptual implications for theoretical neuroscience. Front. Physiol.
**2010**, 1. [Google Scholar] [CrossRef] - Zhao, S.X.; Ye, F.Y. Power-law link strength distribution in paper cocitation networks. J. Am. Soc. Inf. Sci. Technol.
**2013**, 64, 1480–1489. [Google Scholar] - Kaklauskas, L.; Sakalauskas, L. Study of on-line measurement of traffic self-similarity. Cent. Eur. J. Oper. Res.
**2013**, 21, 63–84. [Google Scholar] - Chen, C.-C.; Lee, Y.-T.; Hasumi, T.; Hsu, H.-L. Transition on the relationship between fractal dimension and Hurst exponent in the long-range connective sandpile models. Phys. Lett. A
**2011**, 375, 324–328. [Google Scholar] - Wang, X.J.; Shang, P.J.; Fang, J.T. Traffic time series analysis by using multiscale time irreversibility and entropy. Chaos
**2014**, 24, 032102. [Google Scholar] - Reyes, J.; Morales-Esteban, A.; Martinez-Alvarez, F. Neural networks to predict earthquakes in Chile. Appl. Soft Comput.
**2013**, 13, 1314–1328. [Google Scholar] - Lee, J.S.R.; Ye, S.K.; Jeong, H.D.J. ATMSim: An anomaly teletraffic detection measurement analysis simulator. Simul. Model. Pract. Theory.
**2014**, 49, 98–109. [Google Scholar] - Wang, D.M.; Li, Y.; Nie, P.F. A Study on the Gaussianity and Stationarity of the Random Noise in the Seismic Exploration. J. Appl. Geophys.
**2014**, 109, 210–217. [Google Scholar] - Ye, X.; Xia, X.; Zhang, J.; Chen, Y. Effects of trends and seasonalities on robustness of the Hurst parameter estimators. IET Signal Process.
**2012**, 6, 849–856. [Google Scholar] - Ghizdavet, Z.; Gradinaru, R. Heat balance computation on a clinkering plant over different time steps. Revista Romana De Materiale.
**2013**, 43, 332–338. [Google Scholar] - Uritsky, V.M.; Slavin, J.A.; Khazanov, G.V.; Donovan, E.F.; Boardsen, S.A.; Anderson, B.J.; Korth, H. Kinetic-scale magnetic turbulence and finite Larmor radius effects at Mercury. J. Geophys. Res. Space Phys.
**2011**, 116. [Google Scholar] [CrossRef] - Ghasemi, F.; van Ommen, J.R.; Sahimi, M. Analysis of pressure fluctuations in fluidized beds. I, Similarities with turbulent flow. Chem. Eng. Sci.
**2011**, 66, 2627–2636. [Google Scholar] - Schaefer, A.; Brach, J.S.; Perera, S.; Sejdic, E. A comparative analysis of spectral exponent estimation techniques for 1/f
^{β}processes with applications to the analysis of stride interval time series. J. Neurosci. Methods.**2014**, 222, 118–130. [Google Scholar] - Mandelbrot, B.B. Gaussian Self-Affinity and Fractals; Springer: Berlin, Germany, 2001. [Google Scholar]

**Figure 3.**Prediction results with the sample size 100 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 4.**Prediction results with the sample size 200 at the Station LKWF1in 1999. Solid line: predicted values, dashed line: original values.

**Figure 5.**Prediction results with the sample size 300 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 6.**Prediction results with the sample size 400 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 7.**Prediction results with the sample size 500 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 8.**Prediction results with the sample size 600 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 9.**Prediction results with the sample size 700 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 10.**Prediction results with the sample size 800 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 11.**Prediction results with the sample size 900 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 12.**Prediction results with the sample size 1000 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 13.**Prediction results with the sample size 1100 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 14.**Prediction results with the sample size 1200 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 15.**Prediction results with the sample size 1300 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 16.**Prediction results with the sample size 1400 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 17.**Prediction results with the sample size 1500 at the Station LKWF1 in 1999. Solid line: predicted values, dashed line: original values.

**Figure 18.**Relationship between the prediction error and the past sample size at the Station LKWF1 in 1999.

**Figure 20.**Relationship between the prediction error and the past sample size at the Station SMKF1 in 2003.

**Figure 21.**Relationship between the prediction error and the past sample size at the Station LONF1 in 2003.

**Figure 22.**Relationship between the prediction error and the past sample size at the Station SAUF1 in 2001.

**Figure 23.**Relationship between the prediction error and the past sample size at the Station SPGF1 in 1996.

**Figure 24.**Relationship between the prediction error and the past sample size at the Station VENF1 in 2003.

**Figure 25.**Curve fitting of the Station LKWF1 in 1999. Solid line: original curve, dashed line: fitting curve.

**Figure 26.**Curve fitting of the Station SMKF1 in 2003. Solid line: original curve, dashed line: fitting curve.

**Figure 27.**Curve fitting of the Station LONF1 in 2003. Solid line: original curve, dashed line: fitting curve.

**Figure 28.**Curve fitting of the Station SAUF1 in 2001. Solid line: original curve, dashed line: fitting curve.

**Figure 29.**Curve fitting of the Station SPGF1 in 1996. Solid line: original curve, dashed line: fitting curve.

**Figure 30.**Curve fitting of the Station VENF1 in 2003. Solid line: original curve, dashed line: fitting curve.

Series Name | Record Date and Time | Data Size | Measurement Station |
---|---|---|---|

x_lkwf1_1999(t) | 0:00, 1 Jan.–23:00, 31 Dec. 1999 | 8760 | LKWF1 |

x_smkf1_2003(t) | 0:00, 1 Jan.–23:00, 31 Dec. 2003 | 5851 | SMKF1 |

x_lonf1_2003(t) | 0:00, 1 Jan.–23:00, 31 Dec. 2003 | 8697 | LONF1 |

x_sauf1_2001(t) | 0:00, 1 Jan.–21:00, 31 Dec. 2001 | 8758 | SAUF1 |

x_spgf1_1996(t) | 0:00, 1 Jan.–23:00, 15 Dec. 1996 | 8616 | SPGF1 |

x_ven1_2003(t) | 0:00, 1 Jan.–23:00, 31 Dec. 2003 | 8760 | VENF1 |

Past sample size: n | Number of data to be predicted: m | Past samples used | Predicted data | MSE: e^{2}(n, m) |
---|---|---|---|---|

100 | 40 | 4901st–5000th/4:00, 26 July–7:00, 30 July | 8:00, 30 Jul.–23:00, 31 Jul. | 0.38664 |

200 | 40 | 4801st–5000th/23:00, 21 Jul.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.17422 |

300 | 40 | 4701st–5000th/19:00, 17 Jul.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.09026 |

400 | 40 | 4601st–5000th/15:00, 13 Jul.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.09519 |

500 | 40 | 4501st–5000th/10:00, 9 Jul.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.06699 |

600 | 40 | 4401st–5000th/5:00, 5 Jul.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.04418 |

700 | 40 | 4301st–5000th/1:00, 1 Jul.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.04385 |

800 | 40 | 4201st–5000th/21:00, 26 Jun.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.03291 |

900 | 40 | 4101st–5000th/17:00, 22 Jun.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.02826 |

1000 | 40 | 4001st–5000th/13:00, 18 Jun.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.02601 |

1100 | 40 | 3901st–5000th/9:00, 14 Jun.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.02185 |

1200 | 40 | 3801st–5000th/5:00, 10 Jun.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.02012 |

1300 | 40 | 3701st–5000th/20:00, 5 Jun.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.02028 |

1400 | 40 | 3601st–5000th/16:00, 1 Jun.–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.02313 |

1500 | 40 | 3501st–5000th/12:00, 28 May–7:00, 30 Jul. | 8:00, 30 Jul.–23:00, 31 Jul. | 0.01762 |

Past sample size: n | Number of data to be predicted: m | Past samples used | Predicted data | MSE: e^{2}(n, m) |
---|---|---|---|---|

100 | 40 | 4901st–5000th/13:00, 24 Jul.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.37867 |

200 | 40 | 4801st–5000th/9:00, 20 Jul.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.16937 |

300 | 40 | 4701st–5000th/5:00, 16 Jul.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.11727 |

400 | 40 | 4601st–5000th/0:00, 12 Jul.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.05214 |

500 | 40 | 4501st–5000th/20:00, 7 Jul.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02879 |

600 | 40 | 4401st–5000th/16:00, 3 Jul.–7:00, 30 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.03622 |

700 | 40 | 4301st–5000th/12:00, 29 Jun.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02651 |

800 | 40 | 4201st–5000th/8:00, 25 Jun.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02235 |

900 | 40 | 4101st–5000th/4:00, 21 Jun.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02442 |

1000 | 40 | 4001st–5000th/0:00, 17 Jun.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02077 |

1100 | 40 | 3901st–5000th/20:00, 12 Jun.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02426 |

1200 | 40 | 3801st– 5000th/16:00, 8 Jun.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02265 |

1300 | 40 | 3701st–5000th/11:00, 4 Jun.–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02037 |

1400 | 40 | 3601st–5000th/7:00, 31 May–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.02085 |

1500 | 40 | 3501st–5000th/3:00, 27 May–16:00, 28 Jul. | 17:00, 28 Jul.–8:00, 30 Jul. | 0.01981 |

Past sample size: n | Number of data to be predicted: m | Past samples used | Predicted data | MSE: e^{2}(n, m) |
---|---|---|---|---|

100 | 40 | 4901st–5000th/18:00, 28 Jul.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.15316 |

200 | 40 | 4801st–5000th/14:00, 24 Jul.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.13100 |

300 | 40 | 4701st–5000th/10:00, 20 Jul.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.12900 |

400 | 40 | 4601st–5000th/5:00, 16 Jul.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.11930 |

500 | 40 | 4501st–5000th/1:00, 12 Jul.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.10686 |

600 | 40 | 4401st–5000th/21:00, 7 Jul.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.10962 |

700 | 40 | 4301st–5000th/17:00, 3 Jul.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.10348 |

800 | 40 | 4201st–5000th/13:00, 29 Jun.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.10434 |

900 | 40 | 4101st–5000th/4:00, 21 Jun.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.10594 |

1000 | 40 | 4001st–5000th/9:00, 25 Jun.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.10692 |

1100 | 40 | 3901st–5000th/23:00, 16 Jun.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.10334 |

1200 | 40 | 3801st–5000th/19:00, 12 Jun.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.09628 |

1300 | 40 | 3701st–5000th/15:00, 8 Jun.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.09980 |

1400 | 40 | 3601st–5000th/11:00, 4 Jun.–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.10160 |

1500 | 40 | 3501st–5000th/16:00, 28 May–21:00, 1 Aug. | 22:00, 1 Aug.–13:00, 3 Aug. | 0.09752 |

Past sample size: n | Number of data to be predicted: m | Past samples used | Predicted data | MSE: e^{2}(n, m) |
---|---|---|---|---|

100 | 40 | 4901st–5000th/21:00, 28 Jul.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 3.80104 |

200 | 40 | 4801st–5000th/16:00, 24 Jul.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 2.56308 |

300 | 40 | 4701st–5000th/12:00, 20 Jul.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 1.45882 |

400 | 40 | 4601st–5000th/3:00, 16 Jul.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 1.28306 |

500 | 40 | 4501st–5000th/21:00, 11 Jul.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.96792 |

600 | 40 | 4401st–5000th/17:00, 7 Jul.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.95034 |

700 | 40 | 4301st–5000th/11:00, 3 Jul.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.89086 |

800 | 40 | 4201st–5000th/7:00, 29 Jun.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.63288 |

900 | 40 | 4101st–5000th/3:00, 25 Jun.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.61248 |

1000 | 40 | 4001st–5000th/21:00, 20 Jun.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.48420 |

1100 | 40 | 3901st–5000th/16:00, 16 Jun.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.41888 |

1200 | 40 | 3801st–5000th/12:00, 12 Jun.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.33480 |

1300 | 40 | 3701st–5000th/7:00, 8 Jun.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.33480 |

1400 | 40 | 3601st–5000th/3:00, 4 Jun.–8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.33488 |

1500 | 40 | 3501st – 5000th/23:00, 30 May – 8:00, 2 Aug. | 9:00, 2 Aug.–0:00, 4 Aug. | 0.32614 |

Past sample size: n | Number of data to be predicted: m | Past samples used | Predicted data | MSE: e^{2}(n, m) |
---|---|---|---|---|

100 | 40 | 4901st–5000th/3:00, 21 Jul.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.38766 |

200 | 40 | 4801st–5000th/23:00, 16 Jul.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.18390 |

300 | 40 | 4701st–5000th/14:00, 12 Jul.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.15660 |

400 | 40 | 4601st–5000th/8:00, 8 Jul.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.14834 |

500 | 40 | 4501st–5000th/2:00, 4 Jul.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.13988 |

600 | 40 | 4401st–5000th/21:00, 4 Jul.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.11374 |

700 | 40 | 4301st–5000th/14:00, 29 Jun.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.10464 |

800 | 40 | 4201st–5000th/10:00, 25 Jun.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.08100 |

900 | 40 | 4101st–5000th/6:00, 21 Jun.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.07846 |

1000 | 40 | 4001st–5000th/1:00, 17 Jun.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.07160 |

1100 | 40 | 3901st–5000th/20:00, 12 Jun.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.06674 |

1200 | 40 | 3801st–5000th/16:00, 8 Jun.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.06592 |

1300 | 40 | 3701st–5000th/11:00, 4 Jun.–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.06926 |

1400 | 40 | 3601st–5000th/7:00, 31 May–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.06532 |

1500 | 40 | 3501st–5000th/2:00, 27 May–7:00, 25 Jul. | 8:00, 25 Jul.–23:00, 26 Jul. | 0.06512 |

Past sample size: n | Number of data to be predicted: m | Past samples used | Predicted data | MSE: e^{2}(n, m) |
---|---|---|---|---|

100 | 40 | 4901st–5000th/20:00, 24 Jul.–23:00 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.44356 |

200 | 40 | 4801st–5000th/16:00, 20 Jul.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.12978 |

300 | 40 | 4701st–5000th/12:00, 16 Jul.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.08738 |

400 | 40 | 4601st–5000th/8:00, 12 Jul.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.04698 |

500 | 40 | 4501st–5000th/23:00, 7 Jul.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.03906 |

600 | 40 | 4401st–5000th/19:00, 3 Jul.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.03502 |

700 | 40 | 4301st–5000th/15:00, 29 Jun.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.02538 |

800 | 40 | 4201st–5000th/10:00, 25 Jun.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.02060 |

900 | 40 | 4101st–5000th/6:00, 21 Jun.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.01870 |

1000 | 40 | 4001st–5000th/2:00, 17 Jun.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.01878 |

1100 | 40 | 3901st–5000th/22:00, 12 Jun.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.01894 |

1200 | 40 | 3801st–5000th/18:00, 8 Jun.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.01672 |

1300 | 40 | 3701st–5000th/14:00, 4 Jun.–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.01594 |

1400 | 40 | 3601st–5000th/10:00, 31 May–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.01716 |

1500 | 40 | 3501st–5000th/5:00, 27 May–23:00, 28 Jul. | 0:00, 29 Jul.–15:00, 30 Jul. | 0.01888 |

**Table 8.**Curve fitting results of the six stations. f(n) = e

^{2}(n, m)|m = 40, n is the past sample size.

Station name | Power functions |
---|---|

LKWF1 | $f(n)=78.680{n}^{-1.1550}$ |

SMKF1 | $f(n)=121.600{n}^{-1.252}$ |

LONF1 | $f(n)=0.3194{n}^{-0.1645}$ |

SAUF1 | $f(n)=195.50{n}^{-0.8476}$ |

SPGF1 | $f(n)=9.0370{n}^{-0.6948}$ |

VENF1 | $f(n)=562.50{n}^{-1.5400}$ |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, M.; Li, Y.; Leng, J.
Power-Type Functions of Prediction Error of Sea Level Time Series. *Entropy* **2015**, *17*, 4809-4837.
https://doi.org/10.3390/e17074809

**AMA Style**

Li M, Li Y, Leng J.
Power-Type Functions of Prediction Error of Sea Level Time Series. *Entropy*. 2015; 17(7):4809-4837.
https://doi.org/10.3390/e17074809

**Chicago/Turabian Style**

Li, Ming, Yuanchun Li, and Jianxing Leng.
2015. "Power-Type Functions of Prediction Error of Sea Level Time Series" *Entropy* 17, no. 7: 4809-4837.
https://doi.org/10.3390/e17074809