The paper is devoted to the investigation of the weight function of the Kolmogorov-Wiener filter for forecasting of continuous fractional Gaussian noise with a Hurst exponent Н>0.5. We use a truncated polynomial expansion method for obtaining an approximate solution for the weight function under consideration. The method is based on the Chebyshev polynomials of the second kind. Approximations formed of different numbers of polynomials up to nineteen are investigated. The kernel of the Wiener-Hopf integral equation is a positively defined function in the case of a continuous fractional Gaussian noise, so the method is convergent. In a simple model telecommunication traffic may be described as a continuous fractional Gaussian noise, so the results of the paper may be applied to the telecommunication traffic forecasting.
Introduction and related works
The problem of telecommunication traffic forecast is an important problem for
telecommunications. For example, in [
The telecommunication traffic in systems with data burst transfer is considered to be a fractal process (see, for example, [3, 4]). We consider the traffic as a continuous random fractal process. Such a consideration is reasonable in the case of a large amount of data [5].
There are plenty of telecommunication traffic models, for example: a fractional
Gaussian noise model, a fractional Brownian motion model, wavelet models, etc.
(see, for example, [3, 4]). There are a lot of different approaches to the forecasting of
fractal telecommunication traffic: the ARIMA approach, the FARIMA approach,
approaches based on artificial neural networks, wavelet-based approaches, etc. [
One of the simplest telecommunication traffic models is the model where the telecommunication traffic is described [4, 7] as a fractional Gausian noise which is a stationary random process. The Kolmogorov–Wiener filter may be used for the forecasting of stationary processes [8]. This filter is a rather simple linear stationary filter. So, in our opinion, it is naturally enough to use the corresponding filter for the forecasting of telecommunication traffic in simple models where the traffic is considered to be a stationary random process.
However, we know few works where the Kolmogorov–Wiener filter is used for telecommunication traffic forecasting, and we don’t know any works where the corresponding filter is used for the forecasting of telecommunication traffic in the fractional Gaussian noise model.
The Kolmogorov–Wiener filter was proposed in [5] in order to make the traffic forecast in a model where the traffic is considered as a stationary random process with a power-law structure function. In [5] a simplified Volterra integral equation was proposed in order to obtain the filter weight function which is necessary for the calculation of the filter output. Our previous papers [8–10] were also devoted to the model proposed in [5].
However, the filter weight function should be obtained on the basis of the Fredholm integral equation rather than the Volterra one [8]. The papers [8–10] were devoted to the obtaining of the corresponding weight function based on the Fredholm integral equation of the first kind. A review of the methods of solving the corresponding integral equation is given in [11]. We investigated approximate solutions of the corresponding Fredholm integral equation with the help of a truncated polynomial expansion method (TPEM), which is a special case of the Galerkin method [11]. This method is rather simple and allows one to obtain analytical approximate solutions to the corresponding integral equation. The TPEM is rather popular in different fields of knowledge (for example, see its applications to statistical physics [12, 13]).
In paper [8] we used polynomials orthogonal without weight, and in papers [9, 10] we used the Chebyshev polynomials (CPs) of the second and first kind, respectively. The behavior of solutions is similar for different sets of polynomials, and the method is not necessarily convergent for processes with a power-law structure function. In our opinion, the reason is the following. The correlation function of the process under consideration, which is the kernel of the integral equation, is not a positively defined function. The convergence of a TPEM is guaranteed if the kernel of the corresponding integral equation is a positively defined function (see [14]).
In this paper we consider the TPEM in the model where the traffic is described as continuous fractional Gaussian noise with a Hurst exponent H > 0.5 . The correlation function of the corresponding random process is a positively defined function [15], so the TPEM should be convergent for the model of continuous fractional Gaussian noise. The exact analytical solution of the corresponding integral equation can hardly be obtained, so we use the TPEM which is based on the CPs of the second kind. The goal of the work is to obtain the Kolmogorov–Wiener filter weight function for the forecasting of continuous fractional Gaussian noise with the help of the TPEM and to illustrate its convergence.
The importance of the problem of the telecommunication traffic forecast is described in what precedes. This paper is devoted only to the description of the theoretical fundamentals of the development of the Kolmogorov–Wiener filter for traffic forecasting in the model where the traffic is described as a fractional Gaussian noise. The practical use of the obtained results may the subject of another paper. 2
The Wiener–Hopf integral equation and the truncated polynomial expansion method The Kolmogorov–Wiener filter weight function h(t) obeys the following Wiener– Hopf integral equation [8], which is a special case of the Fredholm integral equation of the first kind: (1) (2) (3) (4) where R(t) is the correlation function of the stationary random process for which the forecast is made and z T is the time interval for which the forecast is made. The data for the input signal are given for t ∈[0,T ] , the non-noisy case is investigated. The correlation function of continuous fractional Gaussian noise in the case H > 0.5 is as follows [15]: where σ2 is the process variance and Н is the Hurst exponent. The substitution of (2) into (1) leads to the following integral equation
T ∫ d τh (τ) R (t − τ) =R (t + z ) 0
R (t ) =2H (2H −1) σ2 t 2H −2 T ∫ d τh (τ) t − τ 2H −2 = (t + z )2H −2 .
0 A search for an exact analytical solution of the integral equation (3) meets difficulties, so we use the TPEM for obtaining an approximate solution.
In the framework of the TPEM the unknown function h(t) is sought as a truncated series in polynomials which are orthogonal on the time interval on which the integral on the left-hand side of (3) is taken. So a polynomial set which is orthogonal on the time interval t ∈[0,T ] is needed. The paper is based on the CPs of the second kind [16]
Un ( x)
[n 2] k =Cn2+k1+1 ∑ xn−2k ( x2 −1) , Cnk = k =0
n!
k !(n − k )!
where [n 2] is the integer part of n 2 . But the CPs of the second kind are orthogonal
on x ∈[
T∫0 Sn ( y ) Sm ( y ) w( y )dy =T4π δmn , δmn =10,,mm=≠nn where
Sn ( y ) =Un 2 y −1 , Sm ( y ) =Um 2 y −1 , w( y ) = 1− 2 y −12 .
T T T The polynomials Sn ( y) are orthogonal on y ∈[0,T ] with the weight w( y) and the unknown weight function h(t) may be sought as a truncated series where gn are the unknown coefficients multiplying the polynomials. Expression (7) is the expression for the Kolmogorov–Wiener weight function h(τ) in the l - polynomial approximation. The coefficients gn are found as follows. First of all we substitute (7) into (3): l−1 T ∑ gn ∫ d τSn (τ) t − τ 2H −2 = (t + z )2H −2 . n≥0 0 (8) h (τ)
l−1 =∑gn Sn (τ)
n≥0 the quantities Gnk are called the integral brackets. In paper [9] it is shown that the integral brackets obey the following properties:
Then we multiply the left-hand side and the right-hand side of (8) by Sk (t) , k =0, l −1 , after which both the left-hand side and the right-hand side are integrated over t on the time interval t ∈[0,T ] . As a result we obtain the following system of linear equations in the coefficients gn : l−1 ∑ gnGnk = Bk , k =0, l −1 n≥0 where the following denotation is used:
Gnk
T T =∫∫ dtd τSn (τ) Sk (t ) t − τ 2H −2 , Bk 0 0
T =∫dt (t + z )2H −2 Sk (t ) , 0 (5) (6) (7) (9) (10) Gnk = Gkn ; Gnk = 0 if n , k are of different parity. The properties (11) are obtained in [9] for the correlation function of a fractal process with a power-law structure function rather than for the correlation function (2). But the only property of the correlation function which was used in [9] for the derivation of (11) is the fact that the correlation function R(t) is an even one. So expressions (11) are valid for the problem under consideration. The first property in (11) takes place in general case, the second one takes place due to the choice of CPs of the second kind.
The properties (11) significantly reduce the computation time. The calculation of the integral brackets takes most of the computation time. With the help of (11) one can conclude that the number of the integral brackets for which a straightforward calculation is needed is equal to
[ l 2] 1 l l f1 (l ) =l + (l − 2) + (l − 4) + ... =∑k=0 (l − 2k ) =2 2 + 1 2l − 2 2 , and in the general case a straightforward calculation is needed for the number of the integral brackets equal to
f2 (l ) =l + (l −1) + (l − 2) + ... + 1 =l (l + 1) 2 .
The ratio of f2 (l) to f1 (l) for different values of l is given in Table 1. l 1 2 3 4 5
A straightforward calculation leads to the following results: (12) (13) (14) Gnk
The following numerical values for the coefficients multiplying polynomials in the l -polynomial approximation are obtained, see Table 2. The calculations are made with the help of the Wolfram Mathematica package. Approximations up to the 19polynomial one are investigated. The integrals (10) are calculated with the help of the standard function NIntegrate built in the package. The calculation of the corresponding integrals by expressions (15) is not faster than the calculation by the NIntegrate function; moreover, the calculation by expressions (15) becomes inadequate for the number of polynomials more than 10. (16) (17) (18) l
The values in Table 2 are rounded off to 3 significant digits. The left-hand side and the right-hand side of (3) are calculated in the Wolfram Mathematica as t T Left (t ) = ∫ d τh (τ)(t − τ)2H −2 + ∫ d τh (τ)(τ − t )2H −2 , Right (t ) = (t + z )2H −2 . (19) 0 t
The coincidence of the corresponding left-hand and right-hand sides is illustrated by the calculation of the mean absolute percentage error (MAPE)
MAPE =Left(t 1 T∫ dt ) − Right (t )
T 0 Right (t )
The results in Table 3 are rounded off to 2 significant digits. Table 3 illustrates that the approximations of small numbers of polynomials are not accurate, but the approximations of rather large numbers of polynomials are accurate. Fig. 1 shows graphs of the functions (19) for the 19-polynomial approximation. We investigate the Kolmogorov–Wiener filter for the forecasting of continuous fractional Gaussian noise. The subject of the investigation is the weight function of the corresponding filter. For simplicity, we restrict ourselves only to the case where the Hurst exponent H > 0.5 . The weight function for the filter under consideration obeys the integral equation (3). A search for an exact analytical solution of (3) meets difficulties, so we use the truncated polynomial expansion method (TPEM) in order to obtain an approximate solution for the unknown weight function. The paper is based on the Chebyshev polynomials (CPs) of the second kind orthogonal on the time interval t ∈[0,T ] , on which the filter input signal is given. The kernel of the integral equation (3) is a positively defined function, so the TPEM is convergent for the problem under consideration (see the corresponding discussion in [14]).
It is shown that the choice of the CPs of the second kind is convenient because the number of the integral brackets for which a straightforward calculation should be made is less than the corresponding number in the general case. It is shown that for the approximations of a rather large number of polynomials the number of brackets is approximately two times smaller than that in the general case.
Numerical approximate solutions are obtained for the parameters (18). The investigation is made up to the nineteen-polynomial approximation. The coefficients multiplying the polynomials and the mean absolute percentage error are calculated; the latter illustrates the accuracy of coincidence of the left-hand and the right-hand sides of the integral equation under consideration. The convergence of the method is illustrated, the accuracy of the approximations increases if the polynomial number increases. However, it should be stressed that one should use a rather large number of polynomials in order to obtain a rather accurate solution; approximations of small number of polynomials are not accurate. The kernel of the Wiener–Hopf integral equation is a positively defined function, so the proposed method should be convergent not only for the parameters (18), but also for other parameters.
The problem of traffic forecasting is an urgent problem for telecommunication
systems. This problem may be important both for the information security and for the
development of the intelligent systems. The description of the problem importance for
information security is given in [
Eastern-European Journal of Enterprise Technologies, Vol. 4, No. 7, 46–49 (2010), in Russian. 5. Bagmanov, V. Kh., Komissarov, A. M., Sultanov, A. Kh.: Teletraffic forecast on the basis of fractal fliters. Bulletin of Ufa State Aviation Technical University, Vol. 9, No. 6 (24), 217–222 (2007), in Russian. 6. Iqbal, M. F., Zahid, M., Habib, D., John, L. K.: Efficient Prediction of Network Traffic for Real-Time Applications. Journal of Computer Networks and Communications, Vol. 2019, Article ID 4067135, 11 pages (2019). doi: 10.1155/2019/4067135 7. Li, M.: Fractional Gaussian Noise and Network Traffic Modeling. Proceedings of the 8th WSEAS International Conference on Applied Computer and Applied Computational Science, 34–39 (2009). 8. Gorev, V. N., Gusev, A. Yu., Korniienko, V. I.: Polynomial solutions for the Kolmogorov–Wiener filter weight function for fractal processes. Radio Electronics, Computer Science, Control, No.2, 44–52 (2019). doi: 10.15588/1607-3274-2019-2-5 9. Gorev, V., Gusev, A., Korniienko, V.: Investigation of the Kolmogorov–Wiener filter for treatment of fractal processes on the basis of the Chebyshev polynomials of the second kind. Ceur Workshop Proceedings, Vol. 2353, 596–606 (2019). 10. Gorev, V., Gusev, A., Korniienko, V.: Investigation of the Kolmogorov-Wiener filter for continous fractal processes on the basis of the Chebyshev polynomials of the first kind.
IAPGOS, No. 1, 58-61 (2020). doi: 10.35784/iapgos.912 11. Polyanin, A.D., Manzhirov, A.V.: Handbook of integral equations. Second edition. Boca
Raton, Chapman & Hall/CRC Press. Taylor & Francis Group (2008). 12. Sokolovsky, S. A., Sokolovsky, A. I., Kravchuk, I. S., Grinishin, O. A.: Relaxation processes in completely ionized plasma in generalized Lorentz model. Journal of Physics and Electronics, Vol. 26(2), 17–28 (2018). doi: 10.15421/331818 13. Sokolovsky, S., Sokolovsky, A.: Mobility of Electrons in Plasma. Proceedings of 2019 IEEE 2nd Ukraine Conference on Electrical and Computer Engineering (UKRCON), Lviv, Ukraine, 783–787 (2019). doi: 10.1109/UKRCON.2019.8880003 14. Ziman, J. M.: Electrons and Phonons. The Theory of Transport Phenomena in Solids. Oxford University Press (2001). doi: 10.1093/acprof:oso/9780198507796.001.0001 15. Quian, Н.: Fractional Brownian Motion and Fractional Gaussian Noise. In: Rangarajan, G., Ding, M. (eds.) Processes with Long-Range Correlations. Theory and Applications. LNP, vol. 621, pp. 22–33, Springer, Heidelberg (2003). doi: 10.1007/3-540-44832-2. 16. Gradshteyn, I. S., Ryzhik, I. M., Geronimus, Yu. V., Tseytlin, M. Yu., Alan, J.: Table of Integrals, Series, and Products. Eights edition. Edited by D. Zwillinger and V. Moll, Amsterdam, Elsevier/Academic Press (2014).