Radio Science

A practical MGA-ARIMA model for forecasting real-time dynamic rain-induced attenuation

Authors


Corresponding author: S. Gong, School of Science, Xi'dian University, Xi'an 710071, China. (ljbrp2003@yahoo.com.cn)

Abstract

[1] The novel and practical modified genetic algorithm (MGA)-autoregressive integrated moving average (ARIMA) model for forecasting real-time dynamic rain-induced attenuation has been established by combining genetic algorithm ideas with the ARIMA model. It is proved that due to the introduction of MGA into the ARIMA(1,1,7) model, the MGA-ARIMA model has the potential to be conveniently applied in every country or area by creating a parameter database used by the ARIMA(1,1,7) model. The parameter database is given in this paper based on attenuation data measured in Xi'an, China. The methods to create the parameter databases in other countries or areas are offered, too. Based on the experimental results, the MGA-ARIMA model has been proved practical for forecasting dynamic rain-induced attenuation in real time. The novel model given in this paper is significant for developing adaptive fade mitigation technologies at millimeter wave bands.

1 Introduction

[2] Developing and applying millimeter wave (MMW)-band wireless systems, for example, multiple input multiple output communication technologies, are hot research areas in future wireless system investigation, because of the many advantages of the MMW system compared to lower frequencies, such as higher capacity, narrower beam, smaller size of terminal, stronger antijamming capability, better electromagnetic compatibility, more easily miniaturizing installation, smaller size of antenna, etc. [Gong, 2008; Xiong, 2000; Xie, 1990; Xu, 2003]. One important aspect of designing the wireless system is the investigation of radio wave propagation characteristics (such as refraction, reflection, scattering, attenuation, phase shifting, depolarization, scintillation, duct, and additional noise), when radio wave propagates through various environments [Ishimaru, 1977; Gong, 2008; Sizun, 2003; Xiong, 2000; Xie, 1990; Xu, 2003; Verma et al., 1989]. The propagation effects, which are more serious at MMW frequencies, directly influence system performance. Those effects restrict the further development of MMW technologies [Ishimaru, 1977; Gong, 2008; Sizun, 2003; Marie Edith Gimonet et al., 2002; Xiong, 2000; Xie, 1990; Xu, 2003; Verma et al., 1989].

[3] One principal technical difficulty for MMW technology is to evaluate and mitigate the attenuation induced by troposphere propagation environments, like rain, cloud, snow, fog, sand, and storm. Among those, rain-induced attenuation is the most severe attenuation [Ishimaru, 1977; Gong, 2008; Sizun, 2003; Xiong, 2000; Xie, 1990; Xu, 2003; Verma et al., 1989]. Rain-induced attenuation can be dozens, even hundreds of decibels. Usually, the attenuation will be more severe at higher frequency. For example, the rainfall, which can induce an inconspicuous effect for C bands, may cause interruption on systems working at Ku and Ka bands [Xiong, 2000; Xie, 1990; Zhao, 2001].

[4] Power reserve technology is one of the most important approaches adopted to mitigate rain-induced fade, which mitigates deep fade by a fixed power margin. A fixed power margin is needed for power reserve technology to mitigate rain-induced fade. The fixed power margin is decided by the results of long-term statistical characteristics of rain-induced attenuation. Many models, which can obtain long-term statistical characteristics, have been proposed: Improved Assis-Einloft Model [Costa, 1983], Australian Model [Flavin, 1996], Brazil Model [Pontes, 1992], Bryant Model [Bryant et al., 2001], Crane Global Model [Crane, 1978, 1980], Crane Two Components Model [Crane, 1982, 1996], EXCELL Model [Capsoni et al., 1987a, 1987b], Garcia Model [García-López et al., 1988; García and Peirò, 1983], ITU-R Model [ITU-R P.618-10, 2009], Karasawa Model [Karasawa and Matsudo, 1990; Karasawa and Matsudo, 1991], Leitao-Watson Model [Leitão and Watson, 1986], Matricciani Model [Matricciani, 1991, 1993], Misme-Waldteufel Model [Misme and Waldteufel, 1980], SAM Model [Stutzman and Dishman, 1982, 1984], Svjatogor Model [Svjatogor, 1985] (all these models mentioned above are also summarized and discussed by Ioan Chisalita et al. [2002]), DAH Model [Dissanayake et al., 1997], Manning Model [Manning, 1986] (the DAH model and Manning model are also summarized and discussed by Ippolito [1999]), UK Model [Doc. 3 M/134, 2005; Doc. 3 M/28, 2003], Japan Model [Karasawa, 1989], and China Model [Zhao, 2001].

[5] While applying power reserve technology, a very large power margin is needed at any time because of the high-level attenuation at higher MMW frequencies. However, the high-level attenuation is a sparse case, because rainfall, especially heavy rain, is a relatively sparse event in time and space. Therefore, one obvious drawback of power reserve technology is the waste of power. For example, the rain rate in Beijing, China, for 0.01% of an average year is about 42 mm/h. The predicted attenuation for 0.01% of an average year using the International Telecommunication Union-Radio (ITU-R) model [ITU-R P.618-10, 2009] is approximately 50 dB, under the following link conditions: 35 GHz frequency, circular polarization state, and 35° link elevation. The fixed power margin of 50 dB must be reserved whenever to ensure the communication reliability of 99.99%, but the fixed power margin is only needed for a small part of an average year. Moreover, the large power can interfere with other wireless systems; it can increase the loading of power-supply system of spacecraft, for example satellites.

[6] Space diversity technology is another main approach to mitigate rain-induced fade. It is based on the spatially uncorrelated character of rain-induced attenuation affecting different links whose Earth stations are located in different regions. While applying space diversity technology, it is necessary to build two or more Earth stations in different regions, which surely increase the investment of human and financial resources. As mentioned, rainfall, especially heavy rain, seldom occurs, so the resources will be wasted for most of an average year.

[7] The fixed power margin for power reserve technology and the investment of human and financial resources for space diversity technology can be seen as a waste for most of an average year. Thus, adaptive fade mitigation technologies, such as adaptive power control (APC) technology, are proposed [de Montera1 et al., 2008]. APC technology is different from power reserve technology. APC is able to increase transmitted power to a specific level, only when the specific rain-induced attenuation is predicted. Therefore, real-time dynamic characteristics of rain-induced attenuation are the urgent needs for APC to predict and trace the real-time changing of rain-induced attenuation [Burgueno et al., 1990; de Montera1 et al., 2008; Willis et al., 2006].

[8] In recent years, some models and methods for forecasting or investigating dynamic rain-induced attenuation have been proposed, including a linear regression model [Dossi, 1990; de Montera1 et al., 2008], a first-order stochastic equation [Manning, 1990, 1991], a Markov-chain method [Castanet et al., 2003; Fiebig, 2002; van de Kamp, 2003], an adaptive linear filter [Grémont et al., 1999], a neural network [Chambers and Otung, 2005; Mallet et al., 2006], a model based on the fade slope [Van de Kamp, 2002], a switching ARIMA process with generalized autoregressive conditional heteroscedasticity (ARCH) errors [de Montera1 et al., 2008], the first- and second-order statistics properties of rain attenuation time series [Burgueno et al., 1990; Baldotra and Hudiara, 2004; Matricciani, 1994; de Montera1 et al., 2008; Willis et al., 2006; van de Kamp, 2003]. In previous publications, the approaches, which are based on data measured in the specific countries or areas, are restricted by their invariable parameters. Some of them merely offer the statistical characteristics of specific measured attenuation time series, or just an idea on how to obtain the statistical characteristics of attenuation time series. Admittedly, those publications indeed prompt related studies on forecasting dynamic rain-induced attenuation.

[9] It is well known that rain-induced attenuation is dependent not only on the determinate factors like link elevation angle, operating frequency, etc., but also on some random factors like rain rate, rainfall type, the shape and size distributions of the raindrops, the space distribution of rainfall rate, etc. Therefore, the result seems not satisfactory if one attempts to self-forecast by applying the statistical characteristics and invariable parameters obtained with a certain measured attenuation time series. Therefore, it is not possible to accurately obtain the change of attenuation at different times during rainfall events in other countries or areas, while applying these statistical characteristics and invariable parameters. The predicted results are not perfect using the ARIMA model with constant parameters, even in the self-forecasting case, an example of this case will be shown in section 2 of this paper. Therefore, the models and methods proposed in previous publications cannot predict accurately enough the next-time attenuation during an on-going rainfall event for an APC system to adjust transmitted power in real time, at the aim of mitigating rain-induced fade. Thus, a practical model or method, adaptive to particular rainstorms, for forecasting real-time dynamic rain-induced attenuation is urgent. It should be able to predict accurately the next-time attenuation based on current or past attenuation during an on-going rainfall event.

[10] In this paper, the novel and practical modified genetic algorithm (MGA)-autoregressive integrated moving average (ARIMA) model for forecasting real-time dynamic rain-induced attenuation has been established by introducing genetic algorithm (GA) ideas into the ARIMA model, with the data measured in Xi'an, China. The model is able to avoid the aforementioned problems, although the data measured in Xi'an, China is used to establish the model. In fact, while applying this model to forecast the attenuation during an on-going rainfall event, the parameters used by the ARIMA model are completely renewed and optimized in real time. The MGA calculates these parameters based on a parameter database, which will be discussed in section 3. It can be found in section 3 that a parameter database with m groups of parameters can give 2m groups of new parameters, and with the process of duplication, overlap, and mutation, new parameters will be generated. It has been proved that this model is adaptive and intelligent, and has the potential possibility to be employed in engineering for any country or area, with MGA generating the optimized parameters of the ARIMA(1,1,7) model.

[11] This paper has given the parameter database of the ARIMA(1,1,7) model based on the data measured in Xi'an, China. However, the proposed model shall be more applicable if more parameters based on attenuation data measured in diverse regions are included in the parameter database. Thus, the methods to establish the parameter databases in other countries or areas are recommended in section 5. The methods are feasible and convenient to obtain the databases in other countries or areas.

2 The Autoregressive Integrated Moving Average Model

[12] The Auto Regressive Moving Average (ARMA) model is widely applied in forecasting random phenomena of physical science, economics etc. Moreover, the ARCH model, spatial ARCH model, and generalized ARCH model are variants of the ARMA model. Besides, the ARMA model is very suitable for predicting a stationary time series [Lan, 1997; Xiang et al., 1986; Yi, 2002].

[13] Suppose that At, t = 1, 2, 3 … N is a group of stationary time series, the ARMA(p, q) model for At is written as (1), which is elaborated by Lan [1997], Xiang et al. [1986], and Yi [2002].

display math(1)

where, φi and θi are parameters of the auto regression model and moving average model, respectively, εt is a Gaussian white-noise series, p and q are orders of the ARIMA model.

[14] The ARIMA model is used to forecast nonstationary time series, optimized from the ARMA model. The improvement is based on an extra operation of the kth order and rth time difference on a nonstationary time series. If the time series At, t = 1, 2, 3 … N is nonstationary, the ARIMA model for At is presented as (2), which is elaborated by Lan [1997], Xiang et al. [1986], and Yi [2002].

display math(2)

where, differential factor inline image, which represents the operation of the kth order and rth time difference on the time series At, t = 1, 2, 3 … N, can eliminate the nonstationarity. If Xt is a time series, inline image returns a new series written as

display math(3)

[15] Then inline image is written as

display math(4)

[16] Usually, k is taken as 1, which means Δr = Δr1, and the model as (2) is presented as the ARIMA(p ,r, q) model [see Lan, 1997; Xiang et al., 1986; Yi, 2002]

display math(5)

[17] Figure 1 gives a flow chart of establishing the ARIMA model. As shown in Figure 1, to establish the ARIMA(p, r, q) model, it is necessary to go with the following three steps: analyzing and giving supposed model, estimating the model parameters, diagnosing and testing the model [Lan, 1997; Xiang et al., 1986; Yi, 2002].

Figure 1.

The flow chart showing the process of establishing the ARIMA(p, r, q) model.

[18] The first step includes estimating the stationarity of a time series, deciding the needed difference level, and finally deciding p and q. The second step, which serves to estimate the model parameters, is to calculate φi and θi. Finally, it is necessary to evaluate the forecasting precision, to diagnose and test the model.

[19] According to the three steps mentioned in the previous two paragraphs, it is necessary to estimate the stationarity first. In this paper, auto correlation function (ACF) is used to estimate stationarity. Figure 2 is ACF schematic of nonstationary and stationary time series. As illustrated in Figure 2, to meet the stationarity condition of a group of time series, their ACF must decrease to zero quickly, which is elaborated by Lan [1997], Xiang et al. [1986], and Yi [2002]. One can also distinguish stationarity by testing the roots of the model equation, which is elaborated by Lan [1997], Xiang et al. [1986], and Yi [2002].

Figure 2.

The ACF schematic of nonstationary and stationary time series.(A) Nonstationary case and (B) stationary case

[20] The ACF of a lag time s during a random process x can be presented as (6) [Lan, 1997]

display math(6)

where, Cov(xt,xt + s) denotes the covariance of a lag time s of x, D(xt) and D(xt + s) show the variance of x at time t and at time t + s, respectively. Equation ((6)) is a theoretical equation of ACF, which cannot be calculated merely with measured time series. Thus, to calculate ACF with measured time series, one needs to apply the following equation [Lan, 1997]:

display math(7)

where, c(i) is presented as (8), if At, t = 1, 2, 3 … N is stationary [Lan, 1997]

display math(8)

where, inline image is presented as (9) [Lan, 1997]

display math(9)

[21] Moreover, ACF can be calculated by the software Statistical Product and Service Solutions (SPSS) [Hao et al., 2003].

[22] Figures 3-5 give an example of estimating stationarity. Figure 3 gives the rain rate data measured on 5 July 2011 in Xi'an, China. Figure 4 is the simulation attenuation time series using the data in Figure 3, the simulation method will be introduced in section 5 of this paper. Figure 5 is the ACF of the data in Figure 4. Figure 5 shows that the time series in Figure 4 is not stationary, because the ACF does not quickly decrease to zero.

Figure 3.

The rain rate data measured on 5 July 2011, in Xi'an, China.

Figure 4.

The simulated attenuation time series using the data in Figure 3.

Figure 5.

The ACF of the simulated attenuation time series in Figure 4.

[23] Furthermore, only if a group of time series is stationary, can the process of diagnosing and testing the supposed model be continued. Otherwise, the difference operation on the time series must be performed, until the time series is stationary. It can be concluded that the difference operation on the data in Figure 4 is necessary. Figure 6 is the differential series of the data in Figure 4 under the condition of k = r = 1, and Figure 7 gives the ACF of the data in Figure 6. Figure 7 shows that the differential series in Figure 6 is stationary, because the ACF quickly decreases to zero within a few samples. Figure 7 implies that the difference operation with k = r = 1 on the data in Figure 6 is enough to satisfy the stationarity condition. Below, it will be shown how the speed at which the ACF decreases to zero determines the model.

Figure 6.

The differential series of the data in Figure 4 under the condition of k = r = 1.

Figure 7.

The ACF of the differential series in Figure 6.

[24] After establishing stationarity of a group time series and deciding the difference level, it is the turn to decide p and q in equations ((2)) or ((5)). Deciding the values of p and q depends on the properties of ACF and partial auto correlation function (PACF) of a time series.

[25] The PACF of a lag time s means the correlation between xt and xt + s under the condition of disregarding the effects induced by xt + 1, xt + 2xt + s − 1, or means the conditional correlation between xt and xt + s with xt + 1, xt + 2xt + s − 1 being given. The PACF of a lag time s during a random process x can be presented as (10) [Lan, 1997].

display math(10)

[26] Equation ((10)) is a theoretical equation of PACF. Also, PACF(s) = φss is proved by Lan [1997], where, φss matches equation ((11)) [Lan, 1997]

display math(11)

ρ(j) in equation ((11)) can be calculated by equation (7), and ρ(j) = ρ(−j) for all j, which is the property of an autocorrelation function. If the system of s equations of equation ((11)) is solved using measured time series, successively for s = 1, s = 2, etc., this gives φss for every value of s. Then, PACF(s) is given by

display math(12)

[27] Also, PACF can be calculated by the software SPSS [Hao et al., 2003]. Figures 7 and 8 are the ACF and the PACF of the data in Figure 6. Figures 5, 7, and 8 are the results calculated by SPSS.16 [Song, 2008], “16” is a version number of this software.

Figure 8.

The PACF of the differential series in Figure 6.

[28] If a new time series inline image or ΔrAt is stationary, p and q in (2) or (5) can be estimated through ACF and PACF of inline image or ΔrAt with the following principle: if ACF(s) and PACF(s) are cut off at lag number q1 and q2 respectively, then one can get q = q1, p = q1 − q2, which is elaborated by Lan [1997], Xiang et al. [1986], and Yi [2002]. Here “cut off” means that all values of ACF(s) and PACF(s) are between the “Upper confidence limit” and the “Lower confidence limit” after a lag number, for example in Figure 7 and Figure 8. The “Upper confidence limit” and the “Lower confidence limit” are given by SPSS.16 [Song, 2008] based on the option of “Confidence Interval Width.” The option of “Confidence Interval Width” is selected as 95% in this paper. Figures 7 and 8 imply that q = 7 and p = 1 in (2) or (5) are reasonable, because the ACF in Figure 7 is cut off at lag number 7 and the PACF in Figure 8 is cut off lag number 6. Up to now, it can be concluded that the ARIMA(1,1,7) model, as the supposed model to predict the data in Figure 4, is reasonable.

[29] Now, the second step, estimating model parameters, is discussed. There are many theories and methods to estimate the parameters of the ARMA model or the ARIMA model, such as, least-squares estimation, moment estimation and direct estimation by ACF, which are elaborated in Lan [1997], Xiang et al. [1986], and Yi [2002]. For example, if q = 0 in (1), equation (1) becomes

display math(13)

[30] Let

display math(14)

where, φ = (φ1, φ2 … φp)T, and N is the length of the time sequence. ξ(φ) must be minimized, so the optimized φ should match the following equation:

display math(15)

[31] Then, the following linear equation set is derived

display math(16)

[32] Equation ((16)) can be simplified as

display math(17)

[33] Solving (17) gives φi in (13).

[34] When q ≠ 0, φi and θi can also be calculated by the least-squares estimation, which is discussed in detail by Lan [1997].

[35] The method to estimate the parameters in (2) or (5) is the same with the one used for (1), except for the usage of different time series: At is used for (1) , while inline image or ΔrAt is for (2) or (5).

[36] Moreover, φi and θi can be also estimated based on measured time series by using various types of software, such as Statistics Analysis System [Liangping, 2000], R (A statistics software that is named after Robert Gentleman and Ross Ihaka, who developed the software) [Dalgaard, 2004], Econometrics Views (Eviews) [Yi, 2002], and SPSS [Hao et al., 2003]. In this paper, φi and θi are estimated by SPSS.16 [Song, 2008].

[37] The final step, diagnosing and testing a supposed model, is discussed in this paragraph. There are two simple ways to diagnose and test a supposed model, which can be found in Lan [1997], Xiang et al. [1986], and Yi [2002]. One way is analyzing the residual series between the forecasted series and the original series. If the residual series is similar to a white-noise series, then the model can be confirmed. The other way is analyzing the fitting degree of the forecasted series and the original series [Lan, 1997]. Figure 9, as an example of diagnosing and testing a supposed model by analyzing residual series, gives the ACF and the PACF of the residual series between the forecasted series under the condition of q = 7, p = 1 and the real (differential) series in Figure 6. It can be concluded that the residual series are similar to a white-noise series, because the ACF and the PACF in Figure 9 are nearly equal to zero. Figure 9 implies that the supposed ARIMA(1,1,7) model is valid for the data in Figure 4. Finally, the process of establishing the ARIMA is introduced.

Figure 9.

ACF and PACF of the residual series between the forecasted series and the real (differential) series in Figure 6.

[38] In this paper, the ARIMA model is established by many attenuation time series, including the attenuation data measured in Xi'an, China in 2010 and the simulated attenuation data using the method given in Gong [2008], van de Kamp [2003], and Van de Kamp [2002], and the simulated attenuation data using the rain rate data measured in Xi'an, China from 2010 to 2011. The process for simulating those attenuation time series is explained in sections 4 and 5 of this paper. Furthermore, we continually measure rain rate in Xi'an, China.

[39] The ARIMA model has been validated with each group of measured rain attenuation time series and simulated time series. Figures 3-9 show the process of establishing the model with an example of simulated time series. Figures 10-15 give one example to demonstrate the stationarity of measured time series of rain attenuation and differential rain attenuation, and to demonstrate the rationality of the ARIMA(1,1,7) model using rain attenuation time series measured on 21 April 2010 in Xi'an, China. The following conclusions can be drawn: rain-induced attenuation time series is not stationary; the differential series of rain-induced attenuation series with k = r = 1 is stationary; q = 7 and p = 1 are reasonable. In other words, it can be concluded that the ARIMA(1,1,7) model is applicable to predict rain-induced attenuation time series.

Figure 10.

The rain attenuation time series measured on 21 April 2010, in Xi'an, China.

Figure 11.

The ACF of the measured attenuation time series in Figure 10.

Figure 12.

The differential series of the data in Figure 10 under the condition of k = r = 1.

Figure 13.

The ACF of the differential series in Figure 12.

Figure 14.

The PACF of the differential series in Figure 12.

Figure 15.

ACF and PACF of the residual series between the forecasted series and the real (differential) series in Figure 12.

[40] Consequently, the resulting ARIMA(1,1,7) model can be written as

display math(18)

[41] Also, the forecasting process using the ARIMA(1,1,7) can be written as

display math(19)

where, φ and θi, i = 1, 2 … 7 can be estimated by statistical software (see before), and εt can be replaced by the residual between the forecasted attenuation and the real attenuation.

[42] In this paper, φ and θi, i = 1, 2 … 7 are estimated using SPSS.16 [Song, 2008]. However, concerned with precision, this software gives an optimized constant μ based on the analyzed data. In fact, the constant μ is also recommended in many publications covering topics of time series analysis, for example Lan [1997], Xiang et al. [1986], and Yi [2002], to improve forecasting precision, because the residual between prediction results and measured results is more near white noise by introducing the constant μ. μ is a constant, which represents average level of a stationary random process or time series. In theory, the constant μ can be regarded as the mean of a stationary random process or time series. With the introduction of μ, inline image needs to be used in the forecasting model, which can be obtained by

display math(20)

[43] Therefore, (19) is replaced by

display math(21)

[44] The constant μ in (20) and (21) is optimized by SPSS software. Tables 1 and 3 (in section 5 of this paper) list the parameters in (21), which are optimized by SPSS based on the measured rain-induced attenuation series. The parameters of the ARIMA(1,1,7) model based on the measured rain-induced attenuation during ten rainfall events in Xi'an, China in 2010 are listed in Table 1, those based on other measured data are listed in Table 3 (in section 5 of this paper).

Table 1. Parameters of the ARIMA(1,1,7) Model Based on 10 Groups Attenuation Data Measured in Xi'an, China, in 2010
Date2010201020102010201020102010201020102010
Param.0314032903310414042004210513051405160517
μ0.00740.00280.0135–0.00010.0110–0.00250.0033–0.0032–0.0002–0.0111
φ–0.8539–0.7514–0.2116–0.99390.30020.40040.88910.74220.4198–0.9387
θ1–0.5715–0.49520.1541–0.79610.45360.67271.00270.90880.5884–0.8684
θ20.37680.394500.25120.34410.08320.07480.08840.0967–0.00500.5681
θ30.13580.24590.03640.19000.0413–0.0948–0.0837–0.11840.07700.5912
θ4–0.02300.0059–0.03430.0641–0.00390.0409–0.06860.0401–0.01320.0470
θ5–0.0613–0.09670.01240.09580.0287–0.03710.0343–0.06400.04070.0864
θ60.01021–0.0096–0.01100.13490.0073–0.07770.01740.01170.07000.2061
θ7–0.01910.03760.07040.0685–0.04210.10500.00870.03990.04210.0467

[45] Although the ARIMA model has been already proved applicable to single-case research, as shown in Tables 1 and 3, the parameters of the model are different for each case, because the rain-induced attenuation is dependent not only on some determinate factors, but also on some random factors like rain rate, rainfall type, the shape and size distributions of the raindrops and the space distribution of rainfall rate.

[46] To further verify the ARIMA(1,1,7) model, a couple of forecasted results have been given in Figure 16-18, using the parameters estimated on the attenuation data measured in Xi'an, on 14 March 2010. However, the forecasted results are not satisfactory: Figure 16 implies that the parameters can not reflect the real changes well in self-forecast case, while Figures 17 and 18 clearly showing a worse result, which implies that the used parameters can not reflect the real changing of other attenuation time series.

Figure 16.

The comparison results of the forecasted series using the model parameters estimated on data measured on 14 March 2010, with the series measured in the same day. Figure 16B is part of Figure 16A. (A) All results and (B) magnified portion of all results.

Figure 17.

The comparison results of the forecasted series using the model parameters estimated on data measured on 14 March 2010, with the series measured on 14 May 2010. Figure 17B is part of Figure 17A. (A) All results and (B) magnified portion of all results.

Figure 18.

The comparison results of the forecasted series using the model parameters estimated on data measured on 14 March 2010, with the series measured on 25 July 2010. Figure 18B is part of Figure 18A. (A) All results and (B) magnified portion of all results.

[47] Admittedly, due to its convenient features, the ARIMA(1,1,7) model has become significant for forecasting dynamic rain-induced attenuation, but this model cannot be directly applied to engineering because of the poor predicted results. Due to the existence of some random factors associated with attenuation, the ARIMA(1,1,7) model with constant parameters is unsatisfactory. Therefore, concerned with engineering applications, it is urgent to propose a way to dynamically adjust the parameters of the ARIMA(1,1,7) model. Thus, in this paper, the novel and practical MGA-ARIMA model has been established by introducing the ideas of GA [Lu and Li, 1997; Ming and Sun, 2002; Wang and Liming, 1998; Guoliang et al., 1996] into the ARIMA(1,1,7) model, which is discussed in the next section of this paper.

3 The MGA-ARIMA Model Based on GA Ideas

[48] The novel and practical MGA-ARIMA model for forecasting dynamic rain-induced attenuation in real time has been established by introducing GA ideas into the ARIMA(1,1,7) model, with the data measured in Xi'an, China. While being used to forecast attenuation during any on-going rainfall event, the parameters of the ARIMA(1,1,7) model are self-adaptive. The parameters of the ARIMA(1,1,7) model are completely renewed and optimized in real time, when the MGA-ARIMA model is employed to forecast attenuation during an on-going rainfall event. Those parameters are calculated by the MGA with a parameter database, which is obtained by the measured data in Xi'an, China.

[49] Certainly, the proposed model shall be more universally applicable if the parameter database includes more parameters based on the attenuation data measured in diverse regions. Thus, an effective way to obtain the parameter databases of other countries and areas has been given in section 5, which can be utilized conveniently by almost all research groups.

[50] The MGA-ARIMA model ensures forecasting precision by adaptively calculating and selecting suitable parameters in real time. Figure 19 gives the flow chart of establishing the MGA-ARIMA model. The steps to establish the MGA-ARIMA model are as follows:

  • Step 1.Build or obtain the parameter database of the ARIMA(1,1,7) model, as listed in Tables  and . Any researching organization can supply the database by the approaches given in section 5.
  • Step 2.Select m sets of parameters from the database as a sub-database using k-nearest neighbors (K-NN) clustering method [Zhang and Srihari, ; Fukunaga and Narendra, ; Moradian and Baraani, ] to save time and reduce complexity of the optimizing process. It is necessary to adopt this method, when the database includes a huge number of parameters. If this is the case, the precondition for the clustering centre is a specific link in a certain area. The clustering centre is some groups of parameters which need to be preobtained by SPSS using the data measured or simulated under the corresponding link conditions in the same area. Note that the input simulated data for SPSS can be preobtained by the way given in section 5. m is named as the length of a “chromosome”, each group of parameters is named as a “gene.”
  • Step 3.Initialize and determine the amount N of “population”. The larger N is, the more complex MGA is, and therefore the higher forecasting precision will be. Suppose the length of a “chromosome” given in Step 2 is m, the maximum value of N is
display math(22)

N can be decided according to GA theories or certain required precision.

  • Step 4.Code. Randomly generate N groups of 1 × m arrays formed by “0” and “1”, which represent the initial state of model parameter. For example, if m = 4 and N = 4 < Nmax = 16, one coding result may be
display math(23)

where, each 1 × m array, named as a “chromosome” or a “scheme”, gives a combination mode, and the ith column of a “chromosome” represents the ith group parameters within the sub-database. “1” indicates adopting corresponding parameters, while “0” means not adopting corresponding parameters.

  • Step 5.Calculate the average of each “chromosome” based on the coding results. The average acts as the initial parameters for the ARIMA(1,1,7) model. Suppose m = 4 and the sub-database is formed by parameters numbered as “20100314”, “20100329”, “20100331”, and “20100414”, which are listed in Table . The “chromosome” 1101 is the average of parameters numbered as “20100314”, “20100329”, and “20100414”. Obeying the rule, the initial parameters based on the coding results in (23) are listed in Table .
  • Step 6.Find a function to test fitting goodness. The fitting goodness can be defined according to AG theories [Lu and Li, ; Ming and Qian, ; Wang and Liming, ; Guoliang et al., ]. The function in this paper is defined as
display math(24)

where, At means the real value of rain-induced attenuation, and ARIMA(At − 1,φ,θi) represents the predicted value using the parameters based on one of N “chromosomes”. All results of the function are ft1, ft2, …, ftN, among which ftj(j = 1, 2 … N) corresponds the jth “chromosome”. If one or more of those results are greater than a prescribed threshold, the selected “chromosome” represents some optimized parameters. Otherwise, an iterative approach from Step 5 to Step 9 must be taken to generate randomly new “chromosomes”, until at least one “chromosome” is obtained that passes the test.

  • Step 7.Duplicate. Compute the sum of ft1, ft2, …, ftN as
display math(25)
Figure 19.

The flow chart of establishing the MGA-ARIMA model.

Table 2. Initial Parameters Based on the Coding Results in (23)
Parameters “Chromosome”μφθ1θ2θ3θ4θ5θ6θ7
11010.0032–0.8664–0.62090.37180.19060.0157–0.02070.04520.0290
10100.0105–0.5328–0.20870.31400.0861–0.0286–0.0245–0.00040.0257
01010.0011–0.8726–0.64560.36930.21800.0350–0.00050.06270.0530
10110.0068–0.6865–0.40450.32400.12080.00230.01560.04470.0399

[51] Randomly generate N values denoted as D1, D2, …, DN, meeting the condition of 0 < Di < Ft. Mark ft1, ft1 + ft2, …, Ft and D1, D2, …, DN on a one-dimensional coordinate axis. If one or more values of D1, D2, …, DN are greater than zero and less than or equal to ft1, duplicate the first “chromosome.” If one or more values of D1, D2, …, DN are greater than ft1 and less than or equal to ft1 + ft2, duplicate the second “chromosome.” In the same way, the entire duplication process is implemented. The duplicated “chromosomes” will be used in next iteration. The “population” amount after duplication process will be N ' and N ' ≤ N, because one or more “chromosomes” may have not been duplicated. Figure 20 gives an example of the duplication process, when m = 4 and N = 4. In the case of Figure 20, the first three “chromosomes” are duplicated and the fourth is not duplicated, because D1 is greater than zero and less than ft1, D2 and D3 are greater than ft1 and less than ft1 + ft2, D4 is greater than ft1 + ft2 and less than ft1 + ft2 + ft3.

  • Step 8.Find an overlapping rule and then overlap. In this paper, the accordance overlapping rule has been adopted, which is an 1 × m array (named as overlapping rule array) formed by “0” and “1.” An overlapping rule array can be found with the similar approach of the duplication process. That is, randomly generate m values denoted as E1, E2, …, Em, meeting the condition of 0 < Ei < m. Mark E1, E2, …, Em on a one-dimensional coordinate axis. If one or more values of E1, E2, …, Em are greater than 0 and less than or equal to 1, the first value of overlapping rule array is “1”, otherwise it is “0.” “1” represents operation “OR”, while “0” represents operation “AND.” If one or more values of E1, E2, …, Em are greater than 1 and less than or equal to 2, the second value of overlapping rule array is “1”, otherwise it is “0.” In the same way, the entire overlapping rule array can be found. Figure  gives an example of array-finding process, when m = 4 and N = 4. The overlapping rule array illustrated in Figure  is
display math(26)
Figure 20.

The schematic diagram of the duplication process under conditions of m = 4 and N = 4.

Figure 21.

The schematic diagram of finding an overlapping ruler array under the conditions of m = 4 and N = 4.

[52] Equation ((26)) means that the operation “OR” is performed on the first three “genes”, while the operation “AND” is performed on the forth “gene.”Next, overlap on the population of N ' “chromosomes” according to the overlapping rule array, each “chromosome” can randomly overlap with any other with equal probability. For example, when the first “chromosome” in (23) overlaps with the second “chromosome” in (23) using the overlapping rule array in (26), the result will be

display math(27)

[53] Thus, after overlap, a new “population” of N ' ' “chromosomes” will be produced. The largest N ' ' is equal to N ' (N ' − 1) in theory, but N ' ' will generally be smaller because some overlapped “chromosomes” can be identical.

[54] After duplication and overlap, some better “chromosomes” may be found. However, it may be possible that the N ' ' “chromosomes” are not the optimized “chromosomes”, because duplication and overlap are good at searching and optimizing in global. In other words, duplication process and overlap process make some of the N ' ' “chromosomes” be close to the optimized “chromosomes.” If the “genes” of the “chromosomes” can be optimized in local, the optimized “chromosomes” may be found more quickly. Mutation process can increase the capability of local searching, which is helpful for optimizing the N ' ' “chromosomes” in local. By the way, the necessity of mutation and the relationships among duplication, overlap and mutation are elaborated by Ming and Sun [2002].

  • Step 9.Mutate on the “population” produced in Step 8. Generate N ' ' random numbers y1, y2, …, yN ' ', meeting the condition of 0 < yj < 1. Assume the mutation probability p, if yj < p, then mutate the jth “chromosome.” In this case, define p/m as the unit probability of mutation of a “chromosome.” Then generate m random numbers k1, k2, …, km, meeting the condition of 0 < ki ≤ 1/m. If ki < p/m, mutate the ith “gene” of the “chromosome”, which means to transform “gene” “1” to “0”, vice versa.

For example, suppose N ' ' = 3 and “chromosomes” are the first three ones in (23), as shown below

display math(28)

[55] Assume p = 0.5, y1 = 0.3, y2 = 0.6, y3 = 0.4, according to the process described above, mutate the first and third “chromosomes” in (28). Therefore, 0.5/4 is defined as the unit probability of mutation of the mutated “chromosomes.” Also assume four random numbers for the first “chromosome” as k1 = 0.1, k2 = 0.2, k3 = 0.08, and k4 = 0.17, mutate the first and third “genes” of the first “chromosome”, so 1101 turns out to be 0111. In the same way, assume four random numbers for the third “chromosome” as k1 = 0.2, k2 = 0.2, k3 = 0.08, and k4 = 0.05, mutate the third and fourth “genes” of the third “chromosome”, so 0101 turns out to be 0110. After mutation, (28) will change into the following equation:

display math(29)

[56] It is the final aim to accurately find a rule for selecting the mutation probability p. However, that is difficult. Here we give our advice. One can assume p as a set of sorted random values in ascending order, which are greater than 0 and less than 1, and whose length is equal to N ' '. Then, try every value by a loop structure, until one finds the best “chromosome” that passes the test. Note that the assumed sorted random values are different for different loop calculations.

[57] After the mutation process, iterate from Step 5, until one or more values of ft1, ft2, …, ftN ' ', which are indicators of testing goodness, are greater than a predetermined threshold η. In that case, the parameters, which are calculated from the first-rank “chromosome” (i.e., corresponding to the maximal function value), are most suitable for forecasting.

[58] Note that, in any case, because the real value of rain-induced attenuation cannot be known beforehand, the MGA parameters adopted at time t have been optimized to represent the dynamics of attenuation at time t − δ. The lag time δ is mainly decided by the needed time of obtaining real attenuation at the next prediction time point, which can be considered as sampling interval of measuring attenuation system, and lightly decided by computing time. However, such delay will not prevent the MGA from improving forecasting precision, because the adaptive process of adjusting parameters brings the forecasting system continuously closer to the real random properties of attenuation varying. Thus, by introducing GA ideas into the ARIMA model, the forecasting results will be more precise, which has been fully verified in the next section.

4 Performance of the MGA-ARIMA(1,1,7) Model

[59] To verify the validity of the MGA-ARIMA(1,1,7) model, the predicted results are compared with the real data measured in Xi'an, China, on 14 March, 14 May, and 25 July 2011. The conditions for forecasting are as follows: 10 groups of parameters listed in Table 1 as the database (m = 10, skipping the usage of the K-NN clustering algorithm), N = 30 < Nmax = 1024, the threshold η = 200, and the mutation probability p is decided by the method given in section 3. The comparison results are given in Figures 22, 23 and 24.

Figure 22.

The comparison results of the forecasted attenuation time series using the MGA-ARIMA model with the attenuation time series measured on 14 March 2010. Figure 22B is part of Figure 22A. (A) All results and (B) magnified portion of all results.

Figure 23.

The comparison results of the forecasted attenuation time series using the MGA-ARIMA model with the attenuation time series measured on 14 May 2010. Figure 23B is part of Figure 23A. (A) All results and (B) magnified portion of all results.

[60] Obviously, the forecasting results in Figures 22-24 are better than the results in Figures 16-18, which implies the high validity of the proposed model in this paper. The prediction results are almost equal to the measured data during most of the time, which is the result that the adaptive process of adjusting parameters brings the forecasting system continuously closer to the real random properties of attenuation varying. The good agreement between the prediction results and the measured data proves that the delay of δ in optimizing the MGA parameters (see end of section 3) does not prevent the MGA from improving forecasting precision.

Figure 24.

The comparison results of the forecasted attenuation time series using the MGA-ARIMA model with the attenuation time series measured on 25 July 2010. Figure 24B is part of Figure 24A. (A) All results and (B) magnified portion of all results.

[61] To prove the applicability of the MGA-ARIMA(1,1,7) model in other countries or areas, an example is given using the model to forecast the data simulated in northwestern Europe, under the aforementioned conditions. The data is simulated by Markov chains, and the validity of the method has been confirmed in Gong [2008], van de Kamp [2003], and Matricciani [1994]. The simulated data A(t) (t = n ⋅ ts, ts denotes sampling time) follow the probability distribution as (30)

display math(30)

where, mA and σA are the mean and variance of an attenuation time series respectively, which vary with time. The fact of mA and σA changing with time further proves the invalidity of any forecasting model with constant parameters.

[62] Suppose the initial value is A(0 × ts) = A0, which determines mA and σA for the first simulated value, and the relationships can be written as (31) and (32) [Gong, 2008; van de Kamp, 2003; Matricciani, 1994]

display math(31)
display math(32)

A[(i − 1)ts] and A[(i − 2)ts] (where i = 2, 3 …) determines mA and σA for the second and other simulated values, and the relationships can be written as (33) and (34) [Gong, 2008; van de Kamp, 2003; Matricciani, 1994]

display math(33)
display math(34)

where, the values of β1, β2, α2, and γ2 relate to sampling time step ts, climate characteristics, link elevation angle, signal frequency and so on [Gong, 2008; van de Kamp, 2003; Matricciani, 1994]. The values of β1, β2, α2, and γ2 found to be representative for the Maritime climate of northwestern Europe, for the elevation angles between 27° and 30°, for the frequencies between 10 GHz and 50 GHz [Gong, 2008; van de Kamp, 2003; Matricciani, 1994], are given in (35)–(38).

display math(35)
display math(36)
display math(37)
display math(38)

[63] The simulated data generated by the model can also be scaled up or down unlimitedly, which is elaborated by van de Kamp [2003]. Figure 25 is an example given in van de Kamp [2003].

Figure 25.

An example in van de Kamp [2003].

[64] Comparing the simulated values with the data forecasted by the MGA-ARIMA(1,1,7) model, a satisfactory result is illustrated in Figure 26, which proves the validity of the proposed model applied in northwestern Europe.

Figure 26.

The comparison results of the values forecasted using the MGA-ARIMA model with the simulated values, the conditions for forecasting are the same with those in Figures 22-24. Figure 26B is part of Figure 26A. (A) All results and (B) magnified portion of all results.

[65] It can be concluded from the good agreement in Figures 22-24 and Figure 26 that the MGA-ARIMA(1,1,7) model is practical in Xi'an, China, and also practical for the conditions for which van de Kamp's model is valid. Also, the good agreement in Figure 26 implies that the MGA-ARIMA(1,1,7) model has the potential possibility to be used in any country or area. Certainly, the model will be more applicable, if one can establish a parameter database from more plentiful resources. Thus, other parameters in the database obtained in Xi'an and the methods to establish the databases in other countries or areas, have been given in the next section.

5 More Parameters and Methods to Build Parameter Databases in Other Countries or Areas

[66] The parameters of the ARIMA(1,1,7) model based on different data measured in Xi'an, China are listed in Table 3. The parameters in both Tables 1 and 3 can be shared by the users of the MGA-ARIMA(1,1,7) model to forecast dynamic rain-induced attenuation in other countries or areas.

Table 3. More Parameters of the ARIMA(1,1,7) Based the More Data Measured in Xi'an, China in 2010
Date2010201020102010201020102010201020102010
Param.0525062606290702070307070708070907130717
μ–0.0398–0.00440.00310.00480.00160.11170.00050.0073–0.00140.0025
φ0.69920.93390.94990.98610.90270.47010.97650.9442–0.4123–0.5459
θ10.95120.94110.98490.94081.02410.33971.08740.8416–0.1428–0.3366
θ2–0.14640.0587–0.15770.05940.09160.0192–0.01410.18270.09070.3576
θ3–0.0234–0.0278–0.06510.0176–0.1529–0.1238–0.0483–0.14190.09140.1639
θ40.0800–0.00520.11820.06380.02630.11800.05430.08860.04570.0738
θ5–0.06610.00210.0879–0.00790.00001–0.0395-0.05790.06850.05330.0241
θ60.06330.06900.1196–0.0700–0.01530.11450.0291–0.09840.12170.0953
θ70.1413–0.0379–0.0879–0.00380.02620.1027–0.05060.0589–0.00430.1467
Date2010201020102010201020102010201020102010
Param.0722072307240725072908220823082409050907
μ0.00030.0022–0.0001–0.03420.0282–0.00010.0020–0.00080.0023–0.0002
φ0.98050.9402–0.98080.53690.9963–0.28750.94710.80180.94540.9275
θ11.10630.8208–0.87960.24800.8512–0.01761.09750.90721.13931.1213
θ20.02150.15060.16220.29370.30410.24720.0256–0.03770.0547–0.0413
θ3–0.13790.07420.10300.0793–0.02460.1632–0.0339–0.0256–0.1588–0.0735
θ40.0372–0.05290.0100–0.0251–0.04530.1996–0.09150.00030.01310.0029
θ50.02500.0413–0.0940–0.09660.00660.01770.0049–0.0040–0.0897–0.0072
θ6–0.11030.0065–0.14790.1857–0.04560.06440.0083–0.00100.0050–0.0016
θ70.0580–0.0405–0.09330.3149–0.0464–0.0739–0.02560.07230.0364–0.0007

[67] Note that more parameters based on attenuation time series measured or simulated in more countries or areas under different link parameters are very important for improving the MGA-ARIMA(1,1,7) model.

[68] This paper recommends two methods to build the parameter database as listed in Tables 1 and 3. One method is to estimate the parameters by SPSS software or other similar software based on measured attenuation time series, while the other method is based on the simulated attenuation time series using measured rain rate. We prefer to use the latter method, because measuring rain-rate time series is simpler and more economical than measuring the attenuation directly, and, the measured rain rate data can give simulated attenuation data under more link parameters. Rain-rate time series can be measured with any rain gauge or similar instruments. In this paper, OTT Parsivel is used, which is an advanced laser-based disdrometer for comprehensive measurement of all precipitation types.

[69] The attenuation time series can be simulated based on the measured rain-rate data by the following equation [Van de Kamp, 2007]:

display math(39)

where, hR is rain height in kilometers [ITU-R P.839-3, 2001], Θ is link elevation angle in degree, l is coordinate along the slant path in kilometers, γ(t,l) is the specific attenuation in dB/km. Equation (39) can be changed to

display math(40)

where, s is the horizontal projection coordinate of location l [see Van de Kamp, 2007]. γ(t,s) can be calculated by (41) [see ITU-R P.618-10, 2009 and ITU-R P.838-3, 2005]

display math(41)

where, R(t,s) is the rain rate at the coordinate s with an integration time of 1 minute, ar and br can be computed by (42) [see ITU-R P.838-3, 2005]

display math(42)

where, τ is polarization tilt angle, τ = 0° represents the situation of horizontal polarization, τ = 90° of vertical polarization, τ = 45° of circular polarization. ah, av and bh, bv are frequency-dependent coefficients for calculating the specific rain attenuation using (41) and (42), which can be found in Gong [2008] and ITU-R P.838-3 [2005].

[70] In (41), R(t,s), which is dependent on the measured rain rate R1 = R(t,0) at s = 0, is derived by Van de Kamp [2007]. The probability distribution of instantaneous rain rate R2 = R(t,s) conditional to R1 is lognormal [Van de Kamp, 2007]

display math(43)

where, m2(t) is the median value of R2 conditional to R1, σ2(t) is the standard deviation of ln R2 conditional to R1. m2(t) and σ2(t) are given by Van de Kamp [2007] as following equations:

display math(44)
display math(45)

[71] In (45), Bs describes the spatial dynamics of rain rate, for example Bs can be given as (46) [Van de Kamp, 2007]

display math(46)

[72] Strictly speaking, mR in (44) is instantaneous (unconditional) spatial median value of rain rate at any coordinate s, and σln R in (45) is instantaneous standard deviation of ln R(s), where R(s) is rain rate at any coordinate s. Van de Kamp [2007] shows that mR can be replaced by long-term median value of rain rate, and σln R can be replaced by long-term standard deviation of ln R(t). Therefore, mR and σln R can be decided by measured rain rate time series. Van de Kamp [2007] also gives an empirical model for calculating σln R, which is

display math(47)

where, vver, in Pa/s, is expressed as the change in pressure that an air package encounters in an updraft [Van de Kamp, 2007].

6 Conclusions

[73] Adaptive fade mitigation technologies are crucial to develop MMW wireless systems, and forecasting dynamic rain-induced attenuation in real time is becoming a subject of growing interest.

[74] A practical model for forecasting dynamic rain-induced attenuation in real time must adaptively adjust model parameters to trace the correlativity of attenuation varying over time. Also, a more applicable model should be employed in any country or area especially. It has been verified that the ARIMA model is significant in the field of forecasting random phenomena. However, it has been proved that the idea using the ARIMA(1,1,7) model with constant parameters to forecast rain-induced attenuation is feasible but still not qualified because of the many random elements associated with the attenuation.

[75] Thus, the novel and practical MGA-ARIMA(1,1,7) model has been presented for forecasting dynamic rain-induced attenuation in real time, by introducing GA ideas into the ARIMA model. With such introduction, the model given in this paper can achieve self-optimizing parameters in real time, even though the parameter database is obtained in a specific region like Xi'an.

[76] It is confirmed that the MGA-ARIMA(1,1,7) model is practical and valid with the parameter database. In addition, the parameters listed in Tables 1 and 3 can be shared with the MGA-ARIMA(1,1,7) model in any country or area.

[77] It is worthwhile mentioning that the MGA-ARIMA(1,1,7) will be more applicable with a database including more parameters of diverse regions. Thus, the approaches have been recommended to establish the databases in other countries or areas.

[78] The MGA-ARIMA(1,1,7) model is significant and practical, and should be an attractive alternative for forecasting dynamic rain-induced attenuation in real time, which helps develop adaptive fade mitigation technologies at MMW bands.

Acknowledgments

[79] This work has been supported by “the National Natural Science Foundation of China (61001065)” and “the Fundamental Research Funds for the Central Universities”. We thank all teachers and students, who gave great contributions for attenuation and rain rate measuring. We thank the reviewers who gave many good advices.

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