Rain Attenuation Time Series Synthesizer based on Copula Functions Charilaos Kourogiorgas1, Arsim Kelmendi2, Athanasios D. Panagopoulos1, Spiros N. Livieratos3, Andrej Vilhar4, George E. Chatzarakis3 1
School of Electrical and Computer Engineering, National Technical University of Athens, Greece,
[email protected],
[email protected] 2 Jožef Stefan International Postgraduate School, Ljubljana, Slovenia,
[email protected] 3 Department of Electronics and Electrical Engineering Educators, School of Pedagogical and Technological Education, Heraklion 14121, Athens, Greece,
[email protected],
[email protected] 4 Department of Communication Systems, Jožef Stefan Institute, Ljubljana, Slovenia,
[email protected]
Abstract—In this paper, a methodology based on Gaussian copulas is examined for generating rain attenuation time series. The methodology lies on the use of Gaussian copula for modeling the joint exceedance probability of rain attenuation for the temporal domain. In this paper, preliminary results for the validity of the use of Gaussian copula for modeling rain attenuation joint statistics on temporal domain are given. Then, a simple methodology is provided for the generation of rain attenuation time series. It is shown that the time series reproduce the first and second order statistics (using Kendall’s tau correlation) of rain attenuation using measurements obtained at Ljubljana from Jozef Stefan Institute. Index Terms—Copulas, Kendall’s tau.
I.
rain
attenuation,
time
series,
kind of distribution of rain attenuation, e.g. lognormal or Weibull and it uses a dependence index which takes into account the dependence of two variables even if these are linked through a nonlinear expression [11]. A first attempt to extend this research on time diversity systems has been presented in [12]. In the latter paper, various Copulas are tested trying to find the Copula which best describes the dependence of shifted rain attenuation time series. In this paper, the Gaussian Copula is tested for the modeling of joint statistics of rain attenuation on temporal domain. Then, a method based on Gaussian copula is presented for generating rain attenuation time series and it is tested with measurements from Jozef Stefan Institute at Ka-band [13].
INTRODUCTION
The adoption of high frequency bands, i.e. Ka-band and above for Satellite Communication Systems can give the desired amount of bandwidth for high speed wireless communications, as also indicated for High Throughput Satellites (HTS) [1]. The severe attenuation, mainly rain attenuation, which is induced to the system at these high frequency bands can deteriorate the system’s performance and so Propagation Impairment Mitigation Techniques have been developed [2]. The evaluation of a system which uses dynamic PIMTS and signal processing techniques, such as interference cancellation, is usually undertaken through end-to-end simulations. In these simulations and in the case that there are no measurements, rain attenuation time series synthesizers are required. One of the first rain attenuation time series synthesizers developed was the Maseng-Bakken model (MB) in which a stochastic differential equation was used for the description of rain attenuation as 1st order continuous Markov process [3]. MB model, as the models developed in [4]-[6], model rain attenuation as a lognormally distributed random variable. However, rain attenuation may not always follow lognormal distribution but also Gamma, Inverse Gaussian or Weibull distribution [7]-[9]. In [10], a method for calculating the joint first order statistics of rain attenuation for spatially separated satellite links has been developed based on Copula functions. The advantage of a copula method is that it does not make any assumption on the
II.
JOINT STATISTICS OF RAIN ATTENUATION
Joint statistics of rain attenuation on temporal domain, i.e. P ( At ≥ Ath , At +∆t ≥ Ath ) are used mostly in time diversity systems. In [12], a first analysis for describing these joint statistics with Copulas has been conducted. Here, the Gaussian copula [11] is examined. The general expression of the Gaussian copula for n random variables is given by:
C (u) = Φ nR ( Φ −1 ( u1 ) ,..., Φ −1 ( un ) ) where
(1)
Φ nR is the normal multivariate Cumulative Distribution
Function (CDF) of Gaussian distribution with zero mean −1
variables and correlation matrix R. The function Φ is the inverse CDF of the standard Gaussian distribution. The first test of the Gaussian copula will be conducted using data from the Jozef Stefan Institute [13]. The ground station is located at Ljubljana and the frequency of the link is 20.2 GHz. The elevation angle is 36.3° and the measurement period used for this paper is one year from 1 September 2013 up to 31 August 2014. In Fig. 1 an example of rain attenuation time series is shown for 2 hours interval measured on 3 June 2014.
Fig. 1. Example of attenuation time series measured in Ljubljana
The test of Gaussian copula is through fitting to measured joint CCDFs of rain attenuation. The Gaussian copula used for the test is taken from (1) but for n=2. In Fig. 1, the Gaussian Copula is tested for the modeling of joint statistics of rain attenuation on time domain to measured data at Ljubljana. It can be observed that the Gaussian Copula fits well to data. Therefore, Gaussian Copula can be used for modeling the joint complementary cumulative distribution function (CCDF) of rain attenuation on temporal domain. The Gaussian Copula is characterized by the parameter ρ for two random variables or from correlation matrix R for multiple dependent random variables. Using joint exceedance probability of rain attenuation for various time delays and link characteristics, the parameter ρ can be modeled as a function of time delay and the link characteristics. Here using the measurements from Ljubljana, the parameter ρ is modeled through the following expression:
ρ ( ∆t ) = ae −b∆t + (1 − a)e− c∆t
2
(2)
where a=0.16, b=0.055 and c=9.4·10-6. In Fig. 2, the ρ parameter from the Gaussian Copula fitted to the experimental joint CCDF and from (2) is given for the various time delays.
Fig. 2. Gaussian copula parameter as a function of time delay from fitting to data and from expression (2)
For this specific copula the copula parameter can be related to the Kendall’s tau rank correlation (τ) through the expression [11]:
τ=
2
π
sin −1 ( ρ )
(3)
Therefore, knowing the parameter ρ, one can obtain the Kendall’s τ correlation. The use of Kendall’s τ correlation for the measure of the dependence between random variables has the advantage that it is a distribution free test, i.e. it does not require a hypothesis on the distribution of random variables. In the previous works most synthesizers use the Pearson correlation coefficient which is optimally derived for normally distributed random variables. Therefore, when considering various distributions, Kendall’s τ correlation can be used. Moreover, the Kendall’s tau correlation has a straight interpretation. Starting from the definition of the coefficient, Kendall’s tau is the difference between the number of concordant pairs (C) and the number of discordant pairs (D) divided by the number of observations (N) [11]:
τ= A
concordant
C−D N
pair
(4) is
when
( xi > x j and yi > y j ) or ( xi < x j and yi < y j ) and a discordant pair is ( xi < x j and yi > y j ) or ( xi > x j and yi < y j ) . So, as interpretation this coefficients measures the relation of the order of the random variables. The Kendall’s tau rank correlation as a function of time delay for the copula parameters derived from (2) is shown in Fig. 3.
Fig. 1. CCDF of rain attenuation for a ground station at Ljubljana from experiments and from copula fitting for 2, 5 and 45 min time delays
IV.
Fig. 3. Kendall’s tau parameter as a function of time delay using the parameter ρ from (2)
III.
RAIN ATTENUATION TIME SERIES SYNTHESIS
In this section, a methodology is presented for generating rain attenuation time series using Gaussian Copulas. The goal of this method of synthesis is to reproduce the CCDF which was given as input and the desired Kendall’s tau correlation coefficient. Since we want to reproduce the Kendall’s tau coefficient as second order statistics of the random variable the Gaussian Copula may be used. In case that τ of (3) is known, then with the inverse of (3), one can obtain the Gaussian copula parameter ρ. With the knowledge of the correlation matrix (R) of the multivariate Gaussian copula, one can proceed to the generation of zero mean Gaussian random variables correlated with matrix R. Then taking the normal CDF of the above generated values the CDF of each sample of the desired random variable is calculated. Finally, taking the inverse CDF of the desired distribution, i.e. Gamma, lognormal, Weibull, etc. the random variables are generated. In Fig. 4, time series of rain attenuation with the proposed methodology are shown for a rain event.
NUMERICAL RESULTS
In this section, the synthesizer is validated for the first order statistics and second order statistics. The Weibull distribution is used for the modeling of first order statistics of rain attenuation for no time delay. The Weibull distribution is fitted to the experimental data from Ljubljana station. In Fig. 5, the CCDF of rain attenuation obtained from measurements and from the fitting of Weibull distribution to the data are shown. In this paper, we do not target to examine which distribution fits better to the measured CCDF and so we will not test other distributions. Using the measured CCDF of rain attenuation and the fitted Weibull distribution, we generate rain attenuation time series using (2) as the Gaussian Copula parameters in the proposed methodology. In Fig. 6, the CCDF of rain attenuation for no time delay obtained from the measurements, the Weibull distribution and the simulated time series are shown. It can be seen that the synthesizer reproduces the first order statistics of rain attenuation, as these are given as input (here the Weibull distribution). Moreover, in the same figure the joint exceedance probability of rain attenuation for 15 min time delay obtained from the measurements and the simulation are shown. In the same figure, the joint CCDF of rain attenuation for 15-min time delay obtained from (1) using (2) is shown. From the joint statistics, it can be derived that the synthesizer reproduces the theoretical joint CCDF, i.e. (1) using ρ from (2), which is close to the actual measurements. Finally, the synthesizer is validated for second order statistics. The expression (2) was used as input to the simulation methodology. From the time series the Kendall’s tau was obtained and it is shown in Fig. 7, with the theoretical one obtained from using (2) in (3). It can be observed that the synthesizer reproduces also the second order statistics.
Fig. 5. CCDF of rain attenuation from measurements (solid line) and from the fitting of Weibull distribution (dashed line)
Fig. 4. Rain attenuation time series generated by the proposed methodology
ACKNOWLEDGMENT This research has been supported and co-financed by the European Association on Antennas an Propagation (EurAAP) and by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic Reference Framework (NSRF) - Research Funding Program: ARCHIMEDES III. Investing in knowledge society through the European Social Fund. REFERENCES [1]
Fig. 6. CCDF of rain attenuation for no time delay from measurements, time series and Weibull distribution and joint CCDF of rain attenuation for 15 min from measurements, time series and Gaussian Copula
[2]
[3]
[4] [5]
[6]
[7]
[8]
Fig. 7. Kendall’s tau correlation from time series and from (2)
V.
[9]
CONCLUSIONS
In this paper, firstly, the Gaussian Copula is preliminarily tested for modeling joint statistics of rain attenuation on temporal domain using measurements from JSI. Then, a methodology based on Gaussian copulas is presented for generating rain attenuation time series which reproduce the first order statistics of rain attenuation and the Kendall’s tau correlation. From numerical results, the synthesizer is validated not only considering the theoretical results but also comparing to measured data.
[10]
[11] [12]
[13]
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