TRANSACTIONS ON EMERGING TELECOMMUNICATIONS TECHNOLOGIES Trans. Emerging Tel. Tech. 2013; 24:636–648 Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/ett.2724
SPECIAL ISSUE – COGNITIVE RADIO
Bayesian spatial interpolation as an emerging cognitive radio application for coverage analysis in cellular networks Berna Sayrac1*, Ana Galindo-Serrano1 , Sana Ben Jemaa1 , Janne Riihijärvi2 and Petri Mähönen2 1 Orange Labs, Issy-Les-Moulineaux, Paris, France 2 Institute for Networked Systems, RWTH Aachen University, Aachen, Germany
ABSTRACT Coverage prediction is one of the most important aspects of cellular network optimisation for mobile operators due to the highly competitive market conditions and the obligations towards the regulatory authorities. Although wireless communications research is considerably diversifying, including new areas, problems and concepts with a notable pace, cellular coverage still remains as a core subject of research for operators because of the obligation to adapt to the continuously evolving wireless ecosystem, with new radio access technologies/architectures, emerging applications and innovative cellular concepts. This paper presents the application of a powerful mathematical tool coming from spatial statistics, Bayesian kriging, to construct a radio environment map (REM) for the purpose of cellular coverage prediction as an emerging application of cognitive radio techniques in cellular networks. The proposed approach provides an efficient alternative to the conventional manual coverage prediction on the basis of drive tests, which are expensive, polluting and slow solutions for obtaining the ‘ground-truth’ information. Bayesian kriging-based REMs allow to estimate the coverage situation in those regions where the operator lacks direct information. Our approach can be directly used by operators for the cellular network coverage optimisation. We evaluate the accuracy of the proposed REM construction approach for a long term evolution network with two mesh sizes using highly realistic data sets. Results show that the Bayesian kriging interpolation technique has a good accuracy for cellular coverage prediction, and this accuracy is directly related with the mesh size. Copyright © 2013 John Wiley & Sons, Ltd. *Correspondence B. Sayrac, 38-40 Rue Du Général Leclerc, 92794 Issy Les-Moulineaux cédex 9, Paris, France. E-mail:
[email protected] Received 10 February 2013; Revised 25 June 2013; Accepted 13 September 2013
1. INTRODUCTION When deploying a cellular technology, the operator first performs coverage estimation through sophisticated planning tools on the basis of terrain profile and building data combined with detailed propagation models. Despite the great effort usually spent during the planning and deployment phase, the existence of coverage holes is almost impossible to avoid because of the complexity of network configuration, unforeseen changes in the propagation environment or equipment failures. Therefore, a second process, called coverage optimisation, is required during the operational phase, where it is first necessary to detect the coverage holes (a process called coverage hole detection) and then deploy a solution, which remedies or removes the coverage problem in the uncovered zones. 636
The deployed solution must be cost-efficient and at the same time effective, that is, providing full coverage without creating excessive pollution to the already covered neighbouring areas. In order to achieve such a solution, we need precise information on the locations and shapes of the coverage holes. Obtaining this information is called coverage prediction. Obviously, the effectiveness of the deployed solution depends highly on the accuracy of coverage hole prediction. This paper focuses on coverage hole prediction and showcases how to use the powerful mathematical tools coming from spatial statistics as a cognitive solution to these problems. So far, operators have adopted the following procedure to deal with coverage prediction: (i) to perform drive tests, which consist of geographically measuring different network metrics and indicators with motor Copyright © 2013 John Wiley & Sons, Ltd.
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vehicles equipped with specialised mobile radio measurement equipments and global positioning system; and (ii) to analyse the collected measurements for coverage prediction. On the one hand, drive tests generate a tremendous amount of data to be processed, allowing the operators to get realistic network information close to the actual user experience. This is a very useful and desired information by operators [1]. With the processed drive test measurements, the operator can have a realistic picture of the network in terms of coverage and service quality and find the right optimisation solutions through the modification of one or several network parameters such as the transmission power, antenna locations and antenna orientations and tilts. On the other hand, drive tests are quite an inefficient means to solve the coverage problems because they (i) imply large operational expenditure; (ii) incur delays in detecting and predicting the coverage holes; (iii) are an undesirable source of pollution; and (iv) provide an incomplete picture of the ‘ground truth’ because they are limited to roads and other regions accessible by motor vehicles. All these disadvantages make it mandatory for operators to make the most of the information collected through the drive tests and to minimise the use of them. For this purpose, the 3rd Generation Partnership Project standardisation body has been working on the minimisation of the use of drive tests for long term evolution (LTE) since Release 9 [2]. In Release 10, a minimisation of drive tests (MDT) work item [3, 4] for Universal Terrestrial Radio Access Network was also included. The main focus of MDT in Release 10 is on coverage optimisation. Release 11 focuses on quality of service verification, further improvements in coverage optimisation, positioning enhancement and the study of other MDT use cases [5]. The key idea of the MDT proposed by 3rd Generation Partnership Project is to take advantage of the (especially geo-location) measurement capabilities of the new advanced user equipments (UEs) as well as of the radio measurements performed as part of the radio resource management procedures. The main characteristic of MDT is that the UEs report their geo-located measurements to the network upon operator request. The collected MDT measurements are at operator’s direct disposition to ease any kind of (automated as well as manual) network operation, management and optimisation task. Hence, MDT also acts as a valuable input to the self-organised network functionalities whose main goals are to decrease operational expenses, increase the autonomy of networks and allow the achievement of an automatic, near optimal network quality in next generation networks [6]. The use of location information for maximising the efficiency of wireless network resource utilisation appears as a fundamental component, called location/environment awareness, in cognitive radio (CR), put forward by Mitola in [7]. The first work where spatial information exploitation appears in wireless networks was the available resource map introduced by Krenik, which is proposed as a real-time map of all radio activities in the network for CR applications in unlicensed wide area Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
networks [8]. Then, it was extended to the radio environment map (REM) concept by Zhao [9], who defined it as an integrated database for enhancement of CR systems, mainly for dynamic spectrum access purposes (such as TV white spaces). Such REM stores environmental information, past experience and radio knowledge with the aim of improving the network performance as well as aiding in self-optimisation and management. Shortly after, a more comprehensive/cognitive version of REMs, also called as interference cartography (IC), has been proposed for a mesh-like REM structure where the location points are found on a rectangular mesh [10–12]. In our work, we consider this broader view of the concept of REM, where the main idea is to (i) spatially interpolate the collected geolocated measurements (i.e. drive test measurements and MDT measurements) in order to predict the measurement values at those mesh points where measurements are not available; and (ii) request additional measurements at intelligently chosen mesh points to enhance the quality of those predictions. This paper falls in this latter line of work, and unless otherwise stated, REM will mean IC. As can be noticed easily, the IC concept fits perfectly into both the MDT and the CR framework. Although the MDT measurements provide the exact input needed for constructing the REM, the REM brings environment awareness to today’s cellular networks as a step forward for tomorrow’s ‘cognitive cellular networks’. Radio environment map has been studied within the framework of the FP7 FARAMIR project, a cooperative research project on CR, where a functional architecture together with several system architectures has been proposed [13]. The functional REM architecture is composed of the following entities: (i) measurement capable devices (MCDs); (ii) REM storage and acquisiton (REM SA); (iv) REM manager; and (4) REM user. MCD performs geolocated measurements and report to REM SA. REM SA communicates with the MCD and stores raw as well as interpolated data. REM manager is responsible of data processing (spatial interpolation) and data evaluation (reliability, quality, validity etc.). REM user sends requests for data acquisition. In the literature on spatial statistics, there are several spatial interpolation techniques as powerful mathematical tools that can be used. Among them, we have selected the Bayesian kriging interpolation, introduced by Kitanidis in [14], as it takes into account the various uncertainties in the models used and does not underestimate the standard errors of predictions. Furthermore, it automatically calculates the interpolation model parameters through a process of sub-settings and simulations, all this at the cost of an increment in the computation complexity as the number of measurements increases [15]. Particularly in our problem, this disadvantage does not entail a drawback because we are dealing with a semi-static REM whose construction is an offline process that in practice would be activated by the operator once per day or even a week (for comparison, classical drive test-based coverage prediction is often performed just once per year because of the amount of efforts 637
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and costs involved). Note that the Bayesian kriging interpolation presented in this paper is carried out in the REM manager. The work presented in this paper is the continuation of the work presented in [16, 17], where the Bayesian kriging method was used for the REM’s generation. This paper gives a presentation of the mathematical details of Bayesian kriging, applied to the problem of coverage prediction in cellular networks, including proofs given in the appendix. We analyse the performance of the interpolation process for an LTE network modelled with two mesh sizes, namely 5 m 5 m and 25 m 25 m, in a dense urban environment. Our aim is to (i) determine whether Bayesian kringing is an effective interpolation technique to be used for cellular coverage prediction task; and (ii) show the influence of the REM mesh size on the performance of coverage prediction. Results obtained with real network measurements show that the chosen spatial interpolation approach is indeed a very efficient and promising means of processing the MDT data, which is expected to overcrowd the operators’ databases in the near future, in terms of improving the quality of the coverage predictions. To the best of our knowledge, coverage prediction based on spatial statistics has been studied extensively so far for sensor networks but not for cellular networks. This paper’s line of work is the first to introduce spatial statistics in cellular coverage studies. In particular, this paper (i) presents in detail how Bayesian kriging is applied to cellular coverage analysis; and (ii) analyses the effect of the REM’s mesh size on the coverage prediction quality. The remainder of this paper is organised as follows. Section 2 presents comprehensively the Bayesian kriging interpolation method for coverage prediction, the considered scenario and the followed coverage prediction algorithm. Section 3 shows the obtained results for the two considered mesh sizes. Finally, Section 4 summarises our main conclusions.
the receivers, y.xi / can be expressed as y.xi / D p0 10˛ log10 di C s.xi / C zi
where p0 is the transmitted power (in dBm), ˛ is the path loss coefficient, di is the distance between the transmitter and the receiver location, xi , s.xi / is the shadow fading factor (in dB) and zi is the zero-mean additive noise term, which incorporates the uncertainties of the measurement process and all other random effects due to the propagation environment. Equation (1) is the well-known large-scale propagation model in the logarithmic scale, which models the wireless channel as the sum of a deterministic linear path loss term and two stochastic terms: shadowing and noise. This model is one of the most widely used wireless channel models due to its simplicity and to its overall ability to represent the main characteristics of the wireless channel behaviour in a variety of important wireless environments. The random noise process is assumed to consist of independent and identically distributed Gaussian samples, which are also independent of the shadowing term. Shadowing is a zeromean Gaussian random variable that is spatially correlated according to the exponential correlation model [18] ˚ dij 1 E s.xi /s.xj / D rij D exp
In this section, we summarise the steps to be followed for applying Bayesian kriging in cellular coverage prediction. First, we present the used network model and our assumptions. Second, we introduce the prediction method. Third, we explain how the parameters used in our model are estimated, and finally, we present the Bayesian kriging-based coverage prediction algorithm.
2.1. Modelling and assumptions We consider the down link transmission of a cellular radio access network with a given base station transmitter equipped with an omnidirectional antenna. Let y.xi / denote the down link received power (in dB) at location xi . Assuming that the fast fading effects are averaged out by 638
(2)
where 1 is the shadowing variance in decibel, dij is the Euclidean distance between locations xi and xj , and controls the correlation distance of the shadowing. We assume that such power measurements are carried out by a set of N receiving terminals, located at the set of locations x D fx1 ; x2 ; : : : ; xN g. Arranging these measurements in a N 1 column vector y.x/, we obtain the vector-matrix relation y D Xˇ C u
2. COVERAGE PREDICTION WITH BAYESIAN KRIGING
(1)
(3)
where 2
1 6 :: X D4 : 1 ˇD
p0 ˛
3 10 log10 .d1 / 7 :: 5 : 10 log10 .dN / 2
3 s.x1 / C z1 6 7 :: and u D 4 5 : s.xN / C zN
(4)
Here, X is a N 2 deterministic matrix of known functions of the measurement locations x, ˇ is the 2 1 parameter vector of the spatial mean and u is a N 1 multivariate Gaussian vector whose covariance matrix is Qyy .; ; / D 1 .Ryy ./ C IN /, where 1 Ryy ./ is the N N covariance matrix of the spatially correlated shadowing term whose .i ; j /th entry is equal to rij of Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
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Equation (2), IN is the N N identity matrix and Œ:T denotes vector/matrix transpose. Note that the variance of the noise process is . Note also that y, X and u are functions of the locations x.
The aim is to predict the received power values at the mesh points where we do not have measurements. Let x0 denote the M 1 vector of the mesh points. The same underlying model is assumed for the predictions, namely, y0 D X0 ˇ C u0
(5)
where y0 is the M 1 vector of received power values at locations x0 , X0 is the M 2 matrix of deterministic effects for y0 and u0 is the M 1 stochastic vector whose covariance matrix is denoted by Q00 .; ; /. Note that for notational convenience, the dependence of y, X and u (y0 , X0 and u0 , respectively) on x (x0 , respectively) is not shown in Equations (3) and (5). Putting the measurements and the predictions together, we obtain the multivariate Gaussian model given by
y0 y
N
n o yO 0 D min E .y0 y0 /T .y0 y0 /jy D E fy0 jyg y 0
2.2. Prediction
loss/risk/cost (also known as Bayes risk). The most commonly used risk function is the squared error risk or the mean squared error (MSE):
X0 Q00 .; ; / ˇ; X Qy0 .; ; /
Q0y .; ; / Qyy ; ;
(6)
where Q0y .; ; / and Qy0 .; ; / are the crosscovariance matrices between y0 and y. Note that Q0y D T . Qy0 The model parameters ˇ, , and are unknown. Therefore, they have to be estimated from the existing measurement dataset. However, those parameters are not completely unknown to us: we have some prior knowledge on their probable values. For example, we can say that the radiated power p0 is close to the power at the antenna feeder, the propagation path loss coefficient ˛ is around 3:5pin urban areas [19] and the shadowing standard deviation typically ranges between 8 and 11 dB for typical outdoor Above RoofTop to Below RoofTop scenarios [20]. Including those prior information into the model helps us enhance the prediction quality. Besides, the Gaussian assumption of the stochastic component vector, u, in the chosen model, provides tractability in the complex analytical derivations of the Bayesian inference framework. However, we emphasise that the Gaussian nature of the problem is firmly grounded on experiments and is not simply assumed for mathematical convenience. The task of predicting the received power values at locations x0 is equivalent to finding an estimator yO 0 of the random vector y0 given measurements y. This estimator is preferably a linear function of measurements y, which minimises a pre-determined loss/risk/cost function. In Bayesian context, this is equivalent to the Bayes estimator, which minimises a pre-determined posterior expected Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
(7)
which is equivalent to the posterior mean E fy0 jyg. For the MSE Bayes estimator of Equation (7), the Bayes risk (or the MSE) is the posterior variance o 1 n E .y0 yO 0 /T .y0 yO 0 /jy M o 1 n E .y0 E fy0 jyg/T .y0 E fy0 jyg/jy D M 1 trace.covfy0 jyg/ (8) D M
MSE D
The used derivations are based on [21]. For the MSE Bayes estimator, we need to calculate the marginal posterior (or Bayesian) probability density function (PDF) p.y0 jy/, or at least its moments such as mean E fy0 jyg and covariance cov fy0 jyg. In the following, we will present a brief derivation. We start by the following expression for the Bayesian PDF: ZZZZ p.y0 jy/ D
;;;ˇ
p.y0 jˇ; ; ; ; y/p.ˇ; ; ; jy/
(9) The first integrand in Equation (9) is the conditional PDF of y0 given y and the model parameters. Assuming that Qyy is non-singular, it can be shown (see Proof 1 in the Appendix) that this PDF is Gaussian with mean 1 .y X ˇ/ and covariance matrix Q X0 ˇ C Q0y Qyy 00 1 Q0y Qyy Qy0 . Note that for notational convenience, the dependence of Q0y and Qy0 on the model parameters is not shown here. The second integrand in Equation (9) is the joint posterior PDF of the model parameters, which can be decomposed into the product of two joint posterior PDFs: p.ˇ; ; ; jy/ D p.ˇ; j; ; y/p.; jy/
(10)
We can express the first joint posterior PDF p.ˇ; j; ; y/ as the product of a likelihood term and a prior PDF given by p.ˇ; j; ; y/ D c1 .y; ; / p.yjˇ; ; ; /p.ˇ; j; / (11) h’ i1 where c1 .y; ; / D ˇ; p.yjˇ; ; ; /p.ˇ; j; / is a normalising constant term, which can be computed numerically, p.ˇ; j; / is the joint prior PDF of ˇ and (with and known) and p.yjˇ; ; ; / is the likelihood 639
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term, which is Gaussian with mean y X ˇ and covariance Qyy : ˇ ˇ1=2 p.yjˇ; ; ; / D .2/N =2 ˇQyy ˇ 1 T 1 exp .y X ˇ/ Qyy .y X ˇ/ 2 (12) An important issue in Bayesian inference is the choice of prior distributions. It is common practice to choose functional forms that facilitate the involved analytical treatments. Such functional forms are called conjugate priors, meaning conjugate to the likelihood function, such that the posterior distribution has the same functional form as the prior distribution. Conjugate priors have been identified for the most widely used distribution functions [15]. In order to find the conjugate prior for p.ˇ; j; /, we rewrite the likelihood p.yjˇ; ; ; / of Equation (12) in the following form (see Proof 2 in the Appendix):
p.yjˇ; ; ; / D
with 0 D rank.H 0 /. This PDF represents a multivariate Gaussian random vector ˇ whose covariance matrix is scaled with a random variable whose inverse is Gamma distributed. The joint posterior PDF p.ˇ; j; ; y/ is obtained by replacing Equations (12) and (20) in Equation (11) and is also a Normal-Gamma-2 density 00
with the following parameters (see Proof 3 in the Appendix):
.2/N =2 jSyy j1=2 =2 exp .ˇ b/T H .ˇ b/ 2 1 (13) =2 exp q 2 where Syy D Qyy D Ryy C IN
(14)
1 H D X T Syy X
(15) q 00 D
Hb D X
T
1 Syy y
DN
(18)
1 q D yT Syy .y X b/=
(19)
Note that the expression in Equation (13) is a proper PDF if D rank.X /. Otherwise, it represents only < rank.X / linear combinations of ˇ. The functional form of Equation (13) calls for the Normal-Gamma-2 density as the conjugate prior for p.ˇ; j; / [15], given by
p.ˇ; j; /D.2/
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jH j
(22)
H 00 b00 D H 0 b0 C H b
(23)
00 D rank.H 00 /
(24)
00 D 0 C 0 C C 00
(25)
1 y C b0T H 0 b0 b00T H 00 b00 0 q 0 C yT Syy
00
(26)
The second joint posterior of Equation (10), p.; jy/, can be expressed as follows: (17)
0 1=2 0 =2
H 00 D H 0 C H
(16)
D rank.H /
0 =2
00
p.ˇ; j; ; y/ D .2/ =2 jH 00 j1=2 =2 exp .ˇ b00 /T H 00 .ˇ b00 / 2 00 00 00 00 =2 q 00 =21 1 2 2 1 exp 00 q 00 (21) 2
p.; jy/ D c2 .y/p.yjˇ; ; ; /p.ˇ; j; /p.; / 1 p.ˇ; j; ; y/ (27) hRRRR c2 .y/ D ˇ;;; p.yjˇ; ; ; /p.ˇ; j; /p.; /1 is a function of y only and can be numerically evaluated. Using Equations (12), (20) and (21) in the aforementioned equation and simplifying the terms, we obtain
0 0 0 0 =2 q 1 0 0 0 T 0 0 1 0 =21 exp .ˇ b / H .ˇ b / exp q 2 2 2 2 (20)
Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
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p.; jy/Dc2 .y/.2/
.N C0 00 /=2
jSyy j
1=2
0 1=2
jH j
00 1=2
jH j
The joint prior p.; / can be written as the product of the marginal densities p./ and p. /, because the noise process is assumed independent of the shadowing process. Furthermore, we assume discrete PDFs for p./ and p. /.
1
0 00 0 0 0 =2 00 00 00 =2 q q p.; / 2 2 2 2 (28)
The inner integral, p.y0 j; ; ; y/, is the marginalisation (with regard to ˇ) of a joint Gaussian PDF and, therefore, is Gaussian with the following mean and variance [14]:
1 1 1 Efy0 j; ; ; yg D X0 Q0y Qyy X H 0 C X T Qyy X H 0 b0 1 1 1 1 1 y C Q0y Qyy C X0 Q0y Qyy X H 0 C X T Qyy X X T Qyy
(32)
1 T 1 1 1 1 X0 Q0y Qyy varfy0 j; ; ; yg D Q00 Q0y Qyy Qy0 C X0 Q0y Qyy X H 0 C X T Qyy X X (33) Then, the Bayesian PDF p.y0 jy/ can be rewritten as: Therefore,
p.; / D
XX k
k l ı. k /ı. l /
(29)
l
p.y0 jy/ D “ Z ;
where ı.:/ is the Dirac delta function. Then, Equation (28) takes the following form:
p.y0 j; ; ; y/p. j; ; y/d p.; jy/d d (34)
The inner integral is recognised as a Student distribution [15] with mean and covariance matrix of
i h 1 1 1 1 1 1 y Efy0 j; ; yg D X0 S0y Syy X H 00 H 0 b0 C S0y Syy C X0 S0y Syy X H 00 X T Syy
covfy0 j; ; yg D 0
T 00 q 00 1 1 00 1 1 S X S S S C X S S X H S S X 00 0y y0 0 0y 0 0y yy yy yy 00 2 00
p.; jy/ D c2 .y/.2/.N C /=2 0 00 0 0 0 =2 q jH 0 j1=2 1 2 2 2 XX k l jSyy .k ; l /j1=2 jH 00 .k ; l /j1=2 k
l 00
q 00 .k ; l / 2
00 =2 ı. k /ı. l /
(30) Note that in the aforementioned equation, Syy , H 00 and q 00 are functions of and . Now, we can return to the Bayesian PDF of Equation (9) and rewrite it as follows: • p.y0 jy/ D ;;
Z
ˇ
p.y0 jˇ; ; ; ; y/p.ˇj; ; ; y/d ˇ
p. j; ; y/p.; jy/d d d (31) Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
(35)
(36)
where S00 D Q00 , S0y D Q0y and Sy0 D Qy0 are the scaled covariance and cross-covariance matrices, respectively. Finally, using Equations (30), (35) and (36), we can calculate the posterior mean of the MSE Bayes estimator as X p.; jy/Efy0 j; ; yg (37) Efy0 jyg D ;
and the covariance is given by covfy0 jyg D
X
p.; jy/
;
Œcovfy0 j; ; yg C .Efy0 j; ; yg
(38)
Efy0 jyg/.Efy0 j; ; yg Efy0 jyg/T 641
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2.3. Model parameter estimation
and
Estimation of the model parameters is carried out by calculating their posterior expectations, that is, evaluating Z E fˇjyg D
ˇp.ˇjy/d ˇ ˇ
“
Z
ˇ
D
p.ˇj; ; y/p.; jy/d d d ˇ
ˇ
;
Z
“ D
ˇp.ˇj; ; y/d ˇ p.; jy/d d
ˇ
;
“
E fˇj; ; yg p.; jy/d d
D
;
E fˇjyg D
X X k ;k
p. jy/d Z “ D p. j; ; y/p.; jy/d d d ;
Z
“ D
p. j; ; y/d p.; jy/d d
E f j; ; yg p.; jy/d d
D
(40)
;
To calculate the aforementioned expressions, we need the marginal posteriors for and ˇ, which become ˇ 00 ˇ 1 ˇ H ˇ 2 00 00 1 00 ˇ ˇ 2 ˇ q 00 ˇ 2 00 C 00 2 00 C 00 00 2 1 00 T H 00 1 C 00 .ˇ b / .ˇ b / q 00
p.ˇ j; ; y/ D
00 2
(41) and 00 00 q 00 2 1 2 00 00 2 1 exp 00 q 00 2 2
p. j; ; y/ D
(42)
Equations (41) and (42) are obtained by integrating Equation (21) with regard to ˇ and , respectively, [21]. Equation (41) is a 00 -multivariate Student distribution with 00 degrees of freedom, scaling matrix q 00 H 001 and mean b00 , whereas Equation (42) is a Gamma-2 distribution with mean q 001 and variance 2=. 00 q 002 /. Then we can rewrite Equations (39) and (40) as “ E fˇjyg D ;
b00 p.; ; y/p.; jy/d d
(45)
1 q 00 .k ; k ; y/
(46)
p.k ; k jy/
2.4. Coverage prediction algorithm
;
“
p.k ; k jy/b00 .k ; k ; y/
These equations form the foundation for our coverage prediction algorithm introduced in the following.
Z E f jyg D
(44)
k ;k
E f jyg D
and
1 d d q 00 .; ; y/p.; jy/
Considering the discrete nature of p.; jy/, the two integrals of Equations (43) and (44) become weighted sums of PDFs; hence, the posterior parameter expectations become weighted averages of the conditional parameter expectations:
(39)
;
642
“ E f jyg D
(43)
On the basis of partial information regarding the network coverage, the operator can use the Bayesian kriging technique, described earlier, to construct a full map over the area under analysis. To do so, we assume the existence of a REM manager, a software framework for carrying out such spatial data interpolation [13], located at the network’s Operation and Maintenance centre. The REM manager has two main tasks, namely (i) to collect the received power information in the analysed area; and (ii) to perform the coverage estimation. With the Bayesian kriging technique, the operator will have an automated and remote representation of the network coverage avoiding or minimising drive tests and the expends and delays they imply. The coverage estimation algorithm proposed in this work assumes that the area of interest is spatially discretised. We consider a regular mesh [11], and we refer to the minimum area of the mesh as a pixel. We assume that the measurements performed by any UE inside this pixel correspond to only one location, for instance, the centre of the pixel. We define p as the minimum percentage of required measurements, which is the minimum required percentage of pixels with available measurements to perform the interpolation process. The coverage estimation algorithm can be summarised then in the following steps: (1) The REM manager sends measurement requests to the UEs until it collects a required percentage, p, of pixels with available measurements to perform the interpolation process. To achieve p, two options are possible: (i) the REM manager broadcasts the pixels where the measurements are required, and only the UEs in these pixels report their measurements; or (ii) the REM manager requests that all the UEs report their locations with additional information such as velocity or battery life and then chooses Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
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the UEs that should perform and report the measurement. Although in the first option, the signalling overhead is minimised, an algorithm that compares the requested location with the current UE locations and makes the decision of performing and reporting the measurement needs to be implemented by the UEs. The second option is simpler, from an implementation point of view, as the UEs are only asked to report their location and measurements. Besides, it is worth pointing out that the delay in waiting for the required UE measurements is typically lower than the required time to perform and process the drive tests. Therefore, we can affirm that the proposed methodology delays for the coverage prediction are likely not significant in comparison with traditional methods. (2) Once the minimum amount of required measurements are gathered, the REM manager performs the Bayesian kriging interpolation, presented in Section 2, to estimate the signal power values in those points where it lacks information. Finally, the REM manager constructs the REM by overlapping the available real measurements with the interpolated ones.
3. EVALUATION OF THE INTERPOLATION PROCESS ACCURACY In this section, we present the results obtained when the previously introduced Bayesian kriging technique is used for cellular coverage analysis. We first introduce the considered scenarios and simulations assumptions, and then we present an analysis on the interpolation process accuracy.
early deployment stage, we first build a 3G coverage map based on 3G received pilot powers [reference signal code power (RSCP)] and then translate these RSCP values to their equivalent metric in LTE RSRP. To translate the 3G RSCP to LTE RSRP, we use the methodology proposed in [22]. The 3G RSCP coverage map is computed with a reliable planning tool with a highly accurate propagation model, which is commonly used for operational network planning [23]. For the calibration of the propagation model, the environmental conditions, that is, terrain profile and buildings, were considered, and they were also validated through repeated drive tests. Therefore, it can be assumed that the LTE coverage map, which we consider as the ‘ground truth’, is very close to the real wireless coverage situation. We consider a dense urban scenario in the south of Paris with an area of 250 m250 m, covered by a macrocell with an omnidirectional antenna. The analysis is performed over RSRP measurements, as presented in Figures 1(a) and 2(a), for two mesh sizes, that is, 5 m 5 m and 25 m 25 m. We define a minimum RSRP ı D 114 dBm. Those pixels where the received RSRP is below this threshold are considered to be out of coverage. In particular, the results presented in this paper are obtained for a random selection of the pixels where the operator lacks information for reasons of simplicity. In reality, the measurement locations are expected to follow certain mobility patterns. The incorporation of realistic mobility patterns in measurement locations is part of our ongoing research work. Figures 1(b) and 2(b) represent the obtained REM after a typical realisation of the interpolation process, as explained in the step two of the coverage estimation algorithm in Section 2.4. Figures 1(c) and 2(c) represent the obtained errors (in decibel). The next section presents a detailed analysis of the obtained error in the prediction process.
3.1. Simulation assumptions 3.2. Simulation results For simulation purposes, we construct an LTE coverage map composed of LTE received pilot powers [reference signal received power (RSRP)]. As we aim to be as close as possible to the ‘ground truth’, and as LTE is in very 500
In this section, we evaluate the accuracy of the proposed interpolation technique. Statistics are computed for 100 snapshots, from whereon the results converge. In order to 500
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evaluate the feasibility of the REM in terms of the influence of the considered mesh size, we compare the results obtained for two considered scenarios. Multiple statistics presented in this section have been measured for different percentages of measurements, p D f50; 80g%, used in the interpolation process. These values have been chosen on the basis of experts’ knowledge. The percentage of available measurements depends on the pixel size and the mobiles density. Also, considering the MDT, operators can ask to specific UEs to report measurements. Figures 3 and 4 represent the average interpolation error for both considered cases, that is, mesh size of 5 m 5 m and 25 m 25 m, respectively. It measures the average amount of pixels incorrectly estimated in the interpolation process. This average error consists of the false alarm and misdetection probabilities. The false alarm probability measures how many pixels are estimated to have a signal power below the threshold when in reality it is above the threshold. The misdetection probability measures how many pixels are estimated to have a signal power above the threshold when the real measurement is below or equal to
the signal power threshold [17]. Results presented in both figures show that the interpolation process performs well, even for low amount of available measurements, that is, p D 50%, with an average error around 2.4% and 3.4% for 5 m 5 m and 25 m 25 m cases, respectively. As was expected, the average error decreases as the amount of available measurements for the interpolation process increases to p D 80%, reaching a value of around 0.8% and 1.2%, respectively. Therefore, when the available measurements of the network increase from p D 50% to p D 80%, the average error decreases, by approximately 67% and 65% for the 5 m 5 m and 25 m 25 m cases, respectively. We can observe that, for the two studied cases, the misdetection probability is higher than the false alarm one. The difference between these two probabilities is because Bayesian kriging is based on the assumption that the underlying stochastic structure is ‘smooth’ [revealed by the exponential model, i.e. covariance matrix entries given by Equation (2)]. At those locations where the real structure has abrupt changes (severe shadowing), the predictions will decay smoothly, and predictions will inevitably be
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above the real values. So the interpolation algorithm will ‘miss’ those uncovered pixels. If information on buildings and other structures affecting the propagation environment is available, however, they can be included as part of the Bayesian interpolation framework as additional shadowing-like components and used to improve prediction accuracy in regions in which such abrupt changes in coverage occur. Figures 5 and 6 present the mean absolute error (MAE) for the 5 m 5 m and 25 m 25 m mesh size cases, respectively. The MAE measures the average magnitude of the errors in a set of forecasts, without considering their direction. As expected, when ‘low’ amount of measurements are available, that is, p D 50%, the obtained errors are higher than for the case of ‘high’ amount of available measurements, that is, p D 80%, because the interpolation process is more accurate. Comparing Figures 5 and 6, it can be observed that for small size meshes, errors are on average smaller than when bigger mesh sizes are considered. This fact evidences the dependence of the interpolation process on the typical distance between points in the mesh. Therefore, if the distance between available measurements and the points where interpolation is performed is smaller, the Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
interpolation is more precise, that is, closer the available information, higher the accuracy of predictions. That is why for the 5 m 5 m case, the interpolation process is more precise and accurate than for the 25 m 25 m. This is also the reason why in Figure 6, even when considering p D 80% of measurements for the interpolation, we obtain some ‘high’ average errors. In particular, for the 5 m 5 m mesh size case, when p D 50%, the MAE goes from around 1.08 to 1.2 dB with a peak in 1.12 dB, and when there is a higher number of available measurements, p D 80%, the MAE goes from 0.83 to 1 dB with a peak in 0.92 dB. Therefore, when p is increased from 50% to 80%, the histogram moves to the left. Thus, we can affirm that in average, the accuracy of predictions improves by 0.2 dB. For the 25 m 25 m mesh size case, the improvement in the average accuracy of the predictions is around 0.3 dB. Finally, Figure 7 represents the cumulative distribution function of the error obtained for p D f50; 80g% measurements in a typical realisation, for 5 m5 m and 25 m25 m mesh sizes. As can be observed, the cumulative distribution functions are smoother for the 5 m 5 m case than for the 25 m 25 m because when the mesh size considered in the area under coverage analysis is smaller, the interpolation is performed in a larger number of pixels, and therefore, there are more error samples. This figure shows that for the 5 m 5 m case, the 80% of the errors are distributed between 2 and 2 dB, meanwhile for the 25 m25 m, 80% of the errors are distributed between 5 and 5 dB.
4. CONCLUSION In this paper, we have presented in detail how a powerful mathematical tool coming from spatial statistics, Bayesian kriging interpolation, can be used to construct a cognitive tool, REM, for cellular coverage prediction. Detailed mathematical derivations with proofs for the developed estimators have been given for this purpose. In order to 645
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find out whether Bayesian kriging is an effective interpolation method for cellular coverage prediction, we have evaluated the prediction performance of the interpolation process using realistic measurement data of an LTE network in a dense urban environment. The impact of the mesh size on prediction quality has also been evaluated by carrying out predictions and comparative evaluations with two mesh sizes, that is, 5 m5 m and 25 m25 m. Results show that Bayesian kriging interpolation is a promising technique, which can be used for cellular coverage prediction. The prediction accuracy increases with decreasing mesh size, at the expense of an increase in computational complexity. This, however, is an affordable cost, because (i) coverage prediction is a semi-static process (coverage characteristics do not change very frequently) that can be carried out in an offline manner at a centralised processing centre (i.e. ‘cloud’-type architectures); and (ii) compared with the conventional process of manual coverage prediction based on drive tests, the required investment for the proposed approach (in terms of the time operational expenditure for processing power) is very low. The trade-off between computational complexity and prediction performance for varying mesh sizes can be exploited in a layered-REM framework [24], where large-scale REMs with high mesh sizes can be used over large geographical areas to have high-level information on the coverage situation over that area. From this high-level information, the sub-areas that need more precise treatment can be detected, and smaller scale REMs with low mesh sizes can be constructed for obtaining finer information with higher accuracy/prediction over that sub-area. Future research work will include enhanced interpolation models with multiple base stations, where spatial statistical techniques for modelling inter-cell interference will be developed. It is also foreseeing to include mobility patterns in the available measurements.
ACKNOWLEDGEMENTS
According to Equation (6), the joint conditional PDF p.y0 ; yjˇ; ; ; / is multivariate Gaussian with mean ŒX0 X T ˇ and covariance matrix
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The marginal PDF p.yjˇ; ; ; /, being equal to the likelihood term of Equation (12), is also multivariate Gaussian with mean y X ˇ and covariance Qyy . Then, the conditional distribution p.y0 jˇ; ; ; ; y/ is also mul1 .y X ˇ/ tivariate Gaussian with mean X0 ˇ C Q0y Qyy 1 and covariance matrix Q00 Q0y Qyy Qy0 [25, 26]. Proof 2. Replacing Qyy by Syy = , the likelihood function of Equation (12) can be rewritten as ˇ ˇ1=2 N =2 p.yjˇ; ; ; / D .2/N =2 ˇSyy ˇ exp .y X ˇ C X b X b/T 2 1 Syy .y X ˇ C X b X b/ (A.3) Splitting the term N =2 into two product terms and rearranging the exponentials, we obtain ˇ ˇ1=2 =2 p.yjˇ; ; ; / D .2/N =2 ˇSyy ˇ 1 exp .X b X ˇ/T Syy .X b X ˇ/ 2 1 .y X b/ .N /=2 exp .y X b/T Syy 2 (A.4)
The authors would like to thank Jean-Baptiste Landre, from Orange Labs, for his helpful comments. We thank EU for providing partial funding of this work through the FARAMIR project (Grant number ICT-248351). J. R. and P. M. also thank RWTH Aachen University and DFG for partial financial support through the UMIC Research Centre.
APPENDIX Proof 1. We can write the conditional PDF p.y0 jˇ; ; ; ; y/ as the quotient of the joint conditional PDF p.y0 ; yjˇ; ; ; / by the marginal PDF p.yjˇ; ; ; /: p.y0 jˇ; ; ; ; y/ D p.y0 ; yjˇ; ; ; /=p.yjˇ; ; ; / (A.1) 646
Factoring out X in the first exponential and recognising 1 X D H , the first exponent is observed to be that X T Syy equal to =2 multiplied by .b ˇ/T H .b ˇ/. As for the second exponential, it yields the following expression if we expand the transposed term: i
h 1 1 .y X b/ bT X T Syy .y X b/ exp yT Syy 2 (A.5) 1 .y The term on the right vanishes because X T Syy X b/ D 0 because b is a solution of the equation 1 y. Finally, replacing the definitions in H b D X T Syy Equations (17), (18) and (19), we obtain
Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
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ˇ ˇ1=2 p.yjˇ; ; ; / D .2/N =2 ˇSyy ˇ =2 exp .ˇ b/T H .ˇ b/ 2 =2 exp q (A.6) 2
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.H b C H 0 b0 /T ˇ ˇ T .H b C H 0 b0 / (A.12)
Proof 3. Using Equations (12) and (20), we can rewrite Equation (11) as follows: p.ˇ; j; ; y/ D c1 .y; ; / ˇ ˇ1=2 1 N =2 ˇ T 1 ˇ Qyy .2/ exp .y X ˇ/ Qyy .y X ˇ/ 2 0 0 .2/ =2 jH 0 j1=2 =2 exp .ˇ b0 /T H 0 .ˇ b0 / 2 0 0 0 0 =2 1 q 0 =21 exp 0 q 0 1 2 2 2 (A.7) Replacing Qyy by Syy = and regrouping the terms, we obtain 0
1 y C b0T H 0 b0 C ˇ T .H C H 0 /ˇ V D yT Syy
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Finally, rearranging and regrouping terms gives
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Adding and subtracting the term b00T H 00 b00 and recalling that H 00 D H 0 C H and H 00 b00 D H 0 b0 C H b, we find
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From Equation (26), the first three terms are recognised as q 00 00 q 0 0 . Then, Equation (A.8) can be rewritten as
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where 0
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V Dy
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C ˇ T H 0 ˇ C b0T H 0 b0 ˇ T H 0 b0 b0T H 0 ˇ (A.11) Rearranging and regrouping the terms yields Trans. Emerging Tel. Tech. 24:636–648 (2013) © 2013 John Wiley & Sons, Ltd. DOI: 10.1002/ett
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