{"title": "GPPS: A Gaussian Process Positioning System for Cellular Networks", "book": "Advances in Neural Information Processing Systems", "page_first": 579, "page_last": 586, "abstract": "", "full_text": "GPPS: A Gaussian Process Positioning System\n\nfor Cellular Networks\n\n\u2217\nAnton Schwaighofer\n, Marian Grigoras\u00b8, Volker Tresp, Clemens Hoffmann\n\nSiemens Corporate Technology, Information and Communications\n\n81730 Munich, Germany\n\nhttp://www.igi.tugraz.at/aschwaig\n\nAbstract\n\nIn this article, we present a novel approach to solving the localization\nproblem in cellular networks. The goal is to estimate a mobile user\u2019s\nposition, based on measurements of the signal strengths received from\nnetwork base stations. Our solution works by building Gaussian process\nmodels for the distribution of signal strengths, as obtained in a series\nof calibration measurements. In the localization stage, the user\u2019s posi-\ntion can be estimated by maximizing the likelihood of received signal\nstrengths with respect to the position. We investigate the accuracy of\nthe proposed approach on data obtained within a large indoor cellular\nnetwork.\n\n1\n\nIntroduction\n\nCellular networks form the basis of modern wireless communication infrastructure. Ex-\namples include GSM and UMTS networks for mobile phones, wireless LAN (WLAN) for\ncomputer networks, and DECT for cordless phones. Within these networks, location-based\nservices (services that are tailored speci\ufb01cally to the current position of the mobile user)\nhave great potential. Examples of such services are guiding the user through a building\nor city, delivering the time-table of buses at the nearest bus stop, or simply answering the\nuser\u2019s query \u201cWhere am I?\u201d. All such services crucially depend on methods to accurately\nestimate the position of the mobile user within the network (\u201clocalization\u201d, \u201cpositioning\u201d).\n\nIn this article, we present a novel approach to obtain position estimates for the mobile user.\nMost importantly, this method is based solely on infrastructure that is already present in\na typical cellular network, and thus leads to minimal extra cost. Furthermore, we focus\non indoor networks, where a number of speci\ufb01c problems needs to be addressed. Since\nour approach relies heavily on Gaussian process models, we call it the \u201cGaussian process\npositioning system\u201d (GPPS).\n\nWe proceed by introducing the localization problem in detail in Sec. 1.1, and by giving a\nbrief overview of previous approaches. Sec. 2 follows with a description of the Gaussian\nprocess positioning system (GPPS). Sec. 3 shows how the required calibration stage of\nthe system can be performed in an optimal manner. Sec. 4 presents an evaluation of the\n\n\u2217\n\nAlso with the Institute for Theoretical Computer Science, Graz University of Technology, Austria\n\n\fGPPS in a DECT network environment. We show that the GPPS gives accurate location\nestimates, in particular when only very few calibration measurements are available.\n\n1.1 Problem Description\n\nOur overall goal is to develop a localization system for indoor cellular networks, that is (in\norder to minimize cost) based solely on existing standard networking hardware. Location\nestimates can be based on different characteristics of the radio signal received at the mobile\nstation (i.e., the laptop in a WLAN network, or the phone in a DECT network). Yet,\nin most hardware, the only available information about the radio signal is the received\nsignal strength. Information like phase or propagation time from the base station requires\nadditional hardware, and can thus not be used.\n\nIn general, estimating the user\u2019s position based only on measurements of the signal strength\nis known to be a very challenging task [7], in particular in indoor networks. Due to re-\n\ufb02ections, refraction, and scattering of the electromagnetic waves along structures of the\nbuilding, the received signal is only a distorted version of the transmitted signal. Noise\nand co-channel interference further corrupt the signals [4]. Furthermore, when using stan-\ndard hardware, we must expect a high level of measurement noise for the signal strength.\nChanges in the environment can also have a strong impact on signal propagation. For ex-\nample, in a WLAN environment [1], it has been noted that shielding by a single person can\nattenuate the signal by up to \u22123.5 dBm. Also, the localization system ought to be robust,\nsince base stations may fail, be switched off, or may be temporarily shielded for unknown\nreasons. In these cases, a sensible localization system should not draw the conclusion that\nthe user is far from the respective base station.\n\nDue to the complex signal propagation behaviour, almost all previous approaches to indoor\nlocalization use an initial calibration stage. Calibration here means that signal strengths\nreceived from the network base stations are measured at a number of points inside the\nbuilding. Systems differ in their ways of using this calibration data.\nIn principle, two\nbasic approaches can be used here. In a \u201cforward modelling\u201d approach, a model of signal\nstrength as a function of position is built \ufb01rst. The localization procedure then tries to\n\ufb01nd the location which best agrees with the measured signal strengths. Alternatively, the\nmapping from signal strengths to position can be modelled directly (\u201cinverse modelling\u201d).\n\nThe RADAR system [1], one of the \ufb01rst indoor localization systems, is an inverse mod-\nelling approach using a nearest neighbor technique. [7] build simple probabilistic models\nfrom the calibration data (forward modelling), in conjunction with maximum likelihood\nposition estimation. Bayesian networks have been considered by [2], with states of node\ncorresponding to different locations (using coarse discretization). Discrete locations, yet\nwith a \ufb01ner grid, are also considered in [5], in an approach inspired by robot navigation.\n\n2 The Gaussian Process Positioning System\n\nThe dif\ufb01culties of indoor localization, as mentioned in Sec. 1.1, call for a probabilistic\nmethod for localization. The key idea of the Gaussian process positioning system (GPPS)\nis to use Gaussian process models for the signal strength received from each base station,\nand to obtain position estimates via maximum likelihood, i.e. by searching for the position\nwhich best \ufb01ts the measured signal strengths.\n\nConsider a cellular network with a total of B base stations. Assume that, for each of base\nstations, we have a probabilistic model that describes the distribution of received signal\nstrength. More formally, we denote by p j(s j |t) the likelihood of receiving a signal strength\ns j from the j-th base station on position t.\n\n\fWith the models p j(s j |t), j = 1, . . . , B given, localization can be done in a straight-forward\nway. The user reports a vector s (of length B) of signal strength measurements for all base\nstations.\nIt may occur that no signal is received from some base stations (indicated by\ns j = /0), e.g., because the user is too far from this base station, or due to hardware failure.\nIn the GPPS, the estimated position \u02c6t is computed by maximizing the joint likelihood1 with\nrespect to the unknown position,\n\n(1)\n\n\u02c6t = argmax\n\nt\n\nj:s j(cid:54)=/0\n\np j(s j |t).\n\nIn the above equation, we only use the likelihood contributions of those base stations that\nare actually received. Alternatively, one could use a very low signal strength as a default\nvalue for each base station that is not received [7]. We found that this can give high errors\nif a base station close to the user fails, since now the low default value indicates that one\nshould expect the user to be far from the base station. Thus, by using the above expression,\nwe also obtain a certain degree of robustness with respect to hardware failures or other\nunexpected effects.\nYet, we still need to de\ufb01ne and build suitable base station models p j(s j |t), j = 1, . . . , B.\nIn the GPPS, we use Gaussian process (GP) models for this task, where each base station\nmodel is estimated from the calibration data. Gaussian processes are particularly useful\nhere for several reasons. Firstly, one obtains a full predictive distribution, as opposed to the\npoint estimate output by other regression approaches. Secondly, GPs are a nonparametric\nmethod that can \ufb02exibly adapt to the complex signal propagation behaviour observed in\nindoor cellular networks.\n\nMind that this approach opens a wide range of possibilities for further extensions. Due\nto particular project requirements, we currently only use the maximum likelihood position\nestimate in Eq. (1) (\u201cone-shot localization\u201d without error estimates). Instead of the implic-\nitly assumed uninformative prior in Eq. (1), one could, for example, specify an informative\nprior based on known previous positions of the user, in conjunction with a motion model.\nSubsequently, the complete posterior distribution p(t|s) can be evaluated for localization.\nIn the following sections, we will describe the GP models in more detail, and also discuss\nthe choice of kernel function, which is of great importance in order to build an accurate\nlocalization system.\n\n2.1 Gaussian Process Models for Signal Strengths\nIn the GPPS, a Gaussian process (GP) approach is used for the models p j(s j |t) that de-\nscribe the signal strength received from a single base station j. Details on GP models can\nbe found, for example, in [6]; we only give a brief summary here.\n\nRecall from Sec. 1.1 that the proposed GPPS is based on a set calibration measurements,\nwhere the signal strength is measured at a number of points spread over the area to be\ncovered. Consider now the calibration data for a single base station j. We denote this cali-\nbration data by D j = {(xi, yi)}N\ni=1, meaning that a signal strength of yi has been measured\non point xi, with a total of N calibration measurements.\nFor simplicity of computation, we use a GP model with Gaussian noise, i.e., the measured\nsignal strength yi is composed of a \u201ctrue\u201d signal strength s(xi) plus independent Gaus-\nsian (measurement) noise ei of variance s 2, with yi = s(xi) + ei. The Gaussian process\nassumption for the true signal s implies that the true signal strengths for all calibration\n\n1Assuming independence of the individual measurements. One could also use a solution inspired\nfrom co-kriging, that takes into account the full dependence between signals received from different\nbase stations. We did not consider this solution for reasons of ef\ufb01ciency.\n\n(cid:213)\n\fpoints (s(x1), . . . , s(xN)) are jointly Gaussian distributed, with zero mean and covariance\nmatrix K. K itself is given by the kernel (covariance) function k, with Kmn = k(xm,xn),\nm, n = 1, . . . , N.\nGiven the calibration data D j, the predictive distribution for the signal strength s j received\non some arbitrary point t turns out to be Gaussian. With v(t) = (k(t,x1), . . . , k(t,xN))(cid:62)\n,\ny = (y1, . . . , yN)(cid:62)\n\nand Q = K + s 2I, mean and variance of the prediction are\n\nE(s j |D j,t) = v(t)(cid:62)\nvar(s j |D j,t) = k(t,t)\u2212 v(t)(cid:62)\n\n\u22121y\nQ\n\n\u22121v(t)\nQ\n\n(2)\n(3)\n\nUsing these expressions for the predictive distribution (a univariate Gaussian) in Eq. (1)\nbecomes straight forward. Also, gradients of the likelihood with respect to the position t\ncan be derived easily [8]. Thus, the position estimate, Eq. (1), can be computed easily using\neither some standard optimization routine, or by evaluating the likelihood grid-based in the\narea of interest.\nAn important issue is also the choice of noise variance s 2 and the parameters q of the kernel\nfunction k (which we have not explicitly denoted above) . We set them by maximizing the\nmarginal likelihood of the calibration data with respect to the model parameters, which\nturns out to be [6]\n\n( \u02c6s 2, \u02c6q) = argmax\n\ns 2,q (cid:16)\u2212logdet Q\u2212 y\n\n\u22121y(cid:17) .\n\n(cid:62)\n\nQ\n\n(4)\n\nThe model parameters ( \u02c6s 2, \u02c6q) are set individually for each base station.\n\n2.2 The Mat\u00b4ern Kernel Function\n\nIn our GPPS application, with a 2-dimensional input space for the GP models, the choice\nof an appropriate kernel function is a more critical issue if compared to typical machine\nlearning applications with many input dimensions. For the commonly used squared expo-\n(cid:48)(cid:107)2), it has been argued [9] that sample paths of\nnential kernel, k(x,x\nsuch GP models are \u201cin\ufb01nitely smooth\u201d, thus often leading to unreasonably low predictive\nvariance. In GPPS, we instead use the Mat\u00b4ern class of kernel functions [9], which allows a\ncontinuous parameterization of the smoothness of the sample paths via its parameter n. Its\nfunctional form is\n\n(cid:48)) = exp(\u2212w(cid:107)x \u2212 x\n\nk(x,x\n\n(cid:48)) = Mn (z) =\n\n(5)\nwhere G( n) is the Gamma function and Kn (r) is the modi\ufb01ed Bessel function of the second\nkind of degree n. The parameter\nn determines the smoothness (fractal dimension) of the\nsample paths and can be estimated from the data using Eq. (4). We use an isotropic kernel\nfunction with length scale w, thus z2 = w(cid:107)x\u2212 x\n\n(cid:48)(cid:107)2.\n\n2(cid:0)\u221a\nnz (cid:1)n\n\nG( n)\n\n\u221a\nnz)\nKn (2\n\n2.3 Learning GP Models with Mat\u00b4ern Kernel\n\nFor ef\ufb01cient solutions of Eq. (4), we require derivatives of the Mat\u00b4ern kernel function\nEq. (5) with respect to all its parameters n, w. Numerical gradients, as used for example by\n[9], require a large number of evaluations of the Bessel functions and thus lead to a huge\ncomputational overhead. To compute the derivatives analytically, we use\n\nG(\n\nn)\n\n= G( n) Y(\n\nn)\n\nand\n\n\u00b6K n (z)\n\n\u00b6z\n\n= \u2212 1\n2\n\n(Kn\u2212 1(z) + Kn+ 1(z))\n\n(6)\n\nwhere Y(\nis the Polygamma function of order 0. To the best of our knowledge, there is\nno closed form expression for the derivative of the Bessel function Kn (z) with respect to its\n\nn)\n\n\u00b6\n\u00b6\nn\n\f\u00b6z\n\n\u00b6K n (z)\n\ndegree n. We approximate this by\nidentities, we \ufb01nd for the gradients of the Mat\u00b4ern function, Eq. (5),\n\u221a\n\u221a\nnz) + Kn+ 1(2\n\n\u00b6M n (z)\n\n= DKn (z) \u2248 e \u22121(Kn+ e (z)\u2212 Kn (z)). Using these\nn (cid:0)\u221a\nnz (cid:1)n\n(cid:0)Kn\u2212 1(2\nn) (cid:19)\nnz (cid:1)\u2212 Y(\n+ log(cid:0)\u221a\n(cid:18)\u2212 z\nn (cid:0)Kn\u2212 1(2\n\nnz) (cid:1) + DKn (2\n\n\u221a\n\u221a\nnz) + Kn+ 1(2\n\nnz) (cid:19) .\n\nnz) (cid:1)\n\n\u00b6M n (z)\n\nz\n\n=\n\n\u221a\nMn (z)\u2212 2\n= Mn (z)(cid:18) 1\n2(cid:0)\u221a\nnz (cid:1)n\n\n+\n\n2\n\nG( n)\n\n\u221a\n\n2\n\nG( n)\n\n(7)\n\n\u221a\n\nBased on the above equations, derivatives of Eq. (4) with respect to the model parameters\ns 2,n, w can be computed using standard matrix algebra, see [6].\n\n3 Optimal Calibration and Model Building\n\nIn order to make the GPPS, as presented in Sec. 2, a practical system, two further is-\nsues need to be solved. Firstly, it must be noted that taking calibration measurements is\na very time-consuming (thus, expensive) task. The number of calibration data must thus\nbe kept as low as possible, while retaining high localization accuracy. This question has\nbeen addressed in the literature under the name optimal design.\n[3] showed that\u2014in a\n2-dimensional space\u2014hexagonal sampling design yields optimal results in terms of inte-\ngral mean square error when the covariance structure of the underlying Gaussian process\nis unknown. We also adopt this optimal design for the GPPS system when evaluating it in\nSec. 4.\n\nSecondly, we assumed a GP model with zero mean in Sec. 2.1, which clearly does not \ufb01t\nthe propagation law of radio signals. In the actual GPPS, the GP mean is a linear function\nof the distance to the base station (when signal strength is given on a logarithmic scale).\nThe overall process of building the GPPS is summarized is follows. Starting point is the\ncalibration data, with a total of C measurements. On calibration point xi, i \u2208 {1, . . . ,C}, we\nreceive a signal strength of ci j from base station j, j \u2208 {1, . . . , B}, or ci j = /0 if base station\nj has not been received at xi (for example, due to signal obstruction). Signal strength is\nmeasured in dB, all model \ufb01tting is thus done on a logarithmic scale.\nThe calibration data is then split into subsets D j containing those points where base station\nj has actually been received, i.e., D j = {(xi, ci j) : ci j (cid:54)= /0}, corresponding to D j introduced\nin Sec. 2.1. For each base station, that is, for each data D j, we proceed as follows:\n\n1. Often, the exact position of base station j is not known.2 In this case, we use a\nsimple estimate for the base station position, that is the average of the 3 calibration\npoints xi with maximum signal strength yi. This estimate is rather crude, yet we\nfound it to give sensible results in all of the con\ufb01gurations we have considered. In\nparticular with sparse calibration measurements, more sophisticated estimates for\nthe base station position are dif\ufb01cult to come up with.\n\n2. Compute the distance of each calibration point to the base station (using either\nthe exact or the estimated position obtained in step 1). As the mean function of\nthe GP model, we \ufb01t a linear model3 to the received signal strength as a function\nof distance to the base station. Subtract the value of the mean function from the\n2When setting up the network, or after modifying the network by moving base stations, the base\n\nstation positions are often not recorded.\n\n3Alternatively, one could also use a procedure similar to universal kriging, and combine \ufb01tting of\n\nthe mean function with learning the parameters of the kernel function, see Eq. (4).\n\n\u00b6\nn\nn\n\u00b6\nn\n\foriginal measurements, and use the modi\ufb01ed values in the subsequent GP model\n\ufb01tting procedure.\n\n3. Use Eq. (4) to \ufb01nd optimal parameters for the GP model, which are the noise\n\nvariance s 2, the Mat\u00b4ern smoothness parameter n and the input length scale w.\n\n4 Evaluation in a DECT Network\n\nWe tested the accuracy of the GPPS in a large DECT cellular network. In a large assem-\nbly hall of 250\u00d7 180 meters, measurements of signal strengths received from DECT base\nstations were made on 650 points spread over the hall. In this environment, moving robots,\nmetal constructions, corridors, of\ufb01ce cubicles, etc., are all affecting the signal propagation.\nWe observed a very high \ufb02uctuation of received signals (up to \u00b110 dB when repeating mea-\nsurements, while the total signal range is only \u221230 to \u221290 dB), both due to measurement\nnoise, and due to dynamical changes of the environment.\n\nWe compare the GPPS with a nearest neighbor based localization system (abbreviated by\nNNLoc in the following), that is quite similar to the RADAR [1] approach.4 This system\n\ufb01nds the calibration measurements that best match the signal strength received at test stage.\nThe best matches are used in a weighted triangulation scheme to compute the location\nestimate. This method requires careful \ufb01ne tuning of parameters, and we have to omit\ndetails for brevity here.\n\nDense Calibration Points\nIn a \ufb01rst experiment, we investigate the achievable precision\nof location estimates when using the full set of calibration measurements. We evaluate both\nthe GPPS and the nearest neighbor based method in a 5fold cross validation scheme. The\ntotal set of measurements is split up into \ufb01ve equally sized parts, where four of these parts\nwere used as the calibration set. The resulting positioning system is tested on the \ufb01fth part\nleft out. This is repeated \ufb01ve times, so that each point is being used as the test point exactly\nonce. We found that, in this setting, the nearest neighbor based method NNLoc works very\n\ufb01ne, and provides an average localization error of 7 meters. The GPPS performs slightly\nworse, with an average error of 7.5 meters. With the GPPS, localization is typically based\non around 15 base stations, that is, 15 likelihood terms contributing to Eq. (1).\n\nUnfortunately, such a high number of calibration measurements is unlikely to be available\nin practice. Taking calibration measurements is a very costly process, in particular if larger\nareas need to be covered. Thus, one is very much interested in keeping the number of\ncalibration points as low as possible.\n\nExperiments with Sparse Calibration Points\nIn the second experimental setup, we aim\nat building the positioning system with only a minimal number of calibration points. Again,\n5fold cross validation was performed. After splitting the data into \ufb01ve parts, we select\nsubsets of \u02dcC = 100,50,25,12 points, either at random or simulating the optimal design,\nfrom the union of four of these parts. The localization system is built based on these\n\u02dcC points and evaluated on the \ufb01fth part of the data. In order to simulate a near-optimal\ndesign (see Sec. 3), we superimpose a hexagonal grid with \u02dcC points on the area under\nconsideration. Out of the given calibration measurements, we select those \u02dcC points that are\nclosest (in terms of Euclidean distance) to the grid points.\n\nIn Fig. 1 we plot the localization accuracy, averaged over the 5fold cross validation,\nof the GPPS and the nearest neighbor based system built on only \u02dcC calibration points,\n\n4We also investigated localization using Eq. (1) with a simplistic propagation model, where the\nexpected signal (on log scale) is a linear function of the distance to the base station. Yet, this approach\nlead to very poor localization accuracy, and is thus not considered in more detail here.\n\n\fFigure 1: Mean localization error of the GPPS and the NNLoc method, as a function of\nthe number of calibration points used. Vertical bars indicate \u00b11 standard deviation of the\nmean localization error. The calibration points are either selected at random, or according\nto an optimal design criterion\n\n\u02dcC \u2208 {100,50,25,12} calibration measurement. It can be clearly seen that the GPPS system\n(with optimal design) achieves a high precision for its location estimates, even when using\nonly a minimal number of calibration measurements. With only 12 calibration measure-\nments, GPPS achieves an average error of around 17 meters, while the competing method\nreaches only 29 meters at best. In this setting, the average distance in between calibration\nmeasurements is around 75 meters. Both the NNLoc system and the GPPS system show\nlarge improvements of performance when selecting the calibration points according to the\noptimal design, instead of a purely random fashion. Also, note that the localization error\nof the GPPS system degrades only slowly when the number of calibration measurements\nis reduced. In contrast, the curves for the nearest neighbor based method show a sharper\nincrease of positioning error.\n\nIt is worth noticing that the choice of kernel functions has a strong impact on the local-\nization accuracy of the GPPS. In Fig. 2(a), we also plot a comparison of the GPPS with\n(cid:48)(cid:107)).\neither the Mat\u00b4ern kernel, Eq. (5), or an RBF kernel of the form k(x,x\nGP models with RBF kernels tend to be over-optimistic [9] about the predictive variance,\nEq. (3), which in turn leads to overly tight position estimates. Thus, the accuracy of GPPS\nwith RBF kernel is clearly inferior to that of GPPS with Mat\u00b4ern kernel. It is also inter-\nesting to consider different methods for selecting the calibration points. Fig. 2(b) plots\nthe accuracy obtained with GPPS, when calibration points are either placed randomly, on a\nhexagonal grid (the theoretically optimal procedure) or on a square grid. Somehow counter-\nintuitively, a square grid for calibration gives a performance that is just as good or even\nworse than a random grid. In contrast, localization with NNLoc performs about the same\nwith either hexagonal or square grid (this is not plotted in the \ufb01gure).\n\n(cid:48)) = exp(\u2212w(cid:107)x\u2212x\n\n5 Conclusions\n\nIn this article, we presented a novel approach to solving the localization problem in indoor\ncellular network networks. Gaussian process (GP) models with the Mat\u00b4ern kernel function\nwere used as models for individual base stations, so that location estimates could be com-\nputed using maximum likelihood. We showed that this new Gaussian process positioning\nsystem (GPPS) can provide suf\ufb01ciently high accuracy when used within a DECT network.\n\n\f(a) GPPS using either the Mat\u00b4ern or the\nRBF kernel function\n\n(b) GPPS with calibration measurements\nplaced either randomly, on a square grid, or\non a hexagonal grid (optimal design)\n\nFigure 2: Average localization error of the GPPS method with different kernel function\n(left) and different methods for placing calibration points (right)\n\nA particular advantage of the GPPS system is that it can be based on only a small num-\nber of calibration measurements, and yet retain high accuracy. Furthermore, we showed\nhow calibration points can be optimally chosen in order to provide high accuracy position\nestimates.\n\nAcknowledgments Anton Schwaighofer gratefully acknowledges support through an\nErnst-von-Siemens scholarship.\n\nReferences\n[1] Bahl, P., Padmanabhan, V. N., and Balachandran, A. 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Springer Verlag, 1999.\n\n\f", "award": [], "sourceid": 2541, "authors": [{"given_name": "Anton", "family_name": "Schwaighofer", "institution": null}, {"given_name": "Marian", "family_name": "Grigoras", "institution": null}, {"given_name": "Volker", "family_name": "Tresp", "institution": null}, {"given_name": "Clemens", "family_name": "Hoffmann", "institution": null}]}