Position Estimation In Uncertain Radio - DiVA Portal

xiiiContents34.118120120121122124D Paper D1Introduction . . . . . . . . . . . . . . . . . . . . . . . . .2Localization Method . . . . . . . . . . . . . . . . . . . .3Gaussian Process for Fingerprint Construction . . . . .3.1Characterizing Spatial Correlation . . . . . . . .3.2Estimate Model Parameters . . . . . . . . . . . .3.3Build New Fingerprints . . . . . . . . . . . . . . .4Kriging for Fingerprint Reconstruction . . . . . . . . . .4.1Characterizing Spatial Correlation . . . . . . . .4.2Estimate Parameters and Build New Fingerprints5Comparison Between Kriging and Gaussian Process . .6Field Campaign . . . . . . . . . . . . . . . . . . . . . . .6.1Data Collection . . . . . . . . . . . . . . . . . . .6.2RSS Map Reconstruction Results . . . . . . . . .6.3Localization Results . . . . . . . . . . . . . . . . .7Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Particle Filtering Algorithm Based on GPExperimental Results . . . . . . . . . . .4.1Setup . . . . . . . . . . . . . . . .4.2Propagation modeling . . . . . .4.3Performance Evaluation . . . . .Conclusions . . . . . . . . . . . . . . . .E Paper E1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2GP Based Flow Modeling and Prediction for a Single Individual .2.1Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2Standard Gaussian Process Regression . . . . . . . . . . . .2.3Grid Based On-line Gaussian Process Regression . . . . . .2.4Kernel Selection . . . . . . . . . . . . . . . . . . . . . . . . .2.5Hyperparameters Determination . . . . . . . . . . . . . . .3Aggregated Flow Modeling and Prediction for Multiple Individuals3.1A Brief Overview of Sequence Clustering . . . . . . . . . .3.2Flow Modeling and Prediction for Clusters of Individuals .4Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1Individual Flow Model and Prediction . . . . . . . . . . . .5.2Aggregated Flow Modeling (Multiple Individuals) . . . . .6Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143145147147147148150151151152152153154154157162

NotationOperators and SymbolsNotationT[·][ · ] 1tr( · )k · k · E( · )X( · )Tln( · )log10 ( · ) θ / θ θθ θ Tθerf( · )IN10MeaningVector/matrix transposeInverse of a non-singular square matrixTrace of a square matrixEuclidean norm of a vectorCardinality of a setStatistical expectationShort-hand notation for XX TNatural logarithmLogarithm to base 10Kronecker productThe gradient operatorThe Laplace operatorThe standard Gaussian error function.Identity matrix of size N NA vector of all 1sA vector of all 0sDistributionsAbbreviationN (µ, σ 2 )Cat( · )MeaningGaussian distribution with mean µ and variance σ 2Categorical distributionxv

OAUEMeaningBest linear unbiased estimatorCumulative distribution functionConfidence intervalConditional mean squared errorCramér-Rao boundForward filtering backward simulationGaussian processGaussian process regressionIndependent and identically distributedLeast squareMaximum likelihoodMaximum likelihood estimatorMean squared errorProbability density functionParticle filterParticle smootherRoot mean squared errorReceived signal strengthSequential importance resamplingSequential Monte CarloState space modelTime difference of arrivalUser equipment

1IntroductionIn estimation theory, two scientific problems are usually formulated. The firstproblem relies on the modeling of the system. In such a case, given a set of specific input states x, the output of the system are noisy observations which can becollected or measured. The target is to train a model of the system, which relatesthe input states with the output observations. The model can be either used to repeat the same experiment or to predict the output observations given a new inputstate, where the latter case is the focus of this thesis. In the second problem, thetrue state is usually invisible or unmeasurable. The aim is to infer the true state(or the latent variables in this case) from the noisy observations. This usuallyconsists of a two-steps procedure. In the first step, after the observation data hasbeen collected, a measurement model imputes relationships between latent variables and observations needs to be selected. Then, state inference is performedto estimate the true state from the noisy observations.To solve the first problem, the process of which is usually known as machinelearning or system identification, both parametric and non-parametric modelingcan be used. The model is usually explicitly given by specifying a finite number of model parameters in the parametric modeling. For example, the linearregression model in Bishop (2006), autoregressive model and state space modelin Ljung (1999). Unlike the parametric modeling, in non-parametric modeling,the number of parameters grows as the number of observations increases. Inaddition, there is no explicit model form such that the parameters in the nonparametric modeling are only determined by the data, not the model. One example of non-parametric modeling is Gaussian process (GP) Rasmussen and Williams(2006), which will be detailed in section 2.2.To solve the second problem, as stated previously, a proper measurement model1

21 Introductionfor the noisy observations should be selected. Then, the inference of latent variables is usually solved in a statistical framework. For the modeling step, it issometimes easy to choose an accurate model for the observations. For instance, ifthe observations are the voltage V measured at the two sides of a resistor R andthe state is the current C flowing in the resistor, we have V R C, according toOhm’s Law.However, in most cases, it is usually hard to find an exact accurate measurementmodel. In addition, for most of the time, the observations at the output of thesystem are disturbed by noise. For instance, let’s assume the observations are thereceived signal strength (RSS) values measured from a reference network nodeat a set of latent positions. Due to the additive noise, shadowing, multi-pathfading, and other effects, there is no explicit expression to impute the relationship between the latent positions and the observed RSS values. In such cases,empirical models have been proposed. For instance, the Hata-Okumura modelHata (1980) and the piecewise linear model Goldsmith (2006). While all thoseare parametric models, non-parametric models such as Gaussian processes canbe used Ferris et al. (2006). Also, the model can be assumed to be known beforehand, otherwise it needs to be determined by solving the first problem at thetraining phase. The latter case is within the scope of this thesis.For the inference step, a statistical framework is always used, one of which isthe Bayesian inference Box and Tiao (1973). Bayesian inference is used to updatethe probability for a hypothesis as more evidence or information becomes available. However, a major problem of Bayesian inference is that there are manyintractable integrals. Hence, there is usually no closed form expression for theseintegrals such that in most cases, approximations are needed.In the 1980s, Monte Carlo methods emerged as a good tool for solving integrationproblems in Bayesian inference. It is known as one of the computational methods which use statistical simulations to approximate the estimates. Monte Carlomethods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution Kroese et al.(2014). Sequential Monte Carlo methods emerged more recent in 1990s, which sequentially takes the observation and updates information about the hidden statesDoucet et al. (2001). Here in this thesis, we will mainly use sequential MonteCarlo methods to solve the problem of inferring the latent states from observations. Detailed background and two examples of sequential Monte Carlo methods will be presented in the next chapter.The main contributions of this thesis are to use different techniques to solve thetwo problems described above in different applications. In the next followingsection, two examples of applications are provided to further illustrate the twoproblems respectively.

3Examples of Applications401.1DataModel MeanUpper CILower CI20010Temperature30o0.00.20.40.60.81.0Normalized TimeFigure 1.1: Gaussian process regression for daily temperature in a year Zhao(2016).1.1Examples of ApplicationsSo far, a general description of the two problems we are aiming to solve has beengiven. However, those problems can be encountered in various research areas andapplications. To be more concrete, some illustrative examples will be provided tofurther assist the understanding and to better nail down the scope of this thesis.We begin with an example of non-parametric modeling in machine learning.Given a dataset which contains one year of the daily temperatures for some placein Japan Zhao (2016), we want to train a non-parametric model which can be usedas a reference to predict the trend of temperature for the next year. The datasethas been plotted in Figure 1.1.This simple example shows a typical applicationof solving the first modeling problem. By using a Gaussian process regressionmethod, we can train a non-parametric model with confidence intervals (CI) asshown in Figure 1.1.For the second problem of inferring the latent variables from the observations,there are many applications in the target tracking area. One example application

41 Introduction1.2Sensor 71Sensor 6Sensor 563 18x2 in meter0.872270.63681Sensor 40.454090.2Target initial position0 Sensor 1Sensor 3Sensor 245-0.200.20.40.60.811.21.41.6x1 in meterFigure 1.2: Target tracking in a sensor network Zhao (2015).can be to perform positioning of a moving device in a sensor network. For instance, we would like to track a device which carrying a speaker that can makequite distinctive sounds Zhao (2015). Then, a sensor network of microphones canbe built up to receive the sound from the speaker. The observations in this caseare the time difference of arrivals (TDOA), which can be computed by correlatingthe received signal at the microphones. The latent variables are the positions ofthe device carrying the speaker. A deployment of the sensor network with thetarget we want to track is plotted in Figure 1.2.Then, the positioning results for the device have been shown in Figure 1.2 by thesolid curve. In this example, the aim is to infer the latent positions of the devicefrom the TDOAs, which corresponds to the second problem.1.2Main ContributionsSo far we have seen some examples of the two problems we are aiming to solve. Ingeneral, the main contributions of this thesis can be summarized into two parts.The first part is to use sequential Monte Carlo methods together with Gaussian

1.3Thesis outline5process modeling for state inference in a novel indoor positioning system. Theposition estimates are further compared with a static solution, where fingerprinting is applied to find the most likely positions given a set of observations. Finallythe last part is to apply non-parametric modeling in sports analytics for trajectorylearning. To be more specific, the contributions are detailed as: A novel indoor positioning system utilizing proximity based reports hasbeen proposed. Advanced modeling with Gaussian process has be appliedto optimize the thresholds which trigger the proximity reports. Sets of optimized thresholds have been obtained by different criteria, which enables anew positioning fashion. (Paper A) Corresponding sequential Monte Carlo methods have been proposed forthe novel proximity based positioning system to estimate the position ofthe moving device in such a network. In addition, theoretical positioningperformance for proximity based system has been derived. This results ina more simple and efficient way of positioning devices with less signalingand bandwidth, while maintaining adequate positioning accuracy. (PaperB) An advanced modeling for proximity reports with Gaussian process hasbeen evaluated with particle filter, one of the most popular sequential MonteCarlo methods. The results well demonstrate that the better modeling ofthe observations help to improve the positioning accuracy. (Paper C) A Gaussian process based algorithm to build receive signal strength databasehas been proposed with comparison to a Kriging method. Then, a statisticalfingerprinting algorithm based on the pre-built RSS database has been designed. The snapshot positions obtained from the fingerprinting algorithmbased on both Gaussian process and Kriging are satisfactory as comparedto the case where the motion of the device is also considered. (Paper D) A novel application of Gaussian processes to trajectory learning in sportsarea. More precisely, flow models for both single trajectory and multipletrajectories haven been described with various Gaussian processes. Resultsof the modeling of different trajectories are proved to be accurate and suchmodels provide valuable insights into sport performance analysis, whichare usually hard to interpret from the video cameras. (Paper E)1.3Thesis outlineThe thesis consists of two parts. The first part introduces the background material regarding the parametric/non-parametric modeling of system and sequentialMonte Carlo methods for state inference. The second part presents the proposedsolutions as a collection of peer-reviewed papers.

61 Introduction1.3.1Outline of Part IIn the first part of this thesis, the background material of modeling is presentedfollowed by two examples of models. Then, sequential Monte Carlo methodsare introduced with highlights on the particle filter and smoother. Finally, theCramér-Rao bound is introduced, which provides a theoretical lower limit forthe estimator.1.3.2Outline of Part IIIn the second part of this thesis, a collection of edited version of papers will bepresented. The summary is given as below.Paper APaper A of this thesis is an edited version of,F. Yin, Y. Zhao, F. Guannarsson, and F. Gustafsson. Received-signalstrength threshold optimization using Gaussian processes. Transactions on Signal Processing, 65(8):2164–2177, 2017.which is an extension of the two earlier contributionsF. Yin, Y. Zhao, and F. Gunnarsson. Fundamental bounds on position estimation using proximity reports. In Proc. IEEE 83rd VehicularTechnology Conference: VTC2016-Spring, 2016.F. Yin, Y. Zhao, and F. Gunnarsson. Proximity report triggering threshold optimization for network-based indoor positioning. In Proc. Int.Conf. on Information Fusion, pages 1061–1069, Washington D.C., USA,July 2015.Summary: This paper presents a generic received-signal-strength (RSS) threshold optimization framework for generating informative proximity reports. Theproposed framework contains five main building blocks, namely the deploymentinformation, RSS model, positioning metric selection, optimization process andmanagement. Among others, we focus on Gaussian process regression (GPR)based RSS models and positioning metric computation. The optimal RSS threshold is found through minimizing the best achievable localization root-mean-squareerror formulated with the aid of fundamental lower bound analysis. Computational complexity is compared for different RSS models and different fundamental lower bounds. The resulting optimal RSS threshold enables enhancedperformance of new fashioned low-cost and low-complex proximity report basedpositioning algorithms. The proposed framework is validated with real measurements collected in an office area where bluetooth-low-energy (BLE) beacons aredeployed.

1.3Thesis outline7Paper BPaper B of this thesis is an edited version of,Y. Zhao, C. Fritsche, F. Yin, F. Gunnarsson, and F. Gustafsson. Sequential Monte Carlo methods and theoretical bounds for state inferencebased on proximity reports. To be submitted to IEEE Transactions onVehicular Technology, 2017a.which is an extension of the earlier contributionY. Zhao, F. Yin, F. Gunnarsson, M. Amirijoo, E. Özkan, and F. Gustafsson. Partile filtering for positioning based on proximity report. InProc. Int. Conf. on Information Fusion, pages 1046–1052, WashingtonD.C., USA, July 2015.Summary: In paper A, a framework of optimizing thresholds for converting RSSto proximity reports has been developed. In this paper, we further consider positioning of devices based on a time series of proximity reports, which are generated using the optimized thresholds, from a mobile device to a network node.This corresponds to nonlinear measurements with respect to the device positionin relation to the network nodes. Therefore, sequential Monte Carlo methods,namely particle filtering and smoothing, are applicable for positioning. Positioning performance is evaluated in a typical office area with Bluetooth-low-energybeacons deployed for proximity detection and report, and is further comparedto parametric Cramér-Rao lower bounds. Accuracy is concluded to vary spatiallyover the office floor, and in relation to the beacon deployment density.Paper CPaper C of this thesis is an edited version of,Y. Zhao, F. Yin, F. Gunnarsson, M. Amirijoo, and G. Hendeby. Gaussian processes for propagation modeling and proximity based indoorpositioning. In Proc. IEEE 83rd Vehicular Technology Conference:VTC-Spring, 2016a.Summary: In paper B, in order to perform the positioning of device using proximity reports, first we assume a linear log-distance model for the RSS observations,which gives linear relationship between the RSS and the distance of device fromthe network node in logarithmic scale. Then, the RSS is converted to proximityreports. However, in practice, the modeling of RSS is always considered as a latent and nonlinear observation model. To address these problems, we use one ofthe powerful tools, namely Gaussian process regression (GPR) for propagationmodeling of RSS. This also provides some insights into the spatial correlation ofthe radio propagation in the considered area. T

where the radio propagation is uncertain. In Bayesian inference framework, it is usually hard to get analytical solutions. In such cases, we resort to Monte Carlo methods to solve the problem numerically. In addition, we apply Bayesian non-parametric modeling for trajectory learning in sport analytics.