Gaussian mixture models for signal mapping and positioning

Highlights

• We propose a new Gaussian mixture model for signal mapping.

• Proposed Gaussian mixture model can model complex signal attenuation patterns.

• When used in positioning the proposed model provides accurate position estimates.

Abstract

Maps of RSS from a wireless transmitter can be used for positioning or for planning wireless infrastructure. The RSS values measured at a single point are not always the same, but follow some distribution, which vary from point to point. In existing approaches in the literature this variation is neglected or its mapping requires making many measurements at every point, which makes the measurement collection very laborious. We propose to use GMs for modeling joint distributions of the position and the RSS value. The proposed model is more versatile than methods found in the literature as it models the joint distribution of RSS measurements and the location space. This allows us to model the distributions of RSS values in every point of space without making many measurement in every point. In addition, GMs allow us to compute conditional probabilities and posteriors of position in closed form. The proposed models can model any RSS attenuation pattern, which is useful for positioning in multifloor buildings. Our tests with WLAN signals show that positioning with the proposed algorithm provides accurate position estimates. We conclude that the proposed algorithm can provide useful information about distributions of RSS values for different applications.

Introduction

Indoor positioning is often based on signals of existing wireless networks due to the lack of GNSS signals, high cost of installing and upkeeping dedicated positioning infrastructure, and wide availability of the wireless networks [1]. However, the use of existing wireless infrastructure that is not designed for indoor positioning, such as WLAN or Bluetooth, requires a map of the signal environment [2], a so-called “radio map”. Also, consumer WLAN systems provide only RSS measurements and, for example, range [3], time-of-arrival [4] or direction-of arrival [5] measurements, cannot be used for positioning.

Because walls attenuate and reflect electromagnetic signals and antenna radiation patterns are not directionally uniform, signal strengths cannot be modeled accurately with omnidirectional models, such as path loss in vacuum. Signal attenuation can be modeled using analytic or numerical physics models that take into account the location and materials of the walls [6]. An alternative to physics based models is an empirical model based on a set of measurements of RSS values collected at known locations in the indoor region of interest. These measurements are commonly called FP, and there exist many fingerprint positioning methods [7], [8], [9], [10], [11].

In this paper, we concentrate on situations where each FP is associated with the location where the measurement was made and the radio map is built first in an offline phase. The map is later used in the online phase for positioning. In the literature, there are also algorithms that map the environment without knowing FP locations [12], [13], [14], [15], doing the mapping and positioning in one phase.

One approach to estimate the radio map from FP is WKNN. Here the position is computed by comparing RSS values with FP in the database; the estimated position is a weighted average of locations of the k FP with the most similar RSS values among received AP [16]. The learning phase, thus, consists of collecting the FP. In the positioning phase, the received RSS values are compared to FP in the database. In this phase, there are several variations of weighting the k nearest points. The weighting may be based on the difference of RSS values [16] or the rank of the RSS [17]. The data may also be preprocessed to improve the positioning performance in a varying signal environment [18]. WKNN is commonly used due to its simplicity, but it has two relevant shortcomings: its accuracy can be improved, it does not provide error boundaries for its estimates, and the database size and computational cost increases as more FP are collected.

Kernel methods and histogram methods [16] extend the WKNN by modeling the distribution of RSS values at calibration points. In the learning phase, multiple measurements are made at each mapped location and the distribution is modeled using a histogram or a kernel method. This increases the required work in the mapping phase significantly and gives only small improvement in the database size or accuracy.

In [19], missing measurement values in FP are filled and then principal component analysis is applied for dimensionality reduction. Then a GMM is fitted to model the reduced dimension FP. Compared to WKNN the computational complexity of the positioning does not increase as the number of samples increase.

In [20], a GMM is proposed to be used to model a the distribution of RSS values inside cells. Compared to the kernel and historgam methods this method allows to track the dependency of RSS values from different AP. In [21] a GMM is used to model the RSS values in the mapped area. The GMM is not used as probabilistic model, but rather as a measurement model with an added Gaussian noise.

PL models are based on a functional relation between signal strength and the distance or path to the receiver [22], [23]. Isotropy may be assumed in open space with omnidirectional antennas. But in built environments, the walls and other structures attenuate and reflect the signals. In particular, the attenuation caused by concrete structures is larger than attenuation from light indoor walls [24]. This anisotropy causes problems if a PL model is to be used in multifloor buildings.

In proximity method the measurement is hard thresholded to provide information whether the measurement was close to an AP [25]. Similarly coverage area methods consider only area where an AP is received with strong enough RSS [26].

GP were used to model radio maps in [27], [28], [29]. Their approach has the advantage that it can model RSS patterns of arbitrary distribution in space. However, they need to assume Gaussian noise and the variance of the noise does not depend on the measured RSS values.

Among the above presented methods, only kernel, histogram and GMM methods in [19], [20] are able to model non-Gaussian variations of RSS values, which is useful for quantifying measurement uncertainty in a probabilistic positioning algorithm. However, these methods can only model this characteristic at the points or cells where multiple measurements where collected.

In this paper, we propose a model that fills the above-described gap in the literature. The model uses a GM for mapping RSS values in buildings and the map is used for indoor positioning. Specifically we propose a GMM for the joint distribution of RSS values and locations. The main benefit, compared to other methods in literature, is that it models the distribution of the RSS values for each AP in the mapped area without the need of making multiple measurements at each FP location. The proposed model allows us to compute location estimates and RSS distributions in closed form, which allows to evaluate models without approximations.

Compared to GMM based algorithm in [20], the proposed algorithm has the benefit that there is no need for collecting samples in cells as the position dimension is considered as a continuum. Similarly [19] treats the measurement points as separate points without using spatial information. Furthermore, the algorithm in [21] uses the GMM only as a measurement model.

Compared to the Gaussian process methods in [27], [28], [29], the proposed model and GP models can model arbitrary signal attenuation patterns inside buildings, but the proposed model can also model arbitrary distributions of RSS values measured at any location close to the area where measurements were made, while GP models rely on the Gaussian distribution.

One important aspect of the model is that it can be used to estimate the distribution of the attenuation and of RSS values in different parts of the building. This is valuable information also for other applications than indoor positioning; for example, RSS maps can help radio infrastructure planners to identify areas that have poor data transfer rates.

The rest of this paper is organized as follows. Section 2 gives the theoretical background of our approach. Section 3 presents the proposed model for building an RSS map. The filtering algorithm based on the proposed model is presented in Section 4. Section 5 presents examples of applying the proposed for WLAN signal mapping and positioning in a four story building. Section 6 concludes the paper.