The radio map for nonparametric methods consists of FP data (see Subsection 3.1 for details). In the positioning phase the UE’s mea- surements are compared with the RM’s FP entries to infer a position estimate[38]. There exist various approaches for determining this estimate, which can be divided in deterministic and probabilistic methods. Here only a brief overview on theses methods is given. For additional information and references the reader is referred to[38], on which this Subsection is based.
Deterministic nonparametric methods
Deterministic nonparametric positioning methods assume the state to be deterministic (hence their name). The estimate ˆx is a convex combination of N FP locations lFPj, this means
ˆx= N X j=1 ωj PN j=1ωj lFPj, (22)
whereωj ≥ 0 for all j . Let y be the vector containing the measured
RSS values and PRSSj be the vector containing the RM-stored RSS values of each transmitter in the j th FP location. A possible weight ωj is the inverse of the norm of RSS innovation that is defined as
ωj =
1 ||y − PRSSj||
. (23)
For the norm|| · ||, for example, Manhattan norm (1-norm), Euclidean norm (2-norm), and Mahalanobis norm are used (see[38] for def- initions of various norms and references in which they are used). In[77] it is pointed out that using (23) finding ˆx can be interpreted as a minimisation problem, because a FP location’s weight is highest if the sum of differences between observed transmitter RSS values in current UE location and that FP location is minimal. An alternative weight could be the number of transmitters that are observed in both current UE and FP locations. This approach can be interpreted as a maximisation problem.
The simplest approach based on (22), the nearest neighbour (NN) method, uses the location of the FP with the highest weight as po- sition estimate. In[P1] the weighted K -nearest neighbour (WKNN)
algorithm, an extension of NN, is used for comparison. The WKNN computes the UE position estimate according to (22) using only the FP locations with the K largest weights. For the remaining FP loca- tions the weights are set to zero. The WKNN is widely applied because it yields good robustness and accuracy while having only medium complexity and cost[55].
The WKNN performs at least as well as all parametric methods for a test in real-world indoor WLANs in[P1] when all data is used. In the test area, two buildings at Tampere University of Technology, the WLAN AP density is high. However, when removing 90% of the APs from the test area the WKNN positioning accuracy deteriorates signif- icantly and the algorithm is outperformed by the parametric methods. This is in line with [28], where it was discovered that range-based positioning techniques outperformed nonparametric FP positioning algorithms in case of limited FP data.
Probabilistic nonparametric methods
Probabilistic nonparametric methods assume the state x to be ran- dom, and the position estimate is computed using Bayes’ rule (1). The idea of probabilistic methods is to divide the area covered by the RM into cells. A natural choice is to use the FP locations as centres of those cells, resulting in N cells b1, . . . , bN.
In[38] a uniform prior is used. Hence the prior is p(x) = PN j=1[x ∈ bj] PN j=1|bj|, (24)
where[·] is the Iverson bracket and |bj| is the volume of the j th cell.
Furthermore, the measurements from one FP are assumed to repre- sent the RSS distribution inside the whole cell. Hence the likelihood inside that cell is constant, and the likelihood is
p(y|x) =
N
X
j=1
p(y|bj)[x ∈ bj]. (25)
The likelihood p(y|bj) can be computed by several approaches.
In[38] suitable methods and references in which they are used are given.
One obvious drawback of probabilistic nonparametric methods is the need to compute the likelihood in each of the N FP locations stored in the RM, which can be computationally demanding. Therefore, in the following subsections (probabilistic) parametric methods are considered.
Another issue that affects all FP positioning methods discussed in this subsection is the generation and maintenance of the RM. When generating the RM data might not be collected in some areas, because they are restricted or inaccessible, which results in gaps in the RM[80]. Furthermore, one of the tests with real-world WLAN data in[P1] shows that FP-based positioning methods (represented by the WKNN in[P1]) suffer significant performance deterioration when the RM is outdated, because the radio environment is constantly changing[38, 86].
In[80] various interpolation and extrapolation techniques for recov- ering missing FP data and filling gaps in the RM are studied using extensive WLAN data. For recovering missing data techniques from two categories are considered. In the first category, linear interpo- lation is used to fill the gaps between known FP locations while for extrapolation minimum method, mean method and gradient method are used to estimate the data in the gaps. In the second category, inter- polation and extrapolation are carried out jointly using NN method and inverse distance weighting.
The methods studied in[80] are mainly meant for RM generation. However, they could also applied for RM maintenance. For example, instead of collecting new FPs for all FP locations in the RM new data could be collected only in some of them and for the remaining lo- cations FP data could be updated using the presented interpolation and extrapolation techniques.
5.2
Coverage area models
The CA-based positioning method is used in[P2], for comparison, and in[P1]. It is developed and explained in detail for positioning in[48]. The method requires a RM with estimated CAs of transmitters. In[P1] and in [P2] the CA of a transmitter is modelled as a multivariate normal distribution with placeµ and shape Σ. The CA-based posi-
tioning method uses only a list ID= {ID1, ID2, . . . , IDK} of transmitter
IDs observed by the UE in its current location x.
Under the assumption of mutually independent observations the likelihood is p(ID|x) = K Y k=1 p(IDk∈ ID|x) ∝ exp −1 2 K X k=1 (x − µk)TΣ−1k (x − µk) = exp−1 2(x − ¯x) TS−1(x − ¯x) + constant (26) with S=PKk=1Σ−1k −1and ¯x= SPkK=1Σ−1k µk .
Koski[48, p. 35] points out that the assumption of mutual indepen- dence is a weakness of the proposed approach and might not always hold. For example, CAs of neighbouring transmitters usually overlap. Therefore, assuming that observing IDi is independent of observ-
ing its neighbour IDj does not hold. However, the assumptions was chosen in order to keep the algorithm simple and computationally light.
The position estimate of the CA-based method given the prior x∼ MVN µ0,Σ0 is p(x|ID) ∝ p(x)p(ID|x) ∝ exp −1 2(x − ¯x0) TS−1 0 (x − ¯x0) , (27)
which is a multivariate normal with covariance S0=
PK k=0Σ−1k −1 and mean ¯x0= S0PkK=0Σ−1k µk
. The prior can be interpreted as one CA itself, because it is also a multivariate normal like the CAs themselves. Then the position estimate is the weighted mean of the CA centres with weights being the inverses of their covariance matrices. The posterior covariance matrix provides an uncertainty measure for the estimate; a measure that is not provided by the WKNN algorithm. In case no prior information on the location is available, Koski[48, p. 35] suggest to use a prior with large covariance. Furthermore, she
describes how information that a transmitter is not observed in the UE location could be used for positioning but points out that this will become impractical (due to high computational demand) for large radio maps.
In the WLAN-based tests of[P1] the CA approach performs well in all 4 test scenarios. When using all data and up-to-date RM its accuracy is only slightly worse than those of WKNN and the more compli- cated parametric methods. For the scenario where only APs with 5 strongest RSS values are used for positioning and in the scenario with low AP density it still performs on a similar level and close to the other parametric methods. Finally, it is amongst the best methods for the test scenario with outdated RM; its performance is on the same level as with up-to-date RM.
Thus, while being a quite simple method it provides reasonable pre- cise position estimates under all tested circumstances, making it a good choice as reference algorithm in[P2].
Both[P1] and [P2] also use a filtered version of the CA approach. The update is done using a standard Kalman filter.