=Paper=
{{Paper
|id=Vol-2498/short6
|storemode=property
|title=A probabilistic fingerprinting method for indoor localization based on RBF network
|pdfUrl=https://ceur-ws.org/Vol-2498/short6.pdf
|volume=Vol-2498
|authors=Yangkang Yu,Ling Yang
|dblpUrl=https://dblp.org/rec/conf/ipin/YuY19
}}
==A probabilistic fingerprinting method for indoor localization based on RBF network==
https://ceur-ws.org/Vol-2498/short6.pdf
A probabilistic fingerprinting method for indoor localiza-
tion based on RBF network
Yangkang Yu 1[0000-0002-8292-1086] and Ling Yang 1[0000-0001-7663-7743]
1 Tongji University, Shanghai, China
lingyang@tongji.edu.cn
Abstract. Received Signal Strength Indication (RSSI) fingerprinting is known
as the most concerned method for indoor localization as its high accuracy and
low cost. Numerous RSSI based methods have shown an attractive performance
but the major drawback is the high dependency on the database construction. In
this paper, we propose a localization method based on radial basis function
(RBF) network. Choosing Gaussian radial basis functions with appropriate
widths, the probability algorithm can be effectively conducted to the RBF net-
work regardless of deficiency of the RSSI data. By further conducting the su-
pervised learning of RBF network the RM database can be calibrated and up-
dated once some new dataset is available, so as to achieve a better localization
performance. Experimental results in a multi-floors building verify that the per-
formance of the proposed RBF network is superior to other common used
methods.
Keywords: Indoor localization, RSSI fingerprinting, RBF network
1 INTRODUCTION
Location-Based Service (LBS) has been widely used in a variety of contexts, such as
health, indoor object search, personal life, etc. Advances in smartphones have made it
feasible to conduct positioning, tracking, navigation, and location-based security
[1][2]. Global Navigation Satellite Systems (GNSS) is used widely in outdoor envi-
ronment for an optimal choice to achieve LBS, but the inability of these signals to
penetrate buildings means other techniques must be explored for indoor positioning.
Nowadays one of the most popular indoor positioning technologies is WLAN posi-
tioning, which is easy to implement on many mobile platforms to achieve a meter-
level localization accuracy.
Algorithms for fingerprint-based localization include deterministic and probabilistic
methods. Deterministic algorithms generally store the mean value of RSSI as the fea-
ture of RPs. It uses the similarity between online signal and database fingerprint to
estimate the location of the user. Traditional deterministic methods could be easily
implemented based on k nearest neighbors [3]. Some other more complex determinis-
tic algorithms such as support vector machine [4] and Deep Neural Networks (DNN)
[5] show better localization accuracies with higher computational costs. However, due
to the random fluctuation of RSSI in indoors measurement errors are inevitable what-
ever in offline or online phase, so only storing mean values of the RSSIs in the RM
2
cannot represent the whole RSSI distribution information at RPs. Therefore, probabil-
istic algorithms, such as Horus, usually record and store the RSSI distribution at each
RP and use the probability distribution information for estimation [6].
However, the fingerprint-based localization still suffers from some defects. The first
issue is the database insufficiency. Offline survey is usually a time-consuming pro-
cess, and for ordinary custom-grade applications, data collected at each RP could be
extremely rare. In these circumstances, most probabilistic algorithms might be invalid
since the requisite RSSI distribution characterization cannot be conducted without
sufficient data. The second issue is the database deviation. The localizations for RPs
in indoors are usually conducted by some type of low-accurate surveys. Also, the
RSSIs from specific APs received at RPs usually fluctuate uncertainly due to the limi-
tations of low-cost sensors. Therefore, deviations on the RM construction are inevita-
ble generally.
Response to these issues, we propose a localization method based radial basis func-
tion (RBF) network [7][8]. Considering the RPs as the basic units and the RSSI mean,
variance, site location on each RP as the network parameters, it is a straightforward
way to implement the RBF network to an indoor localization scene. Choosing Gaussi-
an radial basis functions with appropriate widths, the probability algorithm can be
effective conducted by the RBF network regardless of deficiency of RSSI data. As the
network and RM shares the same features. In addition, compared to other network
such as the DNN, RBF network shows unique physical significance and has simplici-
ty structure as it exploits the radio map topology and the probabilistic model. General-
ly, with the proposed RBF network, the indoor localization accuracy and robustness
would be improved effectively, since the error uncertainty of the RSSIs and RPs co-
ordinate are introduced on both RM construction and real-time localization proce-
dures.
In this paper, in order to conduct a complete and precise localization in different in-
door scenarios, a parallel localization network by using the Gaussian radial basis
functions was proposed. It is designed for both floor detection and location estima-
tion, where the floor detection was considered as a classification problem and the
location estimation was treat as a regression one. In the offline phase, the radio map
construction is the procedure of the network parameters initialization. In the online
phase, when getting a RSSI measurement with an unknown location at an unknown
floor, we use a complete parallel network to determine the floor and then to estimate
the location within the floor.
2 PROBABILISTIC LOCALIZATION MODEL
By considering the distribution characteristics of the RSSI fingerprints on both offline
and online phases, probabilistic algorithms can improve the system accuracy and
stability, compared to most deterministic algorithms. Therefore, more advanced in-
door localization systems have been focusing on optimizing probabilistic algorithms.
In this section, we discuss the probabilistic localization model from two aspects, floor
detection in a building and location estimation on the determined floor.
3
2.1 Floor Detection
Nowadays, floor detection becomes a necessity since multiple floors are quite com-
mon in buildings or other indoor/outdoor venues. In the indoor localization, the floor
misjudgment usually introduces severer biases. Therefore it should be avoided first
and foremost. In this subsection, we present a classification algorithm for floor detec-
tion.
In the localization scenes, we assume that there are K reference points. In the offline
phase, it generally stores the RSSI mean ππ , π = 1, β¦ , πΎ and the RSSI variance
πΊπ , π = 1, β¦ , πΎ from the k-th RP with the response location is π³π , π = 1, β¦ , πΎ. Each
RP k belongs to a unique floor πΉπ , π = 1, β¦ , π½. In the online phase, when get an RSSI
vector X, we can deduce the unknow floor F by a classification.
πΉΜ = argmax π(πΉπ |πΏ) (1)
πΉπ
Where π(πΉπ |πΏ) is the probability of the j-th floor under the condition of RSSI πΏ. It
can be obtained by
πΎ
π(πΉπ |πΏ) = β π(πΉπ , π³π |πΏ) (2)
π=1
As the probability of π(πΉπ , π³π |πΏ) always equal to zero when π β πΉπ . Then we have
π(πΉπ |πΏ) = β π(πΉπ , π³π |πΏ) (3)
πβπΉπ
By applying the Bayes theorem, the posterior probability π(πΉπ , π³π |πΏ) could be written
as
π(πΏ|πΉπ , π³π )π( π³π |πΉπ )π(πΉπ )
π(πΉπ , π³π |πΏ) = , π β πΉπ (4)
π(πΏ)
where π(πΉπ ) is the prior probability of the floor πΉπ . The uniform priors can be used
here that introduce no bias toward any particular floor. Thus πΉπ can be treated as a
constant. π(πΏ) is the distribution of signal strength, which is independent with the
location π³π and floor πΉπ . It can also be treated as a normalizing constant. Assuming πΎπ
is the number of RPs in the floor πΉπ , then
1
π( π³π |πΉπ ) = , π β πΉπ (5)
πΎπ
With respect to (5), Equation (4) can be simplified as
1
π(πΉπ , π³π |πΏ) β π(πΏ|πΉπ , π³π ), π β πΉπ (6)
πΎπ
Combined with (3) and (6), the final expression of floor detection can be written as
1
πΉΜ = argmax β π(πΏ|πΉπ , π³π ) (7)
πΉπ πΎπ
πβπΉπ
Finally, we can calculate the probability for each floor πΉπ separately, and choose the
maximum one as the corresponding floor.
4
2.2 Location Estimation
After floor detection, next step is to find the most likely location π³Μ on the determined
floor F. As we know, location is a continuous value while floors are always presented
discretely. Therefore, instead of classification, regression is a better way to find a
continuous solution.
Let π(π³π |πΏ, πΉ) be the probability of the k-th RP location under the condition of RSSI
πΏ and a known floor F. it is easily to obtain the probable location as the weighted
regression, as
π³Μ = β π³π π(π³π |πΏ, πΉ) (8)
πβπΉ
with
β π(π³π |πΏ, πΉ) = 1 (9)
πβπΉ
where π β πΉ denote the k-th RP which belong to floor F. By applying the Bayes theo-
rem, we can then obtain the so-called posterior probability of the location
π(πΏ|π³π , πΉ)π(π³π |πΉ)
π(π³π |πΏ, πΉ) = ,π β πΉ (10)
π(πΏ|πΉ)
where π(π³π |πΉ) is the prior probability of the location π³π on the floor F. For simplici-
ty we use only uniform priors here that introduce no bias toward any particular loca-
tion. π(πΏ|πΉ) is the distribution of signal strength, which is independent with the loca-
tion π³π and can be treated as a constant. Equation (10) can be simplified as
π(π³π |πΏ, πΉ) β π(πΏ|π³π , πΉ), π β πΉ (11)
with respect to (9), we can normalize (11) as
π(πΏ|π³π , πΉ)
π(π³π |πΏ, πΉ) = ,π β πΉ (12)
βπβπΉ π(πΏ|π³π , πΉ)
Therefore, the location π³Μ can be calculated by the conditional probability of RSSI
under location π³π within a floor F.
βπβπΉ π³π π(πΏ|π³π , πΉ)
π³Μ = (13)
βπβπΉ π(πΏ|π³π , πΉ)
With above two procedures, the probability model of indoor localization has been
theoretically constructed. However, practical application of the theory still faces fol-
lowing challenging issues: One is the deviation of the database. As the uncertainty
of the RSSI fluctuation and the inaccuracy of the indoor localization measurements
at RPs, database features would unavoidably deviate from the true values. The
other one is the data insufficiency of the database. In most custom-grade indoor
localization applications, collecting adequate RSSI measurements at each RP is actu-
ally impracticable since the offline survey usually covers a vast indoor area with
complex layouts. Somewhere, RSSI data collected on some RPs could be extremely
rare and inaccurate. As a result, localization by probabilistic algorithms could become
invalid in practical applications.
5
3 RBF LOCALIZATION NETWORKS
Response to above issues, we propose to combine the probabilistic localization model
with the RBF network. As RBF network shows the characteristic of explicit physical
significance and simplicity structure, the probabilistic algorithm based on RBF net-
work can be well implemented and improved in localization.
3.1 Radial Basis Functions Network
Historically, radial basis functions were introduced for the purpose of exact function
interpolation. Given a set of input vectors {πΏπ β βπ , π = 1, 2, β¦ , π} along with corre-
sponding target values {π
π β βπ , π = 1, 2, β¦ , π}, the goal is to find a smooth function
π(π₯) that fits every target value exactly, so that
π(πΏπ ) = π
π , π = 1, 2, β¦ , π (14)
The radial basis functions (RBF) technique consists of choosing a function F that has
the form
πΎ
π(πΏ) = β ππ π(βπΏ β ππ β) (15)
π=1
where π(βπΏ β ππ β) is the radial basis function of the k-th locally-tuned unit, and βββ
denotes a norm that is usually an Euclidean distance. The ππ β βπ , π = 1, β¦ , πΎ is the
center vector of the radial basis functions and the ππ β βπ , π = 1, β¦ , πΎ is the weight
vector.
There are some different kinds of radial basis π(π) for different fields and the most
commonly used is Gaussian functions π(π) = exp(β π 2 β2π 2 ). Henceforth, we focus
on the use of a Gaussian function as the radial basis function
βπΏ β ππ β2
π(βπΏ β ππ β) = exp (β ) (16)
2ππ2
where ππ is a measure of the width of the k-th Gaussian function with center ππ . We
will discuss to apply the RBF network to the probabilistic localization in next subsec-
tion.
3.2 Localization Network
With the RBF theory, we construct a classification network to detect the floor where
the user is and a regression network to estimate the userβs location at a known floor.
In the offline phase, the radio map construction can be considered as the initialization
of the RBF network parameters. In the online phase, when getting a RSSI measure-
ment with an unknown location at an unknown floor, we use a complete parallel net-
work to determine the floor and then to estimate the location.
Floor Detection Network. It is easily to discover the connection between RBF net-
work and floor detection algorithm. If consider each RP as an independent unit of the
network, and the mean of RSSI ππ at k-th RP as the corresponding center vector, the
conditional probability π(πΏ|π³π , πΉ), π β πΉ can be denoted by the radial basis function
as
π(πΏ|π³π , πΉ) = π(βπΏ β ππ β) (17)
6
When collecting an RSSI vector X, the floor classification function F(x) can be ob-
tained according to (7)
1
π(πΏ) = argmax β π(βπΏ β ππ β) (18)
πΉπ πΎπ
πβπΉπ
where πΎπ is the RPs number on the floor πΉπ . Combined with (16), (17) and (18), the
final expression of floor detection function π(πΏ) can be written as
1 βπΏ β ππ β2
π(πΏ) = argmax β exp (β ) (19)
πΉπ πΎπ 2ππ2
πβπΉπ
where ππ is a measure of the width of the k-th Gaussian function with center ππ . For
convenience, we can set a common width π for all Gaussian unit, and then adjust the
width to make the network achieve a higher performance. By this way we can obtain
an approximate value of π when the training data is insufficient to get the truth value.
Location Estimation Network. Similarly, we can conduct the location estimation by
RBF network. As the Gaussian function π(βπΏ β ππ β) denotes the probability of
RSSI π(πΏ|π³π , πΉ), π β πΉ , it is easily to obtain the probability of the locations
π(π³π |πΏ, πΉ) using a normalization technique as
π(βπΏ β ππ β)
ππ (πΏ, πΉ) = (20)
βπβπΉ π(βπΏ β ππ β)
with
β ππ (πΏ, πΉ) = 1 (21)
πβπΉ
Accordingly, the ππ (πΏ, πΉ) denote the conditional probability of location π³π under the
RSSI measurement X on the floor F. When getting an RSSI vector X, if considering
the RP location π³π as the weight vector, the location estimation function π(πΏ) can be
written as
π(πΏ) = β π³π ππ (πΏ, πΉ) (22)
πβπΉ
Combined with (16), (20) and (22), the final location estimation function can be writ-
ten as
βπΏ β ππ β2
π³π exp (β )
2ππ2
π(πΏ) = β (23)
βπΏ β ππ β2
β
πβπΉ πβπΉ exp (β )
2ππ2
where ππ is a measure of the width of the k-th Gaussian function with center ππ . As
the same, we can set a common width π for all Gaussian unit, and then adjust the
width to make the network achieve a higher performance.
In the offline phase, we can initialize the parameters of RBF location network includ-
ing the unit center ππ , width ππ and location π³π through the RM construction. In the
online phase, when getting a RSSI measurement with an unknown location at an un-
known floor, we make use of the RBF localization network to obtain a most probable
solution.
7
4 EXPERIMENT
In this section, we evaluate the performance of the proposed probabilistic localization
based on RBF network by comparing it to other methods in a specific experiment.
4.1 Experiment environment
The dataset was collected in the Beijing APM Mall with 7 floors (50Γ250m for each
floor). The training set consists of 8673 data collected at 2891 RPs. Validation set and
test set collection was conducted in a few days later. Totally about 2220 data point
were evenly distributed in the whole building. The true locations of these points are
all measured by the total station. Given the high density and large number of RSSI
observations, we were able to evaluate and compare the results of using different
localization algorithms.
4.2 Floor detection result
The performances of the floor detection network by KNN [3] and RBF network are
shown in Table 1. It indicates that the floor missed detection rates are different for
different floors. The miss detection rates for the F3 and F5 floors are the much higher
than other, around 2.7% and 1.0% respectively, and floor detection for locations at the
F1, F2 and F6 floors are all succeed in this experiment. Generally, the overall success-
ful detection rate of two methods are all satisfactory. RBF network still shows a little
superior to KNN due to the more complete probability model.
Table 1. Floor missing rate on different floors
Floors B1 F1 F2 F3 F4 F5 F6 Overall
KNN(K=1) 0.2% 0% 0% 2.7% 0.4% 1.0% 0% 0.60%
RBF network 0% 0% 0% 2.7% 0% 1.0% 0% 0.54%
4.3 Location estimation result
Table 2 shows the mean values of the localization errors at each floor by different
methods, KNN (K=1), KNN (K=5) [3], SVM [4], DNN [5], and RBF network. The
first four methods are commonly investigated in literatures and the last two are pro-
posed in this work. Generally, no matter what methods are used the localization accu-
racies at F1 and F2 are much higher than others, while the localization accuracies at
F5 and F6 are the worst. Compared with other four methods, the RBF networks show
obviously better performance at every floor.
Table 2. Average error of several methods of location estimation (error in meters)
Floors KNN(K=1) KNN(K=5) SVM DNN RBF network
B1 11.19 9.29 10.69 9.59 8.56
F1 7.66 6.27 7.26 6.45 6.06
F2 8.97 7.67 8.57 7.83 7.74
F3 11.16 9.83 10.89 9.84 9.64
F4 11.50 9.58 11.09 10.04 9.60
F5 13.67 12.93 13.08 13.23 12.09
F6 13.87 11.30 13.21 11.86 10.05
8
5 CONCLUSION
In this paper, we introduce the principle and algorithm of probabilistic localization in
detail. We propose to combine the probabilistic localization model with the RBF net-
work, which shows explicit physical significance and has simplicity structure. In the
offline phase, the radio map is firstly constructed by initially training the network. In
the online phase, when obtaining a RSSI measurement, the floor identification and
location estimation are carried out in order.
We compared the performance of the proposed method with others popularly used
indoor localization methods in a seven floors experimental environment. Analysis
results show that RBF network has a satisfactory performance in terms of floor detec-
tion and position estimation. The advantages of the proposed method are analyzed and
summarized as follow. Firstly, it provides an effective probabilistic approach that can
be applied to deficient RSSI dataset. Secondly, by considering the error distribution
better localization accuracy and higher robustness can be achieved.
Acknowledgement
The authors are grateful acknowledge the Wayzi Company for providing the experi-
ment data. This research was funded by the State Key Laboratory of Satellite Naviga-
tion System and Equipment Technology (Grant No. CEPNT-2017KF-10), and the
Open Foundation of Key Laboratory of Precise Engineering and Industry Surveying
of National Administration of Surveying, Mapping and Geoinformation (Grant No.
PF2017-05).
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