Fingerprint Kalman Filter in Indoor Positioning Applications

3rd IEEE Multi-conference on Systems and Control, July 8-10, 2009, Saint Petersburg, RUSSIA
pp. 1678-1683 (c)2009 IEEE
Fingerprint Kalman Filter in Indoor Positioning
Applications
Simo ALI-LÖYTTY, Tommi PERÄLÄ, Ville HONKAVIRTA, and Robert PICHÉ
Tampere University of Technology, Finland
Email: simo.ali-loytty@tut.fi
Abstract—In this paper, we present a new filter, the Fingerprint
Kalman Filter (FKF), and apply it to indoor positioning.
FKF enables sequential position estimation using WLAN RSSI
measurements and fingerprint data. Fingerprints that are collected
beforehand in a calibration phase contain samples of
the radio map in certain points, namely, calibration points.
This means that FKF does not need an analytic formula of
the measurement equation like conventional Kalman-type filters
do (e.g. the Extended Kalman Filter (EKF) and the Unscented
Kalman Filter (UKF)). Like EKF and UKF, FKF is based on the
recursive computation of the Best Linear Unbiased Estimator
(BLUE) and has small computational and memory requirements.
An often-used Kalman-type filter for this problem is so-called
Position Kalman Filter (PKF) that uses static position solutions as
measurements for the conventional Kalman filter. In the test part
of the paper, we compare FKF, PKF and different static location
estimation methods, namely, the Nearest Neighbor (NN) and the
Kernel method. The test data is real WLAN RSSI measurement
data. The results indicate that the filters give more accurate
position estimates than the static methods. FKF performs better
than PKF with NN as the static estimator, and the computational
load of FKF is smaller than PKF with the Kernel method.
I. INTRODUCTION
The idea of fingerprinting is to estimate the position of
the receiver using current measurements and “fingerprint”
data that are collected during the calibration phase. One
fingerprint contains at least the coordinates of the calibration
point and the mean of the data. An example of fingerprint data
is WLAN Received Signal Strength Indicators (RSSI). The
state of the art method of fingerprinting is the Weighted K-
Nearest Neighbor (WKNN) estimate of the position. WKNN
is a convex combination of the calibration points. Usually,
if we have enough measurements, WKNN works well but
when we have few measurements, we do not get accurate
position estimates. One way to improve our position estimate
is to use a filtering framework. Filtering enables the use of
past measurements in the computation of the current position
estimate. Normally, this means that we have a model for the
state evolution available. One possible approach to filtering
is to use static position solutions as measurements for the
Kalman Filter (KF), this kind of filter is called the Position
Kalman Filter (PKF) [1] [2, II.D]. Naturally, PKF works only
when we have reasonably accurate static position solutions
available. In this paper, we consider the situation where we
do not always have the accurate static position solution.
We consider the discrete-time system:
Initial state: x0, (1a)
State model: xk = fk−1(xk−1) + wk−1, (1b)
Meas. model: yk = hk(xk,vk), (1c)
p
where the state x = ∈ R
.
nx contains at least the
position coordinates p ∈ Rd in 2D (d = 2) or in 3D (d = 3).
We assume that the initial state x0 ∼ D(ˆx0, P0), where the
notation D(ˆx0, P0) means that x has distribution with mean
ˆx0 and covariance (matrix) P0. Function fk−1 : Rnx → Rnx is a known differentiable state transition function, and
wk−1 ∼ D(0, Qk−1) is the state model noise. The measurement
vector yk ∈ Rny . Function hk : Rnx × Rny → Rny is a
measurement function, which is known only in the calibration
points. The measurement noise vk is dependent on the position
(state) and is thus not additive. We assume that the state model
noise and the measurement noise are mutually independent
and independent of the initial state. The aim is to compute the
best (in some sense) estimate of the state x using all current
and past measurements.
Because the measurement function hk is not fully known
and the measurement noise is not additive, it is not possible
to use conventional nonlinear KF extensions, such as the
Extended Kalman Filter (EKF) [3, 4, 5] and the Unscented
Kalman Filter (UKF) [6, 7]. Sometimes, if we have enough
calibration points, it is possible to interpolate the necessary
measurement function hk values from the radio map and
to use UKF [8]. Another possibility is to use the Bayesian
framework and general nonlinear filters, such as Particle
Filters [9], Point Mass Filters [10, 11] and Gaussian Mixture
Filters [12, 13, 14, 15, 16], but then we should make more
assumptions about the initial state, noise distributions and the
measurement model. Furthermore, these nonlinear filters have
big memory and computational requirements and they are
quite sensitive to modeling errors.
An outline of the paper is as follows. In Section II, the
mathematical formulation of the radio map is presented. In
Section III, the static Best Linear Unbiased Estimator (BLUE)
is presented. In Section IV, the algorithm and derivation of
the new filter, namely, Fingerprint Kalman Filter (FKF), are
presented. The idea of FKF is originally presented in the M.Sc.
thesis [17]. The test results are presented in Section V.

II. RADIO MAP
Radio map F is the set of all fingerprints, that is
F =
Fi,
i∈IF
where IF is the set of all possible indices and Fi is the ith
fingerprint. The ith fingerprint has the form
Fi = (Ai, āi, Pai), (2)
where Ai ⊂ R d is the ith cell, and the approximate coordinate
of the ith calibration point is denoted by pi. The jth component
of the vector āi is the mean value of the RSSI values
measured from the access point APj and the jth diagonal
element of the diagonal matrix Pai is the variance of the RSSI
values measured from the access point APj. It is possible
that a fingerprint contains more information that might be
useful than what is presented here, see examples [17, 18, 19].
However, the above mentioned information is sufficient for
FKF. An illustrative example of a possible radio map for a
single access point is presented in Figure 1. The red polygons
represent the cells and the stem plot describes the means (ā)
of the RSSI readings recorded in each calibration point (p).
In this particular example, certain cells are empty because no
RSSI readings were obtained from the access point in these
cells.
Figure 1. Example of a radio map
AP
Signal strength
We assume that the measurements collected in the calibration
point pi or at least inside the current cell Ai represent the
RSSI spectrum inside the cell. This means that in every point
of the cell Ai we can use quantities āi and Pai as the estimates
of the mean of RSSI measurement and the covariance of
RSSI measurement, respectively. This assumption holds if
we have small enough cells. We also assume that cells Ai
are disjoint, that is Ai ∩ Aj = ∅ if i = j. Furthermore,
we assume that the cells cover all possible locations, that is
P (∪i∈IAAi) = 1 for every time instant. This is quite a strong
assumption and is not always true. One possibility for the
cases where we have an incomplete radio map is to interpolate
the missing fingerprints using the existing fingerprints or
predict the fingerprints using radio wave propagation models.
However, due to the complexity of indoor environments, filling
the gaps in the radio map might turn out very challenging.
More information about these approaches are in publications
[8] and [20].
III. BEST LINEAR UNBIASED ESTIMATOR
In this section, we present the well-known Best Linear
Unbiased Estimator (BLUE) [5, 16], which is the heart of
the new filter (Sec. IV). In this section we have omitted the
subscripts since we are examining a static estimation problem.
However, in the next section we will use the estimator recursively
and thus denote the time index with a subscript. Let the
expectation and the covariance of the joint distribution of the
state vector x ∈ R nx and the measurement vector y ∈ R ny
be
x ˜x
E =
y ˜y
x
V =
y
Pxx Pxy
Pyx Pyy
and (3a)
, (3b)
respectively. All linear unbiased estimators of the state x have
the form
VK
ˆx = ˜x + K(y − ˜y),
where the gain matrix K ∈ Rnx×ny is arbitrary. The covariance
of the estimator error is
△
= E (x − ˆx)(x − ˆx) T
= Pxx − KPyx − PxyK T + KPyyK T
= Pxx − PxyP −1
yy Pyx
+ (KPyy − Pxy) P −1
yy (KPyy − Pxy) T .
The Best Linear Unbiased Estimator is such that it minimizes
the estimator error covariance (4). It is easy to see (using (4))
that the minimun is obtained by choosing the gain matrix as
KKF = PxyP −1
yy .
This is the same gain matrix as the gain matrix of KF. Here
the minimization means that the matrix
VK − VKKF = (KPyy − Pxy)P −1
yy (KPyy − Pxy) T ≥ 0
is positive semidefinite for all K. So, BLUE and the corresponding
covariance of the error are
(4)
ˆx = ˜x + PxyP −1
yy (y − ˜y) (5a)
VKKF = Pxx − PxyP −1
yy Pyx
(5b)
Furthermore, it can be shown that if the joint distribution of
the state x and the measurement y is Gaussian, the conditional
distribution of the state is Gaussian, and BLUE (5) computes
the parameters of that distribution [21, Theorem 3.2.4] [5].

IV. FINGERPRINT KALMAN FILTER
The new filter, the Fingerprint Kalman Filter (FKF) is a
Kalman-type filter. Kalman-type filter is an umbrella term for
filters that approximate and store only the current estimate
and error covariance [16]. Because of that, Kalman-type filters
usually have small computational and memory requirements.
Many Kalman-type filters, such as the new filter FKF, EKF
and UKF, are based on recursive use of BLUE (Sec. III).
At every time instant the Kalman-type filter approximates
the necessary expectations (3) based on the previous state
and error estimates, and computes the new state and error
estimates using (5). FKF algorithm for the system (1) is
given as Algorithm 1. The symbols used in the algorithm are
described in Table I.
Algorithm 1 Fingerprint Kalman filter
1) Start with the initial estimate ˆx0 = E(x0) and the covariance
P0 = V (x0) of the estimation error. Set k = 1.
2) The prior estimate of state at tk is
ˆx −
k
= fk−1(ˆxk−1),
and the covariance of the prior estimation error is
where
P −
k = Fk−1Pk−1F T k−1 + Qk−1,
Fk−1 = f ′ k−1 (ˆxk−1). (6)
3) The posterior estimate of the state at tk is
ˆx k = ˆx −
k + P xy k P −1
yy k (y k − ˆy k )
and the covariance of the posterior estimation error is
where ˆyk =
Pxxk
Pk = Pxxk − Pxy k P −1
yy k P T xy k ,
i∈IF
=
i∈IF
Pxy = k
i∈IF
Pyy = k
and ˆpk =
i∈IF
i∈IF
βi,kāi,
βi,k Ppi + (pi − ˆpk)(pi − ˆpk) T ,
βi,k(pi − ˆpk)(āi − ˆyk) T ,
βi,k Pai + (āi − ˆyk)(āi − ˆyk) T .
βi,kpi.
4) Increment k and repeat from 2.
The prediction step (Step 2) of the Algorithm 1 is computed
using the linearization of the state transition function like EKF
computes the prediction step. The update step (Step 3) of the
Algorithm 1 is based on BLUE and the expectations (3) are
derived in Appendix A. Perhaps the most problematic part
of FKF is the approximation of the quantities αi ≈ βi,k and
Pxi ≈ Ppi because we do not know the prior distribution
Table I
EXPLANATION OF SYMBOLS USED IN ALGORITHM 1
Symbol Explanation Reference
βi,k Weight of ith calibration point (7)
āi Mean of radio map RSSI values (2)
fk−1 State transition function (1b)
Fk−1 Jacobian of the state transition function (6)
pi Coordinates of the ith calibration point (2)
Pa i Covariance of radio map RSSI values (2)
Pp i Covariance of uniform distribution in cell i (8)
Qk−1 State noise covariance matrix (1b)
yk Measurement vector (1c)
x −
k exactly, but instead, we only know the prior estimate
ˆx −
k and the covariance matrix of the estimation error P−
k .
Using these pieces of information we have to compute βi,k
and Ppi. Naturally, there are many ways to compute these
approximations; our implementation of the FKF algorithm
uses the following approximations [17]
P x −
k
∈ Ai ≈ βi,k =
N ˆx− k
P −
k
i∈IF
(pi)
N ˆx− k
P − (pi)
k
, (7)
where pi is the ith calibration point and N µ
Σ is the density
function of Gaussian distribution N(µ, Σ)
N µ
Σ (p) = exp− 1
2 (p − µ)TΣ−1 (p − µ)
det(2πΣ)
and
V (xi) = Pxi ≈ Ppi = V (˜xi), (8)
where the random variable ˜xi is uniformly distributed in the
cell Ai. If the cell Ai ⊂ R2 is a rectangle
x1
x1,min ≤ x1 ≤ x1,max
then
Ai =
V (˜xi) =
x2
(x1,max−x1,min) 2
12
0
x2,min ≤ x2 ≤ x2,max
V. TESTS
0
(x2,max−x2,min) 2
12
FKF was implemented and tested in Matlab using WLAN
RSSI measurements. The radio map was constructed so that
the receiver’s orientation was varied, and the time spent in each
calibration point was approximately 60 seconds. The test data
was gathered by moving the same receiver in the proximity of
the calibration points around the test building. Altogether, 4
different routes were recorded resulting in approximately 20
minutes of test data. One of the routes is shown in Figure 2.
These four routes are the same routes as in paper [18].
The radio map and the test data were gathered with a
MacBook laptop using an AirPort wireless network adapter
and Wireshark software. To improve the performance of
the position estimation algorithms the weakest signals were
removed from the data. Also, the incomplete normalized RSSI
.

histograms in the radio map were modified by adding small
probability mass to each bin so that no zero bins would occur
inside histograms.
As reported in paper [18], we have found that a stationary
state model gives better results than a constant velocity state
model. So, in this paper filters FKF and PKF use only the
stationary state model
xk = xk−1 + wk−1,
where state x ∈ R2 contains the 2D position coordinates and
the state model noise
8.3 0
wk−1 ∼ D 0,
.
0 8.3
Figure 2. Test case
The test data was processed with FKF and the results compared
to some known static position fingerprinting methods,
as well as Position Kalman Filter, which used the solutions of
the static methods as linear measurements. The static methods
considered in the test were the Nearest Neighbor method (NN)
[19, 22, 18, 17] using the 1-norm as a distance measure in the
RSSI space and the Weighted K-Nearest Neighbor method
(WKNN) [23, 24, 18, 17] using exponential Kernel approximation
of the radio map histograms (WKNN) with kernel
width of 2 and all of the calibration points (K = 77). Note that
the Kernel approximation of the radio map histograms [18, 17]
uses the whole calibration data instead of using only the mean
values like other methods in this paper. Consequently, WKNN
and PKFWKNN have significantly greater computational and
memory requirements than NN, PKFNN or FKF.
The results of the tests are given in Table II. The reported
quantities are the mean of the 2-norms of the error in meters
(ME), the root mean squared error (RMS) in meters, the 95th
percentile of errors (95%) in meters, and the maximum error
(Max) in meters.
Table II
RESULTS OF THE TESTS
ME RMSE 95% Max
(m) (m) (m) (m)
NN 7.7 14.3 21.6 127.4
WKNN 5.9 9.8 13.7 105.9
PKFNN 6.9 11.8 18.4 110.3
PKFWKNN 4.7 5.8 10.1 36.5
FKF 4.7 5.8 11.1 27.2
We see that PKFs give more accurate position estimates
than the corresponding static methods. In this test, FKF
outperforms the static methods as well as PKFNN. PKFWKNN
gives similar results as FKF. PKFWKNN is a little better than
FKF with respect to the 95th percentile and a little worse
with respect to the maximum error, but the difference is not
significant.
Figure 2 shows one of the test routes together with the
calibration cells, and the position solutions given by FKF and
PKFWKNN. As can be seen in the figure, the test cases are
almost one dimensional in the sense that the calibration points
mostly lie on line segments. Therefore, the results might be
somewhat too optimistic.
Also, robust methods could be used to remove the gross
errors to enhance the performance of the filters. Potentially,
this would improve PKF more than FKF. The robust methods
might, for example, examine the size of the normalized
innovations occurring in Kalman-type Filters and re-weight
the measurements accordingly (see e.g. [25]) or completely
discard measurements that seem to be false.
VI. CONCLUSIONS
In this paper, a novel filter for using fingerprints in indoor
positioning was presented. The novel filter, FKF, was tested
using real WLAN RSSI measurements and compared with
the best methods known, namely, PKF with different static
estimators discussed in [18]. In our tests, FKF outperformed
PKFNN and the static estimators. The difference between the
accuracies of FKF and PKFWKNN is not significant but FKF
is faster than PKFWKNN and also uses less information of
the radiomap, and thus FKF might be useful for positioning
in mobile devices with limited computational and memory
capacity. The test case was not as extensive as it might be,
and thus more tests in different areas should be done to verify
the results of this paper.

ACKNOWLEDGMENT
This research was partly funded by Nokia Corporation.
Simo Ali-Löytty also acknowledges the financial support of
Tampere Graduate School in Information Science and Engineering.
APPENDIX A
DERIVATION OF UPDATE STEP OF FKF
The update step of FKF is based on BLUE (Sec. III), so
it is enough to compute the approximations of the necessary
expectations (3)
and
E
x
y
x
V
y
˜x
˜y
≈
ˆp
ˆy
Pxx Pxy
Pyx Pyy
. (9)
First of all, we have to compute Fx,y(x, y), the cumulative
x
density function of the random variable . We use the
y
following assumptions, notations and approximations (see also
Sec. II):
1) We assume that cells Ai are disjoint, that is
Ai ∩ Aj = ∅ if i = j
and the cells cover all possible states so that
P (∪i∈IF Ai) = 1.
2) We define random variable xi which is constrained inside
the cell Ai. The cumulative density function of random
variable xi is
Fxi(x) = P (xi ≤ x) = 1
αi
x
−∞
χAi(x)dFx,
where normalization factor αi = P (x ∈ Ai) =
(assumption αi = 0) and characteristic function
1, x ∈ Ai
χAi(x) =
.
0, x /∈ Ai
Ai dFx
Note that if the cumulative density function Fx is absolutely
continuous, the random variable x has probability
density function (pdf) px and the random variable xi has
pdf pxi, which is proportional to px inside the cell Ai
and it is zero outside the cell Ai.
3) Let the mean and covariance of the random variable xi
be E(xi) = ¯xi and V (xi) = Pxi, respectively.
4) We assume that the mean of the random variable xi
is approximately the same as the coordinates of the ith
(2)
calibration point pi, that is ¯xi ≈ pi. We also assume that
the covariance of the random variable xi is the same as
the covariance of a random variable which is uniformly
(8)
distributed inside the cell Ai, that is Pxi ≈ Ppi, see
Sec. IV. Furthermore, we assume that the normalization
factor αi is approximately the same as the weight of the
ith calibration point, that is (see Sec. IV)
αi ≈ βi = N˜x
P − xx (pi)
N
i∈IF
˜x
P − .
(pi)
xx
5) Define random variable yi = (y|x ∈ Ai) which is the
measurement when the state is inside the cell Ai. We
assume that random variables yi and xi are independent.
6) Let the mean and covariance of the random variable yi
be E (yi) = ¯yi and V (yi) = Pyi, respectively.
7) We assume that the mean of the random variable yi
is approximately the same as the mean value of the
RSSI value measured in calibration point pi, that is
(2)
¯yi ≈ āi. Furthermore, we assume that the covariance
of the random variable yi is a diagonal matrix and its
diagonal components are approximately the variance of
(2)
the measured RSSI values, that is Pyi ≈ Pai, see Sec. II.
The cumulative density function is
Fx,y(x, y) = P (x ≤ x,y ≤ y)
1)
=
P (x ≤ x ∩ x ∈ Ai,y ≤ y)
i∈IF
5)
=
P (x ≤ x ∩ x ∈ Ai)P(yi ≤ y)
i∈IF
2)
=
αiP (xi ≤ x)P(yi ≤ y).
i∈IF
(10)
Thus the mean and parameters of the covariance (9) of the
x
random variable are .
y
⎡ ⎤ ⎡ ⎤
αi E(xi) αi¯xi
x (10) ⎢ ⎥
E = ⎢ i∈IF
3,6) ⎢ ⎥
⎥
y ⎣
αi E(yi)
⎦ = ⎢ i∈IF ⎥
⎣ ⎦
αi¯yi
4,7)
≈
⎡
⎢
⎣
i∈IF
i∈IF
i∈IF
βipi
βiāi
⎤
⎥
⎦
Pxx = E (x − ˜x)(x − ˜x) T
ˆp
ˆy
= E xx T − ˜x˜x T
(10)
= αi E xix T T
i − ˜x˜x
i∈IF
1)
=
i∈IF
i∈IF
αi
E xix T T
i − ˜x˜x
,
3)
=
αi Pxi + ¯xi¯x T i − ˜x˜x T
=
i∈IF
i∈IF
i∈IF
αi Pxi + (¯xi − ˜x)(¯xi − ˜x) T
4)
≈
βi Ppi + (pi − ˆp)(pi − ˆp) T ,

and
Pxy =
αi(¯xi − ˜x)(¯yi − ˜y) T
Pyy =
4,7)
4,7)
i∈IF
≈
i∈IF
≈
βi(pi − ˆp)(āi − ˆy) T ,
i∈IF
αi
i∈IF
Pyi + (¯yi − ˜y)(¯yi − ˜y) T
βi Pai + (āi − ˆy)(āi − ˆy) T ,
Pyx = P T
xy ≈ βi(āi − ˆy)(pi − ˆp) T .
i∈IF
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