This master's thesis presents a new approach for indoor positioning, based on the notion of separating ellipsoids. In order to improve the position estimation algorithm, the technique is combined with the algorithm A*, being applied to binary maps of the examined buildings to take into account obstacles such as walls. The combination of separating ellipsoids and A seems to promise an improvement over previous algorithms based on a probabilistic approaches.

1. WIRELESS POSITIONING
USING ELLIPSOIDAL
CONSTRAINTS
Giovanni Soldi
Lund University

2. i
Acknowledgements
First, I would like to thank my supervisor at the department of Mathematical Statis-
tics at Lund University, professor Andreas Jakobsson, for his continuous support
and commitment throughout this master’s thesis. Furthermore, I am really grateful
to Qubulus AB for providing me with the data and for giving me the opportunity
of a three-month internship.
I would like to thank also my supervisor in Milano, professor Marco Trubian, and
my Erasmus coordinator, professor Kevin Payne, for his help with all the bureau-
cratic issues. Special thanks go to all my international friends that I have known
during these ten months and made my stay in Sweden more fun.
Last but not least, I would like to thank my mother and all my family for their
continuous support and encouragement during all these years of studies.

3. ii
Abstract
Since the advent of the global positioning system in 1999, positioning systems have
been used to deliver location-based services (LBs) in outdoor environments. In
recent years, LBs have become of equal interest in indoor environments in a wide
range of personal and commercial applications. For this reason, recent research has
focused its attention on wireless positioning systems. This master’s thesis presents
a new approach for indoor positioning, based on the notion of separating ellipsoids.
In order to improve the position estimation algorithm, the technique is combined
with the algorithm A∗ , being applied to binary maps of the examined buildings to
take into account obstacles such as walls.
The combination of separating ellipsoids and A∗ seems to promise an improve-
ment over previous algorithms based on a probabilistic approaches.

4. CONTENTS
1 INTRODUCTION 1
1.1 Wireless Positioning Systems 1
2 PROBLEM EXPLANATION AND PREVIOUS WORK 3
2.1 Notation 3
2.2 Problem explanation 3
2.3 Properties of the weights functions 𝑤𝑖 4
2.4 The kernel-based method 5
2.5 A projection-based method 6
3 A NEW APPROACH 9
3.1 Theory 9
3.2 Homogeneous Embedding 11
3.3 SDP Formulations 12
3.4 Classification Rule 13
4 DATA 15
4.1 Plotting the locations on the maps 15
4.2 Data 18
4.3 Version with one slack for location 19
4.4 Iterative version 20
4.5 Variance ellipsoids 22
5 THE A* ALGORITHM 24
5.1 The A* algorithm 24
5.1.1 How A* works 24
5.2 Applied A* star 25
iii

5. iv
6 ANALYSES 28
6.1 Computation of the distance matrix and of the ellipsoids 28
6.2 Classification 31
6.3 Interpolation Step 38
6.4 Results for the first floor of the Hansa Mall 40
6.5 Results for the ground floor of the Hansa Mall 42
7 CONCLUSIONS 45
BIBLIOGRAPHY 46

6. Chapter 1
INTRODUCTION
One must learn by doing the thing;
for though you think you know it
You have no certainty, until you try.
Sophocles
1.1 Wireless Positioning Systems
Since the advent of the Global Positioning System (GPS) in 1999 [1], position-
ing systems have been used to deliver Location-Based services (LBs) in outdoor
environment. In the recent years, LBs have become of equal interest in indoor envi-
ronments as well as in a wide range of personal and commercial applications. These
include location-based network management and security, medicine and health care,
personalized information delivery and context awareness.
Unfortunately, coverage of the GPS system is limited in indoor environments
and dense urban areas. For this reason, recent research has focused on some ex-
isting wireless communication infrastructures such as wireless local area networks
(WLANs) to be a complementary technique. A WLAN is characterized by a num-
ber of access points (APs), which are devices that allow wireless communication
based on the IEEE 802.11 standard and are widespread indoors for Internet access.
Since the power-sensing function is available in every WLAN device, localization
using the received signal strength (RSS) is a relatively cost-effective solution.
Commonly, the procedure to estimate the position using RSS measurements is
based on a two step method known as location fingerprinting. Fingerprinting based
positioning is divided into an offline and an online phase.
1. In the offline phase (also known as training phase), by site-surveying, the RSS
from multiple APs at different points in the building are collected and stored
in a fingerprint database, called a radio map. More specifically, a number
𝑁 of survey locations in the site of interest are chosen and for each location
a number 𝑀 of RSS measurements is collected and stored to constitute the
fingerprint of that location. For each location, the sample mean and the
sample variance of the 𝑀 measurements are computed.
1

7. 2 Introduction Chapter 1
Figure 1.1. Example of a building map with the plotted survey locations.
2. In the online phase, the client’s position is inferred online by comparing the
measured RSS with the previous stored measurements, and the position esti-
mate is a combination of the locations whose fingerprints most closely match
the observation. This approach can be viewed as an application of pattern
recognition.
There are four key challenges in fingerprinting:
1. generation of fingerprints,
2. preprocessing for reducing computational complexity and enhancing accuracy,
3. selection of APs for use in positioning, and
4. estimation of the distance between a new RSS observation and the fingerprints.
When the positioning algorithm is performed on handheld devices, extra care
should be taken due to their constrained resource and, clearly, using all available
APs increases the computational complexity of the algorithm. For this reason, an

8. Section 1.1. Wireless Positioning Systems 3
AP selection method is needed. The recent work of Kushki et al. [2] offers a real-
time AP selection technique that minimizes the correlation between the selected
APs to reduce the complexity and ensure the coverage.
The contribution of this thesis is twofold:
∙ the initial development of a new algorithm based on the separating ellipsoids
method, in order to obtain a better estimate of the user position compared to
the previous methods.
∙ The application of the A∗ algorithm in order to take into account the previous
positions of the user and possible obstacles, such as walls, and to reduce the
computational complexity of the positioning algorithm.
The second chapter of this report explains the mathematical problem of the
wireless positioning and gives a brief overview of the previous work by Kushki et
al. [2] and by Fang et al. [3]. The third and the forth chapter concern the theory
of the separating ellipsoids method and the application of this method to an initial
dataset from the Entr´ mall in Malm¨. The fifth chapter explains the 𝐴∗ algorithm
e o
and how it is used in the context of wireless positioning. Finally, the sixth chapter
presents the obtained results by the carried analyses.

9. Chapter 2
PROBLEM EXPLANATION
AND PREVIOUS WORK
2.1 Notation
The following notation will be used throughout the report:
∙ 𝐿 : number of APs
∙ 𝑁 : number of survey locations in the radio map
∙ p : point in 2-D cartesian space
∙ r : measured RSS vector
∙ F(p𝑖 ) : Fingerprint record at the 𝑖th location of the radio map
∙ A : Multiple Discriminant Analysis (MDA) projection matrix
∙ 𝐷 : number of discriminative components (DCs) generated after the MDA
projection
∙ q : Projected RSS vector by the projection matrix A
ˆ
∙ F(p𝑖 ) : Projected fingerprint record at the 𝑖th location of the radio map
∙ Φ𝑖 : Computed ellipsoid for the location 𝑖
∙ 𝐷𝑖𝑠𝑡(𝑖, 𝑘) : real distance between the location 𝑖 and the location 𝑘
2.2 Problem explanation
One of the fundamental problems in location fingerprinting is to produce a position
estimate using training information on a discrete grid of training points when a new
4

10. Section 2.3. Properties of the weights functions 𝑤𝑖 5
RSS observation is received. That is, we seek a function 𝑔 such that
𝑔 : ℝ 𝐿 × ℝ2 × . . . × ℝ 2 → ℝ 2
𝑁
𝑔 : (r, p1 , . . . , p𝑁 ) → p
ˆ (2.2.1)
where p1 , . . . , p𝑁 are the cartesian coordinates (𝑥, 𝑦) of the 𝑁 survey locations with
respect to a predefined reference point (0, 0). If the function 𝑔 is restricted to the
class of linear functions, the problem is reduced to determining a set of weights such
that
∑𝑁
ˆ
p= 𝑤(r, F(p𝑖 ))p𝑖 , (2.2.2)
𝑖=1
where F(p𝑖 ) = [r𝑖 (1), . . . , r𝑖 (𝑀 )] is an 𝐿 × 𝑀 matrix defined as
⎛ 1 1
⎞
𝑟𝑖 (1) . . . 𝑟𝑖 (𝑀 )
F(p𝑖 ) = ⎝ . .. . ⎟
⎜ . . ⎠
. . . (2.2.3)
𝐿 𝐿
𝑟𝑖 (1) ... 𝑟𝑖 (𝑀 )
The column of the fingerprint matrix are RSS vectors r𝑖 (𝑡) = [𝑟𝑖 (𝑡), . . . , 𝑟𝑖 (𝑡)]𝑇
1 𝑀
that contain readings from 𝐿 APs at time 𝑡 in the 𝑖th survey location p𝑖 . For
convenience the weights 𝑤(r, F(p𝑖 )) can be also denoted with 𝑤𝑖 .
2.3 Properties of the weights functions 𝑤𝑖
The weights 𝑤𝑖 in the estimation function (2.2.2) are required to be decreasing
functions of the distance between an observation vector and the training records.
That is, survey points whose training records closely match the observations should
receive a higher weight. In particular, the functions 𝑤(r, F(p𝑖 ) should satisfy the
following properties:
∑𝑁
1. 𝑖=1 𝑤(r, F(p𝑖 )) = 1 so that the estimated position belongs to the convex
hull defined by the set of survey positions. This can be achieved by including
a normalization term in (2.2.2).
2. 𝑤(r, F(p𝑖 )) is a monotonically decreasing function in both arguments with
respect to a distance measure 𝑑(r, F(p𝑗 )):
𝑑(r, F(p𝑖 )) ≥ 𝑑(r′ , F(p𝑖 )) ⇒ 𝑤(r, F(p𝑖 )) ≤ 𝑤(r′ , F(p𝑖 )), (2.3.1)
𝑑(r, F(p𝑖 )) ≥ 𝑑(r, F(p𝑗 )) ⇒ 𝑤(r, F(p𝑖 )) ≤ 𝑤(r, F(p𝑗 )). (2.3.2)
The inequality (2.3.1) means that, if a RSS measurement r′ is closer than
another RSS measurement r to the training records in a location 𝑖, then, the
survey point 𝑖 should receive a higher weight for the RSS measurement r′ than
the RSS measurement r.

11. 6 Problem Explanation and previous work Chapter 2
The inequality (2.3.2) means that, if a RSS measurement r is further from
the training measurements in a location 𝑖 than the training measurements in
another location 𝑗, then, the survey site 𝑖 should receive a lower weight than
the location 𝑗.
2.4 The kernel-based method
The method developed by Kushki et al. [2] involves the use of kernel functions to
estimate the weights 𝑤𝑖 . In particular, a non-linear mapping 𝜙 : r ∈ ℝ𝑑 → 𝜙(r) ∈ ℱ
is used to map the input to a higher (possibly infinite) dimensional space ℱ where
the weight calculations take place. Once mapped, the weights are calculated in the
following way:
𝑀
1 ∑ ⟨𝜙(r), 𝜙(r𝑖 (𝑡))⟩
𝑤(r, F(p𝑖 )) = . (2.4.1)
𝑀 𝑡=1 ∥𝜙(r)∥ ∥𝜙(r𝑖 (𝑡))∥
where ⟨⋅, ⋅⟩ denotes the scalar product in ℱ. At first glance, the calculation of
weights in a possibly infinite dimensional space may seem computationally in-
tractable. Fortunately, the kernel trick can be used to calculate the inner product
in ℱ without the need for explicit evaluation of the mapping 𝜙. The kernel trick
allows the replacement of inner products in ℱ by a kernel evaluation on the input
vectors. In the WLAN context, the kernel is a function 𝑘 : ℝ𝐿 × ℝ𝐿 → ℝ such
that 𝑘(r, r′ ) = ⟨𝜙(r), 𝜙(r′ )⟩. Since the training data only enter weight calculations
through inner products, the kernel trick can be used to carry out inner products
in ℱ without the need for explicit evaluation of mapping 𝜙. The weights function
𝑤(r, F(p𝑖 )) then becomes
𝑀
1 ∑ 𝑘(r, r𝑖 (𝑡))
𝑤(r, F(p𝑖 )) = √ (2.4.2)
𝑀 𝑡=1 𝑘(r, r)𝑘(r𝑖 (𝑡), r𝑖 (𝑡))
. Mercer’s theorem guarantees the correspondence between a kernel function and
an inner product in a feature space ℱ, given that the kernel is a positive definite
function. Moreover the weights 𝑤(r, F(p𝑖 )) satisfy the properties (2.3.1) and (2.3.2)
with the distance 𝑑 defined as follows:
𝑁
𝜙(r) 1 ∑ 𝜙(r𝑖 (𝑡))
𝑑(r, F(p𝑖 )) = − (2.4.3)
∥𝜙(r)∥ 𝑁 𝑡=1 ∥𝜙(r𝑖 (𝑡))∥
In fact, we have that:
1(
𝑑(r, F(p𝑖 ))2 − 𝐶 ,
)
𝑤(r, F(p𝑖 )) = − (2.4.4)
2
One of the kernels proposed in the article by Kushki et al. [2] is the Gaussian kernel
defined as follows:
𝑁
−∥r − r𝑖 (𝑡)∥2
( )
1 ∑
𝑤(r, F(p𝑖 )) = 𝑒𝑥𝑝 . (2.4.5)
𝑁 𝑡=1 2𝜎 2

12. Section 2.5. A projection-based method 7
One drawback is that the Gaussian Kernel is linearly dependent on the number 𝑁
of the survey locations.
2.5 A projection-based method
The method proposed by Fang et al. [3] is based on the calculation of a projection
matrix A to map the RSS measurements in a feature space of lower dimension where
they are uncorrelated and then to take into account only the most discriminative
information.
More specifically, when AP selection techniques are applied to reduce the com-
putational cost, an additional binary-decision vector v = [0, 1, 0, . . . 0]𝑇 ∈ ℝ𝐿 is
required to indicate which APs are retained. If the number of non-zero components
is 𝐷, the dimension of the RSS measurement r is reduced from 𝐿 to 𝐷.
Unlike the zero-one weighting (binary decision) in the selection of APs, the
projection-based system reduces dimension from 𝐿 to 𝐷 by combining APs with
a discriminative projection A. The projection matrix A and the number 𝐿 is
determined by Multiple Discriminant Analysis (MDA).
The MDA criterion has widely been applied for the problem of multiple-class
classification. In our context, the classes can be viewed as the training measure-
ments in the different reference locations in a fingerprinting problem. The objective
of MDA is to find the components that are useful for discriminating between them.
After the MDA projection, the generated components, which are named discrim-
inative components (DCs), carry the most discriminative information and rank it
by its information quantity. Assuming that r ∈ ℝ𝐿 is the RSS measurement, the
projection-based system extracts the required DCs as follows:
q = A𝑇 r (2.5.1)
where q ∈ ℝ𝐷 , A𝐿×𝐷 is the MDA optimized projection matrix. The value of
𝐷 represents the number of required DCs for the location system and it can be
determined by a system threshold according to different applications.
In classical MDA, two scatter matrices called between-class (S𝐵 ) and within-
class (S𝑊 ) matrices are defined to quantify the quality of the projection, as follows:
𝑁 𝑀
∑∑
S𝑊 = (r𝑖 (𝑡) − u𝑖 )(r𝑖 (𝑡) − u𝑖 )𝑇
¯ ¯ (2.5.2)
𝑖=1 𝑡=1
𝑁
∑
S𝐵 = 𝑀 (¯ 𝑖 − u)(¯ 𝑖 − u)𝑇
u ¯ u ¯ (2.5.3)
𝑖=1
1
∑𝑀
¯
where u𝑖 = 𝑀 𝑡=1 r𝑖 (𝑡) is the mean of the training measurements in the 𝑖th
∑𝑁 ∑𝑀
location and u = 𝑁 1
¯ ⋅𝑀 𝑖=1 𝑡=1 r𝑖 (𝑡) is the global mean of the all measurements.
S𝑊 measures the closeness of the samples within the locations, whereas S𝐵
measures the separation between locations. An optimal projection would maximize

13. 8 Problem Explanation and previous work Chapter 2
Figure 2.1. In this figure it is possible to observe an example of projected RSS
measurement space for three different locations (three DCs).
the between-class measure and minimize the within-class measure. One way to
achieve this result is to maximize the ratio as follows:
ˆ ∣A𝑇 S𝐵 A∣
A𝑀 𝐷𝐴 = arg maxA 𝑇 (2.5.4)
∣A S𝑊 A∣
The numerator of (2.5.4) measures the separation among different locations that
is the RSS spatial separation (between-class), whereas the denominator measures
the compactness of each location that is the RSS temporal variation (within-class).
As a result, the optimal weighting matrix A satisfying (2.5.4) leads to the best
separation between different reference locations, because the discrimination and the
compactness are maximized at the same time. This way, the generated DCs in the
projected space have low temporal variation but high spatial separation to provide
discriminating information as much as possible. The dimension in the projected
space (number of DCs) can be determined by a system threshold according to
different applications.
Once obtained, the projection matrix A and the number of desired DCs, 𝐷, a
probabilistic approach is used to derive the weighting functions 𝑤𝑖 .
First, the observed RSS signal r and the fingerprint matrix F(p𝑖 ) are projected
by the projection matrix A:
q = A𝑇 r (2.5.5)
ˆ
F(p𝑖 ) = [A𝑇 ⋅ r𝑖 (1), . . . , A𝑇 ⋅ r𝑖 (𝑀 )] (2.5.6)
⎛ 1 1
⎞
𝑞𝑖 (1) . . . 𝑞𝑖 (𝑀 )
= ⎝ . .. . ⎟
⎜ . . ⎠
. . . (2.5.7)
𝐷 𝐷
𝑞𝑖 (1) ... 𝑞𝑖 (𝑀 )

14. Section 2.5. A projection-based method 9
For 𝑖 = 1, . . . , 𝑁 , we have:
ˆ
P(q∣F(p𝑖 ))
𝑤𝑖 = ∑𝑁 (2.5.8)
ˆ
P(q∣F(p𝑖 ))
𝑖=1
ˆ
where P(q∣F(p𝑖 )) indicates the likelihood value of the location p𝑖 , given the ob-
servation q = [𝑞 1 , . . . , 𝑞 𝐷 ]𝑇 . The likelihood that the Gaussian model produces is
formulated as:
𝐷
(𝑞 𝑑 − 𝑢𝑖 (𝑑))2
( )
ˆ
∑ 1 ˜
P(q∣F(p𝑖 )) = √ ⋅ 𝑒𝑥𝑝 − (2.5.9)
2𝜋˜𝑖 (𝑑)
𝑠 2˜𝑖 (𝑑)
𝑠
𝑑=1
where 𝑢𝑖 (𝑑) and 𝑠𝑖 (𝑑) are calculated in the following way:
˜ ˜
𝑀
1 ∑ 𝑑
𝑢𝑖 (𝑑) =
˜ 𝑞 (𝑡) (2.5.10)
𝑀 𝑡=1 𝑖
𝑀
1 ∑ 𝑑
𝑠𝑖 (𝑑) =
˜ (𝑞 (𝑡) − 𝑢𝑖 (𝑑))2
˜ (2.5.11)
𝑀 𝑡=1 𝑖
The final position can then be estimated by:
𝑁
∑
ˆ
p = 𝑤𝑖 ⋅ p𝑖
𝑖=1
𝑁 ˆ
∑ P(q∣F(p𝑖 ))
= ∑𝑁 ⋅ p𝑖 (2.5.12)
P( ˆ
q∣F(p𝑖 ))
𝑖=1 𝑖=1

15. Chapter 3
A NEW APPROACH
The way to calculate the weights 𝑤𝑖 using the Gaussian likelihood presented in the
previous chapter can be seen as a problem of statistical learning. In other words,
given a number 𝑁 of classes (in our specific case the 𝑁 survey locations) and a new
point 𝑥 (in our context a new RSS measurement r), for each class 𝑖 = 1, . . . , 𝑁 the
likelihood that the new point 𝑥 belongs to the class 𝑖 is calculated. We then search
the maximum of these and the class that realizes the maximum should be the class
to which the new point 𝑥 belongs. In this context, the method of the separating
ellipsoids [4] might constitute an improving alternative.
3.1 Theory
Suppose that one has a number 𝑁 of distinct classes and a number 𝑀 of points
(𝑥𝑖 , 𝑐𝑖 ), where 𝑐𝑖 denotes the class to which the 𝑖 point belongs. Fixing a class 𝑘,
with 𝑘 = 1, . . . , 𝑁 , one would like to include the points of this class in an ellipsoid,
in order to obtain the best separation ratio from all the points of the other classes.
A way to express this is to ask for two different separating ellipsoids that share the
same center and axis directions, but, the second one is larger by a factor 𝜌𝑘 , which
is subsequently called the separation ratio. The inner ellipsoid encloses all labeled
points 𝑥𝑖 with 𝑐𝑖 = 𝑘, while, all the remaining points lie outside the external one.
The higher the value of 𝜌𝑘 , the better the separation is. In Figure 3.1 it is possible
to observe 10 different classes randomly generated, each with 50 points. For these
classes, the separating ellipsoids have been computed and the separation ratios 𝜌𝑘
with 𝑘 = 1, . . . , 10, have been maximized.
An ellipsoid in ℝ𝑑 can be parametrized by its center 𝜇 and a symmetric, positive
semidefinite matrix P that determines its shape and size
𝜀𝑑 (𝜇, 𝑃 ) = {𝑥 ∈ ℝ𝑑 ∣(𝑥 − 𝜇)′ 𝑃 (𝑥 − 𝜇) ≤ 1}. (3.1.1)
where ’ denotes the transpose. The ellipsoid is degenerate if 𝑃 is singular. A scaled
concentric ellipsoid with the same shape can be obtained by dividing the matrix 𝑃
with a scalar 𝜌 > 0 and the scaled ellipsoid has an equivalent representation as:
𝜀𝑑 (𝜇, 𝑃 ) = {𝑥 ∈ ℝ𝑑 ∣(𝑥 − 𝜇)′ 𝑃 (𝑥 − 𝜇) ≤ 𝜌}, (3.1.2)
10

16. Section 3.1. Theory 11
Figure 3.1. In this figure it is possile to observe 10 different classes randomly
generated, each with 50 points. For these classes, the separating ellipsoids have been
computed and the separation ratios 𝜌𝑘 with 𝑘 = 1, . . . , 10 have been maximized.
√
where the ratio 𝜌 is the ratio between the lengths of the corresponding semimajor
axes of the two concentric ellipsoids. Suppose we are given the labeled training set
{(𝑥𝑖 , 𝑐𝑖 )}𝑀 . For each class 𝑘 ∈ {1, . . . , 𝑁 } the associated problem is to find the
𝑖=1
ellipsoids 𝜀𝑑 (𝜇𝑘 , 𝑃𝑘 ) and 𝜀𝑑 (𝜇𝑘 , 𝑃𝑘 ) such that 𝜌𝑘 is maximized while satisfying the
𝜌
𝑘
constraints
𝑥𝑖 ∈ 𝜀𝑑 (𝜇𝑘 , 𝑃𝑘 ) , ∀𝑖 : 𝑐𝑖 = 𝑘
( )
𝑃𝑘
𝑥𝑖 ∈ 𝜀𝑑 𝜇𝑘 ,
/ , ∀𝑖 : 𝑐𝑖 ∕= 𝑘. (3.1.3)
𝜌𝑘
In other words, the problem is
maximize 𝜌𝑘
′
subject to (𝑥𝑖 − 𝜇𝑘 ) 𝑃𝑘 (𝑥𝑖 − 𝜇𝑘 ) ≤ 1,
∀𝑖 : 𝑐𝑖 = 𝑘
(𝑥𝑖 − 𝜇𝑘 )′ 𝑃𝑘 (𝑥𝑖 − 𝜇𝑘 ) ≥ 𝜌𝑘 ,
∀𝑖 : 𝑐𝑖 ∕= 𝑘,
𝑃𝑘 ર 0
𝜌𝑘 ≥ 1, (3.1.4)
where the optimization variables are 𝜇𝑘 , 𝑃𝑘 , 𝜌𝑘 and 𝑃𝑘 ર 0 denotes the constraint
that 𝑃𝑘 must be a positive semidefinite matrix. This problem is always feasible and
the patterns of class 𝑘 are separable from all others using ellipsoids if and only if
the optimal 𝜌𝑘 > 1.

17. 12 A new approach Chapter 3
To handle also the case when the patterns are not strictly separable using ellip-
soids, the idea of soft margins is used, introducing slack variables 𝜉𝑖 for each of the
pattern inclusion or exclusion constraints, and a weighted penalty term is added to
the objective function. The new problem is then formulated in the following way:
∑
maximize 𝜌𝑘 − 𝛾 𝑖 𝜉𝑖
subject to (𝑥𝑖 − 𝜇𝑘 )′ 𝑃𝑘 (𝑥𝑖 − 𝜇𝑘 ) ≤ 1 + 𝜉𝑖 ,
∀𝑖 : 𝑐𝑖 = 𝑘
′
(𝑥𝑖 − 𝜇𝑘 ) 𝑃𝑘 (𝑥𝑖 − 𝜇𝑘 ) ≥ 𝜌𝑘 − 𝜉𝑖 ,
∀𝑖 : 𝑐𝑖 ∕= 𝑘,
𝑃𝑘 ર 0, 𝜉𝑖 ≥ 0. (3.1.5)
Here, 𝛾 is a positive weighting parameter. Both the problems (3.1.4) and (3.1.5)
are nonconvex but can be turned into convex optimization problems (being SDPs)
using an homogeneous embedding technique.
3.2 Homogeneous Embedding
Any ellipsoid in ℝ𝑑 can be viewed as the intersection of a homogeneous ellipsoid
(one centered at the origin) in ℝ𝑑+1 and the hyperplane
𝐻 = {𝑧 ∈ ℝ𝑑+1 ∣𝑧 = (𝑥, 1), 𝑥 ∈ ℝ𝑑 }. (3.2.1)
An homogeneous ellipsoid in ℝ𝑑+1 can be expressed as
𝜀𝑑+1 (0, Φ) = {𝑧 ∈ ℝ𝑑+1 ∣𝑧 ′ Φ𝑧 ≤ 1}, (3.2.2)
where Φ is a symmetric positive semidefinite matrix. To find the intersection of
𝜀𝑑+1 (0, Φ) with the hyperplane 𝐻, the matrix 𝑃 is partitioned as follows:
( )
𝑃 𝑞
Φ= , (3.2.3)
𝑞′ 𝑟
where 𝑃 ∈ ℝ𝑑×𝑑 , 𝑞 ∈ ℝ𝑑 , and 𝑟 ∈ ℝ. If 𝑧 = (𝑥, 1), then
𝑧 ′ Φ𝑧 ≤ 1 ⇔ 𝑥′ 𝑃 𝑥 + 2𝑞 ′ 𝑥 + 𝑟 ≤ 1. (3.2.4)
Now let
𝜇 = −𝑃 −1 𝑞
𝛿 = 𝑟 − 𝑞 ′ 𝑃 −1 𝑞, (3.2.5)
then
𝑧 ′ Φ𝑧 ≤ 1 ⇔ (𝑥 − 𝜇)′ 𝑃 (𝑥 − 𝜇) + 𝛿 ≤ 1. (3.2.6)

18. Section 3.3. SDP Formulations 13
Since P is positive semidefinite, we always have 𝛿 ≤ 1. In addition, whenever 𝑃
is positive definite, we have 0 ≤ 𝛿 < 1, and
( )
𝑃
𝜀𝑑+1 (0, Φ) ∩ 𝐻 = 𝜀𝑑 𝜇, . (3.2.7)
(1 − 𝛿)
( )
𝑃
In this case, 𝜀𝑑+1 (0, Φ) is called a homogeneous embedding of 𝜀𝑑 𝜇, (1−𝛿) .
Given a nondegenerate ellipsoid 𝜀𝑑 (𝜇, 𝑃 ) its homogeneous embedding in ℝ𝑑+1 is
nonunique, and can be parametrized as 𝜀𝑑+1 (0, Φ𝛿 ), for 0 ≤ 𝛿 < 1, where
( )
(1 − 𝛿)𝑃 −(1 − 𝛿)𝑃 𝜇)
Φ𝛿 = (3.2.8)
−(1 − 𝛿)𝜇′ 𝑃 (1 − 𝛿)′ 𝜇′ 𝑃 𝜇 + 𝛿
The special case 𝛿 = 0, is called a canonical embedding and this case 𝜀𝑑+1 (0, Φ0 )
is a degenerate ellipsoid because the matrix Φ0 is singular.
3.3 SDP Formulations
In this section, the problem of separating patterns in ℝ𝑑 using homogeneous ellip-
soids in ℝ𝑑+1 is considered. At first only two classes: the inner points and the outer
ones. For this purpose, the 𝑀 training data are divided into two sets {𝑥𝑖 }𝑀1 and
𝑖=1
{𝑦𝑗 }𝑀2 , where the 𝑥′ 𝑠 are points belonging to a specific class, and 𝑦𝑗 𝑠 are all the
𝑗=1 𝑖
′
other points and 𝑀1 + 𝑀2 = 𝑀.
First, the training data must be embedded in the hyperplane 𝐻 by letting
( )
𝑥𝑖
𝑎𝑖 = , 𝑖 = 1, . . . , 𝑀1 , (3.3.1)
1
( )
𝑦𝑗
𝑏𝑗 = 𝑗 = 1, . . . , 𝑀2 . (3.3.2)
1
The problem (3.1.5) using homogeneous embedding can be stated as
maximize 𝜌
subject to 𝑎′ Φ𝑎𝑖
𝑖 ≤ 1,
𝑖 = 1, . . . , 𝑀1
𝑏′ Φ𝑏𝑗 ≥ 𝜌,
𝑗
𝑗 = 1, . . . , 𝑀2 ,
Φ ર 0, 𝜌 ≥ 1 (3.3.3)
where the optimization variables are Φ and 𝜌. This is a convex optimization problem,
more specifically an SDP. Once the problem (3.3.3) is solved, the parametrization
in ℝ can be recovered using the transformations in (3.2.4) and (3.2.5) The two
concentric separating ellipsoids are
𝜀𝑑 (𝜇, 𝑃/(1 − 𝛿)), 𝜀𝑑 (𝜇, 𝑃/(𝜌(1 − 𝛿))). (3.3.4)
The next important properties are satisfied:

19. 14 A new approach Chapter 3
∙ If the patterns are separable, then the optimal solution to (3.3.3) is always a
canonical embedding, i.e, 𝛿 = 0.
∙ If the patterns are nonseparable, then the optimal solution to (3.3.3) always
has 𝜌 = 1, and Φ is degenerate such that 𝛿 = 1.
Similarly, the version with slack variables can be formulated as an SDP problem:
(∑ ∑ )
maximize 𝜌 − 𝛾 𝑖 𝜉𝑖 + 𝑗 𝜂𝑗
subject to 𝑎′ Φ𝑎𝑖 ≤ 1 + 𝜉𝑖 ,
𝑖
𝑖 = 1, . . . , 𝑀1
𝑏′ Φ𝑏𝑗 ≥ 𝜌 − 𝜂𝑗 ,
𝑗
𝑗 = 1, . . . , 𝑀2 ,
(3.3.5)
where the optimization variables are Φ, 𝜌, and the slack variables 𝜉𝑖 and 𝜂𝑗 , while 𝛾 is
a weighting parameter. Once the SDP problem (3.3.5) is solved, the transformations
(3.2.4) and (3.2.5) can be also used to find the separating ellipsoids in ℝ𝑑 .
Instead of using the weighting parameter 𝛾, the problem (3.3.5) can be parametrized
using the ratio 𝜌. Switching to notations that are explicit for multiclass problems,
given any fixed 𝜌 ≥ 1, for each class 𝑘 ∈ {1, . . . , 𝑁 }, the problem to solve is:
∑𝑛
minimize 𝑖=1 𝜂𝑖𝑘
′
subject to 𝑧𝑖 Φ𝑘 𝑧𝑖 ≤ 1 + 𝜂𝑖𝑘 ,
∀𝑖 : 𝑐𝑖 = 𝑘
′
𝑧𝑖 Φ𝑘 𝑧𝑖 ≥ 𝜌 − 𝜂𝑖𝑘 ,
∀𝑖 : 𝑐𝑖 ∕= 𝑘
Φ𝑘 ર 0,
𝜂𝑖𝑘 ≥ 0, 𝑖 = 1, . . . , 𝑀, (3.3.6)
where 𝑧𝑖 = (𝑥𝑖 , 1), and the optimization variables are Φ𝑘 and 𝜂𝑖𝑘 .
In Figures 3.2 and 3.3 the problem (3.3.6) has been solved for ten classes ran-
domly generated, each consisting of 50 points, using respectively values of 𝜌 equal
to 1.5 and 2.
3.4 Classification Rule
After solving in the training phase the problem (3.3.6) for each class 𝑘 ∈ {1, . . . , 𝑁 },
let Φ∗ , with 𝑘 = 1, . . . , 𝑁 , be the optimal obtained solution. Then, given a new
𝑘
data point 𝑥 ∈ ℝ𝑑 , we first let 𝑧 = (𝑥, 1) ∈ ℝ𝑑+1 and compute
𝜌𝑘 = 𝑧 ′ Φ∗ 𝑧,
𝑘 𝑘 = 1, . . . , 𝑁, (3.4.1)

20. Section 3.4. Classification Rule 15
Figure 3.2. In this figure the problem (3.3.6) has been solved for ten classes
randomly generated, each consisting of 50 points, using a value of 𝜌 equal to 1.5.
Figure 3.3. In this figure the problem (3.3.6) has been solved for ten classes
randomly generated, each consisting of 50 points, using a value of 𝜌 equal to 2.
then label the data with the class
ˆ
𝑘 = 𝑎𝑟𝑔 min 𝑧 ′ Φ∗ 𝑧 ′ . (3.4.2)
𝑘
𝑘
The quantity 𝜌𝑘 = 𝑧 ′ Φ𝑘 𝑧 for each 𝑘 = 1, . . . , 𝑁 can be seen as a distance (ellipsoid
ˆ
distance) of the point 𝑧 ∈ ℝ𝑑+1 from the class 𝑘. The class 𝑘 that realizes the
minimum of this distance is the class which mostly represents the point 𝑧.
Next chapters will deal with the application of this concept to the problem of
wireless positioning as an alternative to the Gaussian method.

21. Chapter 4
DATA
4.1 Plotting the locations on the maps
The first hurdle to overcome concerns the plot of the survey locations on the map.
Given the building map image, the GPS coordinates of both the top-left corner
(NW) of the map and the bottom-right corner of the map (SE) and the GPS coor-
dinates of the locations, we need to remap these coordinates to the ones to be used
on the display.
For each pair of GPS coordinates (𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒(⋅), 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑒(⋅)) we set:
𝑥𝐺𝑃 𝑆 (⋅) = 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑒(⋅) (4.1.1)
𝑦𝐺𝑃 𝑆 (⋅) = 𝑙𝑎𝑡𝑖𝑡𝑢𝑑𝑒(⋅) (4.1.2)
First, the GPS coordinates (𝑥𝐺𝑃 𝑆 (⋅), 𝑦𝐺𝑃 𝑆 (⋅)) must be converted to a regular coor-
dinate system where the 𝑥 unit distance is no longer dependent on the 𝑦 scale value.
To perform this, it is necessary to fix a common latitude 𝑦𝐶𝑂𝑀 , that should be the
same for all the points that are to be used in calculation. The needed transformation
is the following:
( )
2𝜋
𝑥𝑈 𝐺𝑃 𝑆 = 𝑥𝐺𝑃 𝑆 ⋅ cos ⋅ 𝑦𝐶𝑂𝑀 (4.1.3)
360
𝑦𝑈 𝐺𝑃 𝑆 = 𝑦𝐺𝑃 𝑆 (4.1.4)
Applying this transformation, the uniformed GPS coordinates of the 𝑁 𝑊 and 𝑆𝐸
corners are found.
Let now 𝑃 be a survey location with uniformed GPS coordinates (𝑥𝑈 𝐺𝑃 𝑆 (𝑃 ), 𝑦𝑈 𝐺𝑃 𝑆 (𝑃 )).
By a translation, one put the origin in the 𝑆𝐸 corner and find the new coordinates
16

22. Section 4.1. Plotting the locations on the maps 17
Figure 4.1. A rotation of angle 𝛽 − 𝛼 is needed so the rectangle becomes straight.
of the 𝑁 𝑊 corner and of the location 𝑃 as
𝑥𝑈 𝐺𝑃 𝑆𝑡 (𝑃 ) = 𝑥𝑈 𝐺𝑃 𝑆 (𝑃 ) − 𝑥𝑈 𝐺𝑃 𝑆 (𝑆𝐸) (4.1.5)
𝑦𝑈 𝐺𝑃 𝑆𝑡 (𝑃 ) = 𝑦𝑈 𝐺𝑃 𝑆 (𝑃 ) − 𝑦𝑈 𝐺𝑃 𝑆 (𝑆𝐸) (4.1.6)
𝑥𝑈 𝐺𝑃 𝑆𝑡 (𝑁 𝑊 ) = 𝑥𝑈 𝐺𝑃 𝑆 (𝑁 𝑊 ) − 𝑥𝑈 𝐺𝑃 𝑆 (𝑆𝐸) (4.1.7)
𝑦𝑈 𝐺𝑃 𝑆𝑡 (𝑁 𝑊 ) = 𝑦𝑈 𝐺𝑃 𝑆 (𝑁 𝑊 ) − 𝑦𝑈 𝐺𝑃 𝑆 (𝑆𝐸) (4.1.8)
𝑥𝑈 𝐺𝑃 𝑆𝑡 (𝑆𝐸) = 𝑥𝑈 𝐺𝑃 𝑆 (𝑆𝐸) − 𝑥𝑈 𝐺𝑃 𝑆 (𝑆𝐸) (4.1.9)
𝑦𝑈 𝐺𝑃 𝑆𝑡 (𝑆𝐸) = 𝑦𝑈 𝐺𝑃 𝑆 (𝑆𝐸) − 𝑥𝑈 𝐺𝑃 𝑆 (𝑆𝐸) (4.1.10)
(4.1.11)
The next steps are suggested by the scheme in Figure 4.1. An anticlockwise rotation
of angle 𝛼 − 𝛽 is applied to the points 𝑁 𝑊 and 𝑃 so that the 𝑁 𝐸 and 𝑆𝑊 corners
lie respectively on the 𝑦− axes and on the 𝑥− axes. It is possible then to use these
new coordinates to plot the locations on the image of the map. The angle 𝛽 is
computed using the image of the map. In particular, denoting with 𝑤 the width
and with ℎ the height of the image of the map in pixels, the length of the diagonal
𝑑 of the image in pixels is given by:
√
𝑑 = 𝑤2 + ℎ2 (4.1.12)

23. 18 Data Chapter 4
Figure 4.2. After the rotation we are able to compute the new coordinates for the
𝑁 𝐸 and 𝑆𝑊 corners.
The angle 𝛽 is then given by:
(𝑤)
𝛽 = arcsin (4.1.13)
𝑑
The angle 𝛼 instead is given by:
( )
𝑦𝑈 𝐺𝑃 𝑆 (𝑁 𝑊 ) − 𝑦𝑈 𝐺𝑃 𝑆 (𝑆𝐸)
𝛼 = arctan (4.1.14)
𝑥𝑈 𝐺𝑃 𝑆 (𝑆𝐸) − 𝑥𝑈 𝐺𝑃 𝑆 (𝑁 𝑊 )
An anticlockwise rotation is applied to the points 𝑃 and 𝑁 𝑊 using the following
matrix:
( )
cos(𝛼 − 𝛽) − sin(𝛼 − 𝛽)
𝐴= (4.1.15)
sin(𝛼 − 𝛽) cos(𝛼 − 𝛽)
The new coordinates for the 𝑁 𝑊 and for the 𝑆𝐸 corners are given then by:
( ) ( )
𝑥𝑈 𝐺𝑃 𝑆𝑡𝑟 (𝑁 𝑊 ) 𝑥𝑈 𝐺𝑃 𝑆𝑡 (𝑁 𝑊 )
= 𝐴⋅ (4.1.16)
𝑦𝑈 𝐺𝑃 𝑆𝑡𝑟 (𝑁 𝑊 ) 𝑦𝑈 𝐺𝑃 𝑆𝑡 (𝑁 𝑊 )
( ) ( )
𝑥𝑈 𝐺𝑃 𝑆𝑡𝑟 (𝑃 ) 𝑥𝑈 𝐺𝑃 𝑆𝑡 (𝑃 )
= 𝐴⋅ (4.1.17)
𝑦𝑈 𝐺𝑃 𝑆𝑡𝑟 (𝑃 ) 𝑦𝑈 𝐺𝑃 𝑆𝑡 (𝑃 )

24. Section 4.1. Plotting the locations on the maps 19
Figure 4.3. Change of coordinates of the location 𝑃 to the domain [0, 1] × [0, 1].
The new coordinates 𝑢(𝑃 ) and 𝑣(𝑃 ) represent respectively the ratio of the width 𝑤
and of the height ℎ of the map image, where the corresponding pixel is.
In this new coordinate system it is easy now to find the coordinates of the 𝑁 𝐸
corner and of the 𝑆𝑊 corner as shown in Figure 4.2. In particular:
( ) ( )
𝑥𝑈 𝐺𝑃 𝑆𝑡 (𝑁 𝐸) 0
= (4.1.18)
𝑦𝑈 𝐺𝑃 𝑆𝑡 (𝑁 𝐸) 𝑦𝑈 𝐺𝑃 𝑆𝑡 (𝑁 𝑊 )
( ) ( )
𝑥𝑈 𝐺𝑃 𝑆𝑡 (𝑆𝑊 ) 𝑥𝑈 𝐺𝑃 𝑆𝑡 (𝑁 𝑊 )
= (4.1.19)
𝑦𝑈 𝐺𝑃 𝑆𝑡 (𝑆𝑊 ) 0
(4.1.20)
A final translation of the points is necessary to put the origin in the 𝑆𝑊
corner and get new positive coordinates (𝑥𝑈 𝐺𝑃 𝑆𝑡𝑟𝑡 (⋅), 𝑦𝑈 𝐺𝑆𝑡𝑟𝑡 (⋅)) for the corners
𝑆𝐸,𝑁 𝑊 ,𝑁 𝐸 and the point 𝑃 .
At this point we calculate:
𝑥𝑈 𝐺𝑃 𝑆𝑡𝑟𝑡 (𝑃 )
𝑢(𝑃 ) = (4.1.21)
𝑥𝑈 𝐺𝑃 𝑆𝑡𝑟𝑡 (𝑆𝐸)
𝑦𝑈 𝐺𝑃 𝑆𝑡𝑟𝑡 (𝑃 )
𝑣(𝑃 ) = (4.1.22)
𝑦𝑈 𝐺𝑃 𝑆𝑡𝑟𝑡 (𝑁 𝑊 )

25. 20 Data Chapter 4
Basically, as shown in Figure 4.3, the initial domain is restricted to [0, 1] × [0, 1] and
𝑢(𝑃 ), 𝑣(𝑃 ) are the coordinates of the location 𝑃 in the new domain and represent
respectively the ratio of the width 𝑤 and of the height ℎ of the map image where
the corresponding pixel is. Then, in order to find the corresponding pixels (𝑝𝑥, 𝑝𝑦)
in the map:
𝑝𝑥 = 𝑟𝑜𝑢𝑛𝑑(𝑤 ⋅ 𝑢) (4.1.23)
𝑝𝑦 = 𝑟𝑜𝑢𝑛𝑑(ℎ ⋅ (1 − 𝑣)) (4.1.24)
where 𝑟𝑜𝑢𝑛𝑑 means rounding to the nearest integer.

26. Section 4.2. Data 21
Figure 4.4. Map of the Entr´ shopping mall in Malm¨ with the 61 plotted loca-
e o
tions.
4.2 Data
The data used to carry on the preliminary analyses have been taken in the Entr´
e
shopping mall in Malm¨. The dataset consists of:
o
∙ 70 access points (APs) for the Wireless signal.
∙ 13 access points (APs) for the GSM signal.
∙ 61 survey locations.
∙ 320 measurements of the wireless signal in each of the 61 locations.
∙ 240 measurements of the GSM signal in each of the 61 locations.
The GSM signal is treated in the same way of the Wifi signal. Since the final
results for the GSM measurements are not as good as for the Wifi measurements,
this report will treat only the Wifi signal.
The first step is to project the Wifi measurements in a lower dimensional space
by a projection matrix A, as described in the Section 2.5 to reduce the number of
APs from 70 to 13 DCs. Since, we will perform this step on all the datasets that we
will have to deal with, from now on, to simplify the notation, we use the wording
RSS measurements to denote the projected RSS measurements.
Furthermore, for each location 𝑖 with 𝑖 = 1, . . . , 61 the signature vectors, thus
the mean vector and the standard deviation value for the RSS measurements, have
been calculated.

27. 22 Data Chapter 4
We initially attempt to apply the method of the separating ellipsoids to the
training measurements for each survey location 𝑘 . In other words, fixing a location
𝑘 we want to construct the ellipsoid that contains the RSS measurements of the
location 𝑘 and keeps outside all the measurements of the other sites different from
𝑘. Given q𝑖 (1), . . . , q𝑖 (𝑀 ), the 𝑀 RSS measurements at the location 𝑖 and letting
( )
q𝑖 (𝑗)
z𝑖 (𝑗) = (4.2.1)
1
the SDP problem to solve is the following:
minimize 𝜌
subject to z𝑘 (𝑖)′ Φ𝑘 z𝑘 (𝑖) ≤ 1,
∀𝑖 = 1, . . . , 𝑀
z𝑙 (𝑗)′ Φ𝑘 z𝑙 (𝑗) ≥ 𝜌,
∀𝑙 = 1, . . . , 𝑁 𝑙 ∕= 𝑘
∀𝑗 = 1, . . . , 𝑀
Φ𝑘 ર 0,
𝜌 > 1, (4.2.2)
where the optimization variables are Φ𝑘 and 𝜌. As expected the measurements are
not strictly separable and for this reason the version with all slack variables is used:
∑
minimize 𝑖,𝑗 𝜂𝑖𝑗
subject to z𝑘 (𝑗)′ Φ𝑘 z𝑘 (𝑗) ≤ 1 + 𝜂𝑘𝑗 ,
∀𝑗 = 1, . . . , 𝑀
z𝑙 (𝑗)′ Φ𝑘 z𝑙 (𝑗) ≥ 𝜌 − 𝜂𝑙𝑗 ,
∀𝑙 = 1, . . . , 𝑁 𝑙 ∕= 𝑘
∀𝑗 = 1, . . . , 𝑀
Φ𝑘 ર 0,
𝜂𝑖𝑘 ≥ 0. (4.2.3)
where the optimization variables are Φ𝑘 and all the slack variables 𝜂𝑖𝑗 . Regrettably,
due to the large number of variables required to solve this optimization problem,
the solution requires a vast amount of computational power and memory allocation.
For this reason, in view of larger environments, alternative versions have been
formulated in order to decrease the number of variables and constraints to optimize.
4.3 Version with one slack for location
One may simplify the original problem by restricting the formulation to only allow
for one slack variable for each location. This approach will differ from the version

28. Section 4.4. Iterative version 23
with all slack variables since the points of each location have all the same slack
variable. That is, let 𝑘 be a fixed survey location. The formulation of the problem
is:
∑𝑁
minimize 𝑘=1 𝜂𝑘
subject to z′ (𝑖)Φ𝑘 z𝑘 (𝑖) ≤ 1 + 𝜂𝑘 ,
𝑘
∀𝑖 = 1, . . . , 𝑀
z′ (𝑖)Φ𝑘 z𝑙 (𝑖)
𝑙 ≥ 𝜌 − 𝜂𝑙 ,
∀𝑙 = 1, . . . , 𝑁 𝑙 ∕= 𝑘
∀𝑖 = 1, . . . , 𝑀 (4.3.1)
Φ𝑘 ર 0, (4.3.2)
𝜂𝑖 ≥ 0, ∀𝑖 = 1, . . . , 𝑁, (4.3.3)
where the optimization variables are Φ𝑘 and the 𝑁 slack variables 𝜂𝑘 . The efficiency
in computing the ellipsoid has improved substantially since the number of variables
to optimize is 𝑁 against 𝑁 ⋅ 𝑀 of the version with all the slack variables. The
main drawback is when there is no separability in the classes. In fact, in this case
the ellipsoid, even if they contain the most part of the point of the respective class,
overlap themselves critically and this can lead to a misclassification of the data.
4.4 Iterative version
As an alternative, instead of considering for each location 𝑘 all the measurements
𝑀 , one may initially solve the optimization problem (4.2.3), taking into account a
lower number of measurements for each location. Then, during next iterations, one
may add the same number of new measurements and solve again the optimization
problem, using for the old points the values of the slack variables computed at the
previous steps.
In particular, let 𝑛1 be an integer number such that 𝑛1 < 𝑀 and 𝑀 is a multiple
of 𝑛1 . At the first iteration, the version with all slack variables is applied but with
only 𝑛1 points. That is, let 𝑘 be a fixed location, then:
∑ (1)
minimize 𝑖,𝑘 𝜂𝑘𝑖
(1) (1)
subject to z′ (𝑖)Φ𝑘 z𝑘 (𝑖) ≤ 1 + 𝜂𝑘𝑖 ,
𝑘
∀𝑖 = 1, . . . , 𝑛1
(1) (1)
z′ (𝑖)Φ𝑘 z𝑗 (𝑖)
𝑗 ≥ 𝜌 − 𝜂𝑗𝑖 ,
∀𝑗 = 1, . . . , 𝑁 𝑗 ∕= 𝑘
∀𝑖 = 1, . . . , 𝑛1
(1)
Φ𝑘 ર 0, (4.4.1)
(1)
𝜂𝑖𝑘 ≥ 0, ∀𝑖 ∀𝑘 (4.4.2)

29. 24 Data Chapter 4
Figure 4.5. In this Figure, the iterative version has been solved for ten classes
randomly generated, each consisting of 50 points, using a value of 𝜌 equal to 1.1
and adding 25 points at each iteration.
(1) (1)
where the optimization variables are Φ𝑘 and the 𝑁 ⋅ 𝑛1 slack variables 𝜂𝑖𝑗 . From
the second iteration 𝑛1 points are added to each location. For the old points the
values of the slack variables computed at the previous steps are used. That is, at
iteration (𝑙) with 𝑙 ≥ 2
∑ (𝑙)
minimize 𝑖,𝑘 𝜂𝑖𝑘
(𝑙) (𝑙−1)
subject to z′ (𝑖)Φ𝑘 z𝑘 (𝑖) ≤ 1 + 𝜂𝑖𝑘
𝑘 ,
∀𝑖 = 1, . . . , (𝑙 − 1) ⋅ 𝑛1
(𝑙) (𝑙)
z′ (𝑖)Φ𝑘 z𝑘 (𝑖)
𝑘 ≤ 1 + 𝜂𝑖𝑘 ,
∀𝑖 = (𝑙 − 1) ⋅ 𝑛1 + 1, . . . , 𝑙 ⋅ 𝑛1
∀𝑗 = 1, . . . , 𝑁, 𝑗 ∕= 𝑘
(𝑙) (𝑙−1)
z′ (𝑖)Φ𝑘 z𝑗 (𝑖) ≥ 𝜌 − 𝜂𝑖𝑗
𝑗 ,
∀𝑖 = 1, . . . , (𝑙 − 1) ⋅ 𝑛1
(𝑙) (𝑙)
z′ (𝑖)Φ𝑘 z𝑗 (𝑖)
𝑗 ≥ 𝜌 − 𝜂𝑖𝑗 ,
∀𝑖 = (𝑙 − 1) ⋅ 𝑛1 + 1, . . . , 𝑛1 ⋅ 𝑙
(𝑙)
Φ𝑘 ર 0,
(𝑙)
𝜂𝑖𝑘 ≥ 0, ∀𝑖, ∀𝑘, (4.4.3)
(𝑙−1) (𝑙)
where 𝜂𝑖𝑘 are the slack variables optimized at the previous step while 𝜂𝑖𝑘 are the
ones that correspond to the new added points and must be optimized. After the
last iteration the last slack variables might be used to recompute again the previous

30. Section 4.5. Variance ellipsoids 25
ones.
After testing this method on the simulated data in which each class consists of
50 points it has been noticed that if only two points are added at each iteration the
ellipsoid are inaccurate. They are moved with respect to the points that they should
contain, while increasing the number of points that we add at each iteration,the
obtained ellipsoids are almost equal to the ones computed with the all slack version.
In order to obtain accurate ellipsoids the number of points 𝑛1 should be major or
equal to 𝑀 . However, such a solution will still suffer from the demanding memory
2
allocation similar to the original problem.
In the Figure 4.5, the iterative version has been solved for ten classes randomly
generated, each consisting of 50 points, using a value of 𝜌 equal to 1.1 and adding
25 points at each iteration.
4.5 Variance ellipsoids
A third option consists of building, after fixing a location 𝑘, a normal oriented el-
lipsoid (variance ellipsoid) that contains at least the most part of the measurements
of the survey site 𝑘. For all the remaining locations, it is possible then to check how
many measurements are inside this ellipsoid and take into account, when computing
the separating ellipsoid for the location 𝑘, only the locations 𝑖 whose measurements
mostly overlap the measurements of the site 𝑘.
In particular, the equation of a normal oriented ellipsoid in an 𝑛-dimensional
space is defined by the equation
𝑥𝑇 ⋅ A ⋅ 𝑥 <= 1
where A is a n-dimensional diagonal matrix
⎛ ⎞
𝜆1 . . . 0
⎜ . .. . ⎟
A=⎝ . . . . ⎠
.
0 ... 𝜆𝑛
where 𝜆𝑖 , for 𝑖 = 1, . . . , 𝑛, are the eigenvalues of the matrix. The relation between
the equatorial radii 𝑟𝑗 for 𝑗 = 1, . . . , 𝑛 of the ellipsoids and the eigenvalues 𝜆𝑗 is
given by
1
𝑟𝑗 = √ (4.5.1)
𝜆𝑗
Since for each location 𝑖 it is possible to calculate the mean vector of the RSS
measurements and the vector of the standard deviations 𝜎(𝑖) = [𝜎1 (𝑖), . . . , 𝜎𝐷 (𝑖)]𝑇 ,
it is also possible to construct the ellipsoid that contains most of the points of the
site 𝑖. The matrix which defines the ellipsoid is given by:
⎛ 1 ⎞
2
𝜎1 (𝑖)
... 0
B(𝑖) = 𝐶 ⋅ ⎜ . .. . ⎟
⎜
⎝ . . . . ⎟
. ⎠ (4.5.2)
1
0 . . . 𝜎2 (𝑖)
𝐷

31. 26 Data Chapter 4
where 𝐶 is a chosen constant and it depends on the number of DCs.
Once computed the ellipsoids for each location we perform the following steps:
1. Fix a location 𝑘.
2. For each location 𝑗, with 𝑗 = 1, . . . , 𝑁 , count how many measurements of this
location are inside the variance ellipsoid of the site 𝑘.
3. Repeat the two steps above for each location 𝑘 with 𝑘 = 1, . . . , 𝑁 .
By doing this, it is possible to know, for each site 𝑘, which are the locations 𝑗,
whose measurements mostly overlap the measurements of the site 𝑘. Finally, taking
into considerations only these locations 𝑗 to compute the separating ellipsoid for
the location 𝑘, it is possible to reduce the number of constraints and variables to
optimize. Regrettably, it can happen that far away locations have many measure-
ments that overlap, or many locations overlap each other. For this reason, larger
buildings will still necessitate prohibitive memory requirements.
Figure 4.6. In this Figure, it is possible to observe a set of 3-D points with the
corresponding variance ellipsoid, computed as described in the section 4.5. As it is
possible to observe, almost all the points lie inside the ellipsoid.

32. Chapter 5
THE A* ALGORITHM
5.1 The A* algorithm
In computer science, A∗ is a widely used algorithm for path-finding and graph
traversal, the process of plotting an efficiently traversable path between points,
called nodes. Noted for its performance and accuracy, the algorithm was first de-
scribed by Peter Hart, Nils Nilsson and Bertram Raphael in 1968 [6]. It is an
extension of Edger Dijkstra’s 1959 algorithm and it achieves better performance
(with respect to time) by using heuristic distances.
A∗ uses a best-first search and finds the least cost-path from a given initial node
to one goal node. To achieve this, it uses a distance-plus-cost heuristic function
(usually denoted by 𝑓 (𝑥)) to determine the order in which the search visits nodes
in the tree. The distance-plus-cost heuristic is a sum of two functions:
∙ the path-cost function, which is the cost from the starting node to the current
node (usually denoted by 𝑔(𝑥))
∙ and an admissible ”heuristic estimate” of the distance from each node to the
goal node (usually denoted by ℎ(𝑥)).
In mathematical terms 𝑓 (𝑥) is then given by:
𝑓 (𝑥) = 𝑔(𝑥) + ℎ(𝑥). (5.1.1)
The function ℎ(𝑥) is a mathematical way of using previous knowledge in order to
expand the fewest possible nodes in searching for an optimal path and speed up the
algorithm. For example, inside a building, ℎ(𝑥) might represent the straight-line
distance to the goal (without taking into account the walls), since that is physically
the smallest possible distance between any two points or nodes. The adjective ”ad-
missible” means that ℎ(𝑥) must not overestimate the distance to the goal. Dijkstra’s
algorithm is a particular case of the A∗ algorithm, then using ℎ(𝑥) = 0.
5.1.1 How A* works
∗
The A algorithm requires a starting node 𝑠 and a goal node 𝑑. The steps of the
algorithm are the followings:
27

33. 28 The A* algorithm Chapter 5
1. Add the starting node 𝑠 to the ”open set”. The open set contains the nodes
that must be explored and they might be inserted in the optimal path towards
the goal node 𝑠.
2. Repeat the following:
a) Look for the node in the open set which has the lowest 𝑓 value. We refer
to this as the ”current node”.
b) Switch it to the ”closed set”.
c) For each of the nodes that are reachable from the current node:
∙ If it is in the closed list, ignore it. Otherwise do the following.
∙ If it is not on the open list, add it to the open list. Make the current
node the parent of this node, and record the 𝑓 , 𝑔 and ℎ costs of the
node.
∙ If it is on the open list already, check to see if this path to that square
costs less, using 𝑔 cost as the measure. A lower 𝑔 cost means that this
is a better path. If so, change the parent of the node to the current
node and recalculate the 𝑔 and 𝑓 scores of the node.
d) Stop when:
∙ The target square is added to the closed list. In this case the path has
been found. Or,
∙ the target square has not been found, and the open list is empty. In
this case there is no path.
3. Save the path. Working backwards from goal node 𝑑, go from each node to
its parent node until the starting node 𝑠 is reached.
5.2 Applied A* star
Since the buildings we have to deal with might have many walls or non-walkable
zones, it would be useful to compute the shortest paths from each survey location
to the remaining ones. In fact, during the online phase, the RSS measurement is
taken by the mobile client (MC) at least every two seconds. During this interval,
the user is unlikely to move to a location which is a long way far from the point
in which he or she is. Then, by calculating the real distances (the lengths of the
shortest paths taking into account possible obstacles) from one survey site to the
other ones, we will be able to compute the ellipsoids considering only the locations
that are most likely to reach.
More specifically, fixing a survey point 𝑘 with 𝑘 = 1, . . . , 𝑁 , we compute the
real distance 𝐷𝑖𝑠𝑡(𝑘, 𝑖) from the location 𝑘 to the location 𝑖 for every 𝑖 = 1, . . . , 𝑁 .
Then, fixing a parameter 𝐷𝑀 𝐴𝑋, the locations 𝑗’s with 𝑗 = 1, . . . , 𝑁 such that
𝐷𝑖𝑠𝑡(𝑘, 𝑗) <= 𝐷𝑀 𝐴𝑋. (5.2.1)

34. Section 5.2. Applied A* star 29
are selected. The ellipsoid for the location 𝑘 is then computed, taking into account
only the locations 𝑗’s that satisfy the inequality (5.2.1). This approach solves in
a smart and intuitive way the memory problem since the number of variables and
constraints to optimize is always reasonable also for large environments.
The binary images (black and white) are stored in the computer as binary ma-
trices where 0 represents the black color and 1 represents the white color. In the
case the image represents a map, the black color is used for the non walkable zones
and the white color for the walkable zones.
In order to be able to compute the real distances in meters from one point of the
building to another one, the binary map needs to be resized. To perform this, a small
distance 𝑑 is fixed, depending on the grade of accuracy that we want to keep after
the resizing. Usually, 𝑑 is chosen to be equal to the half of the minimum distance
between all the locations. Given the GPS coordinates of the North-West angle and
the South-East angle of the map, it is possible to compute the GPS coordinates
of the other two angles (North-East and South-West) and the distances in meters
between these four corners. Using this data, we build a grid of the image, in which
each edge of a square represents a distance equal to 𝑑. Furthermore, the resized
binary map is always a binary matrix, in which each element, 1 or 0, represents a
square of the grid.
The A∗ algorithm, implemented in MATLAB, takes as input the resized binary
image, a starting node (𝑖𝑠 , 𝑗𝑠 ) in the image matrix and a destination node (𝑖𝑑 , 𝑗𝑑 )
in the image matrix. The resulting path is a sequence of nodes in the image matrix
(𝑖𝑠 , 𝑗𝑠 ), (𝑖1 , 𝑗1 ), . . . , (𝑖𝑑 , 𝑗𝑑 ). An example can be observed in Figures 5.1 and 5.2.
For the heuristic distance ℎ, it is reasonable to take the distance from the current
element (𝑖, 𝑗) to the destination element (𝑖𝑑 , 𝑗𝑑 ) multiplied by 𝑑, that is:
√
ℎ(𝑖, 𝑗) = 𝑑 ⋅ (𝑖𝑑 − 𝑖)2 + (𝑗𝑑 − 𝑗)2 (5.2.2)
The transition costs are instead simply computed in the following way:
∙ cost of an horizontal movement = 𝑑,
∙ cost of a vertical movement = 𝑑,
√
∙ cost of a diagonal movement = 𝑑 ⋅ 2.
The distance of the path is then computed by summing the costs of the made
transitions from the starting point (𝑖𝑠 , 𝑗𝑠 ) to the final point (𝑖𝑑 , 𝑗𝑑 ). It is possible
then to build the distance matrix 𝐷𝑖𝑠𝑡.

35. 30 The A* algorithm Chapter 5
⎛ ⎞
0 0 0 0 0 0 0
⎜0 1 1 0 1 1 0⎟
⎜ ⎟
⎜0 1 0 1 0 1 0⎟
⎜ ⎟
⎜0 1 0 1 0 1 0⎟
⎜ ⎟ =⇒
⎜0 1 0 1 0 1 0⎟
⎜ ⎟
⎜0 1 1 1 1 1 0⎟
⎜ ⎟
⎝0 1 1 0 1 1 0⎠
0 0 0 0 0 0 0
Figure 5.1. An example of a binary matrix and the corresponding image
⎛ ⎞
0 0 0 0 0 0 0
⎜0
⎜ ⃝ ⃝
1 1 0 1 1 0⎟⎟
⎜0
⎜ ⃝ 0
1 1 0 1 0⎟⎟
⎜0
⎜ ⃝ 0
1 1 0 ⃝ 0⎟
1 ⎟ =⇒
⎜0
⎜ ⃝ 0
1 1 0 ⃝ 0⎟
1 ⎟
⎜0
⎜ ⃝ ⃝
1 1 ⃝ ⃝ ⃝ 0⎟
1 1 1 ⎟
⎝0 1 1 0 1 1 0⎠
0 0 0 0 0 0 0
Figure 5.2. Example of the shortest path from the starting node (2, 3)
to the final node (4, 6). The resulting path is: (2, 3),(2, 2),(3, 2),(4.2),
(5, 2),(6, 2),(6, 3),(6, 4),(6, 5),(6, 6),(5, 6),(4, 6), that corresponds to the red line in
the image.

36. Chapter 6
ANALYSES
6.1 Computation of the distance matrix and of the ellipsoids
To evaluate the performance of the A∗ algorithm and of the ellipsoid method taking
into account the only reachable locations, two different datasets have been consid-
ered. The first comes from the first floor of the Hansa Mall in Malm¨ and it consists
o
of 38 locations, with 160 Wifi measurements for the locations 𝑖 with 𝑖 = 1, . . . , 35
and with 80 Wifi measurements for the locations 𝑖 with 𝑖 = 36, . . . , 38. The sec-
ond comes from the ground floor of the Hansa Mall in Malm¨ and it consists of 30
o
locations, each with 160 Wifi measurements.
In Figure 6.1 it is possible to observe the two maps, of the ground floor and of
the first floor with the plotted locations.
(a) (b)
Figure 6.1. (a) First floor (b) Ground floor
After resizing the map, A∗ has been applied, in order to obtain the distance
matrix 𝐷𝑖𝑠𝑡, where 𝐷𝑖𝑠𝑡(𝑖, 𝑗) is the real distance between the location 𝑖 and the
location 𝑗 taking into account the walls. In the Figures 6.2-6.4, it is possible to
31

37. 32 Analyses Chapter 6
observe a comparison between the distances computed from some locations 𝑗 to
other survey sites without taking into account the walls and the distances computed
taking into account the walls.
In Figure 6.5, it is clearly visible how the distance from the red location to the
yellow one, if walls are taken into account, is much higher (44𝑚) than the case in
which walls are not taken into account (16𝑚). After, for each location 𝑗 only those
sites that are reachable are selected. Here, we set the parameter 𝐷𝑀 𝐴𝑋 equal to
12𝑚, and form a vector 𝑃 𝑇 (𝑘) that contains only the locations 𝑗 that has a distance
less than 𝐷𝑀 𝐴𝑋 meters from the location 𝑘. The ellipsoids are then computed
for each location 𝑘 taking into account only the sites 𝑗 stored in the vector 𝑃 𝑇 (𝑘).
More specifically, the problem to solve is the following:
∑
minimize 𝑖,𝑗 𝜂𝑖𝑗
′
subject to z𝑘 (𝑗) Φ𝑘 z𝑘 (𝑗) ≤ 1 + 𝜂𝑗𝑘 ,
∀𝑗 = 1, . . . , 𝑀 (𝑘)
z𝑙 (𝑗)′ Φ𝑘 z𝑙 (𝑗) ≥ 𝜌 − 𝜂𝑗𝑙 ,
∀𝑙 ∈ 𝑃 𝑇 (𝑘) 𝑙 ∕= 𝑘
∀𝑗 = 1, . . . , 𝑀 (𝑙)
Φ𝑘 ર 0,
𝜂𝑖𝑘 ≥ 0. (6.1.1)
where the optimization variables are the slack variables 𝜂𝑖𝑗 , Φ𝑘 and with 𝑀 (𝑗) with
𝑗 = 1, . . . , 𝑁 we denote the number of measurements at the location 𝑘, typically
being different for every location. Here, the values of 𝜌 used are: 1.075, 1.05, 1.025.
The main difference between the algorithm in (6.1.1) and the one in (4.2.2) is
that the number of the points that have to lie outside the ellipsoid for the location
𝑘 are notably less since we now consider only the locations that are reachable.
On the other hand, as it is possible to observe, there are no constraints for the
points of locations further than 12 meters. It can happen, then, that some RSS
measurements of some location 𝑗 would be classified in ellipsoids of far locations.
However, this does not constitute a problem, since the far locations are already cut
out by selecting the only reachable ones.

38. Section 6.1. Computation of the distance matrix and of the ellipsoids 33
(a) (b)
Figure 6.2. (a) With no walls (b) With walls
(a) (b)
Figure 6.3. (a) With no walls (b) With walls

39. 34 Analyses Chapter 6
(a) (b)
Figure 6.4. (a) With no walls (b) With walls
(a) (b)
Figure 6.5. (a) With no walls (b) With walls. In this Figure, it is clearly visible
how the distance from the red location to the yellow one, if walls are taken into
account, is much higher (44𝑚) than the case in which walls are not taken into
account (16𝑚).

40. Section 6.2. Classification 35
6.2 Classification
The next step in the analysis is the classification. For each location 𝑘, we compare
the percentage of its measurements that are correctly mapped to site 𝑘 when using:
∙ The Gaussian likelihood taking into account all the locations.
∙ The Gaussian likelihood taking into account only the reachable locations in
𝑃 𝑇 (𝑘).
∙ The ellipsoid distance taking into account only the reachable locations.
More specifically, setting the initial counters 𝑅𝑖𝑔ℎ𝑡1(𝑘) = 0, 𝑅𝑖𝑔ℎ𝑡2(𝑘) = 0 and
𝑅𝑖𝑔ℎ𝑡3(𝑘) = 0, for all the measurements q = q𝑘 (𝑖) at the location 𝑘, the following
quantities are computed:
ˆ ˆ
∙ 𝑘1 = 𝑎𝑟𝑔 max𝑖=1,...,𝑁 P(q∣F(p𝑖 )).
ˆ ˆ
∙ 𝑘2 = 𝑎𝑟𝑔 max𝑖∈𝑃 𝑇 (𝑘) P(q∣F(p𝑖 )).
( )
ˆ3 = 𝑎𝑟𝑔 min𝑖∈𝑃 𝑇 (𝑘) (q 1) ⋅ Φ𝑖 ⋅ q
∙ 𝑘
1
ˆ ˆ ˆ
When 𝑘1 = 𝑘, 𝑘2 = 𝑘 or 𝑘3 = 𝑘, we increment 𝑅𝑖𝑔ℎ𝑡1(𝑘), 𝑅𝑖𝑔ℎ𝑡2(𝑘) or 𝑅𝑖𝑔ℎ𝑡3(𝑘).
We perform this for all the locations 𝑘 with 𝑘 = 1, . . . , 𝑁 . Then, it is possible
to obtain the percentage of points, for each method, that are mapped in the right
location. The final results are available in the next tables, both for the ground floor
and for the first floor.
∙ First column: number of the location.
∙ Second column: percentage of points mapped in the right location using the
Gaussian likelihood and taking into account all the locations.
∙ Third column: percentage of points mapped in the right location using the
Gaussian likelihood and taking into account only the reachable ones.
∙ Fourth column: percentage of points mapped in the right location using the
Ellipsoid distance and taking into account only the reachable ones.
The expectations for the ellipsoid method taking into account only the reachable
locations are pretty high. In fact, as it is possible to observe in all the tables, the
percentages of points that are mapped in the right survey site using this method
are much larger than using both the Gaussian method with all the locations and
the Gaussian method with only the reachable locations.
It is also worth to notice that the percentages of points mapped with the Gaus-
sian method with only the reachable locations is higher than the percentages of
points mapped with the Gaussian method with all the locations, since we do not
consider far locations.

41. 36 Analyses Chapter 6
First floor with DMAX = 12 with rho = 1.075
Gaussian Method (All) Ell Method (Reach) Gaussian Method (Reach)
1 68.75 93.75 85
2 36.875 64.375 36.875
3 8.75 73.75 8.75
4 31.875 71.25 34.375
5 16.25 53.75 16.25
6 26.25 38.75 27.5
7 46.25 67.5 46.25
8 21.25 82.5 25
9 85 93.75 85
10 36.25 78.125 36.25
11 42.5 35.625 42.5
12 30 61.25 41.25
13 51.25 55 51.25
14 18.75 63.75 28.75
15 28.75 51.25 30
16 17.5 56.25 27.5
17 23.75 73.125 46.25
18 54.375 78.125 60.625
19 62.5 87.5 68.75
20 43.75 66.25 50
21 21.25 25 21.25
22 37.5 71.25 40
23 17.5 73.75 18.75
24 52.5 70 57.5
25 12.5 40 12.5
26 21.25 66.25 21.25
27 68.75 65 71.25
28 37.5 68.75 40
29 38.75 83.75 45
30 60 76.25 60
31 60 67.5 60
32 59.375 95 60.625
33 32.5 77.5 45
34 63.75 86.25 65
35 76.875 89.375 76.875
36 40 50 40
37 45 97.5 53.75
38 70 85 70

42. Section 6.2. Classification 37
First floor DMAX = 12 with rho = 1.05
Gaussian Method (All) Ell Method (Reach) Gaussian Method (Reach)
1 68.75 95 85
2 36.875 60.625 36.875
3 8.75 73.75 8.75
4 31.875 71.25 34.375
5 16.25 55 16.25
6 26.25 33.75 27.5
7 46.25 67.5 46.25
8 21.25 82.5 25
9 85 93.75 85
10 36.25 78.125 36.25
11 42.5 32.5 42.5
12 30 61.25 41.25
13 51.25 55 51.25
14 18.75 65 28.75
15 28.75 51.25 30
16 17.5 56.25 27.5
17 23.75 73.125 46.25
18 54.375 79.375 60.625
19 62.5 87.5 68.75
20 43.75 66.25 50
21 21.25 26.25 21.25
22 37.5 71.25 40
23 17.5 73.75 18.75
24 52.5 70 57.5
25 12.5 40 12.5
26 21.25 66.25 21.25
27 68.75 65 71.25
28 37.5 68.75 40
29 38.75 83.75 45
30 60 76.25 60
31 60 67.5 60
32 59.375 93.75 60.625
33 32.5 77.5 45
34 63.75 85 65
35 76.875 89.375 76.875
36 40 50 40
37 45 97.5 53.75
38 70 90 70

43. 38 Analyses Chapter 6
First floor DMAX = 12 with rho = 1.025
Gaussian Method (All) Ell Method (Reach) Gaussian Method (Reach)
1 68.75 95 85
2 36.875 61.875 36.875
3 8.75 73.75 8.75
4 31.875 70 34.375
5 16.25 55 16.25
6 26.25 33.75 27.5
7 46.25 67.5 46.25
8 21.25 82.5 25
9 85 93.75 85
10 36.25 78.125 36.25
11 42.5 31.25 42.5
12 30 61.25 41.25
13 51.25 55 51.25
14 18.75 65 28.75
15 28.75 51.25 30
16 17.5 56.25 27.5
17 23.75 73.125 46.25
18 54.375 79.375 60.625
19 62.5 88.75 68.75
20 43.75 67.5 50
21 21.25 30 21.25
22 37.5 71.25 40
23 17.5 73.75 18.75
24 52.5 68.75 57.5
25 12.5 38.75 12.5
26 21.25 66.25 21.25
27 68.75 65 71.25
28 37.5 68.75 40
29 38.75 83.75 45
30 60 76.25 60
31 60 67.5 60
32 59.375 93.75 60.625
33 32.5 78.75 45
34 63.75 85 65
35 76.875 89.375 76.875
36 40 50 40
37 45 97.5 53.75
38 70 87.5 70

44. Section 6.2. Classification 39
Ground floor with DMAX = 12 m and rho = 1.075
Gaussian Method (All) Ell Method (Reach) Gaussian Method (Reach)
1 56 79.2 58.3
2 42 80 52.5
3 No Wifi measurements No Wifi measurements No Wifi measurements
4 104 88.75 65
5 64 90 80
6 56 92.5 70
7 48 100 60
8 58 80 72.5
9 46 85 60
10 12 100 15
11 66 83.75 46.25
12 44 92.5 62.5
13 46 95 57.5
14 38 95 47.5
15 48 95 60
16 50 87.5 62.5
17 28 87.5 35
18 34 95 42.5
19 70 100 87.5
20 24 72.5 30
21 46 90 57.5
22 58 80 72.5
23 22 97.5 40
24 52 97.5 65
25 No Wifi measurements No Wifi measurements No Wifi measurements
26 60 97.5 82.5
27 44 87.5 62.5
28 22 80 30
29 40 87.5 52.5
30 38 87.5 55

45. 40 Analyses Chapter 6
Ground floor with DMAX = 12 m and rho = 1.05
Gaussian Method (All) Ell Method (Reach) Gaussian Method (Reach)
1 56 79.2 58.3
2 42 82.5 52.5
3 No Wifi measurements No Wifi measurements No Wifi measurements
4 104 87.5 65
5 64 90 80
6 56 92.5 70
7 48 100 60
8 58 82.5 72.5
9 46 87.5 60
10 12 100 15
11 66 83.75 46.25
12 44 92.5 62.5
13 46 97.5 57.5
14 38 97.5 47.5
15 48 95 60
16 50 90 62.5
17 28 87.5 35
18 34 95 42.5
19 70 100 87.5
20 24 72.5 30
21 46 90 57.5
22 58 80 72.5
23 22 97.5 40
24 52 97.5 65
25 No Wifi measurements No Wifi measurements No Wifi measurements
26 60 97.5 82.5
27 44 90 62.5
28 22 77.5 30
29 40 87.5 52.5
30 38 87.5 55

46. Section 6.2. Classification 41
Ground floor with DMAX = 12 m and rho = 1.05
Gaussian Method (All) Ell Method (Reach) Gaussian Method (Reach)
1 56 79.2 58.3
2 42 82.5 52.5
3 No Wifi measurements No Wifi measurements No Wifi measurements
4 104 87.5 65
5 64 90 80
6 56 92.5 70
7 48 100 60
8 58 82.5 72.5
9 46 87.5 60
10 12 100 15
11 66 83.75 46.25
12 44 92.5 62.5
13 46 97.5 57.5
14 38 97.5 47.5
15 48 95 60
16 50 90 62.5
17 28 87.5 35
18 34 95 42.5
19 70 100 87.5
20 24 72.5 30
21 46 90 57.5
22 58 80 72.5
23 22 97.5 40
24 52 97.5 65
25 No Wifi measurements No Wifi measurements No Wifi measurements
26 60 97.5 82.5
27 44 90 62.5
28 22 77.5 30
29 40 87.5 52.5
30 38 87.5 55

47. 42 Analyses Chapter 6
6.3 Interpolation Step
The last step is the interpolation step. Let q = q𝑘 (𝑗) be the 𝑗th training measure-
ment in the location 𝑘 and let p𝑘 be the vector of the GPS coordinates (latitude
and longitude) of the 𝑘th location. By using the measurement q and setting
⎧
1
⎨ if 0 < 𝐷𝑖𝑠𝑡(𝑖, 𝑘) < 5 𝑚
𝐶(𝑖) = 0.8 if 5 ≤ 𝐷𝑖𝑠𝑡(𝑖, 𝑘) < 8 𝑚 (6.3.1)

0.6 if 8 ≤ 𝐷𝑖𝑠𝑡(𝑖, 𝑘) ≤ 12 𝑚
⎩
the position
𝑁
∑
ˆ
p= 𝑤𝑖 ⋅ p𝑖 (6.3.2)
𝑖=1
is estimated by using the following weights:
∙ Gaussian Method taking into account all the locations
ˆ
P(q∣F(p𝑖 ))
𝑤𝑖 = 𝑁
∑
ˆ
P(q∣F(p𝑖 ))
𝑖=1
for 𝑖 = 1, . . . , 𝑁 (6.3.3)
∙ Gaussian Method taking into account only the reachable locations
ˆ
⎧
 ∑ 𝐶(𝑖)⋅P(q∣F(p𝑖 )) ∀𝑖 ∈ 𝑃 𝑇 (𝑘)

⎨ ˆ
𝐶(𝑖) ⋅ P(q∣F(p𝑖 ))
𝑤𝑖 = (6.3.4)
 𝑖∈𝑃 𝑇 (𝑘)

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
⎩
∙ Ellipsoid Method taking into account only the reachable locations
⎧ ∑


⎨ 𝐶(𝑖) ⋅ q′ ⋅ Φ𝑖 ⋅ q
𝑖∈𝑃 𝑇 (𝑘)
𝑤𝑖 = ∀𝑖 ∈ 𝑃 𝑇 (𝑘) (6.3.5)
 𝐶(𝑖)⋅q′ ⋅Φ𝑖 ⋅q

⎩0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒