A calibration procedure for environmental parameters for Wi-Fi tracking

University of Amsterdam

Master Thesis

A calibration procedure for

environmental parameters for Wi-Fi

tracking

Author: Tobiasz Kukawka Supervisor: Dhr. prof. dr. S. Klous KPMG Supervisor: Dr. J. Amoraal Second examiner: Dhr. drs. A.W Abcouwer

A thesis submitted in fulfilment of the requirements for the degree of Master of Science

in the

Business Information Systems University of Amsterdam

Declaration of Authorship

I, Tobiasz Kukawka, declare that this thesis titled, ’A calibration procedure for envi-ronmental parameters for Wi-Fi tracking’ and the work presented in it are my own. I confirm that:

 This work was done wholly or mainly while in candidature for a research degree

at this University.

 Where any part of this thesis has previously been submitted for a degree or any

other qualification at this University or any other institution, this has been clearly stated.

 Where I have consulted the published work of others, this is always clearly

at-tributed.

 Where I have quoted from the work of others, the source is always given. With

the exception of such quotations, this thesis is entirely my own work.

 I have acknowledged all main sources of help.

 Where the thesis is based on work done by myself jointly with others, I have made

clear exactly what was done by others and what I have contributed myself.

Signed:

Date:

Contents

Declaration of Authorship i

Contents ii

Introduction 1

1 Wireless System 2

1.1 Performance factors of Wireless Network . . . 2

2 Devices Reconstruction Model 4 2.1 Friis Free-Space Transmission Equation . . . 4

2.2 Distance Approximation . . . 5

2.3 The χ2 _{Method . . . . 6}

2.4 Measurement Model . . . 7

2.5 Newton Raphson Method . . . 8

3 Simulation of environment parameters estimation 10 3.1 Simulation environment . . . 11

3.2 Simulation of fitting for free space model. . . 13

3.3 Simulation of fitting for indoor environment . . . 15

3.4 Procedure for estimation of environment parameter . . . 17

3.5 The simulation results of estimation of environmental parameter per probe 19 3.6 The application of the procedure to the real data from furniture store . . 24

Summary 31

Discussion and Further Studies 32

Bibliography 33

Introduction

The constant technological development and the increasing popularity of Big Data con-cepts resulted in the increased interest on the market in the behaviour and understanding of customers. Moreover the extensive use of mobile devices resulted in almost each po-tential consumer being equipped in transmitters and receivers of electromagnetic waves. These factors contributed to development of Location Aware Services, which aim is to detect customers within the defined areas. However the Global Positioning System(GPS) technologies initially used by the most of companies do not have abilities to penetrate indoor environments.

As a result of demand on indoor location aware services, KPMG developed a Wi-Fi Tracking Service. The task of the system is to reconstruct coordinates of devices within the premisses of the building it is installed in. The analysis of reconstructed data gives insights into the number and positions of detected devices in the store. However the reconstruction model assumes ideal environment conditions in the system, what leads to the question:

Does the estimation of an additional environmental parameter per probe improve the reconstruction performance?

The question is approached by simulating the new environment and proposing an iter-ative procedure to estimate the environmental parameter per probe.

Work starts with the introduction to the topic and the description of the project envi-ronment in Chapter 1.

Chapter 2 is devoted to the theoretical introduction to the detection and reconstruction of devices. That also includes the descriptions of the reconstruction model and its in-gredients.

Chapter 3 presents the simulation of the model in imperfect conditions and proposes a method to improve the reconstruction performance.

Chapter 4 contains Summary, which is followed by the proposal of Further Studies.

Chapter 1

Wireless System

In general a Wireless Tracking Service can be divided into two parts: the environment of the service and the reconstruction process of devices. The first part is described in the current Chapter, which consists of the explanation of the Wireless System ingredients and its limitations.

Within a wireless network the routers and mobile devices act as transmitters and re-ceivers of electromagnetic waves. Moreover both types of devices generate traffic by communicating with each other in order to exchange data packets either on behalf of the user or for the purpose of maintaining the connection. Thus the wireless network en-vironment includes the physical surrounding in which it is situated as well as the devices and generated by them traffic. Moreover the performance of the network in non-ideal environment can be affected by its characteristics described in the next Section.

1.1

Performance factors of Wireless Network

The performance of Wireless Network may vary upon many factors, which in most cases cannot be externally influenced and thus the results should be also interpreted accordingly. There can be distinguished three basic groups of factors affecting signals[1]: • Physical obstruction - Wireless signals can have problems to penetrate solid objects such as buildings, walls or to some extend people. The key rule is that the more obstacles for the signal to travel between the transmitter and receiver, the more the signal will be affected. Indoor environment delivers more obstacles to travel through as well as reflect and thus the signal strength may vary.

• Network Range and Distance Between Devices - The bigger the distance between two devices communicating with each other, the lower the signal strength as it has to travel further way. In the ideal case the strength of the signal is inversely proportional to the squared distance between these objects.

Chapter 2. KPMG Location Analytics Service 3 • Wireless Network Interference - The growth of popularity of Wireless Net-works is associated with growth of transmissions traffic in the air. Thus more signals in the air with similar frequencies mean higher signals interference between each other and resulting lower performance of the network. The use of unlicensed and popular frequency bands such as 2.4 GHz by e.g. microwaves, influences the received signal strength by the probes from mobile devices, resulting in poor qual-ity of fits.

Due to the physical environment and the use of network a received signal strengths by probes are affected and influence the reconstruction results in the system. Thus following Chapter2consists of the explanation of devices reconstruction model and the procedure for the estimation of position of transmitters.

Chapter 2

Devices Reconstruction Model

The explanation of devices reconstruction model starts with the Friis Free-Space trans-mission equation which is a baseline for the fitting process.

2.1

Friis Free-Space Transmission Equation

The relation between RSSI (Received Signal Strength Indication) and distance is given by the Friis equation [2]:

Pr Pt = GtGr( λ 4πR) 2 _(2.1)

where Pr and Pt are received power at receiver and transmission power of transmitter in Watts. The Gt and Gr are the unit-less representation of gain of transmitting and receiving antennas. Finally the λ and R denote the wavelength and distance between two antennas in meters.

Equation 2.1is valid under the following conditions: • antennas must be in far field . i.e.

R > 2d 2

λ . (2.2)

where d stands for the largest physical dimension of the antenna and λ the wave-length. Given the frequency f of 2.4 GHz and antenna’s linear dimension of 0.15 meters, the far field can be expressed as:

R > 2× (0.15) 2 c f = 2× (0.15) 2 0.125 = 0.36[m], (2.3) 4

Chapter 2. KPMG’s devices reconstruction model 5 where c = 3× 108 _{m/s is speed of light.}

The obtained distance of 0.36 meters lines the beginning of area where the devices waves radiate with spherical wave-fronts and with constant amplitude.

• antennas are in unobstructed free space. i.e. the power 2 stands for a free space environmental coefficient ’n’, assuming no obstacles in the line of sight between both antennas.

• bandwidth is narrow enough that a single wave can be assumed • antennas are correctly aligned and polarized

The calculated transmitter’s signal strength is inversely proportional to the squared distance between two antennas, which gives the estimation of how far the device is situated from a probe.

However the measured RSSI are expressed in dBm. Thus the conversion from Watts to RSSI needs to be conducted which is expressed as:

P [dBm] = 10 log(P [W atts]

1mW ),

what results in the following form of Friis equation expressed in dBm: Pr= P 0 t + 10 log( c 4πRf) 2_{, (2.4)} where: P_t0 [dBm] = 10 log(GtGrPt [W atts] 1mW )

Equation 2.4 can be used also for calculation of the maximum and minimum received signal strength in dBm by probes per device. Taking the values from Equation2.3from current section, gains of 1.5 for both antennas, maximum and minimum values of power transmitted of 10mW and 1mW and the maximum distance of 150 meters, the values can vary from the minimum level of -78 to the maximum of -15 dBm. The final Equation 2.4 is used in the procedure of approximation of the position of the device within the investigated area described in the following Section.

2.2

Distance Approximation

Imagine a situation where we know the transmission power and the transmission fre-quency of the device from which a signal was detected by a probe. The RSSI value of that signal is inversely proportional to the distance of the device from the probe, giving the radius on which it can be situated. Described situation is illustrated on the Figure 2.1in the top left corner marked with number one:

Chapter 2. KPMG’s devices reconstruction model 6 0 2 4 6 8 10 x 0 2 4 6 8 10 y x0 r S0 1. Sensor 0 2 4 6 8 10 x 0 2 4 6 8 10 y S0 x0 r S1 x1 r 2. Sensors 0 2 4 6 8 10 x 0 2 4 6 8 10 y S0 x0 r S1 x1 r S2 x2 r 3. Sensors 0 2 4 6 8 10 x 0 2 4 6 8 10 y 4. xt S0 x0 r r0 S1 x1 r r1 S2 x2 r r2 Sensors Transmitter

Figure 2.1: Steps in determining the position of device

where S0 denotes the position of sensor S0, x0r denotes the vector of coordinates of S0 which are known and the spherical shape indicates the theoretical distance within the device is situated from the probe. The situation from the Figure 2.1 Top Left can be expressed in the form of equation 2.4with known transmission frequency, received and transmitted power where the coordinates of transmitter are unknown.

Now second signal arrives from the same device to the sensor S1 with some known coordinates vector x1

r(Figure2.1 Top Right). The same situation repeats itself in stage three for the third sensor. Having three measurements of RSSI expressed in the equation form 2.4 with known sensors coordinates the set of 3 equations with 2 unknowns can be constructed. In order to find the wanted values, the solution for all three equations must meet the given condition.

Next section introduces the the method of χ2 _{which is used for the approximation of the} position of the device.

2.3

The χ

2

Method

The optimal solution of the system described in Section 2.2, i.e. the optimal set of parameters of the detected device can be found with χ2 _{method defined as [3]:}

χ2 = n X i=1 n X j=1 [m− h(p)]iVij−1[m− h(p)]j (2.5)

Chapter 2. KPMG’s devices reconstruction model 7 where ’m’ are the measurements (i.e RSSI values), h(p) the measurement model, p vector of parameters, and V is the variance of measurements, i.e the squared measurement resolution. The estimation takes place by finding the set of p that minimizes χ2_function such that:

dχ2

dp = 0, (2.6)

The Equation2.6 takes form of: dχ2

dp =−2H

T_V−1_{r = 0, (2.7)}

where elements of ’r’ vector are defined as: ri = mi − hi(p) and for matrix H the elements are: Hik =

∂hi(p) ∂pk

for k-th parameter and i-th measurement model. Finally solving the Equation2.6for p gives the least squares estimator ˆpexpressed in the form:

ˆ

p= (HT_V−1_H)−1_HT_V−1_{m (2.8)}

The estimator exists only if the matrix (HT_V−1_{H) is invertible and the errors on the} estimated parameters are the diagonal elements of their covariance matrix (i.e. the variance of estimators), which is expressed as:

Cov(p) = (HT_V−1_H)−1_{= 2(} d dp[

dχ2 dp])

−1_{, (2.9)}

where the error on i-th parameter: ˆ

σpi =pCov(p)ii (2.10)

2.4

Measurement Model

The fitting process applies the theory from previous three sections of the current Chapter 2. The trilateration from Section 2.2 takes place per coincidence, where the measured signal’s strengths expressed in the form of Equation2.4 can be written as:

Pr = ρ− 10 log R2= ρ− 20 log R, (2.11) where ρ = Pt+ 10 log( c 4πf) 2

Chapter 2. KPMG’s devices reconstruction model 8 is proportional to the emitted signal strength. The R represents the euclidean distance between given probe and the position of emitted signal by device, expressed as:

R = q

(xc

t− xcr)2+ (ytc− yrc)2 = r(xct, ytc),

where xc

r and ycr are the known coordinates of receiver and are used to determine the position of transmitter.

Applying the model to the χ2 _{method from Section 2.3 results in the following form of} χ2 _{ingredients where i represents i-th probes unique measurement within coincidence} per device: p= (xc t, ytc, ρ), mi = Pr i, h(p)i= ρ− 20 log r(xct, ytc)i = ρ− 20 log q (xc t− xcr i)2+ (ytc− ycr i)2, with derivatives in single H row that are in the form:

∂h(p) ∂xc t = −20(x c t− xcr) r(xc t, yct)2× ln(10) ∂h(p) ∂yc t = −20(y c t − yrc) r(xc t, yct)2× ln(10) ∂h(p) ∂ρ = 1

However due to the structure of derivatives the model2.11 is not linear in parameters. Thus the following Section2.5presents the procedure for non-linear models.

2.5

Newton Raphson Method

The optimal set of parameters ˆp in the form of Equation 2.8can be obtained with the use of Newton-Raphson iterative method of approximation of the roots of a function. In that method the estimator can be expressed in the form [4]:

xi+1= xi− f (xi) f0_(x

i)

Chapter 2. KPMG’s devices reconstruction model 9 where f (xi) is a value of the function at xi, f0(xi) is value at xi of function derivative and xi is an i-th approximation.

Applying Newton-Raphson method for approximation of set of p parameters for f (p) = dχ2 dp, for which: f0(p) = d dp[ dχ2 dp] = 2H T_V−1_{H, (2.13)} results in: pi+1= pi+ (HTV−1H)−1HTV−1r|pi (2.14)

The first initial guess in the reconstruction process, i.e. the set of parameters p applied to Newton-Raphson method as xi, is the set of coordinates of probe with the highest measured received signal strength, implying the closest distance to this probe in the set. Further in the process the values from the previous iterations are taken to calculate the new estimate.

Equation 2.15can also be expressed as:

4p = (HTV−1H)−1HTV−1r_|pi, (2.15)

which simplifies the process by returning the correction ∆n to the i_{− th iterations value} of n used for calculations.

The iterative process is repeated until fulfilling the following condition:

∆χ2 = 1 2

d2_χ2

dp2 |pi(4p)

2 _{< 1 (2.16)}

The reconstructed devices with the use of presented model in Chapter 2 can be used as a point of reference for further improvements of the reconstruction process. Fol-lowing Chapter 3 validates the model’s ideal conditions assumption and simulates the reconstruction process with imperfect conditions.

Chapter 3

Simulation of environment

parameters estimation

The Friis Free-Space Model in a form of Equation 2.1 assumes the ideal conditions for signal propagation in the space. The number referenced as ’n’ with assigned magnitude of 2 treats the environment as a free space. However the real conditions may differ from those assumed and the resulting measured RSSI also differs in magnitude. Figure 3.1 below presents two situations of the signals path to the probe with the reference to the n value: 0 2 4 6 8 10 0 2 4 6 8 10

signals in free space (n=2)

signals with constructive/destructive interference

Device

Sensor

Figure 3.1: Signals path from device to sensor with regard to n parameter

When n = 2 the model assumes that signals path (marked with blue line) to the sensor does not experience any interference on its way to the probe. Second situation shows

Chapter 3. Simulation of environment parameters estimation 11 when signals path is interfered and reflected by obstacles. The measured signal strength is influenced and the propagation pattern for that signal changes. The differences in the measured signal strength for a signal in different propagation environments is presented on Figure 3.2, illustrating the relation of received signal strength to the logarithm value of a distance between the probe and transmitter.

0.0 0.5 1.0 1.5 log(R) [m] −100 −80 −60 −40 −20 0 Recie ved signal strength [dBm] Measured Power 2.0 2.5 3.0 1.5

Figure 3.2: Graph of received signal strength with respect to log(R) value

The value of n visible on the Figure 3.2 denotes the slope of the signals power decay while travelling in space. Thus a measured power strength (-30 dBm) on the Figure for two different models gives different distance from the sensor. Given the example of n=2.0 and n=2.5 on the Figure illustrates the overestimation of distance for the ideal case measurement model in the environment of the imperfect conditions. Hence the con-clusions that the use of different environmental parameter influences the reconstruction results of the system.

Next sections describe a simulation which task is to validate the assumption of the influence of environmental parameter on the system’s performance.

3.1

Simulation environment

The environment of simulation consists of the devices and probes within defined ge-ometry. Figure 3.3 presents 1000 generated transmitters situated in the simulation’s

Chapter 3. Simulation of environment parameters estimation 12 environment with 12 probes:

20 10 0 10 20 30 40 50 60 x [m] 20 10 0 10 20 30 40 50 y [m ] Probes Devices

Figure 3.3: Geometry of the simulation

The simulation assumes the environment parameter per probe being equal to the value of that parameter in the free space. For each device a set of measurements is generated, where all probes detect each device within the system. The value of transmission power of a device is set up on the level of 20.0 dBm with the frequency of 2.4 GHz. Addition-ally Figure3.4presents the comparison of ideal and imperfect measurements due to the measurement resolution for a transmitter:

0 10 20 30 40 x [m] 0 10 20 30 40 y [m ] 0 10 20 30 40 x [m] 0 10 20 30 40 y [m ]

Figure 3.4: Detection circumference for a transmitter with and without smearing of the received power strength

Each circle indicates the range within the transmitter can be situated based on the measured signal strength. The Figure on the left shows the ideal sensor with infi-nite resolution, where the circles intersect at the position of the transmitter. On the right figure however, each of the values of the measured signal strength were smeared with numbers chosen randomly from a N (0, 4) distribution. The smearing of generated measurements is conducted to simulate the real conditions, where probes resolution influences the measurements.

Chapter 3. Simulation of environment parameters estimation 13

3.2

Simulation of fitting for free space model

First scenario of simulation takes into account the free space model in the reconstruction process, where environmental parameters take their values equal to 2. In order to assess the performance of the system, the pulls for estimated x and y coordinates are composed. Equation of the pull for an estimated parameter α is defined as follows:

P ull(α)_≡ ∆α σα

= α− αtrue σα

, (3.1)

where σα is a standard deviation of α estimation (i.e. Eq. 2.10), α is the estimated parameter and αtrue is the real value of the parameter. The distribution of pull values describes the estimation correctness of parameters and errors, where the ideal distribu-tion takes the value of mean around 0 and the width of 1. If the standard deviadistribu-tion of distribution is greater than 1, the errors are underestimated and smaller than 1, overestimated. The deviation from the mean of 0 indicates the bias in the estimation. Unfortunately the pulls for coordinates can be only calculated when the real positions of the devices are known. However in order to assess the results in the real world the distribution of pull of residuals can be used. The residuals, defined as ri = mi− hi(p), describe the deviations of the models value for estimated parameters from the measured real value by the probes. The standard deviation of residuals is the square root of diagonal values of:

Cov(r) = V _{− HCH}T, (3.2)

where the only new ingredient, C is the covariance matrix of the fitted parameters. The Equation 3.2 projects the correlations between the parameters onto the measurement space.

Figure 3.7presents the distribution of pulls for x, y and resulting residuals for the first scenario of simulation:

Chapter 3. Simulation of environment parameters estimation 14 4 3 2 1 0 1 2 3 4 Pullx 0 10 20 30 40 50 E nt ri es mean 0.012 +/- 0.028 sigma 0.935 +/- 0.028 nEntries 999 underflow 7 overflow 5 min -3.570 max 3.809 mean 0.027 +/- 0.034 std 1.085 +/- 0.024 4 3 2 1 0 1 2 3 4 Pully 0 10 20 30 40 50 60 E nt ri es mean -0.026 +/- 0.029 sigma 0.855 +/- 0.029 nEntries 999 underflow 6 overflow 10 min -3.536 max 3.810 mean -0.039 +/- 0.034 std 1.062 +/- 0.024 4 3 2 1 0 1 2 3 4 Pullresiduals 0 100 200 300 400 500 E nt ri es mean 0.012 +/- 0.010 sigma 1.012 +/- 0.010 nEntries 1000 underflow 6 overflow 19 min -3.865 max 3.981 mean 0.009 +/- 0.032 std 1.016 +/- 0.023

Figure 3.5: Pulls’ distribution for x,y and residuals

All three distributions follow the pattern of mean equal to 0 and the standard deviation of 1. Moreover 999 out of 1000 devices were fitted, where the impossible fits are distin-guished by the status flag of the fit. Missed fits are the fits for which the values did not converge within the iterations.

Additionally to the pulls the resolution on the coordinates can be investigated to validate the improvement of performance of the system. The resolution of parameters can be estimated by taking the width of a distribution of values defined as:

dx = x_{− x}true, (3.3)

where dx are the values of difference between the estimated coordinates and their real values. Figure below presents the composed distributions of dx and dy:

4 3 2 1 0 1 2 3 4 dx 0 5 10 15 20 25 30 E nt ri es mean -0.018 +/- 0.063 sigma 1.657 +/- 0.064 nEntries 999 underflow 27 overflow 27 min -3.980 max 3.990 mean 0.019 +/- 0.051 std 1.623 +/- 0.036 4 3 2 1 0 1 2 3 4 dy 0 5 10 15 20 25 30 E nt ri es mean -0.097 +/- 0.053 sigma 1.571 +/- 0.053 nEntries 999 underflow 24 overflow 23 min -3.995 max 3.957 mean -0.070 +/- 0.050 std 1.570 +/- 0.035

Figure 3.6: Distributions of dx and dy

The resolution on parameters in the ideal case takes the values of 1.6 meters on both parameters. Moreover the lack of bias in the distribution indicates accurate fits.

Chapter 3. Simulation of environment parameters estimation 15 The value of resolution on coordinates itself however could also be related with the num-ber of probes in the environment. The more probes in the environment, the precision, i.e. the resolution should also decrease as more measurements are used to estimate the parameter. Dependency can be illustrated by repeating the fitting process with the de-fined geometry but different number of probes. Following Figure presents the composed resolutions for y parameter versus the change in the number of probes:

10 15 20 25 30 35 40 45 Number of probes 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 st ddy [ m ]

Figure 3.7: Resolution change with respect to number of probes

The visible pattern of decrease of the coordinate resolution value with increase in the number of probes suggests that the precision of the estimation could be improved by adding additional probes to the system.

The case above illustrates the fitting process and results in the environment which follows the free space signal’s propagation model with the random measurement smearing within defined resolution. Following Section presents the simulation of the system where the environmental parameters differ from the free space model assumption.

3.3

Simulation of fitting for indoor environment

In order to examine how the system behaves in the situation where the environmental parameters differ from the ones in the free space model used for fitting process, the parameters per probe were changed. For that purpose 4 sets of probes were defined, where each set is characterized by different values of environmental parameters.

Chapter 3. Simulation of environment parameters estimation 16 Probe n - parameter Receiver 011 left 3.1 Receiver 011 base 3.0 Receiver 021 left 3.1 Receiver 021 base 3.0 Receiver 031 left 3.1 Receiver 031 base 3.0 Receiver 011 top 3.5 Receiver 021 top 3.5 Receiver 011 right 3.3 Receiver 021 right 3.3 Receiver 031 top 3.5 Receiver 031 right 3.3

Table 3.1: Modified environemtal parameters per probe

Current fit scenario in Section 3.3 assumes that the signals propagate as in the free space, reconstructing devices with the environment parameter as stated in the model (n = 2). However the measurements taken by probes are modelled with the use of the new environment parameters assigned as in Table3.1

The results of successfully reconstructed devices for simulated environment are presented as follows on the Figure: 3.8:

10 5 0 5 10 Pullx 0 5 10 15 20 25 30 35 E nt ri es mean -0.846 +/- 0.133 sigma 2.786 +/- 0.133 nEntries 828 underflow 112 overflow 26 min -9.955 max 9.918 mean -1.255 +/- 0.123 std 3.533 +/- 0.087 10 5 0 5 10 Pully 0 5 10 15 20 25 30 35 E nt ri es mean -1.164 +/- 0.204 sigma 2.944 +/- 0.204 nEntries 828 underflow 196 overflow 3 min -10.000 max 9.631 mean -2.186 +/- 0.129 std 3.698 +/- 0.091 10 5 0 5 10 Pullresiduals 0 50 100 150 200 250 300 E nt ri es mean 0.269 +/- 0.080 sigma 3.613 +/- 0.080 nEntries 828 underflow 12 overflow 164 min -9.512 max 9.994 mean 0.113 +/- 0.120 std 3.440 +/- 0.085

Figure 3.8: Pulls distributions for x,y and residuals for reconstructed devices

Both pulls for coordinates experience negative bias on the mean. Moreover all three pulls show that the errors are underestimated by having the standard deviation of dis-tributions at the minimum level of 3.4 for residuals and reaching the level of 3.5 and 3.7 for x ad y coordinates.

Chapter 3. Simulation of environment parameters estimation 17 defined system which does not include signal propagation in non-free space environ-ment. Additionally for 172 devices the reconstruction procedure failed to estimate the parameters.

The simulation in conditions from Section3.3shows decreased performance of the system with reference to the pulls distributions and the number of reconstructed devices. Worse performance of the system suggests that fitting the devices with the free space model assumption in environment where the environmental parameters differ does not deliver good quality results. Thus the next section proposes a method of iterative calibration of the system with regard to the environmental parameters per probe.

3.4

Procedure for estimation of environment parameter

The reconstruction process of coordinates in the system is conducted per device. Addi-tionally the environmental parameters per probe are treated as common for all devices. Thus the residuals r(p) can be extended by the value of environment parameter in the measurement model giving:

r(p)_{→ r(p, n) = m − h(p, n)}

The values of n are considered as global, meaning they apply to all devices reconstructed within the system. On the other hand the values of p are the local, unique per device. For a sample of all devices the optimal set of parameters p and n is obtained by minimisation of following ensemble χ2_:

χ2_n=X d

χ2_d, (3.4)

where d denotes d-th device for which the resulting χ2

d defined as Equation 2.5 and master equation in minimisation in the form:

∂χ2 n

∂n = 0 (3.5)

is calculated. Each value of χ2

d includes simultaneously the obtained optimal set of parameters p and n. The process to obtain the minimum value of Equation 3.4can be

Chapter 3. Simulation of environment parameters estimation 18 divided however into two consecutive steps. First of all as the set of parameters n is common for all devices, the estimation of p0

dper device can be undertaken for an initial guess of n0 _{by minimisation of χ}2

d. After having minimized χ2 per device the values of environment parameters per probe are estimated. That procedure follows the steps of the described least squares method with respect to n, where the estimates of p0

d are

used. The method is conducted per probe giving the set of n1 _{estimates which are then} used as new initial guess to iteratively continue the minimization of χ2 _{per device.} The given procedure of n-estimation is undertaken by computing the ∆n corrections which are then used to calculate nnew which is expressed as nold+ ∆n. The vector ∆n is obtained by minimisation of equation3.4with respect to the n-parameters. Having that in mind as well as the minimisation procedure explained in the Chapter 2 the optimal set of parameters ∆n minimizing the ensemble can be expressed as:

∆n= (X d [GT_V−1_G] d)−1( X d [GT_V−1_r] d), (3.6) where Gim = ∂hi(p, n) ∂nm

, V is the measurement variance matrix, C is the covariance matrix of parameters and ri = r(p, n)i = m− h(p, n) for i-th measurement and m-th environment parameter. Additionally for the estimation of ∆n only ’successful’ fits are taken, i.e. the fits with ’FITTED’ status. The iterative procedure is presented on the Figure 3.15below:

Figure 3.9: System optimization process

Chapter 3. Simulation of environment parameters estimation 19 starts with determining the ∆n per each probe, using values of p as showed in Equation 3.6. Then the estimates of ∆n are used to update values of nprev what gives: nnew = nprev+∆n. The estimation of ∆n takes place until the change in ensemble χ2nis smaller than 1. If the condition is not met by at least one estimate the process starts again by reconstruction of p parameters.

The results of this iterative procedure are analysed in the next Section to validate, whether the estimation of n parameters per probe can improve the resulting performance of the fitting results.

3.5

The simulation results of estimation of environmental

parameter per probe

Each iteration of the procedure proposed in Section3.4produces the corresponding value of ∆n per probe which is used in further estimation process. The obtained results of n estimation can be presented on two graphs, consisting of ∆n change and the obtained nnew values. Following Figure 3.10 illustrates the obtained values from the procedure for a random sample of 1000 transmitters in the geometry defined in Section 3.3 after convergence of ∆n to 0. 0 1 2 3 4 Iterations 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 nne w Receiver_031_left Receiver_011_top Receiver_021_left Receiver_021_top Receiver_011_left Receiver_031_top Receiver_011_base Receiver_021_base Receiver_031_right Receiver_021_right Receiver_031_base Receiver_011_right 0 1 2 3 4 Iterations 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.41.6 ∆ n

Figure 3.10: Calibration results in simulated environment

Chapter 3. Simulation of environment parameters estimation 20 0, meaning that the further changes are very small in numbers(i.e. smaller than 0.001). Moreover the graph illustrating the change of nnew indicates that after 3 iterations the values of n stabilize around the final result.

Additionally the expectation is that the estimated n values should converge to the real values of n per probe assigned at the beginning of the simulation. Thus the next Table 3.2illustrates the obtained values, which converged after 4 iterations, together with the assigned values of environment parameters and estimation errors:

Probe new n n err real n Receiver 031 left 2.976 ±0.0156 3.1 Receiver 011 top 3.877 ±0.0153 3.5 Receiver 021 left 2.954 ±0.0159 3.1 Receiver 021 top 3.851 ±0.0156 3.5 Receiver 011 left 2.978 ±0.0157 3.1 Receiver 031 top 3.871 ±0.0153 3.5 Receiver 011 base 2.755 ±0.0159 3.0 Receiver 021 base 2.725 ±0.0162 3.0 Receiver 031 right 3.430 ±0.0154 3.3 Receiver 021 right 3.418 ±0.0158 3.3 Receiver 031 base 2.745 ±0.0159 3.0 Receiver 011 right 3.444 ±0.0156 3.3

Table 3.2: Estimated values of n-environment parameter

However in order to determine whether the procedure consistently estimates the values of parameters the estimation process was repeated 5 times for different sets of generated transmitters. Figure 3.11 presents the final results for one receiver per environment group of 5 simulations conducted on different sets of transmitters to estimate the values of environment parameters: 1 2 3 4 5 Simulation number 3.0 3.2 3.4 3.6 3.8 4.0 n-value Receiver_021_top real n: 3.5 n_estimates 1 2 3 4 5 Simulation number 2.6 2.8 3.0 3.2 3.4 n-value Receiver_021_base real n: 3.0 n_estimates 1 2 3 4 5 Simulation number 2.6 2.8 3.0 3.2 3.4 3.6 n-value Receiver_021_left real n: 3.1 n_estimates 1 2 3 4 5 Simulation number 2.8 3.0 3.2 3.4 3.6 3.8 n-value Receiver_021_right real n: 3.3 n_estimates

Figure 3.11: Results of 5 simulations for different sets of transmitters

The above set of graphs presents the inconsistency of estimated parameters with the real values. None of the estimates fall within the range of 3 sigma in the range of real values of the parameters. This means that the estimates significantly differ from the real values. On the other hand the procedures results are consistent in terms of the

Chapter 3. Simulation of environment parameters estimation 21 repeatability of the estimated values, where for 5 simulations values tend to fluctuate within 1 sigma from each other.

The inconsistency in estimation of parameters to their real values should be further investigated. Thus the simulation with different sets of generated transmitters was repeated 100 times. Following Figure presents pulls distributions for 4 groups of probes from Figure 3.11 composed from obtained newn values, their errors and the assigned values : 10 5 0 5 10 Pulln 0 5 10 15 20 E nt ri es Receiver_021_top mean 0.196 +/- 0.180 sigma 2.123 +/- 0.180 nEntries 100 underflow 0 overflow 0 min -4.618 max 5.053 mean 0.385 +/- 0.209 std 2.092 +/- 0.148 10 5 0 5 10 Pulln 0 2 4 6 8 10 12 14 16 E nt ri es Receiver_021_left mean 0.299 +/- 0.163 sigma 2.272 +/- 0.163 nEntries 100 underflow 0 overflow 0 min -6.036 max 4.694 mean 0.266 +/- 0.212 std 2.116 +/- 0.150 10 5 0 5 10 Pulln 0 5 10 15 E nt ri es Receiver_021_right mean 0.326 +/- 0.111 sigma 1.999 +/- 0.111 nEntries 100 underflow 0 overflow 0 min -4.685 max 5.733 mean 0.406 +/- 0.194 std 1.940 +/- 0.137 10 5 0 5 10 Pulln 0 2 4 6 8 10 12 14 16 E nt ri es Receiver_021_base mean -0.146 +/- 0.108 sigma 2.073 +/- 0.108 nEntries 100 underflow 0 overflow 0 min -6.371 max 5.200 mean -0.092 +/- 0.202 std 2.022 +/- 0.143

Figure 3.12: Pulls of environmental parameters per group after 100 simulations

For each group of probes the pull distribution indicate the underestimation of errors in the model, where the values of standard deviation fluctuate around 2. The means on the other hand show no bias in the estimation of the values of environmental parameters. However the goal of the proposed procedure is not only to retrieve the real environment parameter value but also to improve the performance of the system. The following Fig-ure 3.15was composed to present difference in fits quality in comparison to pulls from Section3.2: 4 3 2 1 0 1 2 3 4 Pullx 0 5 10 15 20 25 30 E nt ri es mean 0.010 +/- 0.043 sigma 1.040 +/- 0.043 nEntries 995 underflow 207 overflow 146 min -3.984 max 3.990 mean -0.015 +/- 0.043 std 1.342 +/- 0.330 4 3 2 1 0 1 2 3 4 Pully 0 5 10 15 20 25 30 E nt ri es mean -0.093 +/- 0.057 sigma 1.200 +/- 0.057 nEntries 995 underflow 228 overflow 124 min -3.985 max 3.930 mean -0.123 +/- 0.045 std 1.404 +/- 0.331 4 3 2 1 0 1 2 3 4 Pullresiduals 0 100 200 300 400 E nt ri es mean -0.001 +/- 0.009 sigma 1.070 +/- 0.009 nEntries 995 underflow 3 overflow 9 min -3.938 max 3.989 mean 0.008 +/- 0.034 std 1.082 +/- 0.024

Figure 3.13: Pulls of coordinates and residuals for last iteration at calibration proce-dure

Chapter 3. Simulation of environment parameters estimation 22 For the modelled environment the distribution of pulls of residuals presents the im-provement in the fits quality, with the mean of 0 and standard deviation of 1. The underestimation of errors on residuals was eliminated and bias decreased. This dis-tribution is the only one which can be computed without knowing the real values of coordinates.

The pulls of x and y coordinates present smaller bias in comparison to pulls from Sec-tion 3.3 where both means are now close to 0. Additionally both pulls of coordinates with considered errors on standard deviation indicate the distribution close to ideal case scenario.

The improvement of the model can be assessed by comparison of the coordinates resolu-tion of the calibraresolu-tion procedure with the results of simularesolu-tion in ideal case from Secresolu-tion 3.2. The resolution is calculated in the same way with the use of Equation 3.3, where the results are presented on the Figure3.14.

First iteration 4 2 0 2 4 dx 0 2 4 6 8 10 12 E nt ri es mean -2.566 +/- 0.830 sigma 4.651 +/- 0.885 nEntries 828 underflow 360 overflow 160 min -4.968 max 4.928 mean -0.798 +/- 0.092 std 2.650 +/- 0.065 4 2 0 2 4 dy 0 2 4 6 8 10 E nt ri es mean -5.565 +/- 4.335 sigma 6.773 +/- 2.990 nEntries 828 underflow 479 overflow 113 min -4.996 max 4.993 mean -0.859 +/- 0.097 std 2.788 +/- 0.069 Last iteration 4 2 0 2 4 dx 0 5 10 15 20 25 E nt ri es mean -0.014 +/- 0.065 sigma 1.546 +/- 0.065 nEntries 995 underflow 211 overflow 152 min -4.969 max 4.958 mean -0.017 +/- 0.061 std 1.939 +/- 0.043 4 2 0 2 4 dy 0 5 10 15 20 25 E nt ri es mean -0.247 +/- 0.092 sigma 1.646 +/- 0.092 nEntries 995 underflow 231 overflow 142 min -4.959 max 4.915 mean -0.122 +/- 0.062 std 1.969 +/- 0.044

Figure 3.14: Coordinates resolution for first and last iteration

For the first iteration the resolution of x and y coordinates is equal to 2.7 and 2.8 me-ters with means of -0.8 and -0.9 meme-ters. These values indicate a bias on the estimated parameters values. Comparing the results from first and last iteration, for the latter one the resolution of x and y slightly decreased to 1.9 and 2.0 meters, where the bias on the

Chapter 3. Simulation of environment parameters estimation 23 estimated parameters was eliminated. These changes indicate that the accuracy of the system was increased and thus its performance, where parameters are estimated more accurately. The resolution on parameters also slightly decreased, showing improvement in the precision. Comparison of Figure 3.14 with the results of ideal case simulation indicates similarities in the distribution patterns, suggesting that the calibration proce-dure estimates the parameters with the same precision as in the case of the free space propagation model.

By the use of proposed calibration procedure the bias of the system and underestimation of errors is decreased. Additionally to the pulls the performance can be expressed in the change of number of fitted devices per iteration, which is illustrated on the Figure below: 1 2 3 4 5 Iterations 800 850 900 950 1000 1050 1100 Fitted devices

Fitted devices per iteration Number of all coincidences:1000

Figure 3.15: Change in fitted devices per iteration

Graph shows constant increase in values, where the number of fitted devices increased from 828 to 995 devices, indicating the improvement in the performance.

The calibrated fitting model can be also assessed by looking at the distribution of the probability of χ2 _{per fit. The probability of χ}2 _{can tell what are the chances of getting} the value equal or worse than this χ2_{value. In other words the probability gives a chance} of obtained χ2 _{being from the assumed χ}2 _{distribution. For ideal case the probabilities} are uniformly distributed between 0 and 1. The pattern indicates how well the model matches the data behaviour. Following Figure 3.16 compares the distributions of χ2 probability for the first and last iterations for the indoor simulation:

Chapter 3. Simulation of environment parameters estimation 24 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Prob

(

χ

2₎ 0 200 400 600 800 1000

E

nt

ri

es

First iteration nEntries 828 underflow 0 overflow 0 min 0.000 max 0.127 mean 0.001 std 0.005 0.0 0.2 0.4 0.6 0.8 1.0

Prob

(

χ

2₎ 0 5 10 15 20 25

E

nt

ri

es

Last iteration nEntries 995 underflow 0 overflow 0 min 0.001 max 0.999 mean 0.451 std 0.287

Figure 3.16: Distribution of χ2probability at first and last iteration of simulation

For the first iteration the values are focused between 0 and 0.127, with the mean at 0. This distribution of results suggests that the model in most of the cases does not fit to the data. On the other hand, the distribution of probability of χ2_{at last iteration shows} the improvement of the model. The probabilities are more evenly distributed between the values of 0 and 1. Moreover as the mean equals to 0.45, what is close to ideal, where for uniformly distributed values it is equal to 0.5

Summing up the procedure’s results, the improved performance of the proposed method of environmental parameter estimation can be observed by analysis of pulls of residuals and parameters, the dα distributions for coordinates, the number of fitted devices per iteration and the χ2 _{probability distribution.}

The last Section 3.6of the Chapter3 applies the proposed procedure to the data avail-able from one of the furniture store’s where the KPMG Location Analytics system is installed.

3.6

The application of the procedure to the real data from

furniture store

The available data was gathered in the Checkout and Restaurant areas in a furniture store. For both datasets the raw measurements of probes on 4/11/2015 were taken. The reconstruction process of devices coordinates is conducted in the same manner as in the

Chapter 3. Simulation of environment parameters estimation 25 real system(i.e. per coincidence).

Moreover as the initial resolution of the measurements is not known for given sets, it was first estimated for the free space model assumption in devices reconstruction procedure. The estimation of resolution takes form of:

ˆ σm= r χ2_(σ m= 1) ν , (3.7)

where: ν = Nm− Np is the number of degrees of freedom for Nm measurements and Np parameters. The initial resolution is used for obtaining the environmental calibration parameters. However that resolution is valid for the fits with the free space assumption. Thus after obtaining the values of environmental parameters estimates the resolution for these results is recalculated to obtain its value for the new model and next iteration. The available data sets however contain all raw measurements taken by probes, including the coincidences with insufficient number of unique measurements for the fitting process. The minimum number of measurements taken by different probes for successful fit must be equal to or greater than 4, what was applied to the dataset.

Second criteria in data selection consists of the minimum signal strength with regard to data quality. Low signal strengths indicate that the signal might have been interfered significantly with obstacles such as walls, what influences the results and fits. Thus the selection on signal strengths was made, where the margins for Checkout and Restaurant respectively are equal to -70 and -60 dBm.

The Table3.3below presents the composed information consisting of the data selection results summary, the procedure’s fit results, iterations and general performance of the systems after its convergence.

Summarizing Table3.3, the number of iterations of proposed procedure until its conver-gence is 6 for both, Checkout and Restaurant. In both cases the calibration of environ-mental parameter increased the number of fitted coincidences in total. Additionally in case of Checkout 19 more devices were fitted. The calibration of environment parame-ter enriches the fitted datasets per device with the number of reconstructed positions, together with addition of new reconstructed devices.

Chapter 3. Simulation of environment parameters estimation 26

Checkout Restaurant

Selection set summary

Data time window: 1 hour 20 min

Number of devices: 1206 547

Number of coincidences (500ms window): 70967 39638

Mean number of measurements

per coincidence (500ms window): 8 6

Calibration procedure results

Number of iterations to converge: 6 6

Number of fitted devices at first iteration: 1184 547

Number of fitted devices at last iteration: 1203 547

Number of fitted coincidences at first iteration: 69712 38352

Number of fitted coincidences at last iteration: 70675 39299

Table 3.3: Results summary of environment parameter estimation for real data

The effectiveness of the procedure can be also presented on the graphs illustrating the number of fitted coincidences. Following Figure presents the mentioned graphs for Checkout and Restaurant respectively:

1 2 3 4 5 6 Iterations 69000 69500 70000 70500 71000 71500 Fitted Coincidences Checkout

Fitted coincidences per iteration

1 2 3 4 5 6 Iterations 38000 38500 39000 39500 40000 40500 Fitted Coincidences Restaurant

Fitted coincidences per iteration Number of all coincidences:39638

Figure 3.17: Missed fits per device for Checkout and Restaurant

Both areas experience constant increase in the number of fitted coincidences, illustrating the iterative improvement of the system.

Additionally as in the Section 3.5 the improvements in the system are noticeable in analysis of pulls of residuals, which are presented on the Figure3.18:

Chapter 3. Simulation of environment parameters estimation 27 Checkout 4 2 0 2 4 Pullresiduals E nt ri es

Checkout for first iteration

Mean:0.043 +/- 0.013 Std:1.593 +/- 0.289 4 2 0 2 4 Pullresiduals E nt ri es

Checkout for last iteration

Mean:0.043 +/- 0.012 Std:1.225 +/- 0.267 Restaurant 4 2 0 2 4 Pullresiduals E nt ri es

Restaurant for first iteration

Mean:-0.012 +/- 0.008 Std:1.452 +/- 0.174 4 2 0 2 4 Pullresiduals E nt ri es

Restaurant for last iteration

Mean:0.030 +/- 0.008 Std:1.208 +/- 0.182

Figure 3.18: Residuals pulls distribution for Checkout and Restaurant areas for first and last iteration

Presented pulls show improvement in terms of the standard deviation of distributions comparing the fitted results for first and last iteration. Both cases experienced decrease in the value of standard deviation respectively from 1.6 to 1.2 for Checkout and from 1.5 to 1.2 for Restaurant. Moreover the standard deviation with its estimation error at last iteration for both areas fall within the range of 1, indicating correctly estimated errors on residuals. Additionally both last pulls indicate no bias in the estimation.

The final results of the n approximation are presented in the Table 3.4 together with the errors per parameter:

Checkout Results n new n err 150 2.753 ±0.008 151 2.677 ±0.009 152 2.694 ±0.009 153 2.688 ±0.008 154 2.812 ±0.009 155 2.859 ±0.010 156 2.638 ±0.009 157 2.808 ±0.009 158 2.693 ±0.009 159 2.656 ±0.009 160 2.665 ±0.008 161 2.840 ±0.009

Restaurant Results n new n err RestAlpha 2.044 ±0.004 RestBeta 2.002 ±0.004 RestCharlie 2.280 ±0.004 RestDelta 2.095 ±0.005 RestEcho 2.248 ±0.004 RestFoxtrot 2.082 ±0.005 RestGolf 2.003 ±0.005 RestHotel 2.298 ±0.005 RestIndia 1.933 ±0.005 RestJuliett 1.993 ±0.005 RestKilo 1.879 ±0.005

Chapter 3. Simulation of environment parameters estimation 28 The values of obtained calibration parameters indicate that the environments of both areas differ. For Checkout where all values are higher than 2.6, the proportion between the received and transmitted signal strength is bigger, meaning more disturbances of the signal on its way to the probe.

Moreover results from Table 3.4 in convergence process can be also expressed in terms of previously computed graphs for ∆n and nnew which are obtained with every iteration due to the proposed procedure. Following Figure 3.19presents the obtained values per iteration for Checkout and Restaurant respectively:

0 1 2 3 4 5 6 Iterations 1.0 1.5 2.0 2.5 3.0 3.5 nne w Checkout 150 151 152 153 154 155 156 157 158 159 160 161 0 1 2 3 4 5 6 Iterations 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ∆ n 0 1 2 3 4 5 6 Iterations 1.0 1.5 2.0 2.5 3.0 3.5 nne w Restaurant RestAlpha RestBeta RestCharlie RestDelta RestEcho RestFoxtrot RestGolf RestHotel RestIndia RestJuliett RestKilo 0 1 2 3 4 5 6 Iterations 0.15 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 ∆ n

Figure 3.19: Figures for obtained nnewand ∆n values per iteration

The change in nnew also follows the pattern from previous section, where after few iterations the values of nnewtend to converge to some given constant and do not change significantly in values. Moreover the changes in ∆n also follows the assumed trends, where the changes decrease with each iteration and converge to 0.

Finally as in the Section 3.5, the obtained models for both areas can be assessed by looking at the distributions of probability of χ2 _{for first and last iteration, which are} presented on the Figure 3.20(look next page).

In the case of the Checkout results, the improvement in the model structure is visible, where the peak at 0.0 decreased from above 2000 to below 1200 entries. This drop indicates improvement in the model matching the data due to the calibration procedure, where probabilities are more evenly distributed between 0 and 1.

Chapter 3. Simulation of environment parameters estimation 29 Checkout 0.0 0.2 0.4 0.6 0.8 1.0 Prob(χ2₎ 0 500 1000 1500 2000 2500 E nt ri es First iteration Mean: 0.392 0.0 0.2 0.4 0.6 0.8 1.0 Prob(χ2₎ 0 200 400 600 800 1000 1200 1400 E nt ri es Last iteration Mean: 0.447 Restaurant 0.0 0.2 0.4 0.6 0.8 1.0 Prob(χ2₎ 0 500 1000 1500 2000 E nt ri es First iteration Mean: 0.352 0.0 0.2 0.4 0.6 0.8 1.0 Prob(χ2₎ 0 200 400 600 800 1000 1200 1400 E nt ri es Last iteration Mean: 0.453

Figure 3.20: Distribution of χ2 probability at first and last iteration for Checkout

and Restaurant

The patterns in Restaurant results are similar to the Checkout, where the peak at 0.0 with 2000 entries at first iteration decreased to 1200 entries at last iteration. The rest of probabilities of χ2 _{are also more evenly distributed between 0 and 1.}

For both areas the means also increased to values closer to 0.5, what confirms the more evenly distributed frequencies between 0 and 1.

Finally as the calibration was done on the real data, the measurement resolution is not known but can be estimated with the use of Equation 3.7. The resolution indicates how well the values can be measured in the model. Following Figure3.21 presents the calculated resolutions per iteration:

1 2 3 4 5 6 Iterations 2.5 3.0 3.5 4.0 4.5 5.0 Resoltion Checkout

Estimated resolution per iteration

1 2 3 4 5 6 Iterations 1.5 2.0 2.5 3.0 3.5 Resoltion Restaurant

Estimated resolution per iteration

Chapter 3. Simulation of environment parameters estimation 30 Both areas experienced the decrease in resolution values, where the biggest drop is visible for the second iteration. The patterns visible on the Figure 3.21 also indicate the improvements in the performance of the system, where the measurement resolution decreases with each iteration of calibration procedure.

Summarizing the results of proposed procedure for environmental parameter estimation in the real environment it can be said that the performance of the fitting process in-creases due to the calibration. Thus it can be used to improve visit counts, dwell time calculations or customer tracking with the coincidences (i.e. positions) that could not be fitted before. Moreover both areas behaved in similar manner after applying the calibration procedure. The underestimation of errors was decreased, improving the per-formance of the model with regard to number of fitted coincidences and devices. The calibration procedure resulted also in improvement of the model structure.

Summary

The KPMG’s Location System delivers vast set of data available for further analysis. However the used model assumes the free space environment. Thus the aim of the paper was to answer the question whether the consideration of environment around the probes by applying the estimation procedure for environmental parameter per probe can improve the performance of the system.

To answer the question a simulation of environment in reconstruction process was con-ducted. The obtained results show the decrease in performance in comparison to the ideal case scenario. Thus further attempts to improve the fits quality were made. The results of the introduced calibration procedure of the environmental parameter per probe show that the performance of the system can be improved in terms of its accuracy and resolution. Moreover the number of fitted devices increased and the distribution of pulls of residuals and coordinates followed the Gaussian distribution.

After a success of the procedure in simulation environment it was applied to calibrate environmental parameters per probe for the real data within the furniture store. The obtained results via the proposed procedure significantly improved the performance of the system with regard to the number of fitted positions of devices. Moreover the application of the procedure on the real data shows that the results structure is similar. For both areas the residuals distribution followed the Gaussian distribution.

Unfortunately the given procedure of minimizing ensemble χn2underestimates the errors on environmental parameters. On the other hand the results from different simulations of environmental parameters are consistent with respect to each other. The next section proposes further studies which could improve the procedure.

Discussion and Further Studies

The paper proved that the estimation of the environmental parameter per probe can help in the improvement of the performance of the system. However the obtained results of simulation experience the underestimation of errors(look Section3.5). Following Section proposes other methods for the calibration of environment parameter.

Different method for parameter estimation is the calibration of each probe separately manually at the area the system is located in. The procedure consists of taking the mea-surements of the signal received by probe from one device but different known positions. As the positions of probe and transmitter are known together with signals frequency, emitted and transmitted power, the environmental parameter can be estimated. These estimates can be used also as initial guesses in the future calibration procedure if needed. Moreover as noticed in Section3.5(look Figure 3.12) the results indicate the underesti-mation of errors for environmental estimates. Hence the model could be improved by addition of the correlation between the fit and environmental parameters. Taking these correlation into account the master Equation3.5 takes a form:

d dn = ∂ ∂n+ ∂p ∂n d dp

Applying these changes should result in better estimation of errors and the consistency of the procedure in estimation of the real values of environmental parameters. Moreover the procedure should converge within a single pass [5].

Last proposed method is the cleaning and selection process on the data used for envi-ronmental parameters estimation. By removal of outliers from the fitted devices dataset should result in better results of estimation for environmental parameter.

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