Capítulo II. La respiración en microorganismos
2.3 Hongos
As it is possible to see from figures 4.6 and 4.7 already at the third iteration the algorithm reachs convergence and stops improving. This can be also seen in figure 4.8 where the CDF of the fourth iteration overlaps the CDF of the third one. Table 4.1 presents mean and standard deviation of the position error at the third iteration, a higher number of componets Rbslightly improves the precision.
Position error, iteration 3
Rb = 1 Rb = 3 Rb = 5 Error mean [meters] 29.3 28.4 27.5 Error std dev [meters] 34.3 32.6 31.2
Table 4.1: Position error mean and standard deviation at the third iteration for different values of Rb.
Figures 4.9, 4.10, 4.11 present p(γb) at the third iteration for different values of Rb (Rb = 1, Rb = 3, Rb = 5). Figure 4.12 presents the CDF at the first and third iteration for those values of Rb.
4.2. CONVERGENCE AND GAUSSIAN MIXTURE COMPONENTS
Fig. 4.8: CDF for different iterations, (Rb = 3).
Fig. 4.9: p(γb) for BS 1 (iteration 3, Rb = 1).
4.2. CONVERGENCE AND GAUSSIAN MIXTURE COMPONENTS
Fig. 4.10: p(γb) for BS 1 (iteration 3, Rb = 3).
Fig. 4.11: p(γb) for BS 1 (iteration 3, Rb = 5).
4.2. CONVERGENCE AND GAUSSIAN MIXTURE COMPONENTS
Fig. 4.12: CDF at first and third iteration for Rb = 1, Rb = 3, Rb = 5.
4.3 Comparison with TDOA
The learning algorithm was compared with a TDOA multilateration algo-rithm. TDOA multilateration uses three ToA to generate two hyperbolas and find the estimated position as intersection of these hyperbolas. Here TDOA uses only the three smallest ToA, corresponding to the three closest BS. For an honest comparison the LA was tested with both four and three BS (in this case all base stations are used for the learning but only three are used for the position MAP estimate). Figure 4.13 presents the CDF compared to the LA when 4 BS are used, while figure 4.14 when only 3 BS are used. Figures 4.15, 4.16 present the comparison of position error mean and standard deviation over multiple iterations. By iteration three the LA outperforms the TDOA multilateration (even if only three BS are used). The mean error and standard deviation at iteration three are presented in table 4.2.
Position error, iteration 3
TDOA LA, 4 BS LA, 3 BS
Error mean [meters] 46.9 28.4 31.4
Error std dev [meters] 44.5 32.6 26.5
Table 4.2: Position error mean and standard deviation for TDOA, LA with 4 BS and LA with 3 BS (iteration 3, Rb = 3).
4.3. COMPARISON WITH TDOA
Fig. 4.13: CDF of TDOA and Learning Algorithm with 4 BS (Rb = 3).
Fig. 4.14: CDF of TDOA and Learning Algorithm with 3 BS (Rb = 3).
4.3. COMPARISON WITH TDOA
Fig. 4.15: Mean position error at each iteration for TDOA and LA (Rb = 3).
Fig. 4.16: Position error standard deviation at each iteration for TDOA and LA (Rb = 3).
4.4 Range-Aware algorithm results
The Range-Aware Learning Algorithm (RALA) was compared with the nor-mal Learning Algorithm (LA). In this simulation we let the parameter α vary over distance as explained in the previous chapter. The Range-Aware algo-rithm uses multiple distributions for each base stations. This can be seen in figures 4.17, 4.18, 4.19 where the distributions change between a region and another. In this environment the normal algorithm is losing information by using only one distribution for each base station. Figure 4.20 presents the CDF of the two algorithms. Table 4.3 presents the position error precision at iteration three.
Position error, iteration 3
LA RALA
Error mean [meters] 33.0 28.2 Error std dev [meters] 33.9 31.2
Table 4.3: Position error mean and standard deviation for RALA and LA (iteration 3, Rb = 3).
4.4. RANGE-AWARE ALGORITHM RESULTS
Fig. 4.17: p(γb) for BS 1 in Region 1 (iteration 3, Rb = 3).
Fig. 4.18: p(γb) for BS 1 in Region 2 (iteration 3, Rb = 3).
4.4. RANGE-AWARE ALGORITHM RESULTS
Fig. 4.19: p(γb) for BS 1 in Region 3 (iteration 3, Rb = 3).
Fig. 4.20: CDF of the Range-Aware LA and normal LA.
Chapter 5 conclusion
The simulations proved that the learning algorithm is effective in improving the position estimation. The mean position error was reduced by 40% in respect to the TDOA multilateration, reducing the variance at the same time. The Range-Aware variation proved to be effective if the channel bias changes over space, reducing the mean position error by an additional 15%
in respect to the normal learning algorithm. The results showed also that three base stations are enough to obtain good measurements if the learning is already performed.
One important feature that was found is the fast learning speed, in fact 2000 samples divided in two iterations are enough to reach convergence.
Moreover it is important to remind the fact that a real location server using this algorithm needs to perform the learning only once in a while, either to learn the parameters for the first time or to recalibrate them if it is necessary.
The learned parameters can be memorized and then used without the need to keep track of all the measurments collected during the learning.
5.0.1 Future works
The approach presented in this work can be extended in a plethora of ways and adapted to many different scenarios. In first place the algorithm, be-ing based on learnbe-ing and usbe-ing little to no assumption about the network, is prone to adapt to different environments and conditions. Secondly the
Range-Aware variant opens the way to even more capabilities and possible extensions. In this work we opted for range-based regions, but it is possible to define angle-based regions, or even better, a combination of the two. Fur-thermore it is possible to exploit machine learning even more by letting the definition of these regions to the algorithm itself. One approach to achieve this would be by using clustering of the channel bias over space instead of clustering over time (the one used in this work). This approach can be adapted to IoT and UWB signals where frequent reflections easily degrade the position estimation.
Other possible extensions can be done to the probabilistic framework of the algorithm. Including for example a movement tracking feature and consequently a position predictor that can be used to weight the next position estimates.
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Acronyms
BS Base Station.
CSR Cell-specific Reference Signal.
GNSS Global Navigation Satellite System.
LA Learning Algorithm.
LOS Line Of Sight.
LPP LTE Positioning Protocol.
LS Location Server.
LTE Long Term Evolution.
MAP Maximum A Posteriori.
NLOS Non Line Of Sight.
OTDOA Observed Time Difference Of Arrival.
PCI Physical Cell Identity.
PRS Positioning Reference Signal.
RALA Range-Aware Learning Algorithm.
RS Reference Signal.
RSTD Reference Signal Time Difference.
RTD Relative Time Difference.
SINR Signal To Interference plus Noise Ratio.
TDOA Time Difference Of Arrival.
ToA Time of Arrival.
UE User Equipment.
UWB Ultra Wide Band.
Acknowledgments
I want to truly thank prof. Zanella for his insightful advices during the last steps of the thesis and for being particularly understanding during these months.
A special “thank you” go to my parents who always supported me during my studies. I’m thankful and proud of having them.
Final thanks go to Demet, for her support, and to the friends that have been in my thoughts even if I haven’t seen them in a while: Piero, Giacomo, Federico G., Laura, Federico C., Enrica, Leo and all the others from the old telecommunication group.