4.6 A cooperative RSSI-based algorithm for indoor localization in WSN
4.6.3 Experimental results after the refinement phase
The numerical results are obtained by using Algorithm5when the positions are initialized with the B-MLE (4.44). We consider three wireless sensor networks issued to the standard ZigBee IEEE 802.15.4 and operating at 2.4 GHz. In addition, the three real testbeds involve different dimensions and low power devices (two testbeds using the TMote Sky6nodes and one testbed us- ing the WSN430 nodes, both node types include the CC2420 RF transceiver). For each testbed, the procedure is described as follows. Each of the NL sensor nodes selected for the learning
6
Technical details in http://www.eecs.harvard.edu/~konrad/projects/shimmer/
4.6. A cooperative RSSI-based algorithm for indoor localization in WSN 129
phase broadcasts 100 frames. Then, the M landmarks compute the set of the propagation model parameters {(PL0,k, ηk, σ2k}Mk=1as detailed in [54]. We set two different sizes NLof the learning
data set, a small set involving the first 10 sensor nodes’ positions and a large one from the first 25 sensor nodes’ positions. The RSSI values are collected from the set of received frames, and then the parameters are estimated by applying the MLE criterion as in (4.4)-(4.5). The remaining N − NLsensor nodes compute a first estimate of their positions using the B-MLE (4.44) given
the parameters and the RSSI values from theirs corresponding received frames. The refinement phase is subsequently applied assuming the latter positions as the initialization values for the distributed on-line Algorithm5. At each iteration, a single frame is broadcasted by each sensor in order to compute the local estimates and only two sensor nodes randomly selected exchanged their common estimate positions which are finally updated as the average value.
1) Office Scenario
The testbeds considered in this section are the same as those from [55]. The first testbed located in Paris is a small semi-furnished office at LINCS laboratory with dimensions 4 × 3 m2. It involves the positions of 48 sensors and 5 landmarks. The second larger testbed located in Telecom Bretagne is a classroom at Rammus platform hosted by the RSM department whose dimensions are 8.77 × 6.46 m2. It involves the positions of 57 sensor nodes and 8 landmarks. In both testbeds sensor nodes are placed at 1.25 m height from the floor. None of these rooms are electromagnetically isolated and active wireless access points close to the testbeds may create interferences. The presence of two moving persons in both rooms may affect the line-of-sight as obstruction between the sensor nodes and the landmarks during the communication phases. Both testbeds are illustrated in Figure4.14(available and detailed in [3]).
0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 LM1 LM2 LM3 LM4 LM5
Testbed LINCS: N=48 nodes and M=5 landmarks
x (m) y (m) 2 2.5 3 3.5 −80 −70 −60 −50 −40 −30 −20 −10 d (m) Pd (dBm)
RSSI values at node 15
(a) LINCS testbed (left) and RSSI values (right) col- lected at node 15 from the 5 landmarks.
2 3 4 5 6 −80 −75 −70 −65 −60 −55 −50 −45 −40 −35 d (m) Pd (dBm)
RSSI values at node 20
0 2 4 6 8 0 1 2 3 4 5 6 7 x (m) y (m)
Testbed Rammus: N=57 nodes and M=8 landmarks
10 9 8 7 6 5 4 3 2 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 LM1 LM3 LM2 LM4 LM5 LM6 LM7 LM8 48 49 57 56 55 54 53 52 51 50
(b) Rammus testbed (left) and RSSI values (right) col- lected at node 20 from the 8 landmarks.
Figure 4.14. Offices testbeds and RSSI values collected at squared nodes () from data trans- mitted by the M landmarks. The marker (Q) highlights the real RSSI values. The markers (5) and (+) indicate respectively the average and the minimum and maximum values from 100 i.i.d. random samples drawn when considering the theoretical LNSM in (4.2) given the estimated parameters in Table6
Regarding the real RSSI data shown in Figure 4.14, some values received at each of the two nodes (node 15 at LINCS testbed and node 20 at Rammus testbed) are affected by a bias depending on the landmark. For instance, in Figure 4.14 (a), data coming from LM3 which has the highest path loss exponent η (see Table6) suffer a considerable bias from the theoret- ical mean value (around −42 dBm) since the real values are placed around −70 dBm. From Figure4.14 (b) we observe a gap of around 20 dBm between the theoretical and the empirical mean value of the RSSI coming from LM1 which is the closest one to node 20, maybe due to its proximity to the wall since its corresponding parameters are rather soft (see Table6). The corresponding estimated parameters for each landmark are summarized in Table6.
LM_ID LM1 LM2 LM3 LM4 LM5 PL0 40.72 29.97 40.59 57.69 40.38 η 1.47 2.81 3.11 2.06 0.92 σ2 26.91 49.55 12.09 17.76 31.29 LM_ID LM1 LM2 LM3 LM4 LM5 LM6 LM7 LM8 PL0 52.09 49.23 47.83 41.09 42.67 45.69 71.85 50.97 η 1.31 0.81 1.51 1.97 1.38 1.66 -2.19 0.51 σ2 11.89 26.17 27.51 6.69 26.81 23.08 12.69 13.19
Table 6. Estimated parameters from the office at LINCS testbed (up) and the classroom at Rammus testbed (down) when the small data set of 10 sensor nodes is selected.
2) FIT IoT-LAB platform
The testbed at the FIT IoT-LAB’s platform in Rennes of size 5 × 9 m2involves the positions of 44 sensor nodes and 6 landmarks and its configuration is shown in Figure4.15. The WSN430 open nodes available at the platform are located in a big storage room containing different objects and they are placed at the ceil which is 1.9 m height from the floor in a grid organization. There was no one in the room most of the time and there was only a wireless access point located in the corridor which is separated by a cinder wall (no electromagnetically isolating).
On the right in Figure4.15we illustrate the real and the empirical RSSI values collected at node whose Node_id is 240 coming from the 6 landmarks. Note that the most important bias of more than 10 dBm appears on the RSSI corresponding to the closest landmark LM244 which is situated next to the wall of the room and has the highest path loss value PL0(see Table7). The
estimated parameters for each landmark are summarized in Table7.
3) Comparison and discussion
We run both algorithms, the initialization performed by (4.44) and the refinement phase per- formed by the distributed Algorithm5. In order to evaluate and quantify the achieved accuracy of such methods, we define the normalized mean deviation (NMD) as the RMSE over the N
4.6. A cooperative RSSI-based algorithm for indoor localization in WSN 131 2 4 6 8 −110 −100 −90 −80 −70 −60 −50 d (m) Pd (dBm)
RSSI values at node 240
2 3 4 5 6 7 0 2 4 6 8 10 x (m) y (m)
Testbed selection at Rennes FIT IoT−LAB platform N=44 nodes and M=6 landmarks
155 156 158 159 161 173 178 179 180 181 183 193 194 195 196 197 198 200 201 202 209 211 212 213 215 216 218 225 227 229 230 231 232 234 235 237 240 243 247 249 250 251 252 253 LM157 LM163 LM176 LM214 LM236 LM244
Figure 4.15. Network configuration of the 50 sensors selected at the FIT IoT-LAB’s platform in Rennes [1].
LM_ ID LM157 LM163 LM176 LM214 LM236 LM244 PL0 62.19 63.61 58.4 63.33 58.55 67.67
η 1.76 2.83 3.39 1.98 2.80 2.29
σ2 19.06 40.87 37.04 75.62 30.03 18.97
Table 7. Estimated parameters from the FIT IoT-LAB Rennes testbed when the small data set of 10 sensor nodes is selected.
estimated positions normalized by the testbed’s dimensions, i.e. l × h m2. It can be defined as: NMD = 1 N N X i=1 NMDi = 1 √ l2+ h2 1 N N X i=1 kˆzi− zik ,
where {ˆzi}∀i are the set of the N estimated positions. Figure 4.16illustrates the decreasing
RMSE over the N positions along the iteration instant n for each testbed which involves the communication between two randomly selected nodes at each n. Note that as being a real environment, the algorithm converged to an asymptotic error which may depend on the testbed’s parameters.
Regarding the different curves in Figure4.16, the earliest testbed to achieve an improvement (after n = 24 iterations implying 48 communications) is the Rammus testbed. However, this testbed achieves the worst accuracy after the refinement phase since the curve of its RMSE remains always above the other two. The best accuracy is achieved with the LINCS testbed even if the convergence is slower and 89 refinements iterations implying 178 communications are required to improve the mean localization error. As reported in Table8, a RMSE less than 80 cm is obtained for the LINCS testbed.
In order to summarized the results for each testbed, Table8displays the average error, both RMSE in meters and the respective normalized value NMD. The numerical results are reported when considering the two sizes of data sets chosen during the learning phase of the B-MLE
50 100 150 200 250 0 1 2 3 4 5 6 7 8 n (iteration time) RMSE n DSA, LINCS B−MLE, LINCS DSA, Rammus B−MLE, Rammus DSA, FIT IoT−LAB B−MLE, FIT IoT−LAB
Figure 4.16. Convergence of the RMSE sequence generated by Algorithm5along the iteration time n for each testbed when considering the small data learning set (NL = 10). Markers (5
Q M) emphasize the iteration time when the RMSE of the distributed refinement phase is lower than the RMSE computed by the initialization B-MLE.
Testbed LINCS Rammus FIT IoT-LAB
Method B-MLE DSA B-MLE DSA B-MLE DSA
RMSE (m) 1.39 0.77 2.73 1.73 1.85 1.3
NMD 0.28 0.16 0.25 0.16 0.18 0.13
Improvement (%) 44.3 36.7 29.5
Positions improved (%) 76 72 74
Testbed LINCS Rammus FIT IoT-LAB
Method B-MLE DSA B-MLE DSA B-MLE DSA
RMSE (m) 1.35 1.31 2.27 1.64 2.27 1.84
NMD 0.27 0.26 0.22 0.15 0.22 0.18
Improvement (%) 3.8 30.5 19
Positions improved (%) 55 63 68
Table 8. Localization error averaged over the N estimated sensor nodes’ positions for each of the three testbeds when using the small data set of 10 positions (up) and the big data set of 25 (down).
(NL = 10 and NL = 25). In addition, we compute the ratio of the accuracy improvement
as the percentage (1 − ρ) % where ρ defines the ratio between the NMD achieved after the refinement phase described in Section4.5.1and those achieved previously by the B-MLE. We also compute the ratio regarding the number of improved positions after the refinement phase (see the ratio Positions improved in Table 8). From the results reported in Table 8, the best
4.6. A cooperative RSSI-based algorithm for indoor localization in WSN 133
accuracy improvement is obtained in general in the case of the small data learning set, i.e. NL=
10. The best accuracy about 70 cm is achieved for the smallest testbed (LINCS) where there is at the same time the biggest noise factor σ2about 49.55 dB. However, the LINCS testbed requires the highest number of pairwise communications between the sensor nodes during the refinement phase. Our numerical results appeared to be consistent compared to other experiments involving real indoor scenarios with similar testbeds in terms of dimensions and number of sensor nodes. See for instance the accuracy between 1.5 − 2.5 m from the experiments of [145] or the 2.27 m reported in [45].
Figure 4.17. Boxplot of the NMD values obtained over 100 independent runs of the doMLE (Algorithm 5) for each testbed when considering the parameter learning sizes NL = 10 and
NL= 20.
In addition, Figure4.17illustrates the statistical behavior of the localization error (NMD) through the corresponding boxplot representations. Both LINCS and FIT IoT-LAB testbeds have similar behavior: the standard deviation of the NMD error decreases when increasing the learning data size (from NL= 10 to 25) but the mean value of the NMD error increases. Indeed,
more outliers values appear in such case since considering a large number of positions may add some corrupted data to the learning phase affecting the estimated parameters. The smallest testbed (LINCS) has the worst error performance since it gives the highest standard deviation and mean values when considering the big learning data set (NL= 25) and the highest standard
deviation when considering the small one (NL= 10). On the contrary, the FIT IoT-LAB testbed,
which has the biggest dimensions and is rarely occupied by moving people, maintains the lowest standard deviation error for both values of NLand achieves the best accuracy on the NMD when
considering the small learning set. Finally, the middle-sized Rammus testbed has a more regular behavior since it gives a similar performance independently of NLvalue.
The localization error at each sensor node is detailed in Figure4.18. We report the NMD values for each sensor node {NMDi}∀i at each testbed before (given by the B-MLE) and after
the refinement phase (given by doMLE algorithm) when considering the small and the big learn- ing data set. Regarding Figures4.18and the information summarized in Table8, the accuracy
(a) LINCS testbed, NL= 25. (b) LINCS testbed, NL= 10.
(c) Rammus testbed, NL= 25. (d) Rammus testbed, NL= 25.
(e) FIT IoT-LAB testbed, NL= 25. (f) FIT IoT-LAB testbed, NL= 10.
Figure 4.18. NMD values at each testbed before (blue bars) and after (red bars) the refinement phase (Algorithm5 for each estimated sensor position. On the left we set NL = 25 for the
learning data set and on the right when considering NL= 10.
improvement and the number of positions improved are considerably higher when NL = 10.
For the LINCS testbed about 76% positions are improved while for the Rammus testbed the percentage is 72% when considering 10 positions for the learning phase. However, the percent- ages decrease for the LINCS and Rammus testbeds when 25 positions are considered during the learning phase which are 55% and 63% respectively. Moreover, when regarding localization errors in Figure4.18and the network’s configurations (see Figures4.14and4.15), the estimated positions that do not improve the accuracy after the refinement phase are those from sensor nodes located on more dense area or on the middle surrounded by objects and the other sensor nodes (see for instance nodes 13, 14, 28 and 30 at LINCS testbed, nodes 4, 5, 17, 18 and 28 at Rammus
4.6. A cooperative RSSI-based algorithm for indoor localization in WSN 135
testbed and nodes 197, 216 and 234 at FIT IoT-LAB testbed). In some cases, there is no accu- racy improvement on nodes located on the corners as nodes 252 and 253 at FIT IoT-LAB testbed or as nodes 1 and 42 at Rammus tesbed. In conclusion, after the refinement phase, an accuracy improvement of at least 30% is achieved and more than the 70% of positions are improved for different indoor scenarios involving different dimensions and different radio devices.
Conclusions and perspectives
In this thesis, two applications of distributed stochastic approximation in multi-agent systems have been considered: consensus-based distributed optimization and distributed principal com- ponent analysis (PCA).
Regarding consensus-based methods, we addressed the case where a network of agents seek to find the global minimizer of an optimization problem. The aim is to drive the local iter- ates of all agents to a common minimizer. We have concentrated our efforts in the theoretical analysis of a adaptation-diffusion algorithm, where agents iteratively update their local iterates and merge them by communicating with their neighbors. We have demonstrated the almost sure convergence under weak conditions on the communication protocols. Although double stochasticity is generally assumed in past works, our convergence result holds even when the matrix Wn characterizing the networks exchange is non doubly stochastic. This observation
gives rise to the possibility of using simple communication schemes between agents such as the intuitive broadcast protocol, in which agents send information to their neighbors without ex- pecting any instantaneous feedback from the latter. We have also analyzed the convergence rate of the method. More precisely, we have proved that the estimation error between the iterates and the minimizer tends to zero at rate √γn where γn is the step size of the algorithm. The
normalized estimation error is asymptotically normal. The limiting covariance matrix has been characterized. As a consequence of our results, we have shown the price to pay for using simple non-doubly stochastic weight matrices is an increase of the asymptotic variance of the estima- tion error. We have applied our results and tested our algorithms to the problem of statistical inference in wireless sensor networks, with a special focus on self-localization problems. We have also proposed and analyzed a distributed on-line expectation maximization algorithm (see AppendixA) which relies on the same principles.
Regarding distributed PCA, we addressed the case where the i-th agent seeks to estimate the i-th entry of the principal eigenvectors of a given matrix, based on noisy and distributed measurements of the latter matrix. We have proposed an iterative algorithm which is based on sporadic information exchanges in the network and proved the almost sure convergence of the algorithm. The algorithm can be seen as a distributed version of Oja’s algorithm, which is a popular stochastic approximation method to estimate the principal eigenspace of a matrix. We have applied our results to the issue of self-localization in wireless sensor networks. We have considered the case where agents are identified to sensors which are able to collect noisy measurements of the distance to their neighbors. We have proposed a distributed version of a multidimensional scaling algorithm based on PCA which allows to recover the positions of the
sensors as the eigenvectors of a so-called similarity matrix computed from inter-sensor distances. In addition, our algorithm encompasses the context where measurements are gathered in an on- line and sporadic fashion. We have also tested our algorithms on a wireless sensor network platform, i.e. FIT IoT-LAB platform7. The collaboration with N.A. Dieng have made possible our contributions on the localization framework in WSN (see [56]). Besides, we could show the performance on real testbeds from data acquired within different indoor scenarios (see [3] and [2]).
There still remains several open problems in the continuity of this thesis, which have not been addressed due to the lack of time. First of all, it would be interesting to analyze the ef- fect of Polyak-Ruppert averaging methods which are known to increase the convergence rate of stochastic approximation methods (see [98]). Most probably, such methods are as well effective in a distributed setting as discussed in [19] in the special case of doubly-stochastic matrices. Second, this thesis have focused on iterative algorithms with vanishing step sizes. In stochastic contexts where measurements are collected on-line and then deleted, vanishing step sizes are generally required to ensure the convergence of the algorithms. Nevertheless, in signal pro- cessing, it is as well important to consider methods with constant step size. Such methods are generally non convergent, but allow to adaptively track the variations of the environment (e.g. adaptive optimizationfor target tracking [151]). It is particularly well suited to the case of mo- bile sensor networks for instance. Typically, our self-localization algorithm based on distributed PCA would be especially relevant in the case of a constant step size, as it would allow each