4.2 Complex-domain Full SMDS Algorithm
4.2.4 Comparison of Full SMDS and CD-SMDS
In this subsection we highlight the improvements attained by the complex-domain reformulation of the SMDS algorithm described above, both in terms of reduction of computational complexity and localization accuracy. In passing, we also take the opportunity to revisit the issue of information incompleteness discussed in Section 4.2.2, by showing results not only for full-information scenarios, but also for scenarios with partial information.
Unless stated otherwise, the simulation set up utilized to obtain all the results in this and next chapter are as follows. The test area consists of a 10m-by-10m room equipped with 4 anchor nodes, one at each corner, and populated with 16 target nodes located randomly in its interior with x and y coordinates following a uniform distribution.
Distance measurements are modeled as gamma-distributed random variables [94] with the mean given by the true distance and a standard deviation σd. In other words, the probability density function (PDF) of measured distances ˜d associated with a true
distance d is
pD(d; α, β) = 1
βαΓ(α) · ˜d(α−1)· edβ˜, (4.29) with
α = d2/σd2 and β = σd2/d. (4.30) In turn, angle measurement errors δθ are assumed to be Tikhonov-distributed [95–97].
In other words, the PDF of measured angles ˜θ = θ + δθ associated with a true angle θ is
pΘ(˜θ; θ, ρ) = 1
2πI0(ρ) · eρ cos(θ−˜θ), (4.31) where the concentration parameter ρ≥ 0 is inversely proportional to the angular error variance2.
Due to the non-linear relationship between angular error variances and the Tikhonov shape parameter ρ, we capture the influence of angular errors by the quantity εθ, defined as the bounding angle of the 90th centered percentile, i.e.
εθ= θB
Z θB
−θB
pΘ(t; 0, ρ) dt = 0.9. (4.32)
Estimation errors, denoted by ξ, are measured by the Frobenius norm of the difference between the estimates and true positions of the target nodes, i.e.
ξ = 1
Ntk ˆX− XkF. (4.33)
Finally, for the sake of reproducibility, all computation time comparisons were con-ducted based on simulations run on the MathWorks Cloud public platform [131].
First, in Figure 4.2, we compare the localization accuracies of the original SMDS and the new CD-SMDS algorithm here introduced. The results are for a case of full information (i.e, with complete sets of measured distances and angles) and plotted for three different levels of Tikhonov-distributed angle estimation errors [97] and as a function of the standard deviations of Gamma-distributed distance estimates [94].
It is found that CD-SMDS always outperforms SMDS, with the advantage of the new complex-domain method increasingly significant for higher angle estimation errors. The accuracy advantage of CD-SMDS over the original SMDS algorithm is justified by the fact that the CD-SMDS kernel K has rank 1, while the SMDS kernel K has rank
2The Tikhonov PDF [95–97] pΘ tends to a uniform distribution for ρ → 0, and to a Dirac delta function centered at θ for ρ → ∞. Within a wide range 0 ρ ∞, however, pΘ approaches a normal distribution with mean θ and variance 1/ρ.
at least 2, which implies that the separation of signal and noise spaces performed by the low-rank truncation performed in (4.26) under the CD-SMDS approach, is more effective than that achieved by equation (4.9) for the original SMDS algorithm.
The same (lower rank) argument justifies the earlier prediction that CD-SMDS out-performs the original SMDS algorithm also in terms of computational cost, which is indeed confirmed by the results shown in Table 4.1. In fact, the computation of the largest eigenpair performed through our simulations in MatlabTM is based on the power iteration method [110] which is only able to return the largest eigenpair, but is well-known for its superior performance and low complexity. And since rank(K) = 1, the CD-SMDS algorithm takes advantage of this feature. To obtain the results shown in Table 4.1, we temporarily depart from the simulation set up described above by varying the number of target nodes in the network as indicated in Table 4.1, with all other parameters remaining unchanged. It can be seen from these results that in fact CD-SMDS is many times faster than the original SMDS algorithm, especially as the network size increases. We will return to this complexity comparison later in Section 5.3.
Having thoroughly demonstrated the fundamental advantage of CD-SMDS over SMDS, we finally turn our attention to the more practical case of incomplete information. To this end, we return to the basic simulation scenario of 16 target nodes and compare in Figure 4.3 the performances of CD-SMDS and SMDS for a case where 20% of the entries in the kernels K and K are randomly erased. To obtain the final results shown in the plots, the randomly (but identically) erased kernels are first completed using the OptSpace low-rank matrix completion technique of [117] prior to execution of the full CD-SMDS and SMDS algorithms, and averaged over multiple independent erasure realizations.
Table 4.1: Computation time of full SMDS and CD-SMDS
No. Targets SMDS CD-SMDS Ratio
1 28.67× 10−6s 35.19× 10−6s 0.81 5 23.92× 10−5s 64.06× 10−6s 3.73 10 90.46× 10−5s 13.66× 10−5s 6.62 15 28.90× 10−4s 33.36× 10−5s 8.66 20 66.51× 10−4s 67.48× 10−5s 9.86
Although data incompleteness obviously affects the performance of both algorithms, the new CD-SMDS approach is found to be more robust against erasures than the original SMDS, which once again is explained by the lower rank of the complex kernel K compared to the original kernel K, which favors the completion procedure.
To conclude this section, we have shown that reformulating the two-dimensional SMDS framework in the complex domain leads to straightforward advantages in terms of lower computational complexity, better accuracies, and more robustness to information incompleteness. In the subsequent sections we shall show, however, that the new complex formulation here introduced can be further exploited to obtain additional variations of the CD-SMDS algorithm that take even better advantage of the pecu-liarities of complex-domain edge vector representations, as well as of particular data erasure structures that emerge from typical and practical conditions faced by wireless localization systems.
0 0.5 1 1.5 0
0.01 0.02 0.03 0.04 0.05 0.06
Average Estimation Error vs Ranging Error
AverageEstimationErrorξ(m)
Ranging Error σd (m)
CD-SMDS SMDS εθ= 10◦
εθ= 40◦
εθ= 70◦
Figure 4.2: Localization errors achieved by the full version of the original SMDS method [29] as described in Algorithm 1, and its complex-domain CD-SMDS variation presented here and summarized in Algorithm 2.
0 0.5 1 1.5 0.02
0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065
Average Estimation Error vs Ranging Error (εθ= 70◦ and 20% Erasured Kernel)
AverageEstimationErrorξ(m)
Ranging Error σd (m)
Complete Kernel
OptSpace Completed Kernel [31]
SMDS
CD-SMDS
Figure 4.3: Localization accuracies achieved by original SMDS and CD-SMDS al-gorithms with full information and incomplete kernels completed via the OptSpace algorithm [117].