In[P4] an Expectation Maximization algorithm for solving statistical trilateration problems with additive skew-t distributed measurement errors is developed. The EM computes the MAP position estimate ˆx. The measurement model is y
k|(x = x) ∼ ST ξ + hk(x),σ 2,λ,ν, and posterior pdf (18) is px|y(x|y) ∝ px(x) K Y k=1 2 σtν y¯ k σ Tν+1 λy¯k σ v u t ν + 1 ν +y¯k2 σ2 ! , (39)
where ¯yk = yk − hk(x) − ξ. The measurements are ranges defined
as Euclidean distances between transmitter and receiver and are assumed to be mutually independent. Furthermore, the prior is assumed to be a multivariate-normal x∼ MVN (m, P).
Instead of the skew-t distributed measurement model the hierarchi- cal version y k|(x = x, tk= tk,τk= τk) ∼ N ξ + hk(x) + δλtk, 1− δ2 λ τk σ2 , (40)
with hyperparameters tk|(τk= τk) = |σwk/pτk|, where wk∼ N (0, 1), andτk ∼ Gamma (ν/2, ν/2) is used.
The idea behind the EM algorithm developed in[P4] is to iteratively update the position estimate ˆx with fixed hyperparameters in the maximization step (M-step) and update the hyperparameters with fixed ˆx in the expectation step (E-step). Since for fixed hyperparam- eters the measurement errors are normal distributed according to
3BFGS implementations exist for mathematical programs/ programming languages.
(40) the dGN can be used in the M-step. The hyperparameters are updated using their conditional distributions.
Given the K independent scalar measurements y= y1:K, the dGN
computes for the j -th iteration the mode of the conditional posterior p(x(j )|y, t1:K(j −1),τ
(j −1)
1:K ), that is, it solves the minimization problem
ˆx ← arg min x (x − m) T P−1(x − m)+ (41) PK k=1τ(j −1)k yk− hk(x) − ξ − δλtk(j −1) 2 σ2 1− δ2 λ ,
whereδλ= λ/p1+ λ2. The hyperparameters are initilaized as t
k←
−ξ/δλandτk ← 1 to ensure that the dGN finds the minimizer of the
likelihood for normal noise.
The conditional distribution ofτk, that is p(τk|x, y1:K, t1:K,τ−k) with
τ−k = {τ1, ..,τk−1,τk+1, ..,τK}, is a Gamma distribution with shape
ν/2 and and scale parameter
2σ2 1− δ2 λ ν yk− hk(x(j )) − ξ − δλtk(j −1) 2 + 4σ2 1− δ2 λ .
The product of shape and scale is the distribution’s mean, which is used as updatedτk.
For tk its conditional distribution p(tk|x, y1:K, t−k,τ1:K), where
t−k = {t1, .., tk−1, tk+1, .., tK}, is a truncated normal distribution
bounded from below at zero, i.e. tk ≥ 0 has to hold. The trun-
cated normal distribution’s center is µk = 2δ1λ yk− hk(x) − ξ and
its scale is σk = q σ2 1− δ2λ /(2τk). Similarly to τk, its mean
valueµk+ σkφ(−µk/σk)/ 1 − Φ(−µk/σk) is used as updated tk. For
large negative values of µk/σk this equation for the mean gives significant numerical floating point computation errors. There- fore, in [R1] the mean is computed using the scaled comple- mentary error erfcx, which is defined as erfcx(x ) = ex2
· erfc(x ); φ(−µk/σk)/ 1 − Φ(−µk/σk) in the mean equation is replaced by
q2 π/erfcx −µ k/σk p 2 .
Input: c1:K, y, m, P, m ,σ, εEM,εdGN// εEM and εdGN are
threshold value
x(0)← m, t1:K(0) ← −ξ/δλandτ(0)1:K ← 1// initialization
uEM← εEM
while uEM≥ εEMdo
Initialize ˆx← x(j −1), compute ˜y← y − ξ − δλt1:K(j −1)and R← σ2(1 − δ2λ)diag{1/τ(j −1)1 , .., 1/τ
(j −1)
K }
udGN← εdGN
while udGN≥ εdGNdo
// Run outer while loop of dGN as explained in Algorithm 2 end Set x(j )← ˆx Compute t1:K(j ) andτ(j )1:K asτ(j )k ← σ 2(1−δ2 λ)ν2 yk−hk(x(j ))−ξ−δλtk(j −1) 2 +4σ2(1−δ2 λ) and tk(j )← µk+ σk q2 π/erfcx −µ k/σk p 2
where erfcx← ex2erfc(x ) is the scaled complementary error function,µk←2δ1λ yk− hk(x (j )) − ξ and σ k← v tσ2(1−δ2λ) 2τ(j )k Compute uEM← ||x(j )− x(j −1)|| end
Algorithm 3: Expectation Maximization algorithm
The EM for statistical trilateration is shown in Algorithm 3. Parame- tersεEMandεdGNare set in[R1] to 1 meter for the test with realistic emulated LTE data. This means, that if the position estimates of two consecutive iterations in either EM or dGN are closer than 1 meter apart then the corresponding algorithm is terminated and its current MAP estimate is returned. In[P4] and in the numerical experiment in[R1] fixed number of iterations, nEMfor the EM and ndGNfor the
dGN, are used instead, meaning that the while loops are replaced by for loops in Algorithm 3. For the test with emulated LTE data in[R1] in addition upper limits of nEM= 50 and ndGN= 50 are used to avoid
endless while loops. Furthermore, the number of iterations to findα in the dGN is limited to 5[69] in [P4] and [R1].
Computational complexity of EM and dGN
The computational complexity of EM and dGN is analysed in both[P4] and [R1]. In [P4] it is shown that the number of iterations used by EM nEMand dGN ndGNspecify the two algorithms’ computa-
tional requirements, and that the EM has nEMtimes higher computa- tional complexity than the dGN for the five operation classes addition, subtraction, multiplication, division, and other. Furthermore, the analysis shows that the E-step in the EM algorithm is computationally light compared with the M-step.
In[R1] it is analysed how many iterations both EM and dGN use when Algorithms 2 and 3 are used, i.e. while loops are used and nEMand ndGNonly act as upper limits to avoid endless loops. For the test with
emulated LTE data, the dGN uses on average 4 to 6 iterations. The EM uses on average 5 to 8 iterations, and the dGN run inside the EM uses on average 2 to 3 iterations.
Positioning accuracy of EM and dGN
In the numerical experiment of[P4] the EM clearly outperforms the standalone dGN; it reduces mean and median positioning errors by approximately 25% and the 95th percentile of all errors by approxi- mately 13%.
The numerical experiment is repeated in[R1] under the assumption that ranging errors are skew-t distributed with parameters fitted by a GS to ranging error data from the two LTE channel models EPA5 and ETU70. The experiments support the finding of[P4]. Reductions of 4% to 39% in the error quantiles are observed.
However, the test with emulated LTE data in[R1] draws a different picture. While the EM outperforms the dGN for the EPA5 channel, the dGN performs better for the ETU70 channel. The analysis in the paper shows that the EM’s positioning accuracy is more strongly in- fluenced by the signal-to-noise ratio (SNR) than the dGN. For the test parameters of skew-t and normal distributions are fitted to training data with high SNR (offline phase), but in the online phase position- ing is done in locations with low, medium and high SNR. If the SNR is high the EM outperforms the dGN, for medium SNR both methods perform similarly, and for low SNR the dGN has better positioning
accuracy than the EM. The online data of the ETU70 channel is dom- inated by medium and low SNR values, while for the EPA5 channel medium SNR values dominate. For decreasing SNR the error distri- butions for both channel models change significantly. The slopes at both sides of the error distribution’s peak get flatter, and the er- rors resemble more closely a normal distribution. This explains why the EM performs better than the dGN for EPA5 test, and the dGN performs better for ETU70 test.
In[R1] it is concluded that for high SNR training data the EM algo- rithm is unable to correctly refine the channel parameters for low SNR scenarios in the online phase, and that the EM should outperform the dGN if SNR values would be similar in training and positioning data. Thus, the ability of modelling ranging errors more precisely (by skew-t distribution) and providing better positioning accuracy than the dGN comes at the cost of higher vulnerability to different error distributions in training and positioning data. However, being able to have a more realistic model for the measurement errors was the reason for using a skew-t distributed error model in the first place, so this vulnerability should not be interpreted as weakness. It rather shows that the skew-t model in fact is more realistic than the normal model for the tested data.