Improved Least Median of Squares Localization for Non-Line-of-Sight
Mitigation
Qiao, T., & Liu, H. (2014). Improved Least Median of Squares Localization for
Non-Line-of-Sight Mitigation. IEEE Communications Letters, 18(8), 1451-1454.
doi:10.1109/LCOMM.2014.2327952
10.1109/LCOMM.2014.2327952
IEEE - Institute of Electrical and Electronics Engineers
Accepted Manuscript
http://cdss.library.oregonstate.edu/sa-termsofuse
1
Improved Least Median of Squares Localization for
Non-Line-of-Sight Mitigation
Tianzhu Qiao and Huaping Liu, Senior Member, IEEE
Abstract—Non-line-of-sight (NLOS) propagation is one of the
major problems that cause errors in localization systems. There
exist many algorithms to reduce NLOS caused errors. Most of
these algorithms require a priori information about the NLOS
links, which limits their application. Least median of squares
(LMedS) method is an exception and is thus attractive for its
simplicity. LMedS selects only a subset of the anchors with the
smallest residue for location calculation, but it tends to discard
more range measurements than necessary. Its performance can be
significantly improved, especially when the percentage of NLOS
links is not high. In this paper, we present an improved LMedS
method, which maximizes the subset of reliable anchors to be
used. We show via simulation that compared with the existing
LMedS method, the proposed algorithm has a significantly higher
localization accuracy and is more stable.
Index Terms—Least median of squares method (LMedS), localization, non-line-of-sight (NLOS) propagation.
I. I NTRODUCTION
In localization systems, non-line-of-sight (NLOS) propagation causes a positive ranging offset [1] that drastically
reduces location accuracy. There are two common ways to
deal with this problem [2]–[4]: NLOS identification and NLOS
mitigation. NLOS identification methods try to identify range
estimates that contain NLOS errors, for example, with prior
statistical information about the NLOS and LOS links [2].
Then, such links could be discarded. Generally it can be
done during the range estimation phase. When NLOS offsets
exist in the range measurements, methods such as non-linear
least squares (NLLS) and method of moments (MOM) [5]–[9]
do not work well. Many algorithms have been developed to
mitigate NLOS errors [2]–[4], [10]–[18]. Some of them do
not require a priori knowledge about the LOS and NLOS
links (e.g., the quadratic programming method [12]). Most of
the mitigation methods require some prior information. The
Kalman filtering methods developed in [14]–[16] utilize prior
statistical information about the NLOS and LOS range measurements (e.g., the range error variance); the machine learning
algorithm developed in [17] assumes that the characteristics of
the NLOS and LOS links are known; the linear programming
method [18] requires perfect detection of the NLOS links.
The least median of squares method (LMedS) [10] is of
particular interest in this paper, since it does not need any a
priori information about the LOS or NLOS links, and thus
can be applied for most scenarios. LMedS searches for the
subset of anchors with the least median of squares error
(residue) to improve localization performance when NLOS
links exist. However, it uses only a subset of the available range
Manuscript received May 10, 2014. The associate editor coordinating the
review of this letter and approving it for publication was W. Gifford.
The authors are with the School of Electrical Engineering and Computer
Science, Oregon State University, Corvallis, OR 97331 USA (e-mail: {qiaoti,
hliu}@eecs.oregonstate.edu).
This work is supported by the National Science Foundation under Grant
IIS-1118017
measurements to estimate the position, and has drawbacks: 1)
To accommodate possible cases of a high percentage of NLOS
links, the size of the subset chosen must be small. This tends
to cause the algorithm to discard many range measurements
that could be utilized for better performance; 2) It may result
in ‘outliers’ in the final estimation due to range measurement
errors. This issue will be discussed in detail in Sec. II.
In this paper we develop an improved LMedS algorithm that
resolves the above two issues. The improved algorithm uses
residues to weight all the range measurements and adaptively
searches for the largest subset with small residues, rather than
setting a fixed subset. We will provide extensive simulation
results to compare the accuracy and stability of the original
and the improved algorithms.
II. I MPROVED LM ED S A LGORITHM
A. System model
Let θ be the unknown location of the target, and v m
(m = 1, 2, · · · , M ) the known locations of the anchors. The
measured distances between the target and the mth anchor for
TOA systems are written as:
rm (i) = dm + bm + nm (i), i = 1, · · · , N,
(1)
where dm = kθ − v m k is the true distance between the
mth anchor and the target, i is the index of the observation set, N is the total number of observation sets, bm is
a positive offset caused by NLOS propagation between the
mth anchor and the target, and nm (i) is the random range
measurement error, which can be modeled as independent and
identically distributed zero-mean Gaussian random variables;
2
that is, nm (i) ∼ N (0, σm
) [10].
B. Analysis of the conventional LMedS
The LMedS algorithm [10] is summarized as follows.
1) Randomly select a subset of L anchors, denoted by Sk ,
from a total of M anchors to determine the position
estimate θ̂ k .
2) Repeat 1) until K estimates, θ̂ 1 , · · · , θ̂ K , are obtained.
3) For each θ̂ k , calculate the residue as
RSk = median (rk,1 − r̂k,1 )2 ,· · · ,(rk,L − r̂k,L )2 , (2)
where rk,l is the observed range measurement between
the target and the lth anchor in subset Sk , r̂k,l is the
distance between the estimated position θ̂ k and the lth
anchor in subset Sk .
4) Find the estimate with the smallest residue, and designate
it as the final estimate.
The method does not need any prior information about the
LOS and NLOS links, which makes it easy to be applied to
various scenarios. The algorithm works very well when the
2
NLOS offsets exist in some of the range measurements [10].
However, it has some drawbacks:
1) It uses only a subset of all available range measurements.
Since scenarios where many NLOS links exist are possible, the fixed subset size L must be small. Thus for cases
when most of the links are LOS, its accuracy can be
improved if the number of reliable range measurements
used in the estimation is maximized.
2) LMedS might choose an ‘outlier’ as the final estimate.
For example, due to the range measurements error
(nm (i), bm ), for one particularly subset Sk , the residue
calculated by using Eq. (2) could be the smallest one
even if the subset Sk contains large range measurement
errors. Thus the estimated position can be far off the
true one. An example is illustrated in Fig. 1, where
v 2 , v 3 , andv 4 are chosen for position estimation as these
result in the smallest residue (0). However the estimate
θ 0 is far away from the true target θ.
v2
r2 (i)
θ′
θ
r3 (i)
v3
r4 (i)
increases the estimation stability and accuracy for both LOS
and NLOS cases.
D. Outlier probability
The total number of links M can be written as M = MLOS +
MNLOS , where MLOS and MNLOS are the number of LOS
MLOS
and NLOS links, respectively. Let K = M
,
L , K1 =
L
K2 = K − K1 , and RSLOS /RSNLOS be the residues from the
subset with LOS/NLOS range measurements. We have P1 =
P (RSLOS ≤ RS ) ≈ 1. Similarly, let P2 = P (RSNLOS ≤ RS ).
It is reasonable to assume that P2 P1 .
For the conventional LMedS method, the probability of
reaching an outlier as the final estimate is written as
P2 K 2
P2 K2
≈
.
(3)
P (outlier) =
P1 K1 + P2 K2
K1 + P2 K2
For the proposed method, if the link between anchor v m
LOS −1
and the target is LOS, then v m will be in K10 = ML−1
subsets, which only include
LOS range measurements. Also
K0
P (Fv (vm ) ≥ K10 ) ≥ P1 1 . Thus it is reasonable to set the
threshold Fth = K10 . On the other
hand, if this link is NLOS,
−1
then v m will be in K 0 = M
L−1 subsets, which include at least
one NLOS range measurement. Therefore,
K0 0
X
0
K
0
P (Fv (vm ) ≥ K1 ) =
P2k (1 − P2 )K −k .
(4)
k
0
k=K1
Fig. 1. Illustration of reaching an ‘outlier’ estimate with the LMedS algorithm,
where θ is the true target position and θ 0 is the estimated target position.
C. Proposed algorithm
We propose an improved LMedS algorithm, which resolves
all the above problems:
1) Use the existing LMedS method to compute θ̂ k and RSk
(k = [1, 2, · · · , K]), where each subset Sk contains L
anchors. Here L can be as small as possible.
2) Find all subsets So,p , p = 1, 2, · · · , P that satisfy
RSo,p ≤ RS , where RS is a predefined threshold. In
practice, RS can either be fixed given an expected accuracy, or can be updated dynamically by using the range
measurement error information (e.g., sample variance of
the range measurement). If such subset does not exist,
use the conventional LMedS method.
3) Let Sv denote the set of all anchors appear in So,p .
Calculate the frequency of each anchor appearing in So,p .
For example, suppose So,1 = [1, 2, 3], So,2 = [2, 3, 4],
then Sv = [1, 2, 3, 4]. The corresponding frequency is
Fv = [1, 2, 2, 1]; that is the 1st anchor appears only once
in So,p , the 2nd anchor appears twice in So,p , and so on.
4) Put the L anchors that have the highest frequencies in
Sv,o . Then check the frequencies of all other anchors
in Sv : if the frequency of any anchor is greater than
Fth = max (Fv )−F , where F is a predefined threshold,
then put it in Sv,o ; otherwise, it is discarded.
5) Use the range measurements between the target and the
anchors in Sv,o to determine the final position estimate.
Because the proposed algorithm maximizes the number of
reliable range measurements used in localization, it drastically
Consequently, the upper bound of reaching an outlier as the
final estimate is
P (outlier) ≤ MNLOS × P (Fv (vm ) ≥ K10 ).
(5)
The probabilities of reaching an outlier with the LMedS and
the proposed methods are shown in Fig. 2; when the number
of NLOS links does not exceed 50% of the total number of
links, the latter is orders of magnitude smaller than the former.
5
10
LMedS
0
10
−5
10
P(outlier)
v4
−10
10
−15
10
−20
10
Proposed method
−25
10
P2=0.01
P2=0.02
−30
10
1
P2=0.005
1.5
2
2.5
3
3.5
NLOS links number
4
4.5
5
Fig. 2. Probability of reaching an outlier as the final position estimate with
the LMedS and the proposed methods (M = 8, L = 3 and K = 56).
As the number of NLOS links further increases (e.g, ≥ 5
NLOS links out of 8), of all the K subsets, there is only one
subset at most that contains only LOS links. So the LMedS
algorithm will be very likely to choose the outlier as the final
estimate. The proposed method also performs similarly as it
does not have enough LOS subsets to weight all the anchors.
For unrealistic cases with an even greater M value (e.g., M =
16), however, the proposed method might still work even if the
number of NLOS links exceeds M/2, as long as the number
of LOS links is sufficient to weight all the anchors.
3
1.5
E. Computational complexity
A. Estimation outliers
Fig. 3 shows the estimated position with the LMedS method;
approximately 2% of the final estimates are outliers. With the
proposed method, all outliers are eliminated.
0.5
y
0
−0.5
−1
−1.5
−1.5
−1
−0.5
0
x
0.5
1
1.5
Fig. 3. The estimated position with LMedS algorithm (σ(1%), NLOS offset
(5%), 1 NLOS link).
−1
10
NLLS
Wang
LMedS
Proposed
Position Error
Simulation setup is shown in Fig. 3: 8√anchors are uniformly
placed on a circle with a radius of 2 and a total of 25
targets are placed on the grid {x, y} = {0, ±0.3, ±0.6}. The
total number of range measurements is set to N = 50 and
the subset size is chosen to√be L = 3. Position error is
defined as Position Error = MSE. For each subset, NLLS
estimator is applied for position estimation. Other parameters
are defined as follows: “σ(a%)” denotes the AWGN with
nm (i) ∼ N (0, (dm × a%)2 ); “NLOS offset(b%)” denotes
the NLOS offset with a uniform distribution, that is, bm ∼
U(0, dm × b%); dm defined in Eq. (1) is the true distance
between the target and the mth anchor.
The definition of error characterization described in [10]
is applied here. For Type I error (b% = 0%), a LOS path
exists between the target and all anchors. For Type II error
(b% ≤ 5%), the target is slightly blocked by the objects around
it. For example, the objects between the target and the anchor
are small and not close to either of them. Thus NLOS effects
exist but are not severe. For Type III error (b% ≤ 100%),
the target is substantially blocked by objects around it. For
example, there are many large objects between the target and
the anchor, causing a significant NLOS bias in the range
measurement. For Type IV error, since the target-anchor path
is completely blocked, the corresponding range measurement
does not provide any information about the target position and
is thus discarded during the range estimation phase.
Anchor
Estimation
Target
1
−2
10
−3
10
1
1.5
2
2.5
3
σ (%)
3.5
4
4.5
5
Fig. 4. Performances of four algorithms: NLLS, Wang’s method [12], LMedS,
and the proposed method (Type I range measurement error; 0 NLOS links).
σ (1%)
−1
10
NLLS
Wang
LMedS
Proposed
Position Error
The LMedS method needs to run one estimation per subset.
If NLLS estimator is applied, it needs approximately 230LK
multiplications [7] for the estimation and 5LK multiplications
to calculate the residues for 3-dimensional localization. Thus
it requires a total of about 235LK multiplications.
The proposed method needs additional calculations to compute the weight for each anchor, which generally can be done
with simple comparisons and additions. If NLLS is used for
the final estimation, the proposed method requires a maximum
of 235LK + 230M multiplications. For almost all scenarios,
LK M (e.g., K = 56, L = 3, M = 8 as chosen in simulation:
KL = 168 M = 8). Thus the increase in computational
complexity with the proposed method is negligible.
III. S IMULATION
−2
10
−3
10
1
1.5
2
2.5
3
3.5
NLOS offset (%)
4
4.5
5
Fig. 5. Performances of four algorithms: NLLS, Wang’s method [12], LMedS,
and the proposed method (Type II range measurement error; 1 NLOS link).
B. Type I error
The performances versus the variance of the range measurements of NLLS, Wang’s method [12], LMedS, and the
proposed method are shown in Fig. 4. Since the LMedS method
only includes one subset of range measurements, its accuracy
is worse than those of NLLS and the proposed methods.
C. Type II error
Fig. 5 compares the performances versus the NLOS offset
of NLLS, Wang’s method, LMedS, and the proposed method
when one anchor and the target suffers from this type of error.
The proposed method works slightly better than the LMedS
algorithm, as it tends to include more anchors with LOS links
for position estimation.
D. Type III error
Figs. 6, 7, and 8 show the performances versus the number
of NLOS links of the various methods for cases when NLOS
offset reaches 10%, 50%, and 100% of the actual distance.
With Type III range measurement errors, NLLS does not work
well since it does not have any mechanism to deal with NLOS
effects. The proposed method works significantly better than all
other algorithms. As discussed in Sec. II, when there are many
NLOS links, the subset size L cannot be chosen very large
with LMedS. The proposed method maximizes the number of
reliable anchors for the final estimation, and thus performs
much better than LMedS. Additionally, by comparing Figs.
4
6, 7, and 8, it is observed that NLOS does not appear to
affect the accuracy of the proposed method, since it inherently
results in discarding the NLOS links. When NLOS situation is
severe (e.g., NLOS offset reaches 100% of the actual range,
3 NLOS links), the estimation accuracy with the proposed
algorithm is even sightly better than the case of less severe
NLOS conditions (e.g., NLOS offset (10%), 3 NLOS links).
This is because a large NLOS offset (bm ) will dominate the
range measurement errors, making it less likely to be included
in the final estimation.
R EFERENCES
Position Error
NLLS
Wang
LMedS
Proposed
−2
10
−3
1
1.5
2
2.5
3
NLOS links number
3.5
4
Fig. 6. Performances of four methods: NLLS, Wang’s method, LMedS, and
the proposed method (Type III range measurement errors, NLOS offset 10%).
σ (1%), NLOS offset (50%)
0
10
NLLS
Wang
LMedS
Proposed
−1
Position Error
10
−2
10
−3
10
1
1.5
2
2.5
3
NLOS links number
3.5
4
Fig. 7. Performances of four methods: NLLS, Wang’s method, LMedS, and
the proposed method (Type III range measurement errors, NLOS offset 50%).
σ (1%), NLOS offset (100%)
0
10
NLLS
Wang
LMedS
Proposed
−1
Position Error
10
−2
10
−3
10
1
An improved LMedS method that is effective for NLOS
mitigation is presented in this paper. The key advantage of this
algorithm over the conventional LMedS method is that it maximizes the number of reliable range measurements to be used
in position estimation in an adaptive manner. Consequently,
compared with the conventional LMedS method, the proposed
algorithm achieves a higher estimation accuracy under both
LOS and NLOS cases. Furthermore, it significantly decreases
the probability of reaching an outlier position estimate.
σ (1%), NLOS offset (10%)
−1
10
10
IV. C ONCLUSIONS
1.5
2
2.5
3
NLOS links number
3.5
4
Fig. 8. Performances of four methods: NLLS, Wang’s method, LMedS, and
the proposed method (Type III range measurement errors, NLOS offset 100%).
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