Robust Localization in Wireless Networks From Corrupted Signals

We address the problem of timing-based localization in wireless networks, when an unknown fraction of data is corrupted by nonideal signal conditions. While timing-based techniques enable accurate localization, they are also sensitive to such corrupted data. We develop a robust method that is applicable to a range of localization techniques, including time-of-arrival, time-difference-of-arrival and time-difference in schedule-based transmissions. The method is nonparametric and requires only an upper bound on the fraction of corrupted data, thus obviating distributional assumptions of the corrupting noise distribution. The robustness of the method is demonstrated in numerical experiments.

Several approaches to robust localization exist. A conservative approach is to model the NLOS effects as bounded additive measurement errors or bias [15,17] and then estimate the node location that minimizes the worst-case error. This approach, however, can be overly conservative and sensitive to the user specified error bound. Another approach from the robust statistics literature is to consider the Huber contamination model [10], in which an fraction of data samples are corrupted by a NLOS-data distribution. The NLOS distribution is either assumed to have a specific parametric form -e.g. shifted Gaussian or exponential [12,19] in which case the positions of nodes are estimated via maximum likelihood -or is modelled nonparametrically [7,8,18], a case which is tackled using semiparametric or iterative maximum likelihood methods. An alternative approach is to use a different, robust loss function that is insensitive to outliers [21]. While it is often assumed that the fraction of corrupted data is known, in practice this fraction is an unknown user parameter.
In this paper, we propose a robust localization methodology for data obtained in the contamination setting with an unknown fraction of NLOS data, drawing upon the principles of robust risk minimization in [13]. We only assume that the user is able to set an upper bound ≥ . We demonstrate the methodology for three distinct localization techniques: time-of-arrival (TOA), time-difference-of-arrival (TDOA) and time-difference in schedule-based transmissions (TDST).

Problem formulation
For a general problem formulation, consider a wireless network consisting of N + 1 nodes at locations Only the anchor node locations are known. The anchors and auxiliary nodes all transmit signals that propagate through space with constant velocity. These nodes can be thought of as base stations with known and unknown locations, respectively. The signals carry signatures that enable the identification of the transmitting node.
The self-localizing node, located at x 0 , can either be an active transceiver or a passive receiver, depending on the localization technique considered below. We let θ * denote the set of locations of interest and our goal is to estimate this set using timing measurements at node 0.

TOA: two-way ranging anchor nodes
In TOA, node x 0 is a transceiving node. At a given time, node 0 initiates a transmission to the anchor nodes. Upon receiving the signals, anchor nodes return signals, after a possible fixed delay. This leads to time-of-arrival measurements at self-localizing node 0. The TOA measurement with respect to anchor node i is then where c is the signal propagation velocity and δ ≥ 0 is a transmission delay, see Figure 2a. The unknown location is

TDOA: synchronous anchor nodes
In TDOA, the anchor nodes are synchronized so that they broadcast signals with respect to a common clock. Consider a pair of anchor nodes (i, j) transmitting signals. Their interarrival time at the passive self-localizing receiver node 0 is then where δ ≥ 0 is a (possible) transmission delay at node i, see Figure 2b. The unknown location is

TDST: asynchronous anchor nodes
In TDST, the anchor nodes operate asynchronously so that the transmitted signals are only coordinated through a sequence of observable signal events. Consider a pair of transceiving nodes (i, j), such that node j transmits only after receiving the signal from i and a given delay. The interarrival time at the passive node is then where δ > 0 is the transmission delay at node j, see Figure 2c. As above, we consider However, we can readily accommodate unknown locations of auxiliary transmitting nodes in θ * as demonstrated in the experimental section.

Measurements in LOS and NLOS
We let s ⊆ {1, 2, . . . , N } denote a set of transmitting nodes whose signals are observed by the self-localizing node. For a given s, the ideal interarrival times, from either (1), (2) or (3), can be arranged into a vector expressed as where × 1 vector of distances between all nodes, M(s) is a selection matrix of integers and δ is a known delay parameter. When s contains at least three anchor nodes that are not coaligned, θ * is uniquely determined from the set of interarrival times, see [14]. When s contains at least three anchor nodes that are not coaligned, θ * is uniquely determined from the set of interarrival times, see [6]. When s contains at least three anchor nodes that are not coaligned, θ * can be uniquely determined from the set of interarrival times, see [20, sec. 3.3] and [3,4].
In ideal LOS conditions, a set is drawn as s ∼ p LOS (s), where the distribution is possibly degenerate. For each observed set s, the self-localizing node obtains noisy measurements z of the interarrival times µ(s, θ * ). That is, where the mean of p LOS (z|s) equals µ(s, θ * ) in (4).
In practical wireless environments, however, the pair (s, z) is not always observed in ideal conditions, but rather drawn from where is an unknown fraction of corrupted data and p NLOS (s, z) is an unknown corrupting distribution, that generates outlier noise and biases such that its mean may differ from µ(s, θ * ) [10,21]. The problem under consideration is to estimate θ * given data {(s 1 , z 1 ), . . . , (s n , z n )} drawn i.i.d. from (6). We merely assume that the quality of the data can be specified by an upper bound ≥ [9, ch. 1].

Method
Assuming a set of anchor locations and transmission sequences that yield identifiable locations, we have that where the expectation is with respect to the unknown distribution p LOS (s, z). Note that the right-hand side of (7) assumes that z has finite second-order moments. The standard estimator of θ * is the nonlinear least-squares method, where (6). Under standard regularity conditions θ is consistent when = 0, and corresponds to the maximum likelihood estimate assuming white Gaussian noise [11]. For > 0, however, θ is not robust to corrupted samples that arise in NLOS conditions, as described by (6).

Robust localization
As an alternative to (7), consider the following fitting criterion where, in lieu of p LOS (s, z) in (7), we use the distribution with probability weights π = [π 1 , . . . , π n ] ∈ Π, where Π is the probability simplex. This distribution has an entropy It follows that the standard method (8) corresponds to minimizing (9) using the empirical distribution p(s, z; n −1 1), which attains a maximum entropy of ln n.
Given the bound ≥ , however, we expect at least (1 − )n uncorrupted samples and that the support of p(s, z; π) should cover them. In this case, the maximum entropy would equal ln[(1 − )n] and the search over the unknown support turns (9) into the following joint optimization problem Intuitively, the above minimization problem estimates θ and simultaneously assigns weights π i to each point such that the overall weighted squared-error loss is minimized under the entropy constraint. Smaller weights are assigned to datapoints which are corrupted due to NLOS and larger weights are assigned to uncorrupted points. In this way, (12) enables robust localization of the auxiliary and receiver nodes without distributional assumptions. In the next subsection, we give a blockwise algorithm for solving (12).

Blockwise minimization algorithm
For a fixed θ, define and, similarly, for a fixed π, define θ( π) = arg min Together (13) and (14) form a blockwise coordinate descent algorithm which we summarize in Algorithm 1, that is guaranteed to converge to a critical point of (12) under fairly general conditions [5]. The weighted nonlinear least-squares problem in (14) can be solved using standard search methods (e.g. gradient-based or Newton search), while (13) requires solving a convex problem, whose solution can be obtained using a barrier method that is more efficient than general purpose numerical packages such as cvx [?].

Experimental results
In this section, we illustrate the wide applicability of the proposed robust localization method using synthetic TOA, TDOA and TDST data. The performance is evaluated using the localization error where x is the node location of interest.
We observe n = 100 measurements from (6). The LOS distribution is where adjacent timing measurement errors have correlation structure given by Q. The uniform distribution U(s) draws s from a set S. The corrupting NLOS distribution is where we consider an exponential distribution with measurement bias µ NLOS . We set σ LOS = 3 ns and µ NLOS = 75 ns. Unless otherwise specified, the unknown corruption fraction is set to = 15%.
Remark: The code for the experiments is available at github.com/Muhammad-Osama/RobustLocalization.

TOA: two-way ranging anchor nodes
Consider a wireless network consisting of N = 8 nodes as shown in Figure 1a. Since the TOA measurements are uncorrelated, Q = I. The unknown location of interest is The set S consists of two sequences s 0 = {6, 5, 7, 8} and s 1 = {4, 3, 2, 1} where the node numbers are given in Figure 1a. The sequences have been selected so that the anchor nodes are not coaligned in either s 0 or s 1 .
In the case of TOA, the measurement model (1) admits an (overparameterized) linear form which is ideal for classical methods in robust statistics, such as the Huber method [21]. The Huber method is thus tailored for the task of TOA in which it provides a useful benchmark. We compare it to the standard nonlinear least-squares method (8) and the proposed method (12).  Figure 3b shows the RMSE for the different methods, where we see that the robust method is insensitive to , with a graceful rise in RMSE when exceeds . In sum, the proposed method outperforms the standard nonlinear least-squares method and is close to the benchmark provided by the TOA-tailored Huber method.
The results are corroborated also in Figure 4 which shows RMSE as a function of x 0 . It can be seen that the robust method is more sensitive than the benchmark only near the edges of the vertical boundaries, where the resolution of time-differences decreases.

TDOA: synchronous anchor nodes
We consider again the network in Figure 1a, but now the self-localizing node is a passive receiver. Since TDOA measurements are correlated, we set Q with 1s along the diagonal and 1/3 along the off-diagonals. The unknown location of interest is θ * = {x 0 } = [5,5] .
Note that the measurement model (2) is nonlinear in θ and does not admit a linear re-parametrization. Thus the Huber method is not readily applicable for TDOA as it requires tuning an alternative numerical search techniques. We therefore compare only the standard nonlinear least-squares method (8) and the proposed method (12). Figure 5a shows CDFs of the localization error ∆( x) estimated using 100 Monte Carlo simulations, setting = 20% in Algorithm 1. The performance characteristics are similar to those in Figure 3a, but the absolute error levels are not directly comparable due to different measurement setups. The sensitivity to the unknown corruption fraction is also shown in Fig. 5b when RMSE is plotted against for both methods. We use a very conservative upper bound = 50% in Algorithm 1. The proposed method is consistently robust and insensitive to corrupted data in contrast to the standard method for which the errors rise drastically with . Figure 6 shows that the robust method yields substantial error reduction across space. Finally, we illustrate the ability of the proposed method to effectively isolate corrupted NLOS samples. Since the data is generated synthetically we can classify each sample from p NLOS (z|s) as C(z) = 1 for 'corrupted' or C(z) = 0 for 'normal'. The method solves (12) and learns the probability weights π. If a weight is below a certain threshold, we may classify the corresponding sample z as an outlier, i.e., C(z) = 1. We set the weight threshold to 10 −5 and show the resulting probability of correct detection Pr{ C(z) = 1|C(z) = 1} as well as the probability of false alarm Pr{ C(z) = 1|C(z) = 0} in Figure 7. We use = 50% and vary the unknown fraction , using 50 Monte Carlo runs for each value of . It can be seen that the proposed method can effectively isolate NLOS samples with a low false-alarm rate.
Similar to TDOA, the Huber method is not readily applicable to TDST. We therefore compare only the standard nonlinear least-squares method (8) and the proposed method (12). Figure 8a shows the CDFs of the localization error ∆( x). The characteristics are similar to those in Figure 3a. The sensitivity to the unknown corruption fraction is shown in Fig. 8b where RMSE is plotted versus for a very conservative upper bound = 50% in Algorithm 1. It can be seen that the proposed method also robustifies self-localization in TDST.
Finally, we consider a more challenging wireless network configuration, where one anchor node is replaced by an auxiliary node (N a = 1) at an unknown location as shown in Figure 9a. The goal is to passively localize auxiliary nodes using asynchronous anchor nodes [3,4,20] in adverse NLOS conditions. The locations of interest are θ * = {x 0 , x 1 } = [5,5] , [−10, 10] .