UFIR Filtering for GPS-Based Tracking over WSNs with Delayed and Missing Data

. In smart cities, vehicles tracking is organized to increase safety by localizing cars using the Global Positioning System (GPS). The GPS-based system provides accurate tracking but is also required to be reliable and robust. As a main estimator, we propose using the unbiased finite impulse response (UFIR) filter, which meets these needs as being more robust than the Kalman filter (KF). The UFIR filter is developed for vehicle tracking in discrete-time state-space over wireless sensor networks (WSNs) with time-stamped data discretely delayed on 𝑘 -step-lags and missing data. The state-space model is represented in a way such that the UFIR filter, KF, and 𝐻 ∞ filter can be used universally. Applications are given for measurement data, which are cooperatively transferred from a vehicle to a central station through several nodes with 𝑘 -step-lags. Better tracking performance of the UFIR filter is shown experimentally.


Introduction
Accurate target tracking is one of the key problems in urban areas [1], which especially arises in smart cities design [2].If a target is equipped with the Global Positioning System (GPS) tracker, then measurement data can be transferred to a central station through one or several nodes of a wireless sensor network (WSN) [3].The problem which arises here is associated with information latency and missing data [4] due to the following main causes: high maneuverability of the target [1], failures in measurements [5], network congestion [3], non-line-of-sight (NLOS) problems [6,7], and accidental loss of some collected data [8].Furthermore, latency naturally occurs due to the limited bandwidth, finite propagation time [9], complexity of very large-scale integration and microelectromechanical systems [10], and time required to complete operations such as signal conditioning and storage [11].In networks, communication delays go along with data loss called dropout or intermittence [12,13].Also the delay between the measurement and its availability to the filter causes the problem of out-of-sequence measurement [6,14].
Two basic models have been created for delayed data.The delays are assumed to be known when sensors are able to detect the delays or data are time-stamping [15,16].In many other applications [17,18], the delays are considered to be random.The problem becomes more complex in uncertain systems [19].The best estimate is obtained here by combining delayed and nondelayed data with different probabilities.
The Kalman and  ∞ state estimators are most widely used to deal with latency and associated issues [20].The linear Kalman filter (KF) is optimal when it matches the system perfectly, noise is white Gaussian and uncorrelated, and the noise statistics are known along with the initial values.When such conditions are not obeyed, the KF may demonstrate poor performance [13,21].The robust  ∞ filter bounds the mean square error (MSE) for admissible parameter perturbations and delays [19,22], which allows for minimizing errors with less information required than for the noise statistics [20,23].
Another way to achieve better robustness is to process most recent finite data [24] using finite impulse response (FIR) filters [25].Such filters have been developed during decades by many authors in signal processing [26][27][28][29][30][31][32][33] and control [34][35][36].However, only a few authors have proposed FIR solutions for models with delays [37][38][39][40].Let us notice that the available iterative unbiased FIR (UFIR) algorithm [28,[41][42][43] is most robust among other FIR solutions owing to an ability to ignore the noise statistics and initial values.This filter is bounded-input bounded-output (BIBO) stable and blind on given horizons of  points, but is still not developed for observations with delayed and missing data.
In this paper, we develop the UFIR filter for GPSbased vehicle tracking over WSNs with time-stamped data discretely delayed and missing data.The rest of the paper is organized as follows.In Section 2, we consider the model and formulate the problem.In Section 3, we develop the UFIR filter for observations with delayed and missing data.Section 4 discusses the estimation errors.Section 5 gives an experimental example of applications to GPS-based tracking and concluding remarks are drawn in Section 6.

Tracking Model and Problem Formulation
A typical scenario of GPS-based vehicle tracking in WSNs is sketched in Figure 1.The vehicle current coordinates are measured by the GPS tracker.The time-stamped data are transferred to a central station (CS) via one or several nodes of the WSN.Because each node may discretely delay timestamped data at least on one-step, the vehicle location is observed in CS with a time varying   -step-lag depending on the vehicle location and interaction with the WSN.

Tracking Model.
For   ⩾ 0, the vehicle dynamics and its observation can be represented in discrete-time state-space as where  is the discrete-time index,   ∈ R  is the vehicle state vector,   ∈   is the observation vector,  ∈ R × is not singular, and  ∈ R × .All data are time-stamped, so that   is known at each .Regardless of the delay, the initial state  −1 is supposed to be known.The uncorrelated noise vectors,   ∈ R  and V  ∈ R  , are white Gaussian with known covariances,  = {     } and  = {V  V   }, and the property {  V   } = 0 for all  and .The UFIR filter can be applied if to transform model ( 1)-(2) to have no latency.That can be done if to represent  −  using (1) via   as and then substitute (3) into (2) and arrive at where and the covariance   = {V  V   } of V  is given by In compact matrix forms, ( 6) and ( 7) can be represented as where and Here,  n = 0 and    , = 0 when   = 0 and   > .
Any standard estimation technique can now be applied to models (1) and (4).However, the KF and  ∞ filter were most developed for data with latency.Therefore, below we will introduce in brief only these filters and then compare them to the UFIR filter based on examples of tracking.We will use the following measures: x ≜ x| is the estimate of   over data taken from past up to and including time index , x−  is the prior estimate,   = {(  − x )(  − x )  } is the error covariance matrix, and  −  = {(  − x−  )(  − x−  )  } is the prior error covariance matrix.

Kalman Filter.
For our purposes, we will exploit an alternative form of the KF algorithm given in [44].This algorithm starts with the prior error covariance matrix and then recursively updates the following values: where   is given by ( 5) and   by ( 7) for any   ⩾ 0.
Journal of Electrical and Computer Engineering 3 The  ∞ filter has been derived in [44] in the form (13) of the KF using the game theory.For   ⩾ 0, the  ∞ filtering algorithm becomes where the user-given symmetric positive definite matrices  0 , Q, and Ř have different meanings than in the KF and P −  can be computed via  0 using (12) with  = Q.To keep (14) positive definite, the positive definite matrix   ∈  × is subject to If equal weights are required for all errors, matrix   must be set identity,   = .The performance criterion for this filter is   < 1/  , in which a scalar   must be small enough in order for the filter to be efficient.It then follows that the tuning factor   is not allowed to be negative, even though its negative values may reduce errors when the weighting matrices are not maximized.For Gaussian noise, zero   transforms the  ∞ filter to the KF.For any other noise, small   > 0 may result in better robustness.
The problem now formulates as follows.Given ( 1) and ( 4) with time varying   ⩾ 0 and missing data, we would like to develop the UFIR filter and find its fast iterative form for GPS-based tracking of a moving vehicle as shown in Figure 1.We also wish to know how the UFIR filter, KF, and  ∞ filter measure to each other in applications to tracking.

UFIR Filter for Tracking with Delayed and Missing Data
To develop the UFIR filter for   > 0, we extend models ( 1) and ( 4) on a horizon [, ] of  points, from  =  −  + 1 to  that referring to [43] yields with the following extended vectors and matrices: where  represents a set of {  ,  +1 , . . .,   }.

3.1.
Batch UFIR Filter Form.The UFIR filtering estimate x of the vehicle state   can be obtained at  in the batch form as [27,34] x = H ,  , , where H , is the UFIR filter gain and  , is a vector of real data (22), if to satisfy the unbiasedness condition in which {} means averaging of  and   can be represented with the last row vector in (19) as where  ()  is the th row vector in (25) given by By combining ( 29)- (32) and following [27], one arrives at the unbiasedness constraint in which Now multiplying the both sides of ( 33) with (C  , C , ) −1 C  , C , yields the UFIR filter gain and the batch UFIR filtering estimate becomes where the generalized noise power gain (GNPG) [43] is The batch form (36a) may not suite real-time tracking and we go on with its fast iterative algorithm.

3.2.
Iterative UFIR Filter Form.Provided ( 1) and ( 4), the standard iterative UFIR filtering algorithm [25] can be applied straightforwardly, if to substitute matrix  with  given by (5).The UFIR filtering estimate (36a) can then be computed iteratively using recursions beginning with  =  +  and ending when  = .The initial values for (38) and ( 39) are obtained at  =  +  − 1 in the batch forms as where  , is a vector (22) of real data.When some data are lost, inaccurate, or unavailable,   can be predicted as   =  x−1 , in which case first data on the horizon [0,  − 1] must be available.It is known that the linear UFIR filter is BIBO stable and not prone to divergence.However, latency in information delivery may require an ability to predict lost values that inevitably cause extra tracking errors, which we will consider next.

Tracking Errors Caused by Latency, 𝑘 𝑛 ⩾ 0
Error produced by the UFIR tracker can be defined as   =   − x , where   is specified by (31).Provided   , the error covariance matrix   = {     } can also be represented in two forms.
In the batch form, matrix   appears if to substitute   with (31) and employ x = H ,  , with  , given by (20).That yields where   = diag [  ⋅ ⋅ ⋅ ] and   = diag [  ⋅ ⋅ ⋅ ] are square matrices with  nonzero diagonal elements.It can be shown that the deterministic case of   = 0 and   = 0 makes   = 0 and the UFIR tracker has thus the deadbeat property.
4.1.Iterative Computation of   .Matrix (42) can also be computed iteratively, if to substitute   with (1) and x with (39).Provided the averaging, the recursion for (42) can be found as where  −  is given by ( 12),  ranges as in (38) and (39), and   is taken when  = .Recursion (43) suggests that the tracking error grows with   , because the sum containing  grows with   .However, the same cannot be said about , which does not accumulate the effect of   .
With no latency, the sum in (43) becomes identically zero and one arrives at the error covariance   of the standard UFIR filter [43], which also holds for the KF, if to substitute     with the Kalman gain   .Note that the minimization of tracking errors will require an optimal number  opt − 1 of iterations for the UFIR filter.At the test stage, the optimal horizon  opt can be found for the known ground truth   by minimizing the MSE via the trace of P  [30] depicted as tr P  .Because the ground truth is unavailable in real tracking,  opt can be estimated via the measurement residual as shown in [30].

GPS-Based Tracking of a Moving Vehicle
We will now consider the case shown in Figure 1 when the GPS tracker measures the vehicle coordinates of location at each time index  and transfer time-stamped data cooperatively to a CS of a WSN via one or several nodes.We admit that each node may introduce latency and data will thus arrive at the CS with known delay on   > 0 points.At different time instances, a vehicle may interact with a different number of the nodes that will make the   -step-lag time varying.We will base our investigations on data obtained in the Cook county of Illinois and available for free use from the University of Illinois at Chicago.To simplify the problem, in this paper we will consider the case of a constant latency,  =   .
Concerned with the tracking errors and not with the actual vehicle location, we will conventionally place the start point at zero coordinates as shown in Figure 2.

State-Space Model.
To investigate the trade-off between the estimators, we will suppose that a vehicle is represented with two states in each directions and assign the state vector,

𝐾 = 4, as
, where  1 =   ,  2 = ẋ  ,  3 =   , and  4 = ẏ  .Accordingly, the system matrix attains the form of where the sampling time is  = 1s for the considered database.The GPS tracker provides measurements of the vehicle coordinates,  and .Therefore, the measurement matrix is Provided  and , matrix C , required by the UFIR filter to compute the initial values (40) and (41) for  =  + 3 becomes The only tuning factor  opt = 5 required by the UFIR filter was found for  = 0 by minimizing the derivative of the trace of the mean square value of the residual   − x  (), as shown in [30].Because  opt depends on , we will apply  opt = 5 in the worst case for the UFIR filter.
Having no information about the process noise, we observe similar trajectories and estimate the average vehicle speed by about 10 m/s or 36 km/hour.Next, accepting the speed standard deviation of about 20%, we set  2 = 2 m/s to the second state along each of the coordinates, ignore the unknown noise in the first state,  1 = 0, and describe matrix  as The GPS standard positioning service provides navigation with an error of less than 15 meters with the probability of 95% in the 2-sigma sense.Referring to this value, we assign the standard deviation of the measurement noise in each direction as  V = 15/4 = 3.75 m and obtain Because the above provided matrices  and  are overestimated, we set Q =  and Ř =  for  ∞ .It is expected that the tuning factor  will improve the performance of the  ∞ filter by minimizing the MSE for the maximized errors.However, the ground truth is not available in tracking.Therefore, we will find  for the measured trajectory and consider it as the best case for  ∞ , which is unfeasible.

Effect of Latency on the Estimation Accuracy.
We start with learning the effect of  on the estimation accuracy, which is illustrated in Figure 3 with the root MSEs (RMSEs) in the north direction (-RMSE).The KF is self-tuned to .Therefore, we consider its RMSE as a benchmark.A special feature of the KF is that the RMSE grows with  nonlinearly and faster than in the UFIR and  ∞ filters.
The UFIR filter produces a bit more errors than in the KF with small  and lesser with larger .A special feature is that the UFIR estimate is of low sensitivity to , in which optimal value  opt = 5 holds for 0 ⩽  ⩽ 4, increases to  opt = 6 for 5 ⩽  ⩽ 6, and reaches  opt = 7 when 7 ⩽  ⩽ 10.Of practical importance is that setting  optimally for each  does not improve the performance essentially against the worst case when  opt = 5 is set for all .
The  ∞ filter outperforms both the UFIR filter and KF, provided that  is set properly for each lag .However, this filter is highly sensitive to , in which optimal value  opt ranges from 1.8 × 10 −2 for  = 0 to 1.066 × 10 −5 for  = 10 in a nonlinear way.Unlike in the UFIR filter, a constant  is unacceptable for all .An example is given in Figure 3, where  opt = 7.1 × 10 −4 found for  = 2 is applied in a wide range of .As can be seen, it is only when  = 2 that the  ∞ filter improves the KF performance.For  < 2, there is no improvement and, when  > 2, the  ∞ filter rapidly diverges.

Tracking over Data
Delayed on  = 3.We now suppose that data are transferred from a vehicle to a CS with  = 3 and investigate tracking errors in the north and east directions.Filters will be tuned as follows: UFIR in the worst case of  opt = 5 being valid for  = 0; KF as near optimal; and  ∞ being in the best (unfeasible) case of known ground truth.(ii) Filters temporarily lose an ability of tracking and go along the delayed data when a vehicle quickly changes the direction, as on 340 ⩽  ⩽ 343 in Figure 4(c).

Tracking in the North
(iii) Responding to fast maneuvers, all filters produce dynamic errors such that the UFIR filter comes up with larger excursions but shorter transients, KF with shorter excursions but longer transients, and  ∞ filter demonstrates inbetween properties; see on 80 ⩽  ⩽ 100 in Figure 4(b) and 344 ⩽  ⩽ 360 in Figure 4(c).
Because errors are unacceptably large in all filters when they temporarily lose an ability of tracking, a key question arises of how fast each of them returns back to the normal mode.In this regard the UFIR filter looks better with its shortest transient.

Tracking in the East Direction.
Tracking in the east direction (Figure 5) does not reveal any essential features.The filters still track well the trajectory when a vehicle travels with a near constant velocity as, for example, in a span of 120 ⩽  ⩽ 160 in Figure 5(b).Here, all filters also temporarily lose an ability of tracking and go along the delayed data when the trajectory quickly changes as, for example, in a span of 560 ⩽  ⩽ 570 in Figure 5(c).Finally, the UFIR filter still demonstrates larger excursions but shorter transients, KF shorter excursions but longer transients, and  ∞ filter is inbetween as, for example, in

Tracking with Temporary Lost Data.
We finally admit that some data points can be lost during the transmission and remove 5 data points at  = (615 ⋅ ⋅ ⋅ 620) s and 10 at  = (665 ⋅ ⋅ ⋅ 675) s as shown in Figure 6 for  = 3.To predict lost data, we augment each algorithm with the prediction block as mentioned below (41) and run the filters.As can be seen, the filters act consistently with, however, some specifics.The estimates do not get away essentially from each other and the actual trajectory.However, when a vehicle maneuvers during the prediction, all filters diverge and return back to the actual trajectory with similar transients as in Figures 4 and 5.The latter again speaks in favor of the UFIR filter, which has shorter transients.

Conclusions
The UFIR filter developed in this paper for GPS-based vehicle tracking over WSNs with time-stamped discretely delayed and missing data has demonstrated better performance than the KF and  ∞ filter.The main benefits of using the UFIR filter are that it (1) does not require any information about noise and initial conditions, (2) becomes blind on given horizons, and (3) has shorter transients.The latter can be considered as an important practical advantage in all situations when the trajectory changes rapidly and estimators temporarily loses an ability of tracking.Applications to GPS-based vehicle tracking with known discretely delayed and missed data have proved a better performance of the UFIR filter.

Figure 1 :
Figure 1: Transferring the time-stamped vehicle coordinates measured by a GPS tracker to a central station (CS) via several nodes of a WSN.Latency with a   -step-lag is caused by delays in the node.

Figure 2 :
Figure 2: GPS-based vehicle trajectory measured in the north () and east () coordinates, both in km, with the start point at {0, 0}.Measurements are provided each second at 858 data points.

Figure 3 :
Figure 3: Effect of the -step-lag on the -RMSE of the UFIR filter, KF, and  ∞ filter with different tunings.

Figure 4 (
Figure 4(c).Several observations can be made from Figure 4: (i) All filters track well the trajectory when a vehicle travels with a near constant velocity in one direction, as on 140 ⩽  ⩽ 180 in Figure 4(b).

Figure 6 :
Figure6: Vehicle tracking with temporary missing data in the east direction  by the UFIR filter, KF, and  ∞ filter, all augmented with the prediction option.Tuned improperly, the  ∞ filter becomes unstable and prone to divergence, as in Figure3.