Weighting Strategy for MDS Tracking in LoSConditions

Davide Macagnano and Giuseppe Thadeu Freitas de AbreuCentre for Wireless Communications - University of Oulu

P.O.Box 4500, 90014-Finland[macagnan,giuseppe]@ee.oulu.fi

Abstract— In this paper we propose a low-complexity non-parametric pre-filtering/weighting technique for Time of Arrival(TOA)-based target tracking systems. The output of a wavelettransform computed on each one of the anchor-to-target links isused to estimate the noise and the dynamic for the correspondinglink. Subsequently this information is used to weight and adap-tively decide on the reliability of pre-filtered data. To take full ad-vantage of the pre-filtering scheme proposed, this is used togetherwith the optimization algorithm (SMACOF), a low-complexitysolution to the multidimensional scaling (MDS) problem. Sinceboth the pre-filtering scheme and SMACOF don’t rely on anya-priori knowledge, their combination results in a completelynon parametric approach to the target tracking problem. Theaforementioned technique is investigated and compared against aBayesian state variable formulation of the problem implementedthrough an Extended Kalman Filter (EKF). The MDS-basedtracking algorithm with the wavelet pre-filtering scheme hereproposed, while remaining completely non-parametric is shownto overcome the EKF algorithm under different target dynamicsand noise conditions.

I. INTRODUCTION

Sensor mobility is a topic of growing interest in wirelesssensor networks (WSNs). Specific difficulties vary with the ap-plication, but some of the major challenges for target trackingscenarios are the requirement for low-complexity algorithmsand the ability to track a potentially large number of sensorswith different dynamic simultaneously.

As previously proposed [1], a possible solution to theaforementioned problem is represented by a data analysistechniques that goes under the name of MultidimensionalScaling (MDS). Two approaches to solve the metric MDSproblem are the Classical and the Least-Squares (LS) one. Theformer, while being a completely algebraic solution, is basedon the eigen-decomposition of a kernel (positive semidefinitetransformation) of a dissimilarity matrix. In the localizationcontext this dissimilarity matrix is equivalent to the EuclideanDistance Matrix (EDM) D containing the pairwise distancesamongst all the objects in the scenario [2]. The latter approachinstead, involves the optimization of a stress (generally least-squares) function [3]. In some of the multi-source localiza-tion applications, metric MDS can be a suitable centralizeddistance-based localization technique.

It was shown that for star-like target-to-anchors topologies,classic MDS technique can be modified into a tracking algo-rithm by performing the eigen-decomposition in its solutioniteratively, such that the eigen-pairs corresponding to the

observation at time (t) can be computed as update of theeigen-pair at time the (t− 1) [1] [4]. Being such a techniquenon-parametric, when compared against standard approaches,e.g. Kalman Filter (KF), it has the advantage of having itsperformance independent on the target dynamics.

However, Classic MDS-based tracking algorithm performspoorly if the primary source of difficulty is neither targetdynamics (slow targets) nor their numbers (few targets), butrather the noise affecting the distance estimates [1].

The objective is to design an algorithm capable to weight thesingle anchor-to-target ranging information received at eachtime accordingly to their reliability. Unfortunately, the factthat in the classic MDS the single entries of the EDM cannot be weighted, makes this approach inadequate to cope withthe aforementioned challenge. Consequently a least-squares(LS) formulation of the metric MDS that goes under the nameof SMACOF is here considered. In the LS-MDS formulationweights can determine the relative contribution of the rangingto the embedding error, allowing for a soft exclusion ofthe links affected by larger errors. To remove part of noiseaffecting the measured anchor-to-target time series a wavelet-based pre-filter is here used [5].

Scope of this paper is to show how the aforementioned pre-filtering technique already provides a suitable solution to theaforementioned weighting problem. The weighted SMACOFalgorithm is here compared against a Bayesian formulation ofthe problem using an extended Kalman Filter (EKF).

The reminder of this article is as follow. In section II theformulation of the SMACOF technique is briefly reviewed.

In section III-A the wavelet-based scheme used to smoothand adaptively decides on the reliability of the filtered ob-servations is described. In the same section it is shown howthe same algorithm, while being completely non-parametric,provides suitable weights for the SMACOF algorithm at theonly expense of a minor lag in the computation. Concludingsection IV compares the performance of the SMACOF andEKF algorithm together with the wavelet-based pre-filteringscheme proposed. It is shown that, differently from typicaltarget tracking systems, where prior knowledge on the tar-get dynamic and the statistical characterization of the noiseaffecting the ranging [6] are used, the SMACOF algorithmtogether with the pre-filter block here described achievesbetter performance while still remaining a completely non-parametric solution to the problem.

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II. LS-MDS BASED ALGORITHM

MDS is a multivariate method that provides a spatial repre-sentation of the data under investigation [7]. Given the matrixcontaining all the similarities-dissimilarities δi,j between Npoints, MDS maps their configuration in an embedding spacesuch that the difference between δi,j and relative distances di,jis minimal. One metric often used to describe this error is thefollowing STRESS function

S (X) =N∑j=1

N∑i=1

wij (δi,j − di,j (X))2, (1)

with X as the [N × µ] matrix containing the set of coordinatesfor the N objects in the µ dimensional embedding space anddi,j = [D̃]i,j .

In [8] it was assumed that not all the anchors-to-targetsdistances are measured at each time, making only a subsetof D̃AM(t) available and giving rise to a distance completionproblem. This difficulty was solved minimizing (1) througha weighted Least Square algorithms, rather than its algebraicsolution [2].

Here we use the same algorithm to benefit of the weightsprovided by the pre-filtering algorithm. Equation (1) is provedto be non-convex in X, therefore a reliable starting point isrequired to make the solution robust to local minima [9]. In thetracking context this starting point solution at (t) is providedby the solution computed at time (t− 1).

The minimization technique here considered is based onthe majorization algorithm initially proposed by Jan de Leeuw[3], subsequently refined by Groenen [10] and known underthe name of ”Scaling by Majorizing a Complicated Function”(SMACOF). The idea is to minimize rather than a complicatedfunction f(x), here represented by (1), an auxiliary functiong(x, y), such that for all x in the domain of f , g is themajorizing function of f , and for a particular y in the domainof g, called support point, the two functions coincide, namely{

f(x) ≤ g(x, y)f(y) = g(y, y) . (2)

If (2) holds, than the minimization problem expressed by (1)can be solved through the majorizing function g. As shownin [7], in the case of metric MDS problems, g can be easilyfound from (1). Indeed S (X) can be rewritten as

S (X) = η2δ + η2 (X)− 2ρ (X) , (3)

with η2δ =

∑i≤j wijδ

2i,j , η

2 (X) =∑i≤j wijd

2i,j (X) and

ρ (X) =∑i≤j wijδ

2i,jdi,j (X).

In matrix form the equation above can be rewritten as [10]

S (X) = η2δ + tr

(XTVX

)− 2tr

(XTB (X) X

), (4)

with V constructed as:

vi,j ={−wij i 6= j∑k 6=i wik i = j

, (5)

and the matrix B as:

bi,j =

−wijδi,jd−1

i,j (X) i 6= j di,j (X) 6= 00 i 6= j di,j (X) 6= 0−∑k 6=i bi,k i = j

. (6)

As shown in [7], equation (4) is majorized by

Γ (X) = η2δ + tr

(XTVX

)− 2tr

(XTB (Y) Y

). (7)

This function can be minimized by imposing ∇Γ (Y) =2VX − 2B (Y) Y = 0, from where it derives the followingupdating equation

Xk+1 = V†B(Yk)Yk, (8)

also known as the Guttman transform (with † indicating theMoore-Penrose pseudo-inverse).

The majorized function (7) could be also solved throughdifferent, more complex kind of optimization, e.g. Gradient,Newton or Quasi-Newton type solutions [9]. However equation(8) has the advantage of giving rise to a non-decreasingsequence of the stress value, that provides higher robustnessversus local minima. Concluding, the last step necessary torecover the position of the topology, is to reorient it ac-cordingly to the reference system used, done by procrustestransformation [11].

III. WAVELET-BASED PRE-FILTERING

A. Wavelet Transform

The Wavelet Transform (WT) makes use of a unique dilatedwindow (the wavelet function) to analyze signals. Doing so itallows arbitrarily good time resolution (short window) at highfrequency and good frequency resolution at low frequency(long-window).

This decomposition is done using the family of functions√sψ(s(x − u))(s,u)∈R2 , corresponding to the translated and

dilated version of a function ψ(x), called wavelet, with s andu identifying the scaling and translation factors. Necessarycondition for ψs(x) to define a wavelet, is that its Fouriertransform, ψ̂s(ω), while ψ̂s(ω) being small enough in theneighborhood of ω = 0, must satisfy ψ̂s(0) = 0, allowingto interpret ψ̂s(ω) as a band pass filter [12].

The continuous wavelet transform of f(x), here denoted asWf(s, u), corresponds to the inner product 〈f(x), ψs(x− u)〉,meaning the cross-correlation between the original functionand the scaled wavelet shifted at u. As shown in [13], choosingψ(x) = dφ(x)/dx, with φ(x) as smoothing function, then it ispossible to characterize the shape of irregular functions f(x)through Wf(s, u). Using the properties of the convolutionoperator it follows that

Wfs(x) = f ∗(sdφsdx

)(x) = s

d

dx(f ∗ φs) (x). (9)

From above it is evident that Wf(s, u) can be interpretedas the derivative of a local average of f(x), with smoothingdegree depending on the scale factor s. Choosing the scalefactor as (αj)j∈Z , with α as the elementary dilation step, and

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a discrete dilation factor αj/β, then it is possible to computethe WT on a discrete grid on the time-scale plane.

For α = 2 the WT has scaling and translation coefficientsare sampled over the dyadic grid in time-scale plane [12],giving origin to the discrete wavelet transform (DWT) [14].To compute the DWT of the discrete data we use the socalled Algorithme a Tours [12]. As briefly shown here after,this algorithm is similar to a biorthogonal wavelet transformwithout subsampling, and can that be computed with a fastfilter bank algorithm, for details refer to [12].

Defining as a0 [n] the discrete signal to be analyzed, andassuming its value in n equivalent to the local averagebetween the original continuos function f (x) and a kernelfunction φ (x− n) (namely 〈f(x), φ(x− n)〉), then at anyscale j > 0, a smoothed version of a0 [n] is computed as〈f(x), φ2j (x− n)〉, with

φ2j (t) =1√2jφ( x

2j). (10)

The function φ (x) is called scaling function and corre-sponds to a low-pass filter. The coefficient for the dyadicWT are obtains as z2j [n] = Wf (s, n) = 〈f(x), ψ2j (x− n)〉,with ψ2j (x − n) defined similarly to equation (10). Clearly,accordingly to what discussed above and as shown by equation(9), Wf

(2j , n

)represent the high frequency component of

a0 [n] at the scale j. The full dyadic wavelet representation ofa0 [n] is given by all the wavelet coefficients up to the scale 2J

(J = log2 (N)) together with the low frequency componentaJ [n].

Once the low-pass filter h [n] and high-pass filter g [n] aredesigned accordingly to φ (x) and ψ (x) respectively [12],then a0 [n] is decomposed by cascading, for 0 < j < J , theconvolutions

aj+1 [n] = aj ∗ h̃j [n] , (11)zj+1 [n] = aj ∗ g̃j [n] ,

with hj [n] obtained from h [n] inserting 2j−1 zeros betweeneach sample of the filter (similarly for gj [n]).

From equation (11) it is evident how the low-pass filteringof the signal at the resolution j is closely related to the low-pass version of the signal at the resolution j−1. It is importantto outline the simplicity to computed the transform as well asits multi-scale nature.

Since interested in a wavelet ψ (x) with small support andsatisfying equation (9) the impulse responses h [n] and g [n]are chosen as the one defined in [13], where g [n] is defined asantysimmetrical, as regular as possible and with small support.Although the Algorithme a Tours is highly redundant repre-sentation, in denoising and signal reconstruction application,it has been shown to be preferable compared to the classicaldecimated bi-orthogonal wavelet (OWT) [15].

B. Pre-filtering Strategy

The DWT is used to analyze each one of the anchor-to-target time series collecting the time of arrival (TOA)observations f [n], here equivalent to a0 [n] defined above.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

!

Std Estimate

! = 1

! = 0.1

TrueEstimeted

1 2 3 4 5 6 7 8 90

0.1

0.2

0.3

0.4

0.5

0.6

0.7

" [m/sample]

RMSE

Error

! = 1

! = 0.1

Fig. 1. Estimated standard deviation σ̂d, and RMSE for the estimate, as afunction of the target dynamic ν̄ under different values of σd.

Since the wavelet coefficients zj [n] computed at the differentscales j represent the high frequency components of f [n], itis obvious to try to characterize σd using these coefficients,similarly the output of the scaling function is used to infer onthe dynamics ν.

The changes in the target trajectory affect the shape off [n]. Depending on the velocity and occurrence of suchchanges, f [n] undergoes different transients whose shape canbe characterized by the DWT [12]. Assuming that most ofthe noise is captured by the coefficients z1 [n], to remove thenoise corrupting the measurements we can simply use equation(11) (no reconstruction) together with the selection strategydescribed hereafter.

In [16] a boolean multiresolution support for images wasproposed in order to discern the contribution of informationprovided by the WT coefficients.

This support was computed comparing all the coefficientsof the WT versus a significance level (threshold), function ofthe noise perturbing the image. Similarly, here we borrow theidea to construct a support to distinguish, amongst subsequentobservations of f [n], the one belonging to the same trend.To do so, we check whether the coefficient zj [n] at n, retainthe same polarity across the scales 2 < j < J . Whetherthe condition above is satisfied, then we assign to the nthvalue of our support ±1, accordingly to the polarity of z1 [n],differently we assign 0. In this way we are able to use themultiresolution representation provided by zj [n] (2 < j < J)to better characterize the trend of f [n] assigning 0 to thepoints corresponding to significant discontinuity on f [n] e.g.directional changes on the target trajectory.

Thus, computing such a support at each time n and foreach anchor-to-target link, it is possible to distinguish amongstsubsequent measurements the one belonging to the same setand consequently to estimate σ̂d as the standard deviationcomputed for those subsets in z1[n] characterized by the same

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polarity value in the support, which also correspond to thecoefficients in z1[n] mainly affected by the noise process.Since z1[n] is computed convolving f [n] with g1[n], thenthe estimated σ̂d needs to be compensated accordingly to thevalues of g[n]. As suggested in [16], this is done consideringσ̂ej = σ̂dσj , with σ̂ej as the value estimated directly from zj [n],and σj a reference value pre-computed and dependent on thewavelet used, namely the filter g [n].

Therefore σ̂d is computed from z1[n] from the pointsbelonging to the same subsets on the support as σ̂d = σ̂ej/σj ,this value is then used to weight the corresponding anchor-to-target link as wij = 1/σ̂d.

To distinguish the long term time process associated to thereal TOA measurements (low-pass filtered version of f [n]),the averaged version of a1[n] is used, meaning that at eachsampling time we compute the DWT on a window of size2J and centered at n, as already proposed in [5]. From thisaveraged a1[n] we estimate the perceived dynamic ν at thespecific anchor and for the considered target.

At this point we control whether f [n] can be decomposedin high frequency components of short duration and a lowfrequency part of relatively long duration, meaning that wetest whether if is feasible to approximate f [n] using its low-pass filtered version (a1[n]), using the the boolean operator Θdefined as {

δ > σd → Θ = 1,δ < σd → Θ = 0.

(12)

Using Θ, computed at each time n and for each anchor-to-target link, we decide whether the TOA observation is betterapproximated by the measured ranging (f [n]) or its low-passfiltered version (a1[n]). The only price paid using this waveletsmoothing based on Θ, is the introduction of a lag in thecomputation. However, as discussed later in section IV, thisis a minor delay since equal to 2J − 1. The estimate of thestandard deviation computed on the pre-filtered samples as afunction of the target dynamics ν is shown if figure 1 for twodifferent values of σd.

IV. PERFORMACE

In this section we compared the simulated performance ofthe MDS algorithm versus an EKF formulation of the trackingproblem. A scenario of 4 anchors placed at the corners of a2D square with edges 20[m] long is considered. Each anchorperforms Time of Arrival (TOA) measurements between itselfand the targets in the scenario, that moves in the space accord-ingly to trajectories generated using autoregressive processes(AR) of the first order.

The observations are subject to error distributed asN(0, σ2

). Figure 2 shows the performance for the two al-

gorithms together with the wavelet pre-filter and the adaptiveOnOff smoothing strategy as a function of the target dynamicν̄ and the noise σd. The figure shows how the SMACOF algo-rithm, together with wavelet pre-filtering scheme described insection III being a fully non-parametric approach overcomesthe EKF for almost all the range of dynamics and the different

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Target Dynamic: ! [m/sample]

RMSE [m]

Performance SMACOF versus EKF Algorithm

"d=1

"d=0.1

SMACOF + On−Off SmoothingEKF + On−Off Smoothing

(a) Performance for the SMACOF and EKF algorithms with pre-filteredobservations as a function of ν̄ under different values for σd.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

Noise: !d

RMSE [m]

Performance SMACOF versus EKF Algorithm

"=0

"=2SMACOF (Raw Data)EKF + On−Off SmoothingSMACOF + On−Off Smoothing

(b) Performance for the SMACOF and EKF algorithms with pre-filteredobservations as a function of σd under different dynamic ν̄.

Fig. 2. Performance for the SMACOF and EKF algorithms using the wavelet-based pre-filter together with the adaptive OnOff smoothing strategy obtainedapplying Θ to f [n]. Results as function of the target dynamic ν̄ and the noiseσd.

values of σd cosidered. To improve the EKF performance itwould be necessary to include different degrees of maneuverfor the targets, such as solutions based on multiple models[6], which would anyway affect the system complexity dra-matically.

Following we extend, only for the SMACOF algorithm, theapplication scenario to include the case of anchors subject todifferent levels of perturbation (σd).

Figure 3-4 show the performance for the algorithm with andwithout application of the adaptive OnOff smoothing strategyobtained by Θ on f [n]. In addition the algorithm fed with thesame pre-filtered observations weighted by wij is included. In

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0 0.5 1 1.5 2 2.5 3 3.5 40.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Noise: !d

RMSE [m]

Performance SMACOF Algorithm

"=2

"=0

SMACOFSMACOF + On−Off smoothingW. SMACOF + On−Off smoothing

Fig. 3. Performance of the SMACOF algorithm with pre-filtered/weightedobservations. Results for three anchors with fixed σd = 0.5 and one anchorsubject to σd variable between 0 and 4.

particular Figure 3 shows the case of 3 anchors with σd fixedto 0.5 and the 4th anchor subject to a variable perturbationbetween 0 and 4. The results are shown in case of staticas swell as dynamic ν̄ = 2 scenarios. From the figure it isevident that the application of the weighting strategy hereproposed allows to softly exclude the effect of the faultymeasurements provided by the fourth anchor, with clear benefiton the algorithm performance.

Differently from above, Figure 4 consider the case of all thefour anchors subject to σd randomly chosen between {0.5 −σd}, under both the cases of static and dynamic scenarios(ν̄ = 2).

It can be appreciated how the application of the weightswij , together with the smoothed observations satisfying thecondition imposed by equation(12) on Θ, improves further theoverall performance independently on the target dynamic ν̄.

It is worthy to remind that the weights wij are a side productof the wavelet based pre-filter, therefore their computationdoes not add complexity to the final algorithm. As mentionedin section III-B, to be able to applying the wavelet transforma lag proportional to J is introduced in the system. However,to exploit most of the gain achievable by the multiresolutionsupport of dj [n] necessary to compute Θ, only a small numberof scale is necessary. All the results shown are obtain forJ = 3, namely a lag of 23 − 1 samples in the computation.

Concluding, we remind that the boolean variable Θ iscomputed at each time and for each anchor-to-target link,making the approach robust to sudden changes in the statisticsof σd as well as the dynamics of the target ν̄. The resultsindicate that the SMACOF algorithm if used together with thepre-filter and weighting scheme here described, represents avalid candidate to solve the tracking problem in LoS, in acompletely non parametric fashion.

0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

Noise: !d

RMSE [m]

Performance SMACOF Algorithm

"=2

"=0

SMACOFSMACOF + On−Off smoothingW−SMACOF + On−Off smoothing

Fig. 4. Performance of the SMACOF algorithm with pre-filtered andweighted observations as function of σd. The four anchor-to-target links aresubject to a perturbation randomly chosen between {0.5− σd}.

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