Target Trajectory Estimation within a Sensor Network

Transcription

1 Target Trajectory Estimation within a Sensor Network Adrien Ickowicz IRISA/CNRS, 354, Rennes, J-Pierre Le Cadre, IRISA/CNRS,354, Rennes,France Abstract This paper deals with the estimation of the trajectory parameters for a target moing within a sensor network We are especially interested by fusing binary information at the network leel This binary information is related to the local target behaior; ie its distance from a gien sensor is increasing -) or decreasing +) In this domain, seminal contributions include [3] Howeer, in this rich framework we choose to focus on een simpler obserations so as to put in eidence the limits and the difficulties of the decentralized binary framework More specifically, the binary sequences {, +} can be locally) summarized by the times of closest point approach cpa) So, we consider that the aailable obserations, at the network leel, are the estimated alues of the cpa times The analysis is also greatly simplified if we assume that the target motion is rectilinear and uniform or a leg-by-leg one First, we examine the obserability requirements for the trajectory parameters Though the obserations do not permit a complete obserability, this study allows us to determine the obserable part of the state ector Moreoer, we show that obserable and unobserable parts are separated Thus, it is possible to deelop simple and efficient methods for estimating the obserable parameters In the case of a single-leg trajectory, we resort to a simple maximum-likelihood estimator, while for the case of multiple-leg trajectories other methods are presented It is then possible to gie confidence interals for the unobserable components of the state ector Finally, the constant elocity assumption is relaxed through diffusion process, whether continuous or discrete-time I INTRODUCTION We consider a sensor network, made with N sensors eg ideo),with known) positions Each sensor can only gies us a binary {,+} information [3], ie whether the target-sensor distance is decreasing ) or increasing +) Een if many seminal contributions deals with proximity important papers [], [], we decide here to focus on the binary {, +} information [3] From this binary sequence, it is possible to infer the time-period for which the target-sensor distance is locally minimum For the i-th sensor, we denote t i cpa this time-period Of course, it may be argued that the relatiely) rich binary {, +} information has been considerably reduced Howeer, it has the great adantage to put in eidence the basic limitations of the treatments, the effects of the process noise, etc All the deriations are made within a unique and elementary framework This paper is organized as follows First, the geometric framework is introduced Though quite elementary, this section will be of constant use subsequently Then, elementary linear algebra is used to inestigate obserability requirements when obserations are only made of the t i cpa Een, if complete obserability cannot be achieed by using these obserations it is shown that the obserable part can be separated from the unobserable one, which is generally one-dimensional It is then possible to deelop methods for estimating the obserable part of the target state ector First, this is done for a deterministic target If the target trajectory is deterministic, the difficulty we hae to face is to estimate the maneuer time-periods T i It is shown that when the sensor network is sufficiently dense the error for estimating them can be bounded aboe Then, we relax the hypothesis of deterministic elocity and allow the target to hae a diffusie motion In a first paragraph, we present a continuous-time modeling, and the corresponding estimator of the t cpa The accuracy of a continuous-time estimator is considered, based upon the general framework of stochastic processes Then, we turn toward a discrete-time modeling, and more specifically a hierarchical Marko chain modeling We conclude this paper by simulation results, illustrating the accuracy of the estimators and the pertinence of our deriations and modelings II FROM CARTESIAN TO CPA COORDINATES CONSTANT VELOCITY) Consider that the reference sensor is located at the origin O The target starts from the M point and follows a rectilinear and uniform trajectory : elocity ector) Then, the closest-point-approach cpa) point M cpa is characterized by: the ectors OM cpa OM = x ) and M M cpa are orthogonal, so that we hae see fig ): x + t cpa x ) x + y + t cpa y ) y =, so, that : t cpa = x,, with: = x, y ) T, x = x,y ) ) Similarly, we obtain the following expression of the closest distance from the sensor, cpa = OM cpa :

2 M M t θ Target M cpa CPA t cpa O x O β t Figure The CPA geometry a) The case of a constant elocity ector: Assume that the target follows a rectilinear trajectory, with a constant elocity ector, then, from the preious section, we know that: τ = t j t i, 5) Now, the question is: is-it possible that two elocity ectors say and ) produce the same set of {τ }, ie τ ) = τ ) ) This would mean that: t j t i, =, ) 6) x OM cpa = = [ x + t cpa x ) + y + t cpa y ) ] / ), [ ] x x, /, = x sinx,) = detx,) So there is a : mapping between Cartesian and cpa coordinates defined as as follows: detx,) ) detx,) = cpa ref x, θ θ, = x, = t ref cpa 3) where θ is the target heading Considering that O is the reference ref) position and that t i = OOi and denoting t i cpa the cpa time for the i-th sensor, the following relation will be of constant use subsequently: t i cpa = x t i,, 4) = t ref cpa + t i, Let us stress that the ector t i is assumed to be known and that eq 5 is only conditionnally alid These conditions will not be detailed here, but will be considered in the next sections III OBSERVABILITY ANALYSIS We consider that the obseration is restricted to the t cpa Various assumptions about target trajectory) will be considered, but in all the cases the approach we take here is purely deterministic with a binary answer yes or no) For reasons that will appear clearly through the obserability analysis, let us denote τ = t j cpa t i cpa, the difference of cpa times The aim of this section is to find out whether it is possible to determine the target trajectory parameters If span [{t j t i }] = R, then we deduce from eq 6: =, 7) so that = Indeed, considering eq 7 we deduce from the equality of norms that =, the ector equality = then follows straightforwardly The elocity ector is obserable, but not the position ector x This is quite obious since t cpa can also be written as t cpa = x + λ,, where is a ector orthogonal with Reciprocally, if the t cpa are equal altogether, then under the same conditions span [{t j t i }] = R ), we hae x, = x, which means that x = x + λ Thus, we see that the dimension of the obserable space is equal to 3 The only unobserable parameter is the λ parameter Of course, unobserability of the λ parameter can be drastically mitigated if proximity sensors are added in the sensor network Howeer, we shall focus now on multileg target trajectory b) The case of a multileg trajectory: Assume now that the target follows a multileg trajectory For instance, we shall first restrict to a two-leg trajectory and assume that the maneuering time T is known This means that we hae: st leg : t i cpa, = x+ti, nd leg : t j cpa, = x T +tj, where i are sensor index for cpa arising on the -st leg, and j for the -nd leg It is known see preious paragraph) that and are obserable on each leg Assume now that eq 8 holds true for two ectors x and x, this would mean that: st leg : nd leg : x +t i, x T +t j, = x +ti,,,, 8) = x T +tj, 9)

3 Subtracting right member from left member on each row of eq 9, we deduce from eq 9: { x x, =, ) x x, = Thus, if and are not colinear, then we deduce that x = x This reasoning can be easily extended to a general multileg trajectory Consider now the obserability of both T and x, from eq 9 we hae: st leg : x x, =, nd leg : x x+ T, + T =, with: T = T T ) ) Practically, this means that we hae: { x = x + λ, λ + T, = T ) This equation has a certain importance since it shows that we hae only a single equation for determining the two parameters λ and T T ) Following the same idea with a 3-leg trajectory, we obtain the necessary conditions: x = x + λ, λ + T, = T, λ + T + T, 3 = T + T ) 3, 3) Again, we note that we hae two linear equations for determining 3 unknown parameters λ, T T ), T T )), and the same problem whateer the number of legs Denoting l the number of legs, the dimension of the target state ector is l + ), while under mild conditions the dimension of the obserable space is l + ) Of course is x is known, then the λ parameter is zero, which means that the target trajectory is completely obserable Moreoer, if only r = x is aailable, then the problem becomes fully obserable, which means that the only ambiguity which remains is x = x x,, which is the symmetric wrt ) trajectory Howeer, een if we cannot hae the complete obserability of the target state ector, we can infer conenient estimates of T, T from the t cpa sequences ia the obserability of the i ectors We shall try now to inestigate more precisely the uncertainty in the target trajectory we can infer from eq 3 Thus for a -leg trajectory eq 3 yields: λ, + T, ) = 4) If =, the aboe equation becomes: λ cosθ/) + T sinθ/) =, θ =, ) Thus, we see that it is impossible to separate the uncertainty we hae in x λ) in the first hand and in T in the second one Eq 5 gies us the equation of the domain -here a segment of a straight line- where the maneuer can occurs Practically, we hae bounds about T time between t cpa ) and/ or λ which alow us to bound this domain For a p leg-by-leg trajectory, and denoting T = T,, T p ), we hae: A T = λ W, with:,, 3 3, 3 3 A =, p p, p p p, p p and: W =,,, 3,,, p ) T 5) Haing studied the obserability, we shall now turn toward the estimation IV MAXIMIZING THE LIKELIHOOD: Let us denote ˆτ i the estimated alue of t i cpa on the i-th sensor We suppose that ˆτ i is normally distributed, ie: ˆτ i = τ i x,) + e i, e i N,σ ), where: τ i x,) = x +t i, 6) As seen preiously, it is not possible to infer both x and from the {ˆτ i } Howeer, if we are concerned with the estimation of only, it is worth considering the differences of the cpa times, ie ˆτ, with: ˆτ = τ ) + e i e j, with: τ ) = ti tj, 7) It must be emphasized that the ectors t i t j ) relatie positions of sensors) are assumed to be known and that the τ ) do not inole the x ector Obseration is now made of the ector of ˆτ differences, ie: ˆτ = τ) + e ;, τ) =,τ ), ), e =,e i e j, ) 8) The noise ector e is still normally distributed with zero mean, but with a non-diagonal coariance matrix R The likelihood function Lˆτ ), then stands as follows: Lˆτ ) = cstdet R) / exp ) ˆτ τ) R 9) The MLE estimation of the ector is defined by = arg max Lˆτ ) and can be done by any iteratie method, or een an MCMC one if prior and constraints are added Now, the likelihood functional L) = ˆτ τ ) ) is not conex in general see the Hessian of Lˆτ ) ) Howeer, assuming that the ˆτ are tightly estimated, which means that ˆτ = τ ), where is the exact elocity ector, then the following

4 property holds true: Proposition : Under the aboe assumption, the following implication is alid: L) = = [τ ) τ ) ] =, where is the true target elocity Thus, under identifiability conditions, we hae = Proof: Let us recall the expression of L): L) = τ τ ) ) t i t j ) ) 4 t i t j, ), where, for the sake of breity, we denote τ for τ ) Thus, L) = is equialent to: τ τ ) ) t i t j ) = τ τ ) ) t i t j, 4 ) For the rest of the proof, we denote, a ector for which L ) = The aboe equality eq ) is a ectorial equality, which implies the following equality ia a scalar product with the ector τ τ ) ) t i, t j, ) = τ τ ) ) t i, t j, ), so that, we hae: τ τ ) ) τ ) = ) The same kind of result is obtained if we replace the scalar product with the ector, by the scalar product with the ector, yielding: τ τ ) ) t i, t j, ) = τ τ ) ) t i t j, 4,, or: ) τ τ )) τ =, τ τ ) ) τ ) 3) Now, we know from eq that τ τ ) ) τ ) is zero, hence we hae also: τ τ )) τ = 4) Gathering eqs and 4, we obtain: τ τ )) τ ) τ τ )) τ =, or: τ τ )) =, 5) which means that τ = τ ), whateer the couple ) Finally, we hae obtained that the equality L) = implies the equality τ = τ ), ), and under identifiability condition we hae then = The preious reasoning can be easily extended to a multileg scenario: {x,,, k }, with the same conclusion V ESTIMATING THE MANEUVER TIME PERIODS We hae seen in the preious sections the importance to hae a conenient estimation of the maneuer time periods for a leg-by-leg trajectory) For the sake of simplicity, we shall focus here on a unique elocity change To that aim, an original method has been deeloped Let us consider the sensor network and more specifically the sensor pairs see fig ) We define an equialence relation R on the sensors pairs ), by: Pair ) R k,l) t i t j ) = t k t l ) 6) True Trajectory False Trajectory Figure Sensor T T The target changes its direction We then consider each pair of sensors belonging to gien class of equialence As each pair is oriented in the same way, the difference of the t cpa between the two sensors of each pair should be the same for each pair of the class, for a gien elocity ector Restricting to two classes and ), the situation can be modelled as follows: t cpa ) + ε t cpa ) + ε t cpa ) + ε t cpa ) + ε t cpa ) + ε p t cpa ) + ε p 7) t cpa ) + ε p+ t cpa ) + ε p+ Pair

5 along with: t cpa ) = t, t cpa ) = t, 8) Notice that these quantities eg t cpa )) does not necessarily exist at each time-period The errors ε I are normally distributed Rougly speaking, we hae a time series for the first class left column of eq 7) and the second class right column of eq 7) Thus, if the target maneuer then each time series can be separated in two parts, from which we can infer the target elocity and an estimation of the maneuer time-period A usual method we can resort is to model each time series ia a mixture of normal densities There are many methods to estimate the parameters of the mixture density eg EM [7], moments [6], etc) Here, we choose to focus on the moment method We thus consider the following modeling of the time series: qx,θ) = π px,θ ) + π) px,θ ), 9) and our aim is to estimate the parameter ector Θ = π,θ,θ ) By writing the likelihood of our modeling, we can express the theoretical moments of our sample, and from the data we can determine the alues of the empirical moments [6] Then, by equalizing the empirical and the theoretical moments, we deduce a non-linear system, and,soling it, an estimation of the Θ ector [5] More precisely, we use the following equations, where M i is the i-th theoretical) moment of qx,θ), m i is the empirical moment and x i = m i M i : = π x + π)x, 3) M = π x + π)x + η, M 3 = π x 3 + π)x 3 +3η π x + π)x ), M 4 = π x 4 + π)x 4, +6η π x + π)x ) + 3η 4, where η is the ariance of ε For the uniqueness in the method of moments in the case of mixtures law of two normal distributions, see [4] VI STOCHASTIC VELOCITY MOTION A Diffusie continuous-time process Up to now, it was considered that the target motion was piecewise)-deterministic, we shall now extend our analysis to a random motion, first modelled by a continuoustime stochastic process For the sake of breity, we restrict to a mono-dimensional analysis: eg a target eoling on a known road network dx t = dt + σ dw t, 3) where x t ) t [,T] is the target position, w t ) t [T] a Brownian motion also known as a Wiener Process); is a constant and σ is the ariance of the instantaneous elocity Such an Ito process has a quite simple solution: x t = x + t + σ w t 3) Let x c be the position of the closest point cpa) for the i-th sensor Since the network topology is assumed to be known, x c is perfectly known We would like to hae an estimator of the t cpa time-period, for the stochastic process described in eq 3 For any sensor, a natural estimator ˆt is defined by: ˆt = inf {t x t x c } 33) The object of this paragraph is to inestigate the statistical properties of ˆt conergence, bias, ariance, etc) To that aim, we first express the likelihood of ˆt For a continuoustime process, it is simply: We then hae to deal with : Lˆt ) = t Pˆt t) 34) Pˆt t) = Pˆt t,x t x c ) + Pˆt t,x t x c ) = Pˆt t,x t x c ) see the footnote = Px t x c ) because {ˆt t} {x t x c } 35) So, to determine exactly the likelihood of our modeling, we must calculate Px t x c ) The right calculation is the following one: Px t x c ) = Px + t + σ w t x c ), = P w t x ) c x t ) 36) σ Now, when t is fixed, w t follows a centered Gaussian law, with a standard deiation equal to t So, we hae: Px t x c ) = with : b t πt e u /t) du, 37) b t = x c x t ) = σ σ t cpa t) To continue the calculation, we need to perform a differentiation of an improper) parametrized integral whose both integrand and lower bound depend on the parameter t) So, let us remind the Leibniz formula: assume gt) = βt) ft,u)du, then under appropriate conditions: αt) g t) = βt) αt) t P x t x c ) = f t fu,t)du + β t)fβt),t) α t)fαt),t) 38) Applied to eq 38, Leibniz rule yields: t πt b t u t + tcpa t) σ πt e tσ )e u t du, Assuming σ is ery small, then the target has a ery negligible probability to come back under x c, and then Pˆt t, x t x c) becomes negligible

6 To simplify that expression, we first split the aboe integral, yielding: I t = = t πt t πt t πt b t bt u ) t e u t du, 39) u ) t e u t du, u ) t e u t du The first integral is zero Thus, it is sufficient to consider the second one: bt u ) I t = t πt t e u t du, 4) bt = t u πt t e u t du, + bt t e u t du πt Then, performing an integration by part, we obtain: I t = b t t b t πt e t du, 4) so that, we finally obtain for L t an expression as simple as: L t = t ) cpa + t) σ t exp b t 4) πt t To ensure our estimator is a good one require additional calculations Using eq 4, it is easily shown that the likelihood is maximum for a alue of t say t max gien by : t max = t cpa t cpa ) σ The proof of eq 43, is sketched below: Differentiate wrt t) the density L t, + oσ ) 43) The deriatie is zero for alues of t which are roots of a certain 3-rd order polynomial, Obtain the exact expression of the real positie root ia the Cardan method, eq 43 is deduced from the aboe step ia an expansion around t cpa Een if the density of ˆt does not seem to be symmetric see eq 4), we see that t max is close to t cpa as soon as the ratio σ / is sufficiently small A last step is to calculate the expectation of ˆt Under the same assumption and using eq 4, we hae: E [ˆt ] where: α τ cpa is deterministic = t cpa α + 3α 8 t cpa + o σ ), 44) = σ + 8π + 6σ ) σ Noticing that α is usually quite smaller than t cpa, we conclude that the bias of the estimator of t cpa is equal to σ The expression of the ariance of our estimator can be calculated by the same way, which we do not proide here Howeer, simulation results proide interesting results about the effects of σ on arˆt ) B Discrete-Time Marko modeling We can also consider a discrete-time modeling of the time-arying target elocity Instead of considering a continuous Ito process, we turn now toward a hierarchical Marko chain model Let us define a N-state elocity Marko chain Assuming the elocity is one dimensional, we hae a N-by-N transition matrix for the elocity chain Then, the position chain is implicitly defined as in figure 3 Then, the first part of our approach deals with the choice of Velocity Position Figure 3 Link between the position Marko chain and the elocity Marko chain the probabilities of transition for the elocity They should be chosen so that the stationary law would be as close as possible to a normal law, whose mean is the right alue of the target elocity say i ) More precisely, we choose: if i = i P n+ = j n = i) = α k if i j = k, if i < i P n+ = j n = i) = β k if j i + ) = k, if i > i P n+ = j n = i) = γ k if j i ) = k, where we hae the following assumptions: αi = β i = γ i =, 45) and, α i < α j ) [,k],i > j β i < β j γ i < γ j The position chain is then straightforwardly deduced : k N { Psn+ = i + k s n = i) = P n = k), Ps n+ = i k s n = i) = Similarly to eq 33, we define the discrete-time t cpa estimator by: and the speed estimator : 7 ˆk = inf{k s k > s cpa }, 46) ˆ = cpa ˆk 47) We refer to the Simulations section for experimental results of that method N

7 VII SIMULATION RESULTS We shall now inestigate the preious deelopments ia simulations The simple maximum likelihood estimation will be presented in a first part, in which we will deal with a comparison between two sensor networks Then we will present few results about a diffusie target A Basic MLE To erify the effect of the sensor distribution, we test two scenarios In the first one, the sensors are uniformly distributed oer the sureillance domain In the second one, the sensors are put at random oer the same domain Then, considering a constant speed target motion, the effects of the changes in direction are examined On fig Value of the likelihood maximum Angle of the direction 5 Likelihood maximum for the estiamtion of the elocity Regular Case: Blue, Random Case: Red Figure 6 3 x 5 Maximum of Likelihood Estimator Variance Regular Case in Blue, Random Case in Red Number of estimator Distribution of the estimator of the elocity depending on the direction Regular Case Value of the ariance Angle of the direction 6 4 Angle of the direction Figure 7 Variance of the estimators Figure 4 Uniformly distributed network 4 the distribution of the elocity estimator in the case of a deterministic regular distribution of the sensors is presented In fig 4, we can notice that it is for a middle heading angle around 45 deg ) that the distribution seems to be the less peaky though unbiased) This result can be compared with fig 5 which presents the results for a randomly distributed sensor network This problem no longer holds This is also illustrated by fig 6, where the Number of estimator Distribution of the estimator of the elocity depending on the direction Random Geography Case 6 4 Angle of the direction Figure Randomly distributed network target heading is arying from to 8 deg Not surprisingly, looking at the ariance of the estimators see fig 7), we notice that we hae a 4% increase of the ariance between the degrees direction and the 4 degrees one for a regular network, while this phenomenon does not appear if sensors are put at random Thus, the statistical 9 behaior of the maximum likelihood estimator is tightly dependent of the sensor distribution Howeer, this result should be seriously mitigated for a a multi-leg trajectory, since it has been shown that a regular repartition has definite adantages for estimating both elocity, position and maneuers More generally, optimizing the topology of the sensor network seems to be an important direction of future research in this context B Continuous-time diffusie target Assuming that the trajectory follows now a diffusie process, our aim is to examine the statistical behaior of the t cpa estimator we defined, and whose density has been theoretically deried as a function of the process noise In fig 8, we present the empirical histogram of the t cpa estimator obtained ia simulation and compared with the exact density see eq 4) As we could expect, the two distributions present a good agreement Two important facts hae to be underlined The first one is that een if the mean of the distribution is not the actual alue of t cpa, the maximum of the likelihood corresponds to the real alue The second one is that the distribution of the speed estimator can be straightforwardly inferred from the ˆt cpa distribution, since both are deterministically related As in the deterministic section, we would like to know the influence of the ariance of the stochastic process on the ariance of the t cpa estimator see fig 9), and of the elocity estimator see fig ) Looking at both figures, the effect of the σ parameter is quite isible Howeer, while ar [ˆt cpa σ) ] depends almost) linearly of σ; the

8 8 Distribution of the t CPA estimator in the continuous time process case Red Line: Theoretical Density, Blue Hist: Simulation Results Distribution of the Velocity estimator in the continuous time process case Distribution of the n estimator in the discrete time process case 5 Posterior Distribution of the Velocity in the discrete time process case Number of alues Number of alues Figure 8 Left:Distribution of the estimator = m/s, σ =, x c = m, x = m) Right: Distribution of the elocity estimator same parameters) Value of the ariance of the estimator τ Ealuation of the ariance of the estimator of the t CPA depending on the alue of the ariance of the noise σ Value of the ariance of the trajectory process σ Figure 9 Variance of ˆt cpa as a function of σ = m/s, x c = m, x = m) effect of σ on arˆσ)) is quite non linear So, a great attention should be paid on the estimation of the ariance σ of the t cpa modeling Value of the ariance of the estimator of the elocity Ealuation of the ariance of the estimator of the elocity depending on the alue of the ariance of the noise σ Value of the ariance of the trajectory process σ Figure Variance of the elocity estimator C Diffusie Discrete-time Process Finally, let us consider a discrete-time modeling of the target elocity A single result is proposed, but which underline the accuracy of our discrete-time estimator This time, unlike the preious results, een the mean of the estimator seems to fit the correct alue Once again, the maximum of the likelihood corresponds to the correct alues of both the t cpa, and the speed Number of alues Number of alues Value of the elocity Figure Left: Distribution of the estimator of the time n discrete-time t cpa) Right: Posterior distribution of the elocity Modeling parameters: = 3, N = 5, N c = 5 VIII CONCLUSION In this paper, we chose to focus on the use of the t cpa estimates at the leel of information processing for a sensor network Though this information is rather poor, it has been shown that it can proide clear insights about the limitations of the trajectory estimation ia binary informations Three different models of the target trajectory hae been considered; ranging from the deterministic one to the continuous-time and discrete time random models The limitations due to the rough nature of the measurements hae been carefully considered The fundamental uncertainty about target position can be seriously reduced ia proximity sensors Another way is to consider a sufficiently dense network and using the information gien by the target maneuers A large part of this paper is deoted to what happens when the target departs from a deterministic model For the continuous-time model, it has been possible to perform a theoretical analysis The pertinence of the estimators has been shown and original methods hae been deeloped Perspecties for future work should be centered around the use of the binary informations [3] for multitarget tracking REFERENCES [] X WANG and B MORAN, Multitarget Tracking Using Virtual Measurements of Binary Sensor Networks Proc of the 9-th Int Conf on Information Fusion, Jul 6 [] L LAZOS, R POOVENDRAN and JA RITCEY, Probabilistic Detection of Mobile Targets in Heteregeeous Sensor Networks Proc of the 6-th IPSN, Apr 7 [3] J ASLAM, Z BUTLER, F CONSTANTIN, V CRESPI, G CY- BENKO, D RUS, Tracking a moing object with a binary sensor network Proc of the st international Conference on Embedded Networked Sensor Systems, No 5, pp 5 6 [4] E MONFRINI, Uniqueness in the method of moments for mixtures of two normal distributions Compte Rendu Academie des Sciences, Paris, Ser I, ol 336 3), pp [5] C-T BHATTACHARYA, A simple method of resolution of a distribution into Gaussian components Biometrics, ol 3, 967), pp 5 37 [6] B-G LINDSAY and P BASAK, Multiariate Normal Mixtures: A Fast Consistent Method of Moments JASA, ol 88, no 4, 993), pp [7] AP DEMPSTER, N LAIRD and DB RUBIN, Maximum likelihhod from incomplete dat ia the EM algorithm Journal of the Royal Statistical Society, B-39, 977), pp 38