self adaptive particle filter

8/12/2019 Self Adaptive Particle Filter

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Self Adaptive Particle Filter

Alvaro Soto

Pontificia Universidad Catolica de ChileDepartment of Computer Science

Vicuna Mackenna 4860 (143), Santiago 22, Chile

[email protected]

Abstract

The particle filter has emerged as a useful tool forproblems requiring dynamic state estimation. Theefficiency and accuracy of the filter depend mostlyon the number of particles used in the estimationand on the propagation function used to re-allocate

these particles at each iteration. Both features arespecified beforehand and are kept fixed in the reg-ular implementation of the filter. In practice thismay be highly inappropriate since it ignores errorsin the models and the varying dynamics of the pro-cesses. This work presents a self adaptive versionof the particle filter that uses statistical methods toadapt the number of particles and the propagationfunction at each iteration. Furthermore, our methodpresents similar computational load than the stan-dard particle filter. We show the advantages of theself adaptive filter by applying it to a synthetic ex-ample and to the visual tracking of targets in a realvideo sequence.

1 Introduction

The particle filter is a useful tool to perform dynamic state es-timation via Bayesian inference. It provides great efficiencyand extreme flexibility to approximate any functional non-linearity. The key idea is to use samples, also called particles,to represent the posterior distribution of the state given a se-quence of sensor measurements. As new information arrives,these particles are constantly re-allocated to update the esti-mation of the state of the system.

The efficiency and accuracy of the particle filter dependmainly on two key factors: the number of particles used toestimate the posterior distribution and the propagation func-tion used to re-allocate these particles at each iteration. Thestandard implementation of the filter specifies both factors be-forehand and keeps them fixed during the entire operation ofthe filter. In this paper, we present a self adaptive particle fil-ter that uses statistical methods to select an appropriate num-ber of particles and a suitable propagation function at eachiteration.

This paper is organized as follows. Section 2 providesbackground information about the standard particle filter.

Section 3 presents our method to adaptively estimate the num-ber of particles. Section 4 presents our method to adaptivelyimprove the propagation function. Section 5 shows the re-sults of applying the self adaptive filter to a visual trackingtask. Finally, Section 6 presents the main conclusions of thiswork.

2 Particle Filter

In Bayesian terms, the posterior distribution of the state canbe expressed as:

p(xt/yt) = p(yt/xt)p(xt/yt1) (1)

whereis a normalization factor; xt represents the state ofthe system at time t; and yt represents all the informationcollected until time t. Equation (1) assumes thatxt totallyexplains the current observationyt.

The particle filter provides an estimation of the poste-rior in Equation (1) in 3 main steps: sampling, weighting,and re-sampling. The sampling step consists of taking sam-

ples (particles) from the so-called dynamic prior distribution,p(xt/yt1). Next, in the weighting step, the resulting parti-cles are weighted by the likelihood term p(yt/xt). Finally,a re-sampling step is usually applied to avoid the degeneracyof the particle set. The key point that explains the efficiencyof the filter comes from using a Markovian assumption andexpressing the dynamic prior by:

p(xt/yt1) =

p(xt/xt1)p(xt1/yt1)dxt1. (2)

This expression provides a recursive implementation of thefilter that allows it to use the last estimation p(xt1/yt1)to select the particlesxit1 in the next iteration. These par-

ticles are then propagated by the dynamics of the processp(xt/xit1)to complete the sampling step.At each iteration the operation of the particle filter can be

seen as an importance sampling process[Tanner, 1996]. Im-portance sampling provides an efficient way to obtain sam-ples from a densityp(x), in cases where this function can beevaluated, but it is not affordable or possible to sample fromit directly. The basic idea in importance sampling is to usea proposal distributionq(x), also called importance function,to obtain the samplesxi, and then weigh each sample usinga compensatory term given by p(xi)/q(xi). It is possible to


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show[Tanner, 1996]that under mild assumptions the set ofweighted-samples resembles the target distributionp(x).

The sampling and weighting steps of the particle filter cor-respond to the basic steps of an importance sampling pro-cess. In this case, given that the true posteriorp(xt/yt) isnot known, the samples are drawn from an importance func-tion that corresponds to the dynamic priorp(xt/yt1). Usingthis importance function, the compensatory terms are exactly

the non normalized weights used in the weighting step of theparticle filter. The methods presented in this paper use re-sults from the theory of importance sampling to provide aself adaptive version of the particle filter.

3 Adaptive Selection of the Number of

Particles

The selection of the number of particles is a key factor in theefficiency and accuracy of the particle filter. The computa-tional load and the convergence of the filter depend on thisnumber. Most applications select a fixed number of parti-cles in advance, using ad hoc criteria, or statistical methods

such as Monte Carlo simulations or some standard statisticalbound[Boers, 1999]. Unfortunately, the use of a fixed num-ber of particles is often inefficient. The dynamics of mostprocesses usually produces great variability in the complex-ity of the posterior distribution1. As a consequence, the initialestimation of the number of particles can be much larger thanthe real number of particles needed to perform a good esti-mation of the posterior distribution or, worse, at some point,the selected number of particles can be too small causing thefilter to diverge.

The effect of the number of particles on the accuracy ofthe filter is determined by two factors: the complexity of thetrue density and how closely the proposal density mimics thetrue density. Both factors are very intuitive; the estimation

of a more complex pdf requires a greater number of sam-ples to correctly represent the less predictable shape of thefunction. Also, a greater mismatch between the proposal andthe true densities produces many wasted samples located inirrelevant parts of the true distribution. Previous works toadaptively determine an adequate number of particles havefailed to consider these two factors together[Foxet al., 1999;Koeller and Fratkina, 1998; Fox, 2001].

Here, we propose two methods based in the theory ofStatistics that can be used to adaptively estimate the num-ber of particles to represent the target posterior distributionwithout adding a significant load to the normal operation ofthe filter. At each cycle of the particle filter, these techniquesestimate the number of particles that, with a certain level ofconfidence, limits the maximum error in the approximation.

3.1 KLD-Sampling Revised

The KLD-Sampling algorithm [Fox, 2001] is a method toadaptively estimate the number of samples needed to boundthe error of the particle filter. The error is measured bythe Kullback-Leibler divergence (KL-divergence) between

1Here complexity is understood in terms of the amount of infor-mation needed to code the distribution.

the true posterior distribution and the empirical distribution,which is a well known nonparametric maximum likelihoodestimate. KLD-Sampling is based on the assumption that thetrue posterior can be represented by a discrete piecewise con-stant distribution consisting of a set of multidimensional bins.This assumption allows the use of the 2 asymptotic conver-gence of the likelihood ratio statistic to find a bound for thenumber of particles N:

N > 1

22k1,1 (3)

where is the upper bound for the error given by the KL-divergence, (1 )is the quantile of the2 distribution withk 1 degrees of freedom, and k is given by the number ofbins with support.

The problem with KLD-Sampling is the derivation of thebound using the empirical distribution, which has the implicitassumption that the samples comes from the true distribution.This is not the case for particle filters where the samples comefrom an importance function. Furthermore, the quality of thematch between this function and the true distribution is one ofthe main elements that determines the accuracy of the filter,hence the suitable number of particles. The bound given byKLD-Sampling only uses information about the complexityof the true posterior, but it ignores any mismatch between thetrue and the proposal distribution.

To fix the problem of KLD-Sampling we need a way toquantify the degradation in the estimation using samples fromthe importance function. The goal is to find the equivalentnumber of samples from the importance and the true densitiesthat capture the same amount of information about the latter.

In the context of Monte Carlo (MC) integration,[Geweke,1989]introduced the concept of relative numerical efficiency(RNE), which provides an index to quantify the influence ofsampling from an importance function. The idea behind RNE

is to compare the relative accuracy of solving an integral us-ing samples coming from both the true and the proposal den-sity. Accuracy is measured according to the variance of theestimator of the integral.

If we use MC integration to estimate the mean value ofthe state (EMC(x)), the variance of the estimator is given by[Doucetet al., 2001] 2 :

V ar[ENMC(x)] = V arp(x)/N (4)

whereNis the number of samples coming from the true dis-tributionp(x), and the subscriptpexpresses that the varianceinvolved is computed using the target distribution.

When the samples come from an importance function q(x),

the variance of the estimator corresponds to the varianceof Importance Sampling (IS), which is given by [Geweke,1989]:

V ar[ENIS(x)] = Eq((xEp(x))2 w(x)2)/NIS=

2

IS/NIS,(5)

wherew(x)corresponds top(x)/q(x)the weights of IS andNISis the number of samples coming from the importancefunction.

2In the rest of the section, we concentrate on one iteration of thefilter and we drop the subscript t.


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To achieve similar levels of accuracy, the variance of bothestimators should be equal. This allow us to find a relationthat quantifies the equivalence between samples from the trueand the proposal density,

N =NISV arp(x)

2IS(6)

Replacing (6) in (3) allows us to correct the bound given

by KLD-Sampling when the samples do not come from thetrue distribution but from an importance function:

NIS> 2IS

V arp(x)

1

22k1,1 . (7)

Using MC integration,V arp(x)and2

IScan be estimatedby:

V arp(x) = Ep(x2) Ep(x)

2

Ni=1x

2

i wiNi=1wi

Ep(x)2

and

2IS

Ni=1x

2

i w2

iNi=1wi

2N

i=1xiw2

i Ep(x)Ni=1wi

+

Ni=1w

2

i Ep(x)2

Ni=1wi

(8)withEp(x) =

ni=1xiwi/

ni=1wi. Equation (8) shows

that using appropriate accumulators, it is possible to calculatethe bound incrementally keeping theO(N)complexity of thefilter.

3.2 Asymptotic Normal Approximation

Usually, the particle filter keeps track of the posterior densitywith the goal of estimating the mean or higher order momentsof the state. This suggests an alternative error metric to deter-mine the number of particles. Instead of checking the accu-racy of the estimation of the posterior, it is possible to checkthe accuracy of a particle filter in the estimation of a momentof this density.

Under weak assumptions and using the strong law of largenumbers, it is possible to show that at each iteration the esti-mation of the mean given by the particle filter is asymptoti-cally unbiased[DeGroot, 1989]. Furthermore, if the varianceof this estimator is finite, the central limit theorem justifiesan asymptotic normal approximation for it[DeGroot, 1989],which is given by:

EPF(x) N(Ep(x), 2

IS/NIS) (9)

whereN(, 2)denotes the normal distribution with mean and standard deviation.

Using this approximation, it is possible to build a one sidedconfidence interval for the number of particles that limits theerror in the estimation of the mean:

P(|EPF(x) Ep(x)

Ep(x) | ) (1 ) (10)

where| | denotes absolute value; Ep(x) is the true meanvalue of the state;corresponds to the desired error; and (1 )corresponds to the confidence level.

Following the usual derivation of confidence intervals,Equation (10) produces the following bound for the numberof particles:

NISZ21/2

2

IS

2 Ep(xt)2 (11)

3.3 Testing the Bounds

Figure 1 shows the distributions used to test the bounds.The true distribution corresponds to p(x) = 0.5N(3, 2) +0.5N(10, 2) and the importance function to q(x) =0.5N(2, 4) + 0.5N(7, 4). In all the trials, the desired errorwas set to0.01and the confidence level to 95%.

5 0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

x

pr(x)

Samples from p(x)

True distribution p(x)___

Samples from q(x)

Proposal q(x)

Figure 1: Inefficient allocation of samples due to a mismatch be-tween p(x)and q(x).

Figure 2a shows the number of particles predicted by thedifferent bounds. The predicted number of particles is highlyconsistent. Also, the revised versions of KLD-Sampling con-sistently require a larger number of particles than the originalalgorithm. This is expected because of the clear mismatchbetween the proposal and the true densities.

Figure 2b shows the resulting KL-divergence for eachcase. It is interesting to note how the practical results matchclosely the theoretical ones. Using the original version ofKLD-Sampling, the error in the estimation is significantlygreater than the one specified. However, when the samepredicted number of particles is sampled from the true dis-tribution (solid-line), the resulting error matches closely the

one specified. This shows clearly that constraining the sam-pling to the right assumption, the original bound predictedby KLD-Sampling is correct. In the case of the revised ver-sions of KLD-Sampling the resulting error using Equation (7)matches closely the one specified. In the same way, the errorprovided by the bound in Equation (11) also matches closelythe level specified.

4 Adaptive Propagation of the Particles

The regular implementation of the particle filter propagatesthe particles using the dynamic prior p(xt/yt1). This strat-egy has the limitation of propagating the samples withoutconsidering the most recently available observation, yt. Im-

portance sampling suggests that one can use alternative prop-agation functions that can provide a better allocation of thesamples, for example, a suitable functional of yt. Unfor-tunately, the use of an arbitrary importance function signif-icantly increases the computational load of the particle filter.In this case, as opposed to the standard particle filter, the esti-mation of each weight requires the evaluation of the dynamicprior. This section shows a method to build an importancefunction that takes into account the most recent observationytwithout increasing the computational complexity of the fil-ter.


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0 5 10 15 20 25 30 35 400.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Original KLDSampling

KLDSampling revised using Eq.(7)

Using asymptotic normal approximation

Independent Runs

NumberofParticles

a)

0 5 10 15 20 25 30 35 400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Original KLDSampling

KLDSampling using samples from true distribution_____

KLDSampling revised using Eq.(7)oo Using asymptotic normal approximation...

Independent Runs

KLDivergence

b)

Figure 2: a) Number of particles given by each bound. b) Resulting KL-divergence.

4.1 Previous Work

In the literature about particle filters and importance sam-pling, it is possible to find several techniques that help toallocate the samples in areas of high likelihood under the tar-

get distribution. The most basic technique is rejection sam-pling [Tanner, 1996]. The idea of rejection sampling is toaccept only the samples with an importance weight above asuitable value. The drawback is efficiency: there is a highrejection rate in cases where the proposal density does notmatch closely the target distribution. In[West, 1993], Westpresents a kernel-based approximation to build a suitable im-portance function. However, the computational complexityof the method is unaffordable. In the context of mobile robotlocalization, Thrun et al.[Thrunet al., 2000]propose to sam-ple directly from the likelihood function, but in many appli-cations this is not feasible or prohibitive.

Pitt and Shephard propose the auxiliary particle filter[Pittand Shephard, 1999]. They augment the state representationwith an auxiliary variable. To sample from the resulting jointdensity, they describe a generic scheme with computationalcomplexity ofO(N). The disadvantage is the additional com-plexity of finding a convenient importance function. Pitt andSheppard provide just general intuitions about the form of athis function. In this paper we improve on this point by pre-senting a method to find a suitable importance function.

4.2 Adaptive Propagation of the Samples

Sampling from the dynamic prior in Equation (2) is equiva-lent to sample from the following mixture distribution:

p(xt/yt1)

nk=1

kp(xt/xkt1) (12)

where the mixture coefficients k are proportional top(xt1/yt1). The key observation is that under this schemethe selection of each propagation density depends on the mix-ture coefficientsks, which do not incorporate the most re-cent observationyt. From an MC perspective, it is possible toachieve a more efficient allocation of the samples by includ-ing yt in the generation of the coefficients. The intuition isthat the incorporation ofyt increases the number of samples

drawn from mixture componentsp(xt/xkt1)associated with

areas of high probability under the likelihood function.Under the importance sampling approach, it is possible to

generate a new set of coefficients kthat takes into account ytby sampling from the importance function

p(xt1

/yt). In this

way, the set of samples xitfrom the dynamic priorp(xt/yt1)is generated by sampling from the mixture,

nk=1

k p(xt/xkt1) (13)

and then adding to each particle xit a compensatory weightgiven by,

wit =p(xkt1/yt1)

p(xkt1/yt) , with xit p(xt/x

kt1) (14)

The resulting set of weighted samples {xit, wit}ni=1still comes

from the dynamic prior, so the computational complexity of

the resulting filter is stillO(N). The extra complexity of thisoperation comes from the need to evaluate and to draw sam-ples from the importance function p(xit1/yt). Fortunately,the calculation of this function can be obtained directly fromthe operation of the regular particle filter. To see this clearly,consider the following:

p(xt, xt1/yt) p(yt/xt, xt1, yt1)p(xt, xt1/yt1)

p(yt/xt)p(xt/xt1, yt1)p(xt1/yt1)

p(yt/xt)p(xt/xt1)p(xt1/yt1) (15)

Equation (15) shows that, indeed, the regular steps of theparticle filter generate an approximation of the joint densityp(xt, xt1/yt). After re-sampling fromp(xt1/yt1), prop-agating these samples with p(x

t/x

t1), and calculating the

weightsp(yt/xt), the set of resulting sample pairs (xit,xit1)

with correcting weights p(yt/xit) forms a valid set of sam-

ples from the joint densityp(xt, xt1/yt). Considering thatp(xt1/yt)is just a marginal of this joint distribution, the setof weighted-samplesxit1are valid samples from it.

The previous description provides an adaptive algorithmthat allows the particle filter to use yt in the allocation ofthe samples. First, Nparticles are used to generate the im-portance function p(xt1/yt). Then, starting from this im-portance function, another Nparticles are used to generate


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the desired posterior p(xt/yt). The relevant compensatoryweights are calculated according to Equation (14) and thelikelihood termP(yt/xt). The resulting filter has a computa-tional complexity ofO(2N).

In the previous algorithm the overlapping between a reg-ular iteration of the regular particle filter and the processof generating the importance function provides a convenientway to perform an online evaluation of the benefits of up-

dating the dynamic prior with information from the last ob-servation. While in cases of a poor match between the dy-namic prior and the posterior distribution the updating of thedynamic prior can be beneficial, in cases where these distri-butions agree, the updating does not offer a real advantage,and the extra processing should be avoided. To our currentknowledge, this issue has not been addressed before.

The basic idea is to quantify at each iteration of the par-ticle filter the trade-off between continuing drawing samplesfrom a known but potentially inefficient importance functionp(xt1/yt1) versus incurring in the cost of building a newimportance functionp(xt1/yt)that provides a better alloca-tion of the samples under the likelihood function. The impor-tant observation is that, once the regular particle filter reachesan adequate estimate, it can be used to estimate both the pos-terior distributionp(xt/yt)and the updated importance func-tionp(xt1/yt).

The last step of the algorithm is to find a metric that pro-vides a way to quantify the efficiency in the allocation of thesamples. Considering that the efficiency in the allocation ofthe samples depends on how well the dynamic prior resem-bles the posterior distribution, an estimation of the distancebetween these two distributions is a suitable index to quan-tify the effectiveness of the propagation step. We found aconvenient way to estimate the Kullback-Leibler divergence(KL-divergence) between these distributions, and in generalbetween a target distributionp(x)and an importance function

q(x):KL(p(x), q(x)) log(N) H( wi). (16)

Equation (16) states that for a large number of particles,the KL-divergence between the dynamic prior and the poste-rior distribution can be estimated by calculating how far theentropy of the distribution of the weights,H( wi), is from theentropy of a uniform distribution (log(N)). This is an intu-itive result because in the ideal case of importance sampling,wherep(x) = q(x), all the weights are equal. In consequencethe entropy of the weights is a suitable value to quantify theefficiency in the allocation of the samples.

5 Application

To illustrate the advantages of the self adaptive particle filter,we use a set of frames of a video sequence consisting of twochildren playing with a ball. The goal is to keep track of thepositions of the ball and the left side child. Each hypothesisabout the position of a target is given by a bounding box de-fined by its height, width, and the coordinates of its center.The motion model used for the implementation of the parti-cle filter corresponds to a Gaussian function of zero mean anddiagonal covariance matrix with standard deviations of20forthe center of each hypothesis and0.5for the width and height.

Figure 3 shows the results of tracking the targets using theself adaptive particle filter. The bounding boxes correspondto the most probable hypotheses in the sample set used toestimate the posterior distributions of the states. In the esti-mation of the number of particles, we just consider thex andy coordinates of the center of the bounding boxes, assumingindependence to facilitate the use of Equation (7). We setthe desired error to0.01and the confidence level to95%. A

minimum number of 1000 samples is always used to ensurethat convergence has been achieved. In the adaptation of thepropagation function we set the threshold for the entropy ofthe weights in 2.

Figure 4-left shows the number of particles needed to esti-mate the posterior distribution of the ball at each frame with-out adapting the propagation function. Figure 4-right showsthe number of particles in the case of adapting the importancefunction. The tracking engine decides to adapt the importancefunction at all the frames where the ball travels from one childto the other (Frames 3-7).

In the case of tracking the child, the result shows that thereis not a major difference between the self adaptive particle

filter and the regular filter. The self adaptive filter needs aroughly constant number of particles during the entire se-quence without needing to adapt the importance function.This is expected because the child has only a small and slowmotion around a center position during the entire sequence.Therefore the stationary Gaussian motion model is highly ac-curate and there is not a real advantage of adapting the num-ber of particles or the propagation function.

In the case of the ball, the situation is different. Duringthe period that the ball travels from one child to the other(Frames 3 to 7), it has a large and fast motion, therefore theGaussian motion model is a poor approximation of the realmotion. As a consequence there is a large mismatch betweenthe dynamic prior and the posterior distribution. This pro-

duces an inefficient allocation of the samples and the estimatewithout adapting the importance function needs a larger set ofsamples to populate the relevant parts of the posterior. In con-trast, when adapting the importance function during Frames3 to 7 it is possible to observe a significant reduction in thenumber of samples due to a better allocation of them.

6 Conclusions

In this paper we present a self adaptive version of the parti-cle filter that uses statistical techniques to estimate a suitablenumber of particles and to improve the propagation function.In terms of the estimation of the number of particles, the vali-dation of the bounds using a synthetic example shows that theempirical results match closely the theoretical predictions. Inparticular, the results indicate that by considering the com-plexity of the true density and how closely the proposal den-sity mimics the true density, the new bounds show a clear im-provement over previous techniques such as KLD-Sampling.

The mechanisms used by the self adaptive filter to adaptthe importance function and to identify when the adaptationof the importance function may be beneficial proved to behighly relevant. Using these mechanisms to track targets ina real video sequence, the self adaptive filter was able to ef-


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StereoColoH

Stereo

ColorH

Stereo ColorH

ColorH

Stereo

Figure 3: Tracking results for the ball and the left side child for frame 1, 5, and 14. The bounding boxes correspond to the mostprobable hypotheses in the sample set used to estimate the posterior distributions.

0 2 4 6 8 10 12 14500

1000

1500

2000

2500

3000

3500

4000

4500

NumberofParticles

Frame Number

X dimensionxx

Y dimensionoo

2 4 6 8 10 12 140

500

1000

1500

2000

2500

3000

NumberofParticles

Frame Number

X dimensionxx

Y dimensionoo

Figure 4: Number of particles used at each iteration to track the ball. Left: without adapting the importance function. Right:Adapting the importance function.

ficiently track targets with different motions using a generalGaussian motion model. Furthermore, by avoiding the needof overestimating the number of particles and by allocatingthese particles in areas of high likelihood, the self adaptive fil-ter proved to operate with a similar computational complexityto the regular particle filter.

References

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[DeGroot, 1989] M. DeGroot. Probability and Statistics.Addison Wesley, 2nd edition, 1989.

[Doucetet al., 2001] A. Doucet, N. de Freitas, and N. Gor-don. An introduction to sequential Monte Carlo methods.InSequential Monte Carlo Methods in Practice, pages 314. Springer, 2001.

[Foxet al., 1999] D. Fox, W. Burgard, F. Dellaert, andS. Thrun. Monte Carlo localization: Efficient position es-timation for mobile robots. InProc. of the nat. conf. on AI(AAAI), 1999.

[Fox, 2001] D. Fox. KLD-Sampling: Adaptive particle fil-ters. InAdvances in Neural Information Processing Sys-tems 14 (NIPS), 2001.

[Geweke, 1989] J. Geweke. Bayesian inference in econo-metric models using Monte Carlo integration. Economet-rica, 57:13171339, 1989.

[Koeller and Fratkina, 1998] D. Koeller and R. Fratkina. Us-ing learning for approximation in sthocastic processes.

In Proc. of the international conf. on machine learning(ICML), 1998.

[Pitt and Shephard, 1999] M. Pitt and N. Shephard. Filteringvia simulation: Auxiliary particle filters. Journal of the

American Statistical Association, 94(446):590599, June1999.

[Tanner, 1996] M. Tanner. Tools for Statistical Inference.Springer-Verlag, 3nd edition, 1996.

[Thrunet al., 2000] S. Thrun, D. Fox, and W. Burgard.Monte Carlo localization with mixture proposal distribu-tion. InAAAI National Conf. on Artificial Intelligence,pages 859865, Austin, Tx, 2000.

[West, 1993] M. West. Approximating posterior distribu-tions by mixtures. Journal of the Royal Statistical Society,Serie B, 55(2):409422, 1993.

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