Evolutionary particle filter: re-sampling from the genetic algorithm perspective
ABSTRACT The sample impoverishment problem in particle filters is investigated from the perspective of genetic algorithms. The contribution of this paper is in the proposal of a hybrid technique to mitigate sample impoverishment such that the number of particles required and hence the computation complexities are reduced. Studies are conducted through the use of Chebyshev inequality for the number of particles required. The relationship between the number of particles and the time for impoverishment is examined by considering the takeover phenomena as found in genetic algorithms. It is revealed that the sample impoverishment problem is caused by the resampling scheme in implementing the particle filter with a finite number of particles. The use of uniform or roulette-wheel sampling also contributes to the problem. Crossover operators from genetic algorithms are adopted to tackle the finite particle problem by re-defining or re-supplying impoverished particles during filter iterations. Effectiveness of the proposed approach is demonstrated by simulations for a monobot simultaneous localization and mapping application.
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ABSTRACT: A general framework for using Monte Carlo methods in dynamic systems is provided and its wide applications indicated. Under this framework, several currently available techniques are studied and generalized to accommodate more complex features. All of these methods are partial combinations of three ingredients: importance sampling and resampling, rejection sampling, and Markov chain iterations. We deliver a guideline on how they should be used and under what circumstance each method is most suitable. Through the analysis of differences and connections, we consolidate these methods into a generic algorithm by combining desirable features. In addition, we propose a general use of Rao-Blackwellization to improve performances. Examples from econometrics and engineering are presented to demonstrate the importance of Rao-Blackwellization and to compare different Monte Carlo procedures. Keywords: Blind deconvolution; Bootstrap filter; Gibbs sampling; Hidden Markov model; Kalman filter; Markov...04/1998;
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ABSTRACT: David Goldberg's Genetic Algorithms in Search, Optimization and Machine Learning is by far the bestselling introduction to genetic algorithms. Goldberg is one of the preeminent researchers in the field--he has published over 100 research articles on genetic algorithms and is a student of John Holland, the father of genetic algorithms--and his deep understanding of the material shines through. The book contains a complete listing of a simple genetic algorithm in Pascal, which C programmers can easily understand. The book covers all of the important topics in the field, including crossover, mutation, classifier systems, and fitness scaling, giving a novice with a computer science background enough information to implement a genetic algorithm and describe genetic algorithms to a friend.Addison-Wesley, Reading, Massachusetts. 01/1989;
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ABSTRACT: A mobile robot requires the perception of its local environment for position estimation. Ultrasonic range data provide a robust description of the local environment for navigation. This article presents an ultrasonic sensor localization system for autonomous mobile robot navigation in an indoor semi-structured environment. The proposed algorithm is based upon an iterative non-linear filter, which utilizes matches between observed geometric beacons and an a-priori map of beacon locations, to correct the position and orientation of the vehicle. A non-linear filter based on a genetic algorithm as an emerging optimization method to search for optimal positions is described. The resulting self-localization module has been integrated successfully in a more complex navigation system. Experiments demonstrate the effectiveness of the proposed method in real world applications. PublicadoJournal of Intelligent and Robotic Systems 01/2002; 34:135-154. · 0.83 Impact Factor
Evolutionary Particle Filter: Re-sampling from the
Genetic Algorithm Perspective∗
N. M. Kwok1, Gu Fang2and Weizhen Zhou1
1ARC Centre of Excellence for Autonomous Systems
University of Technology, Sydney
Broadway, NSW, 2007, Australia
2School of Engineering and Industrial Design
University of Western Sydney
Penrith, NSW, 2747, Australia
Abstract—The sample impoverishment problem in particle
filters is investigated from the perspective of genetic algorithms.
The contribution of this paper is in the proposal of a hy-
brid technique to mitigate sample impoverishment such that
the number of particles required and hence the computation
complexity are reduced. Studies are conducted through the use
of Chebyshev inequality for the number of particles required.
The relationship between the number of particles and the
time for impoverishment is examined by considering the take-
over phenomena as found in genetic algorithms. It is revealed
that the sample impoverishment problem is caused by the re-
sampling scheme in implementing the particle filter with a
finite number of particles. The use of uniform or roulette-wheel
sampling also contributes to the problem. Crossover operators
from genetic algorithms are adopted to tackle the finite particle
problem by re-defining or re-supplying impoverished particles
during filter iterations. Effectiveness of the proposed approach
is demonstrated by simulations for a monobot simultaneous
localization and mapping application.
Index Terms—particle filter, re-sampling, genetic algorithms,
Particle filters  had been widely applied in estimation
problems containing non-linear system and non-Gaussian
noise models. The particle filter (PF) is, in principle, a
sample based implementation of Bayesian estimation .
Applications of PFs include those in mobile robot localization
and mapping, , fault diagnosis in nonlinear stochastic sys-
tems , user detection in wireless telecommunications ,
speaker tracking in auditory applications  and many other
areas. Although the application of PFs to non-linear/non-
Gaussian systems has demonstrated satisfactory results, the
implementation complexity is prohibitively high for systems
with limited computing resources especially for dynamical
and high-dimensional systems. This difficulty arises because
∗This work is supported by the ARC Centre of Excellence programme,
funded by the Australian Research Council (ARC) and the New South Wales
of the so-called sample impoverishment, i.e., the loss of
diversity for the particles to adequately represent the solution
space. An obvious solution to this problem is to use a large
number of particles at the beginning of the filtering process.
However, this increases the computational complexity.
In order to reduce the impoverishment effect, or the num-
ber of particles required, several approach had been proposed
in the literature. In , it is suggested to include, as imple-
mentation measures, sample boosting, smoothing and prior
editing. In sample boosting, the number of particles is in-
creased in an order of magnitude during an intermediate cal-
culation stage, then re-sampled to restore to the original size.
This method rather increases the computational complexity.
For the smoothing technique, particles are perturbed as the
virtual result of sampling from a continuous approximation
of the discrete states represented by the particles. However,
there may be difficulties in designing the continuous approx-
imation when the system is highly non-linear or cannot be
accurately modelled. In prior editing, the number of particles
is increased in regions of high likelihoods. This approach is
an advance from sample boosting by concentrating on the
promising solution regions, but the lack of knowledge on
locations and numbers of high likelihood regions may hinder
the success of this method. Since sample impoverishment is
mostly contributed from re-sampling, a test of the effective
particle number is checked before re-sampling is carried out,
see . This method partially avoids the lost of particle
diversity, but there is still no recovery of particles once they
It has been noted in many occasions in the literature, that
particle filters bear certain similar implementation charac-
teristics to that of genetic algorithms (GA),  and .
In , the application of sampling algorithms is treated
as the survival of the fittest inspired by the Theory of
Evolution from which the GA is developed. Furthermore, in
, the connection between PF and GA was established
from the Monte Carlo simulation view point. On the other
hand, incorporation of Bayesian framework into evolutionary
computation was proposed in  with performance im-
provements in function optimizations. More recent work in
the hybridization of PF and GA can be found in  and 
where function optimization problems are addressed. In these
research works, the application of hybridizedGA/PF has indi-
cated an attractive research direction in combining estimation
and optimization. Within the mobile robotics research area,
application of GA was found in . It adopted the GA
to enhance the estimations from an extended Kalman filter
(EKF) but the implementation of the GA was not specifically
addressed. In , a GA with a simple fitness function design
was applied in mobile robot localization and mapping but
insights into the algorithm were not reported either.
In the PF implementation,uniform re-sampling or selection
is frequently employed. This scheme, unavoidably, intro-
duces the sample impoverishment problem and an analysis
is available in . This scheme also contains unlimited
error spread as proved in  where the stochastic universal
sampling (SUS) was proposed which bounds the sampling
error. Apart from uniform re-sampling, there are alternative
selection methods available in the GA literature including
the tournament and truncation selection schemes . It is
also noted that the complexity of a PF depends critically
on the number of particles required. This observation was
considered in , where the choice of the number of parti-
cles was guided by the Chebyshev inequality. Another major
problem in PF implementation is that particles are not being
supplied in the high probability regions as needed. Although
there may not be known a priori on where the region is in
the solution space, the GA approach re-supplies or re-defines
particles via the crossover and mutation operators . These
techniques may suggest an attractive hybridization approach
by combining the advantages of PF and GA. In this paper,
the operation of a particle filter will be re-studied from the
GA perspective. The major contribution of this paper are
in characterizing the sample impoverishment problem and
proposing an alternative re-sampling scheme.
The rest of the paper is organized as follows. In Section
II, the implementations of the particle filter and genetic
algorithm are briefly reviewed. The combination of the two
techniques in mobile robot localization is developed in Sec-
tion III. Simulation results are presented and discussed in
Section IV. A conclusion is drawn in Section V.
II. PARTICLE FILTER AND GENETIC ALGORITHM
A. System Description
Assume a mobile robot being deployed in its operation
area. The robot moves along a straight line with two land-
marks being observed1. In state space description, the robot
1This is a simple one-dimensional problem and the robot is termed a
transition is given by the process model,
xv,k+1= xv,k+ vk∆T + ηv,k,
where xv is the robot state, k is the time index, v is the
velocity control, ∆T is the discrete time interval and ηvis the
process noise assumed as ηv∼ N(0,Q) and is a stationary
sequence. The complete system state is
Note that the landmarks are assumed stationary, so they
are not included into the process model for presentation
simplicity. While the robot moves, it observes or measures
the distance (range) from the landmarks. The measurement
zi,k= xmi,k− xv,k+ ηz,k,
where ziis the range measurement to the i−th landmark xmi,
ηzis the measurement noise assumed to be ηz∼ N(0,R).
B. Particle Filter
The particle filter is developed on the basis of Bayes’ Rule
which states that the posterior is proportional to the product
of the likelihood and prior, given as
where p(·) is a probability density function (pdf), xk is the
current state to be estimated and z1:k is the measurement
up to time index k. The operation of the particle filter is as
follows, see  and .
• Generate xi
initial location of the robot (assumed at the origin of the
coordinate frame), where N is the number of particles.
• Generate random numbers xi
distributed, (a 2 × N matrix) representing the initial
unknown location of landmarks. xmaxis the maximum
• Make range measurements to landmarks, giving z1,2,
which are corrupted by noise.
• Calculate the importance weights,
v= 0, for i = 1···N, representing the
m1,2in [0,xmax] uniformly
j= zj− (xi
superscript (T) stands for transpose.
• Calculate the normalized overall importance weight2,
j),j = 1,2;
v) is the innovation and
˜ wi= Π2
2Note that the product and summation is performed component-wise for
the weights and there are 2 landmarks assumed.
• Perform re-sampling to form new particles ˜ xi, such that
the probability of selection is proportional to the weights
• Calculate the estimate and the uncertainty covariance
ˆ x = ΣN
P = ΣN
i=1¯ wi˜ xi,
i=1¯ wi(˜ xi−ˆ x)(˜ xi−ˆ x)T. (7)
C. Genetic Algorithm
The genetic algorithm, as a stochastic searching algorithm,
is widely treated as a function optimizer and is developed
by the inspiration from Darwin’s Theory of Evolution. GA
simulates the evolution of individuals in competing for sur-
vival. Fitter3individuals cross-breed and produce better off-
springs hence promoting the fitness of the whole population.
Moreover, mutation also occurs during the production of
off-springs. In computer implementations, GA is governed
by the Schema Theorem4originally derived from binary
string representation of the genes of a chromosome within
an individual5. The Schema Theorem can be expressed as
(see  for details),
m(?,t + 1) ≥ m(?,t)f(?)
(1 − pc
L − 1)(1 − pm)o(?), (8)
where m(?,t) is the number of schema ? at generation t,
f(?) is the average fitness of chromosomes having the same
schema,¯f is the average fitness of the whole population, pc
is the crossover probability, δ(?) is the length of a schema,
L is the chromosome length, pmis the mutation probability
and o(?) is the order of a schema.
The Schema Theorem says that the fitness of individuals
having contributing characteristics will increase over genera-
tions (iterations) and finally converge to the optimal solution.
The implementation procedures of the GA are described in
• Generate random numbers (chromosomes) describing
the solution, the number corresponds to the size of the
population and bounded within the solution space.
• Calculate the fitness of chromosomes based on measure-
ments made (equivalent to equ. 6 in PF implementation).
• Select chromosomes into an intermediate population
according to their fitness.
• Perform crossover and mutation to mix/perturb the in-
3Fitness can be viewed as the closeness of a candidate solution from the
4A schema is a defining characteristic of the encoding that contributes to
the optimal solution.
5A chromosome is an encoding of the solution to an optimization problem
which is equivalent to a particle in the particle filter. Hence, the two terms
will be used interchangeably in the rest of the paper.
• If some termination condition is met
. Calculate the estimation (e.g., equ. 7)
. Otherwise, repeat iterations.
III. EVOLUTIONARY PARTICLE FILTER
Based on the similarities and differences between PF and
GA, an evolutionary particle filter is proposed (EPF). This
algorithm complements PF and GA. In particular, solves the
sample impoverishment problem found in the PF. The number
of particles required, the cause of sample impoverishmentand
the time for impoverishment will be investigated in the sequel
from the perspective of the GA selection process.
A. Number of Particles Required
Consider the one dimensional case, a particle x which
is the one left after impoverishment. It was initialized by
drawing a sample from a distribution with mean µ and
variance σ2. The final probability of estimation error ? is
bounded by the Chebyshev inequality given by
P(|x − µ| ≥ ?) ≤σ2
When the particles were initialized to cover a certain range
in the solution space, the chance of the particle that falls in
the vicinity of the true solution is increased by generating
more samples. Now consider that there were N particles
initialized identically independently distributed, and arrange
the particles in a sequence x. Then the Chebyshev inequality
Hence, for a specified error ?, the error probability is inversely
proportional to the number of particles required. However,
there is always a limitation in the computational resources
and the use of a small number of particles is very desirable.
N− µ| ≥ ?) ≤
B. Impoverishment from Re-sampling
The sample impoverishment phenomenon may be studied
via the gambler’s ruin problem. Consider two particles as
gamblers A and B. When they were initialized, capitals,
cA and cB respectively, were assigned according to their
closeness to the true solution. In most particle filter re-
sampling process, a pointer is generated from a uniform
distribution emulating a spin from a roulette wheel. However,
if a small number of particles are used, a true uniform
distribution cannot be guaranteed in practice. In the gambler’s
view, this becomes an unfair game as p ?= q, where p is
the winning probability for gambler A and q is the winning
probability for B.
Consider when a particle is duplicated for gambler A
and a particle is removed from B, which corresponds to
the winning and losing outcomes. The probability that, say,
gambler A eventually wins (sample impoverished) can be
derived from the theory of random walks as
where cA is the initial capital of gambler A and a similar
expression applies to gambler B.
It is evident that as far as the game is unfair, say, gambler
A is favorable, then A will ultimately win all the wealth of
gambler B. Since exact uniform distribution for the selection
pointer is not available in practice, this stochastic effect will
accelerate the impoverishment process.
From the genetic algorithm literature , a linear spaced
pointer is generated in the selection process called the sto-
chastic universal sampling (SUS). These pointers satisfy a
uniform distribution and guarantee the same interval between
pointers. Hence, reduces the selection bias and the adverse
effect of impoverishment is reduced. A set of pointers are
generated by a single roulette wheel spin as follows,
Pt= N−1((1···N) − r),
where r ∈ [0,1] is a random number.
C. Impoverishment Time
In the PF re-sampling procedure, particles are copied or
removed according to their weights ¯ wi, the change in copies
is proportional to multiples of 1/N. If the weights are sorted
and cumulatively summed, they can be approximated by the
power law such as uc, where u is the normalized index, u ∈
[0,1], resulted from sorting and c is the power constant (see
details in ). For example, Fig. 1(a) shows the normalized
weights of particles in the space range ±2.5m. The weights
are generated from a Gaussian distribution with µ = 0 and
σ2= 0.2. The corresponding cumulative sum in normalized
index is depicted in Fig. 1(b) which is the approximation by
the power law (here, c = 6.5 is determined experimentally).
−2.5−2 −1.5−1 −0.500.511.522.5
(a) Distribution of particles
0 0.10.2 0.30.40.50.60.7 0.80.91
(b) Cumulative sum of weights
Fig. 1.Power law approximation
By considering a general range in u and u − 1/N, the
proportion of particles at iteration k is
Pu,k= uck+1− (u − 1/N)ck+1.
This expression indicates the growth of particles within the
range around u. Sample impoverishment occurs when the
particle with the highest weight dominates, i.e., u = 1. The
trace of its growth becomes
?N − 1
Setting this proportion to N − 1)/N, which represents the
highest ranked weight, and after some manipulations, the
impoverishment time k∗is approximately
P1,k= 1 −
k∗≈ c−1(N lnN − 1).
For re-sampling or selection to be effective, the power law
constant must be c > 1 and a larger c imposes larger
selectivity. The above equation shows that in this case,
the impoverishment time k∗is finite and is extended by
the number of particles in proportional to N lnN. Hence,
impoverishment is inevitable when the re-sampling process
is adopted in implementing a particle filter.
D. Proposed Approach
In order to mitigate the sample impoverishment problem
when implementing a particle filter with re-sampling, an
evolutionary approach is proposed where re-sampling is
conducted implicitly thus avoiding the impoverishment. The
algorithm hybridizes the particle filter and genetic algorithm
procedures while complements the advantages of each other.
In the monobot scenario, the system state initially contains
the robot location and the landmark states are augmented
when they are firstly observed to form an overall system
state. The system state is partitioned such that floating point
numbers are used to represent individual state and there are N
copies, or chromosomes, to form a population. The states are
modifies by the genetic crossover operator depending on the
range measurements made between the robot and landmarks
during each time step. The algorithm can be described as
1) System initialization at time k = 0:
• Generate N chromosomes6to represent the robot state,
all are set to zero representing the origin of the coordi-
• Make range measurements from the robot to landmarks,
giving zifor each landmark.
• If the landmarks are firstly seen, generate sets of chro-
mosomes for each landmark7.
• Otherwise, calculate and normalize the fitness of each
chromosome such that they sum to unity.
6Floating point numbers are used in this work to gain a better resolution
of the estimation.
7The chromosomes are locations around the first range measurement with
some arbitrary distribution, e.g., uniform distribution.
3) Update at k > 0:
• Select chromosomes using stochastic universal sam-
• Compute the estimates of the robot and landmark states.
• Re-normalize fitness to f ∈ [0,1].
• Loop through N times.
- Randomly pick two chromosomes c1 and c2 with
fitness f1and f2.
- If f1< γ, then set
c1← c1+ r∆c,
- If f2< γ, then set
c2← c2+ r∆c,
where γ ∈ [0.1] is some fitness threshold (e.g., γ =
0.05), r ∈ [0,1] is a random number, ∆c is the
distance between the chromosomes.
• Repeat from the measurement step until user specified
termination of the filtering process.
∆c = c1− c2
∆c = c2− c1
In the proposed approach, the initialization and measure-
ment stages follow that of standard PF or GA implementa-
tions. In the update stage, chromosomes are selected accord-
ing to their fitness. The resultant chromosomes need to be
separated to prevent impoverishment. A pair of chromosomes
are manipulated when the fitness of one chromosome in
the pair that is lower than some threshold γ. The distance
between the two chromosomes is calculated. The adjusted
chromosome is then repelled from the one of higher fitness.
Moreover, the adjustment is also moderated by the distance
and the weighting given by r. This technique may be viewed
as re-supplying chromosomes or particles to locations in
the solution space not yet being explored while preventing
Simulations were conducted for a monobot initially located
at the coordinate origin then moves repeatedly from left to
right and vice versa in a 5m range. Two landmarks were
placed at 3m and 4m respectively. The robot moves at 0.2m/s
and the odometer measures the velocity with an error of
standard deviation at 0.03m/s, the range measurement to
landmarks carries an error of 0.1m standard deviation, the
noises are assumed Gaussian.
Two cases were simulated: 1) standard particle filter imple-
mentation with re-sampling and 2) the proposed evolutionary
approach. In both cases, the use of a small number of
particles, 500, and a relatively larger number of 5000 are
tested. Fig. 2(a) plots the particles corresponding to the robot
and landmarks and their associated weights in case 1. Due
to sample impoverishment, particles concentrated on discrete
locations. The improvement from adopting the evolutionary
approach is illustrated in Fig. 2(b) with the use of 500
(a) PF re-sampling approach
(b) Proposed approach
Fig. 2.Particles distribution vs. weights: top - robot, bottom - landmarks
particles. It is clear that particles are able to represent the pdf
which can be noted from a trace of the envelope. A more
concentrated region of particles is also observable which
indicates the convergence to the solution.
Time traces of the spread of the particles in case 1 are
plotted in Fig. 3(a) and 3(b) respectively. The top trace is
for the robot location error while the lower two are for the
landmarks, the corresponding 3σ error bound is also shown.
It is clearly seen that for 500 particles used, the particles
collapsed to a single one at about 250 time steps. The location
estimations becomes un-reliable. The sample impoverishment
is also noticeable even when 5000 particles are used, see Fig.
Results from case 2, which adopts the proposed evolution-
ary particle filter approach, are depicted in Fig. 4(a) and 4(b).
The results from the use of 500 particles show acceptable
results while the use of 5000 particles clearly removes the
sample impoverishment problem.
In this paper, the sample impoverishment problem in a par-
ticle filter is resolved by hybridizing techniques using genetic
algorithms. It has been shown by analysis and simulations
that the proposed method produces better estimation results
than the conventional particle filter. This is because that the
proposed method maintains the diversity of particles in the
re-sampling process. Further work will be done on a real
0 50100 150200250300350 400450500
0 50100 150200250300350 400450500
0 50100150200 250300350400450 500
(a) 500 particles
0 50100150 200250 300 350400450500
050 100 150200 250 300350400 450500
(b) 5000 particles
Fig. 3.Simulation results from standard PF implementation
0 50100150200 250300350400450 500
(a) 500 particles
050 100150200250300 350400450500
050100150200 250300350400450 500
(b) 5000 particles
Fig. 4.Simulation results from evolutionary particle filter approach
 N. J. Gordon, D. J. Salmond and A. F. M. Smith, ”Novel approach to
nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc.-F, Vol.
140, No. 2, April 1993, pp. 107-113.
 M. S. Arulampalam, S. Maskell, N. Gordon and T. Clapp, ”A tutorial
on particle filters for online nonlinear/non-Gaussian Bayesian track-
ing,” IEEE Trans. on Signal Processing, Vol. 50, No. 2, February 2002,
 N. M. Kwok and G. Dissanayake, ”Bearing-only SLAM in indoor
environments using a modified particle filter,” Proc. Australasian Conf.
on Robotics and Automation, December 2003, CD-ROM.
 P. Li and V. Kadirkamanathan, ”Particle filtering based likelihood ratio
approach to fault diagnosis in nonlinear stochastic systems,” IEEE
Trans. on Systems, Man and Cybernetics - Part C, Vol. 31, No. 3,
August 2001, pp. 337-343.
 E. Punskya, C. Andrieu, A. Doucet and W.J. Fitzgerald, ”Particle
filtering for multi-user detection in fading CDMA channels,” Proc. 11th
IEEE Signal Processing Workshop on Statistical Signal Processing,
August 2001, pp. 38-41.
 Y. Chen and Y. Rui, ”Real-time speaker tracking using particle filter
sensor fusion,” Proc. of the IEEE, Vol. 92, No. 3, March 2004, pp.
 N. J. Gordon, ”Non-linear/Non-Gaussian filtering and the bootstrap
filter,” IEE Colloquium on Non-linear Filters, May 1994, pp. 1-6.
 J. S. Liu and R. Chen, ”Sequential Monte Carlo methods for dynamical
systems,” Journal of the American Statistical Society, Vol. 93, 1998,
 J. H. Holland, ”Adoption in natural and artificial systems,” Ann Arbor:
The University of Michigan Press, 1975.
 D. E. Goldberg, ”Genetic algorithms in search, optimization and
machine learning,” Addison-Wesley Pub. Co., Massachusetts, 1989.
 K. Kanazawa, D. Koller and S. Russell, ”Stochastic simulation algo-
rithms for dynamic probabilistic networks,” Proc. 11th Annual Conf.
of Uncertainty and AI, 1995, pp. 346-351.
 T. Higuchi, ”Monte Carlo filter using the genetic algorithm operators,”
Journal of Sstist. Comput. Simul., Vol. 59, 1997, pp. 1-23.
 B. T. Zhang, ”A Bayesian framework for evolutionary computation,”
Proc. of the 1999 Congress on Evolutionary Computation, 1999, pp.
 K. Uosaki, Y. Kimura and T. Hatanaka, ”Evolution strategies based
particle filters for state and parameter estimation of nonlinear models,”
Proc. 2004 Congress on Evolutionary Computation, June 2004, pp.
 M. M. Drugan and D. Thierens, ”Evolutionary Markov Chain Monte
Carlo,” in P. Liardet et al. (Eds.), EA 2003, LNCS 2936, pp. 63-76,
Springer-Verlag, Berlin Heidelberg, 2004.
 L. Moreno et al., ”A genetic algorithm for mobile robot localization
using ultrasonic sensors,” Journal of Intelligent and Robotic Systems,
Vol. 34(2), June 2002, pp. 135-154.
 T. Duckett, ”A genetic algorithm for simultaneous localization and
mapping,” Proc. 2003 IEEE Intl. Conf. on Robotics and Automation,
September 2003, pp. 434-439.
 D. G. Goldberg and K. Deb, ”A comparative analysis of selection
schemes used in genetic algorithms,” in Foundations of Genetic Algo-
rithms, B. M. Spatz (Eds.), Morgan Kaufmann Publ., San Mateo, CA,
1991, pp. 69-93.
 J. E. Baker, ”Reducing bias and inefficiency in the selection algorithm,”
Genetic Algorithms and Their Applications, MIT, Cambridge, MA,
1987, pp. 14-21.
 T. Blickle and L. Thiele, ”A comparison of selection schemes used
in genetic algorithms,” Technical Report: Swiss Federal Institute of
Technology, TIK-Report Nr. 11, December 1995.
 Y. Boers, ”On the number of samples to be drawn in particle filtering,”
IEE Colloquium on Target Tracking: Algorithms and Applications,
November 1999, pp. 5/1-5/6.
 F. Herrera, M. Lozano and J. L. Verdegay, ”Tackling real-coded genetic
algorithms: operators and tools for behavioural analysis,” Artificial
Intelligence Review, 1998, Vol. 12(4), pp. 265-319.