particle set after resampling is biased. Since the re-
sampling scheme is repeated in the scaling domain,
the skewed particles are discarded completely, al-
though the total variance obtained by our method is
minimized compared to other resampling methods.
Since these sample points are biased when used for
target tracking, it will affect the accuracy of the tar-
get to some extent. Alternatively, if we give a certain
permissible error threshold for target localization and
tracking, we can also try the feasibility of our method
in this threshold, because it will be solved in a shorter
time with the minimum total variance compared to
other methods.
The broader impact of this work is that it can
speed up existing sequential Monte Carlo applications
and allow more precise estimates of their objectives.
There are no negative societal impacts, other than
those arising from the sequential Monte Carlo appli-
cations themselves.
REFERENCES
Arulampalam, M. S., Maskell, S., Gordon, N., and Clapp,
T. (2002). A tutorial on particle ﬁlters for on-
line nonlinear/non-Gaussian Bayesian tracking. IEEE
Transactions on Signal Processing, 50(2):174–188.
Billingsley, P. (1995). Measure and probability.
Casarin, R., Trecroci, C., et al. (2006). Business Cycle and
Stock Market Volatility: A Particle Filter Approach.
Universit
`
a degli studi, Dipartimento di scienze eco-
nomiche.
Chopin, N. et al. (2004). Central limit theorem for se-
quential Monte Carlo methods and its application to
Bayesian inference. Annals of Statistics, 32(6):2385–
2411.
Dahlin, J. and Sch
¨
on, T. B. (2019). Getting started with
particle Metropolis-Hastings for inference in nonlin-
ear dynamical models. Journal of Statistical Software,
88(2):1–41.
Dai, X. and Baumgartner, G. (2023a). Chebyshev particles.
arXiv preprint arXiv:2309.06373.
Dai, X. and Baumgartner, G. (2023b). Operator-
free equilibrium on the sphere. arXiv preprint
arXiv:2310.00012.
Dai, X. and Baumgartner, G. (2023c). Optimal camera con-
ﬁguration for large-scale motion capture systems.
Dai, X. and Baumgartner, G. (2023d). Weighted Riesz par-
ticles. arXiv preprint arXiv:2312.00621.
Del Moral, P., Doucet, A., and Jasra, A. (2006). Sequential
monte carlo samplers. Journal of the Royal Statistical
Society Series B: Statistical Methodology, 68(3):411–
436.
Del Moral, P., Doucet, A., and Jasra, A. (2012). On adap-
tive resampling strategies for sequential monte carlo
methods.
Douc, R. and Moulines, E. (2007). Limit theorems
for weighted samples with applications to sequential
Monte Carlo methods. In ESAIM: Proceedings, vol-
ume 19, pages 101–107. EDP Sciences.
Doucet, A., De Freitas, N., Gordon, N. J., et al. (2001). Se-
quential Monte Carlo methods in practice, volume 1.
Springer.
Doucet, A., Godsill, S., and Andrieu, C. (2000). On se-
quential Monte Carlo sampling methods for Bayesian
ﬁltering. Statistics and Computing, 10(3):197–208.
Doucet, A. and Johansen, A. M. (2009). A tutorial on parti-
cle ﬁltering and smoothing: Fifteen years later. Hand-
book of Nonlinear Filtering, 12(656-704):3.
Flury, T. and Shephard, N. (2011). Bayesian inference
based only on simulated likelihood: particle ﬁlter
analysis of dynamic economic models. Econometric
Theory, pages 933–956.
Fox, D. (2001). Kld-sampling: Adaptive particle ﬁlters and
mobile robot localization. Advances in Neural Infor-
mation Processing Systems (NIPS), 14(1):26–32.
Gilks, W. R. and Berzuini, C. (2001). Following a moving
target—Monte Carlo inference for dynamic Bayesian
models. Journal of the Royal Statistical Society: Se-
ries B (Statistical Methodology), 63(1):127–146.
Gordon, N., Ristic, B., and Arulampalam, S. (2004). Be-
yond the Kalman ﬁlter: Particle ﬁlters for tracking ap-
plications. Artech House, London, 830(5):1–4.
Gordon, N. J., Salmond, D. J., and Smith, A. F. (1993).
Novel approach to nonlinear/non-Gaussian Bayesian
state estimation. In IEE Proceedings F (Radar and
Signal Processing), volume 140, pages 107–113. IET.
Kitagawa, G. (1996). Monte Carlo ﬁlter and smoother for
non-Gaussian nonlinear state space models. Journal
of Computational and Graphical Statistics, 5(1):1–25.
K
¨
unsch, H. R. et al. (2005). Recursive Monte Carlo ﬁl-
ters: algorithms and theoretical analysis. The Annals
of Statistics, 33(5):1983–2021.
Liu, J. S. (2008). Monte Carlo Strategies in Scientiﬁc Com-
puting. Springer Science & Business Media.
Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo
methods for dynamic systems. Journal of the Ameri-
can Statistical Association, 93(443):1032–1044.
Montemerlo, M., Thrun, S., and Whittaker, W. (2002). Con-
ditional particle ﬁlters for simultaneous mobile robot
localization and people-tracking. In Proceedings 2002
IEEE International Conference on Robotics and Au-
tomation (Cat. No. 02CH37292), volume 1, pages
695–701. IEEE.
S
¨
arkk
¨
a, S. (2013). Bayesian Filtering and Smoothing. Cam-
bridge University Press.
S
¨
arkk
¨
a, S., Vehtari, A., and Lampinen, J. (2007). Rao-
Blackwellized particle ﬁlter for multiple target track-
ing. Information Fusion, 8(1):2–15.
Smith, A. (2013). Sequential Monte Carlo Methods in Prac-
tice. Springer Science & Business Media.
Thrun, S. (2002). Particle ﬁlters in robotics. In UAI, vol-
ume 2, pages 511–518. Citeseer.
Welch, G., Bishop, G., et al. (1995). An introduction to the
Kalman ﬁlter. Technical report, University of North
Carolina at Chapel Hill, Chapel Hill, NC, USA.
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