IEEE SIGNAL PROCESSING LETTERS, VOL . XX, NO. Y, MONTH YEAR 1
Optimal Particle-Filter-Based Detector
Yvo Boers and Pranab K. Mandal
Abstract—In this letter, we propose and prove the asymptotic
optimality of a particle-filter-based detection scheme. The de-
tection method can be used in a general nonlinear/nonGaussian
signal detection problem. The proposed detection mechanism is
based on likelihood ratio (LR) and thus optimal in the Neyman-
Pearson (NP) sense, but we approximate the LR based on a
particle filter (PF). We show the asymptotic optimality by proving
that the PF-based approximation of the LR converges to the
true LR as the number of particles increases to infinity. We also
discuss the practical and operational implications of the result,
the main one being that it is optimal in the sense that no other
processing and detection mechanism can have higher probability
of detection while have same or lower false alarm rate.
Index Terms—Particle Filters, Detection Problem, Hypothesis
Testing, Neyman Pearson Optimality
I. INTRODUCTION
We consider the detection problem for a possibly nonlin-
ear and/or non-Gaussian state-space system. In a previously
published letter [1], a mechanism has been proposed which
combines a particle filter (PF) and a detection mechanism
based on the likelihood ratio (LR) test, where the likelihoods
were estimated/approximated from the running PF. The au-
thors conclude with a “remaining question” about a possible
convergence result for the approximate LR [1, Section III]. To
the best of our knowledge, no explicit convergence results for
the PF-based approximated LR can be found in the existing
literature.
In this letter, we fill that gap by presenting a proof of the
convergence of the PF-based LR to the true LR, for which a
closed form expression is usually not available. In the proof,
we make use of the existing convergence results of a particle
filter, e.g., [2]. The proposed detection mechanism is thus
asymptotically NP optimal in the detection part and when
combined with the PF, it becomes Bayes optimal for the state
estimation part. The NP optimality is appealing not only from
a theoretical point of view, but also from the practical point of
view, because the optimized quantities, namely the detection
probability and the false alarm rate, are very important and
relevant in real life situations. For example, in radar, sonar and
infrared systems or combined sensing systems, a guaranteed
false alarm rate is of paramount importance while minimizing
the missed detection probability. This is precisely what an LR-
test provides. Also, we verify the convergence result through
a numerical simulation with a linear-Gaussian system, where
the true likelihood is known and can be easily calculated.
Numerical results show that the particle filter does provide a
more accurate estimate of the signal as well as the likelihood
ratio, as the number of used particles increases.
Yvo Boers is an unaffiliated/independent and Thales Netherlands (retired)
research scientist, Pranab K. Mandal is an assistant professor of Applied
Mathematics at the University of Twente, Enschede, The Netherlands.
The proposed detection mechanism can, in principle, be
used in a (radar) track-before-detect (TBD) processing chain
for a radar system that needs to be able to detect and track
very small objects in adverse conditions. However, we do
not pursue this as this is not the main purpose of this
correspondence.
Contributions of this letter are, thus:
1) A proof that the particle-filter-based LR converges to the
true, possibly unknown, LR, as the number of particles
tends to infinity.
2) A numerical illustration of the above statement by means
of a simulation example.
3) A discussion on the practical relevance of the result
and how the proposed detector differs from some of the
current state-of-the-art.
The organisation of the remainder of this letter is as follows:
first, in Section II, we define precisely the system that we
consider and we present some preliminary results that will be
needed later on. We present our main result in Section III. In
Section IV, we present the simulation results that support the
results from Section III. Finally, in Section V, we draw some
conclusions. Relevant remarks and discussions can be found
throughout the letter.
II. SYSTEM AND PRELIMINARIES
Let us consider the following general nonlinear system: for
k = 0, 1, 2 . . .,
(state:) s
k+1
= f(s
k
, w
k
) (1)
(measurement:) z
k
= h(s
k
, v
k
) (2)
with
(initial condition:) s
0
∼ p(s
0
), (3)
where the states s
k
∈ S ⊂ R
n
, the measurements z
k
∈ R
p
,
the state (or process) noises w
k
∈ R
m
, with w
k
∼ p
w
, the
measurement noises v
k
∈ R
q
with
v
k
∼ p
v
(4)
and f and h are mappings of appropriate dimensions.
We denote the a posteriori density corresponding to the
system (1) – (4), by:
p(s
k
| Z
k
), (5)
where Z
k
contains the measurement history up to and includ-
ing time k. Moreover, we denote the (cumulative probability)
distribution associated with p(s
k
| Z
k
) by P (s
k
| Z
k
).
In our analysis, we assume that the detection scheme uses
a Sequential Importance Resampling (SIR) particle filter with
the importance density to be the state predictive density and
resampling performed at each time step k; see, e.g., [3].
IEEE SIGNAL PROCESSING LETTERS, VOL . XX, NO. Y, MONTH YEAR 2
Adopting the notation of [1] we denote the output of the
particle filter by:
{s
i
k
, ˜s
i
k
, q
i
k
, ˜q
i
k
}
i=1,...,N
with ˜q
i
k
:= p(z
k
| s
i
k
), (6)
where s
i
k
∈ S represents a particle before resampling, ˜q
i
k
∈ R
+
and q
i
k
∈ [0, 1] are the corresponding unnormalized and
normalized weights, respectively, and ˜s
i
k
∈ S denotes a
particle after resampling. The weights after resampling are,
of course, all equal to
1
N
. Then the empirical a posteriori
distribution is given by (see, e.g., [2] and [1]):
ˆ
P
N
(s
k
) :=
1
N
N
X
i=1
u(s
k
− ˜s
i
k
), (7)
where u(.) is the Heaviside step function [4]. Let us also
define the empirical one-step-ahead prediction distribution
(i.e., corresponding to p(s
k
| Z
k−1
)) as:
ˆ
P
N
(s
k
| Z
k−1
) :=
1
N
N
X
i=1
u(s
k
− s
i
k
). (8)
Remark 1. It is worth noting that the empirical a posteriori
distributions defined by (7) and (8) are stochastic, in the sense
that they depend on the (stochastic/random) outcome of the
(stochastically simulated) particles (˜s
i
k
or s
i
k
). This is, e.g.,
also stated in [2, section V]. Furthermore, the particles are
not independent, whereas in standard statistics literature, when
one considers an empirical distribution of the form (7), the
samples (particles) are, usually, considered to be so.
The following theorem provides convergence results for the
empirical distributions (posterior and one-step-ahead predic-
tion) obtained from a SIR particle filter.
Theorem 1. Consider the dynamical system (1) – (4). Suppose
that the likelihood p(z
k
| s
k
) is bounded and continuous as a
function of s
k
and the transition density p(s
k
| s
k−1
) satisfies
the so-called Feller property (see, e.g., [2]). Let
ˆ
P
N
(s
k
) and
ˆ
P
N
(s
k
| Z
k−1
) be as given in (7) and (8), obtained from a
SIR-PF applied to the dynamical system. Then, as N → ∞,
(a)
ˆ
P
N
(s
k
) converges to P (s
k
| Z
k
) almost surely, and
(b)
ˆ
P
N
(s
k
| Z
k−1
) converges to P (s
k
| Z
k−1
) almost surely.
Here, almost surely refers to the randomness in the simulated
particles.
Proof. Part (a) coincides with Theorem 1 of [2].
For Part (b), note that
ˆ
P
N
(s
k
| Z
k−1
) = c
N
◦ b
k
(π
N
k|k−1
)
and P (s
k
| Z
k−1
) = b
k
(π
k|k−1
), where the notations
c
N
, b
k
, π
k|k−1
are as in [2]. The result then follows from [2,
Eqn. (14)].
We note furthermore (see, e.g., [2, IV-B.1]) that the state-
ment in Theorem 1(b), for example, should be interpreted as
follows. For any continuous, bounded function φ on S
E
ˆ
P
N
(s
k
|Z
k−1
)
φ
N→∞
−→ E
P (s
k
|Z
k−1
)
φ. (9)
For additional background information on the convergence of
a particle filter, we refer the readers to [5], [6] and [7].
III. DETECTION PROBLEM AND THE LR-DETECTOR
The problem of determining whether or not an object/signal
is present in an observed situation/data amounts to a detection
problem, see, e.g., [8]. This can also be formulated as a
statistical hypothesis testing problem. Suppose, we want to
perform the detection over a fixed finite horizon M ∈ N
+
. In
other words, the null hypothesis is that there is no object/signal
present in the last M time points and the alternative is that
there is one.
Note that the measurement model (2) holds if there is an
object present. Without any object, the measurements will be
only noise. Thus, at each time step k, we want to test:
H
0
: No signal present, i.e.,
z
j
= v
j
, j = k − M + 1, . . . , k
versus
H
1
: There is a signal present, i.e.,
z
j
= h(s
j
, v
j
), j = k − M + 1, . . . , k.
A. Main Result / Convergence of LR
Under the Neyman-Pearson (NP) paradigm, an optimal test
procedure is based on the likelihood ration (LR) given by
L(z
k−M +1
, . . . , z
k
) =
p(z
k−M +1
, . . . , z
k
| H
1
)
p(z
k−M +1
, . . . , z
k
| H
0
)
. (10)
The likelihood ratio test rejects the null hypothesis if L > τ ,
where τ is chosen in such a way that the probability of false
alarm is below a certain acceptable threshold. The Neyman-
Pearson Lemma (see, e.g., [8]) states that this procedure has
the highest detection probability among all the procedures with
false alarm probability below the threshold.
The authors in [1] have exploited the running particle filter
to approximate the LR as
L(z
k−M +1
, . . . , z
k
) ≈
ˆ
L
N
:=
Π
k
j=k−M +1
(
P
N
i=1
˜q
i
j
)
N
M
Π
k
j=k−M +1
p
v
(z
j
)
(11)
where ˜q
i
j
and p
v
(·) are as given in (6) and (4), respectively.
The authors have subsequently used the approximated LR to
perform the hypothesis test. However, as pointed out in the
introduction, an explicit asymptotic property of
ˆ
L
N
is missing
in the literature on this topic.
The following theorem provides such an asymptotic result.
Theorem 2. Consider the dynamical system (1) – (4) and
suppose the conditions of Theorem 1 hold. Let the PF output
be given by (6) and
ˆ
L
N
≡
ˆ
L
N
(z
k−M +1
, . . . , z
k
) be as given
in (11). Then
lim
N→∞
ˆ
L
N
(z
k−M +1
, . . . , z
k
) = L(z
k−M +1
, . . . , z
k
). (12)
Proof. Observe that under the null hypothesis H
0
p(z
k−M +1
, . . . , z
k
| H
0
) = Π
k
j=k−M +1
p
v
(z
j
), (13)
and under the alternative hypothesis
p(z
k−M +1
, . . . , z
k
| H
1
) = Π
k
j=k−M +1
p(z
j
| Z
j−1
), (14)
IEEE SIGNAL PROCESSING LETTERS, VOL . XX, NO. Y, MONTH YEAR 3
where p(z
j
| Z
j−1
) can be determined from the measurement
equation (2) as follows:
p(z
j
| Z
j−1
) =
Z
S
p(z
j
| s
j
) p(s
j
| Z
j−1
) ds
j
= E
P (s
j
|Z
j−1
)
p(z
j
| ·)
= lim
N→∞
E
ˆ
P
N
(s
j
|Z
j−1
)
p(z
j
| ·) [using (9)]
= lim
N→∞
N
X
i=1
Z
S
1
N
δ(s
j
− s
i
j
) p(z
j
| s
j
) ds
j
[by (8)]
= lim
N→∞
1
N
N
X
i=1
p(z
j
| s
i
j
) = lim
N→∞
1
N
N
X
i=1
˜q
i
j
[by (6)].
The result then follows from the fact that the limit of the
product of (finite number of) sequences equals the (finite)
product of the limits.
We have thus proven that the particle-filter-based likelihood
ratio is asymptotically equal to the true likelihood ratio.
B. Optimality
The particle-filter-based LR-detectors are especially useful
for a (radar) Track-Before-Detect application; see, e.g., [9].
Note that in our results/model, it is (implicitly) assumed that
an object/signal is either present or absent during the entire
M-sized observation window. This is fundamentally different
from, e.g., the assumptions made for a Bernoulli filter (see
[10]) where objects/signals can appear/disappear at any time
instant. The Bernoulli filter is Bayes optimal for the joint
estimation of object position and object existence. On the other
hand, the combination of a running particle filter together with
the PF-based LR-detector, as described above, will be Bayes
optimal
1
for the position estimation part and NP optimal for
the detection part. We stress that the NP optimality is highly
desired in a (radar) sensor application, because it keeps the
false alarm probability fixed at a specific (low) level and
maximizes the probability of detection. The latter quantities
are used, e.g., in (commercial) radar systems, see [12] and are
well known and well understood by the designers and end-
users alike. We can thus conclude that it is not possible to
construct another detection mechanism that performs strictly
better than the proposed LR-detector in both aspects.
It is worthwhile to remark here that one could design a de-
tector from the Bernoulli filter, by thresholding the probability
of existence of an object. See, for example, [13]. However, as
mentioned in [13, 1st paragraph, Section IV.D], the false alarm
probability generated by the detector cannot be guaranteed
to be fixed (at a low level). This could be considered as a
drawback of the Bernoulli-filter-based detector, if a guaranteed
(low) level of false probability is required. A comprehensive
comparison study between the two detectors remains as an
intersting future work.
1
Bayes optimality here refers [11] to the fact that the method provides
the posterior distribution of the object “positions”, and also, in a sequential
manner so that the posterior at the previous time instant can be used as the
prior in calculating the posterior for the current time. In the case of a PF,
the a posteriori density is reconstructed approximately, but in a sequential
manner.
IV. SIMULATION RESULTS
In this section, we validate the results through a simple
numerical example, namely, a linear-Gaussian system. The
reason is that for such a system we can determine the like-
lihood ratio exactly/analytically. In particular, the likelihoods
p(z
k
| Z
k−1
) can be calculated/evaluated by using the expres-
sions for the innovations in a running Kalman Filter (KF),
see e.g. [14]. This ground truth can then be compared to the
PF-based approximation. In particular, we take the absolute
difference between the true p(z
k
| Z
k−1
), as obtained from
the KF, and its PF-based approximation, as given in the proof
of Theorem 2. We call this difference ∆p(z
k
| Z
k−1
), see also
below. We are fully aware that our numerical simulation study
is not representative for a true and realistic target tracking
situation. Applying our technique to such an example and
comparing it to alternative methods, remains a future task for
now.
The linear-Gaussian system we consider is:
s
k+1
= s
k
+ w
k
(15)
z
k
= s
k
+ v
k
(16)
s
0
∼ N(µ, σ
2
0
) (17)
where the process and measurement noises are assumed to be
independent and zero mean Gaussian random variables with
standard deviations: σ
w
k
= 0.5 and σ
v
k
= 0.5, respectively.
Once a set of synthetic data set is generated, we run a Kalman
filter and a few particle filters with varying number of particles
on the (same) measurement data.
The simulation results are presented below. Table I provides
the approximation error of the particle-based likelihood as a
function of the number of particles used. The likelihoods – the
true (from KF) and the PF-based approximations with N =
100 and N = 5000, for the first 10 time steps – are presented
in Fig. 2 and Fig. 4, respectively. Fig. 1 and Fig. 3 show the
corresponding PF-based state estimates, together with the KF
estimates (the true posterior mean).
Clearly, the figures and the table support the main conver-
gence result of Section III. Indeed, as stated in Theorem 2,
the PF-based LR converges to the true LR, as the number of
particles tends to infinity.
approximation error of p(z
k
| Z
k−1
)
N ∆p(z
k
| Z
k−1
)
100 0.012
1000 0.0047
5000 0.0014
20k 0.0010
50k 0.00047
100k 0.00028
500k 0.00013
TABLE I
ERROR OF THE PARTICLE-BASED LIKELIHOOD AS A FUNCTION OF THE
NUMBER OF PARTICLES
IEEE SIGNAL PROCESSING LETTERS, VOL . XX, NO. Y, MONTH YEAR 4
Fig. 1. Particle filter output – individual particles (N = 100) and PF-means
(black squares) – and Kalman filter estimates (red diamonds). The horizontal
axis denotes time-step k and vertical axis: (simulated) state.
Fig. 2. True, Kalman innovation based, likelihood p(z
k
| Z
k−1
) (red squares)
and particle-based approximation (black diamonds) with N = 100. The
horizontal axis denotes time-step k.
Fig. 3. Particle filter output – individual particles (N = 5000) and PF-means
(black squares) – and Kalman filter estimates (red diamonds). The horizontal
axis denotes time-step k and vertical axis: (simulated) state.
V. CONCLUSION
This letter deals with the detection problem in possibly
nonlinear and/or non-Gaussian systems. A particle filter com-
Fig. 4. True, Kalman innovation based, likelihood p(z
k
| Z
k−1
) (red squares)
and particle-based approximation (black diamonds) with N = 5000. The
horizontal axis denotes time-step k.
bined with a likelihood ratio detector is considered. This is
very appealing, e.g., in radar tracking of stealth/dim objects,
since the NP optimal detector directly relates to operationally
relevant quantities such as the probability of detection and the
probability of false alarm. The most important contributions
and conclusions of this letter are the following.
• It provides a proof of the convergence of the particle-
filter-based detector to the true NP-optimal LR detector
for nonlinear and/or nonGaussian systems/signals.
• It provides a numerical illustration/verification of the
main result by showing, via a simulation example, that
the PF-based likelihoods converge to the true likelihoods.
An interesting topic for further research is to compare the
proposed method, which is based on an NP-optimal detector,
to one which is based on a Bernoulli filter [10], [13]. This,
however, would require some adjustments in the theory and/or
the application to allow for the appearing/disappearing targets.
Furthermore, one would also require a carefully constructed
criterion to be used for comparison, because, as mentioned in
Section III-B, the two mechanisms optimize different quanti-
ties/aspects of the dynamical system under consideration.
VI. ACKNOWLEDGEMENTS
• The authors wish to thank Mrs. Maryann Bjorklund for
proofreading and correcting the document, where needed.
Any errors left, language-wise or otherwise, are the sole
responsibility of the authors.
• The authors wish to thank Prof. Thomas Sch
¨
on for
the valuable discussions on the topic of convergence of
particle filters
• The first author also wishes to thank Prof. Fredrik
Gustafsson for inviting him to present the initial ideas
for this paper at the Department of Automatic Control,
Link
¨
oping University, Sweden.
IEEE SIGNAL PROCESSING LETTERS, VOL . XX, NO. Y, MONTH YEAR 5
REFERENCES
[1] Y. Boers and H. Driessen. “A particle-filter-based detection scheme”,
IEEE Signal Processing Letters, vol. 10, issue 10, pp. 300-302, 2003.
[2] D. Crisan and A. Doucet. “A survey of convergence results on parti-
cle filtering methods for practitioners”, IEEE Transactions on Signal
Processing, vol. 50, no. 3, pp. 736-746, 2002.
[3] A. Doucet, N. de Freitas an N. Gordon. (Eds.) Sequential Monte Carlo
Methods in Practice. Springer Verlag, 2001.
[4] H. Q. Liu, H. C. So, F. K. W. Chan, and K. W. K. Lui. “Distributed
particle filter for target tracking in sensor networks”, Progress In
Electromagnetics Research C, vol. 11, pp. 171-182, 2009.
[5] N. Chopin. “Central limit theorem for sequential Monte Carlo and its
application to Bayesian interference”, Annals of Statistics, vol. 32, no. 6,
pp. 2385-2411, 2004.
[6] X. Hu, T. Sch
¨
on and Lennart Ljung. “A General Convergence Result
for Particle Filtering”, IEEE Transactions on Signal Processing, vol. 50,
no. 3, pp. 1337-1348, 2002.
[7] P. Del Moral and M. Ledoux. “Convergence of Empirical Processes for
Interacting Particle Systems with Applications to Nonlinear Filtering”,
Journal of Theoretical Probability, vol. 3, issue 1, pp. 225-257, 2000.
[8] H.V. Poor. An Introduction to Signal Detection and Estimation. Springer,
1994.
[9] S. J. Davey, M. G. Rutten and N. J. Gordon. “Track-before-detect
techniques”, Chapter 8 in Integrated Tracking, Classification and Sensor
Management, Theory and Applications, M. Mallick, V. Krishnamurthy
and B. N. Vo (Eds.), IEEE Press, 2013.
[10] B. Ristic, B.-T Vo, B.-N. Vo and A. Farina. “A Tutorial on Bernoulli
Filters: Theory, Implementation and Applications”, IEEE Transactions
on Signal Processing, vol. 61, no. 13, pp. 3406-3430, 2013.
[11] B. Ristic, S. Arulampalam and N. Gordon. Beyond the Kalman Filter:
Particle Filters for Tracking Applications. Artech House, Boston, 2004.
[12] M. I. Skolnik. Introduction to Radar Systems. McGraw Hill, 2001.
[13] B. Li, S. Li, Y. Nan, A. Nallanathan, C. Zhao and Z. Zhou. “Deep
sensing for next-generation dynamic spectrum sharing: More than de-
tecting the occupancy state of primary spectrum”, IEEE Transactions
on Communications, vol. , no. 7, pp. 2442-2457, July 2015.
[14] B.D.O Anderson and J.B. Moore. Optimal Filtering. Englewood Cliffs,
NJ: Prentice Hall, 1979.
