Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks
Arnaud Doucet
Nando de Freitas
Kevin Murphy
Stuart Russell
Engineering Dept.
Cambridge University
[email protected]
Abstract
Particle filters (PFs) are powerful samplingbased inference/learning algorithms for dynamic
Bayesian networks (DBNs). They allow us to
treat, in a principled way, any type of probability distribution, nonlinearity and non-stationarity.
They have appeared in several fields under such
names as “condensation”, “sequential Monte
Carlo” and “survival of the fittest”. In this paper, we show how we can exploit the structure
of the DBN to increase the efficiency of particle filtering, using a technique known as RaoBlackwellisation. Essentially, this samples some
of the variables, and marginalizes out the rest
exactly, using the Kalman filter, HMM filter,
junction tree algorithm, or any other finite dimensional optimal filter. We show that RaoBlackwellised particle filters (RBPFs) lead to
more accurate estimates than standard PFs. We
demonstrate RBPFs on two problems, namely
non-stationary online regression with radial basis function networks and robot localization and
map building. We also discuss other potential application areas and provide references to some finite dimensional optimal filters.
1 INTRODUCTION
State estimation (online inference) in state-space models is
widely used in a variety of computer science and engineering applications. However, the two most famous algorithms
for this problem, the Kalman filter and the HMM filter, are
only applicable to linear-Gaussian models and models with
finite state spaces, respectively. Even when the state space
is finite, it can be so large that the HMM or junction tree
algorithms become too computationally expensive. This is
typically the case for large discrete dynamic Bayesian networks (DBNs) (Dean and Kanazawa 1989): inference requires at each time space and time that is exponential in the
Computer Science Dept.
UC Berkeley
jfgf,murphyk,russell @cs.berkeley.edu
number of hidden nodes.
To handle these problems, sequential Monte Carlo methods, also known as particle filters (PFs), have been introduced (Handschin and Mayne 1969, Akashi and Kumamoto 1977). In the mid 1990s, several PF algorithms
were proposed independently under the names of Monte
Carlo filters (Kitagawa 1996), sequential importance sampling (SIS) with resampling (SIR) (Doucet 1998), bootstrap
filters (Gordon, Salmond and Smith 1993), condensation
trackers (Isard and Blake 1996), dynamic mixture models
(West 1993), survival of the fittest (Kanazawa, Koller and
Russell 1995), etc. One of the major innovations during the
1990s was the inclusion of a resampling step to avoid degeneracy problems inherent to the earlier algorithms (Gordon et al. 1993). In the late nineties, several statistical improvements for PFs were proposed, and some important
theoretical properties were established. In addition, these
algorithms were applied and tested in many domains: see
(Doucet, de Freitas and Gordon 2000) for an up-to-date survey of the field.
One of the major drawbacks of PF is that sampling in
high-dimensional spaces can be inefficient. In some cases,
however, the model has “tractable substructure”, which
can be analytically marginalized out, conditional on certain other nodes being imputed, c.f., cutset conditioning in
static Bayes nets (Pearl 1988). The analytical marginalization can be carried out using standard algorithms, such
as the Kalman filter, the HMM filter, the junction tree algorithm for general DBNs (Cowell, Dawid, Lauritzen and
Spiegelhalter 1999), or, any other finite-dimensional optimal filters. The advantage of this strategy is that it can
drastically reduce the size of the space over which we need
to sample.
Marginalizing out some of the variables is an example of
the technique called Rao-Blackwellisation, because it is
related to the Rao-Blackwell formula: see (Casella and
Robert 1996) for a general discussion. Rao-Blackwellised
particle filters (RBPF) have been applied in specific contexts such as mixtures of Gaussians (Akashi and Kumamoto 1977, Doucet 1998, Doucet, Godsill and Andrieu
2000), fixed parameter estimation (Kong, Liu and Wong
1994), HMMs (Doucet 1998, Doucet, Godsill and Andrieu
2000) and Dirichlet process models (MacEachern, Clyde
and Liu 1999). In this paper, we develop the general theory
of RBPFs, and apply it to several novel types of DBNs. We
omit the proofs of the theorems for lack of space: please
refer to the technical report (Doucet, Gordon and Krishnamurthy 1999).
2 PROBLEM FORMULATION
Let us consider the following general state space
model/DBN with hidden variables and observed variables . We assume that is a Markov process of initial distribution and transition equation .
are assumed
The observations to be conditionally independent given the process of
marginal distribution . Given these observations,
the inference of any subset or property of the states relies on the joint posterior distribution
! " . Our objective is, therefore, to estimate this
distribution, or some of its characteristics such as the filtering density ! or the minimum mean square error
(MMSE) estimate #%$ & . The posterior satisfies the
following recursion
the alternative recursion
2>3/: 8 9 : 1 7 8@? 10A 5 67 8 ; 243B5 8 9 5 *8 ? 1 ;2>3/5 67 8*? 1 9 : 1 7 8@? 1 ;
243/567 89 : 1 7 8<;=
2>3/: 8 9 : 1 7 8*? 1 ;
(2)
If eq. (1) does not admit a closed-form expression, then eq.
(2) does not admit one either and sampling-based methods
are also required. Since the dimension of !+ " C is
smaller than the one of !"+ " , , we should expect
to obtain better results.
In the following section, we review the importance sampling (IS) method, which is the core of PF, and quantify the
improvements one can expect by marginalizing out , " i.e. using the so-called Rao-Blackwellised estimate. Subsequently, in Section 4, we describe a general RBPF algorithm and detail the implementation issues.
3 IMPORTANCE SAMPLING AND
RAO-BLACKWELLISATION
! ! ')( ! !*
! i.i.d.
random samIf we were able to sample D
S
(
R
L
L
J
"
C
P
N
O
Q
ples (particles), EGF+HJILK ,MHLIJK
DUT , according to
"
0
"
.
!+
,
, then an empirical estimate of this distribution would be given by
WVU"+ 0 , " C)(
(1)
If one attempts to solve this problem analytically, one obtains integrals that are not tractable. One, therefore, has to
resort to some form of numerical approximation scheme. In
this paper, we focus on sampling-based methods. Advantages and disadvantages of other approaches are discussed
at length in (de Freitas 1999).
The above description assumes that there is no structure
within the hidden variables. But suppose we can divide the hidden variables into two groups, + and , ,
such that "-( , + , *+ + and,
conditional on + " , the conditional posterior
distribution
!, " + . is analytically tractable. Then we can
easily marginalize out , " from the posterior, and only
need to focus on estimating !/+ , which lies in a
space of reduced dimension. Formally, we are making use
of the following decomposition of the posterior, which follows from the chain rule
!+ 0 , " ')( !", " " + . !+ " " '
The marginal posterior distribution !*+ " satisfies
1
The problem of how to automatically identify which variables should be sampled, and which can be handled analytically,
is one we are currently working on. We anticipate that algorithms
similar to cutset conditioning (Becker, Bar-Yehuda and Geiger
1999) might prove useful.
D
R XV
@i+ i , '
Z\[C]b'^`_Jc d'a e f b.^g_Lc d*a h
ILY
"
@i+ i , ' denotes the Dirac delta
where
Zj[C] b'^`_Jc da e f b.g^ _Jc ad h
function located at F +"HLIJ K ,M"HLIL K N . As a corollary, an
estimate of the filtering
distribution !+ , is
V
V "+ , k( Vml
*i+ i, . Hence
ILY Z [C] d^g_La e f d^n_Ja h
one
o can easily estimate the expected value of any function
o
of the hidden variables w.r.t. this distribution, pq . , using
o
p"V . U(sr
(
o " '
+
,
qVt+ " 0 , " . i+ i , R XV
o
F+HJIJ K ,M"HLIJ KCN
LI Y
This estimate is unbiased and,
o . from the strong law of
p
V
large numbers (SLLN),
converges almost surely
o .
D
vxw . If y z d (a.s.)
towards p4
as
o " " &S u
var{ ] b'c d e f b.c d'| }~c d $ *+
,
vxw , then a central
H
K
limit theorem (CLT) holds
€
o
o
†
z
D‚ p"Vt 'ƒ p4 '„…VG(‡‰
ˆGŠŒ‹@ y dŽ
where  †‘ denotes convergence in distribution. TypiD
cally, it is impossible to sample efficiently from the “target” posterior distribution P"+ " , at any time ’ .
So we focus on alternative methods.
o
One way to estimate !+ , and p4 consists of using the well-known importance sampling method
(Bernardo and Smith 1994). This method is based on the
following observation. Let us introduce an arbitrary importance distribution + " , . , from which it is easy
to get samples, and such that !+ " , " " '
 implies
"
"
.
"+
,
.
Then

'
b
c
d
.
b
# H ] e f c d | } ~Cc d K o *+ " , " + " , '
o
p (
# H ] b'c d e f b.c d'| }~Cc d K + " , .'
where the importance weight is equal to
!+ 0
+ " *+ , " (
, " , " i.i.d. samples EGF +"HJIL K ,MHJIJ K N T distributed accordo
ing to + , " , a Monte Carlo estimate of p4 is given by
V o
F + " BH IJ K , "HJIL KCN F + HJIL K , "HBIJ K.N
ILY
V F +"HLIJ K ,M"HJIL K N
l
ILY
X V (
HJIL K o F + "HJIL K , HJIL K N
ILY
where the normalized importance weights HBIJ K are equal to
o
p V )(
V V o ' (
o .
"HBIJ K (
l
F + LH IL K , " LH IJ K N
V
l Y
F + H K ,M"H K N
WVU"+ " 0 , .)(
XV
o
p V "(
(
o . V
o . V o
V N N
# { f b'c d | } ~c d e 'b c d h F F + HJIJ K , " q
l
[
] n^ _Ja
ILY
V F +" JH IL K N
l
ILY
F+ "HJIL K N
where
!+ " '
+ " *+ " U(
+ " .
(
r "+ " , " ' i, o
Intuitively, to reach a given precision, p V . will require
o
as we only
a reduced number D of samples over p V need to sample from a lower-dimensional distribution. This
is proven in the following propositions.
Proposition 1 The variances of the importance weights,
the numerators and the denominators satisfy for any D
var b'c d ~c d 3
3/5 67 8 ;C;
$%&'( 3)<8<;*
var b'c d ~c d
$ %,-( ' 3) 8 ;*
var b'c d ~c d
var b'c d b'c d! ~Cc d" 3
3/5 67 8 A"#
+
$
&
%
var b'c d b'c d! ~Cc d" (1 3)@8;*
var b'c d b'c d! ~Cc d" $ ,% ( 1 3) 8 ;.*
67 8 ;C;
o
A
sufficient
condition
for
to
p
o " V 
satisfy a CLT is var{ ] b'c d e f b.c d | } ~Cc d
+
,
vxw
and + " , "  vxH w for any K + " , " (Bernardo and
o
Smith 1994). This trivially implies that p V also satisfies a CLT. More precisely, we get the following result.
This method is equivalent to the following point mass approximation of !+ , " .
Given D
closed-form expression, then the following alternative imo
portance sampling estimate of p can be used
HJIL K
@i+ " i , " '
Z [C] b.^n_Lc d.a e f b'^n_Jc d@a h
LI Y
For “perfect” simulation, that is + , " Œ(
!+ " , " , we would have "HBIJ K ( D
for any Q .
In practice, we will try to select the importance distribution as close as possible
target distribution in a given
too the
sense. For D finite, p V is biased (since
ito is a ratio of
estimates), but according to the SLLN, p V converges
o
asymptotically a.s. towards p4 . Under additional assumptions, a CLT also holds.
Now consider the case where one can marginalize out , analytically,
then we can propose an alternative estimate
o
for p ' with a reduced variance. As !+ " , " ' (
!+ " " ' !", " + " . , where !", " + " . is
a distribution that can be computed exactly, then an
approximation of !+ " ' yields straightforwardly
an approximation of !+ " , " . Moreover, if
o
#q{ f b'c d| }~Cc d e ] b'c d @+ , " can be evaluated in a
H
K
Proposition 2 Under
o
the assumptions given above, p V and
a CLT
o
€
o
(†
D F p V Mƒ p> N G
V ‡‰ˆ Š
o
€
o
( †
D F p V Mƒ p> N VG
‡ ˆŠ
‰
where y 0/
'
o
p V satisfy
‹@ y
‹@ y
Ž
Ž
y , y and y being given by
1 1 =32 4 b'c d b'c d ~c d65 3C3) 8 3/5 67 8 A!# 67 8 ;87:9 3) 8 ;C;; 3/5 67 8 A!#
'
1 ' =32 4 b'c d ~Cc d" 5 3C3=2?> b'c d ~Cc d b'c d" 3)<8 3/567 8 A# 67 8;C;
'<
7@9 3) 8 ;C; 8 3/5 67 8 ;C;
67 8 ;C;
'<
o
The Rao-Blackwellised estimate p V is usually
o . computationally more extensive to compute than p V so it is
of interest to know when, for a fixed computational complexity, one can expect to achieve variance reduction. One
has
y ƒ y (
#
o " ,
.b c d "~c d
b'c d  +
]H 'b c d | } C~ c d K 
ƒ p4 o '^ + , " a „ „
so that, accordingly to the intuition, it will be worth generally performing Rao-Blackwellisation when the average
conditional variance of the variable , " is high.
4 RAO-BLACKWELLISED PARTICLE
FILTERS
Given D particles (samples) + "HJIL K ,MHJIL K at time ’ ƒ
R , approximately distributed according to the distribution
*+ HJIJ K , "HJIL K " , RBPFs allow us to compute D
particles +HJIL K ,MHJIJ K Ž approximately distributed according
‹
to the posterior *+"HJIL K ,MHJIL K , at time ’ . This is accomplished with the algorithm shown below, the details of
which will now be explained.
4.1 IMPLEMENTATION ISSUES
4.1.1 Sequential importance sampling
If we restrict ourselves to importance functions of the following form
0
"+ " .)(
+ + 1 1 + 1 (3)
1 Y
we can obtain recursive formulas to evaluate + " (
*+ " and thus . The “incremental weight” is given by
32 !" + " P "+ " + "+ +
denotes the normalized version of , i.e. HLILK (
5 V !
H K
HBIJK . Hence we can perform importance
l
Y
sampling online.
4
Choice of the Importance Distribution
Generic RBPF
1. Sequential importance sampling step
For = AWA , sample:
$ 5 * 3/589 5 : 7 8;
8
67 8@? 1 A 1
and set:
For $ 5 67 8 * ! 5 8 A 5 6 7 8@?
1"
AjA# , evaluate the importance
=
weights up to a normalizing constant:
8 For =
2 3 5 6 7 8 9 : 1 7 8;
3 5 8 9 5 67 8@? 1 A : 1 7 8<;L2 3 5 6 7 8@? 1 9 : 1 7 8@? 1 ;
=$ AWA# , normalize the importance
weights:
% 8 & =
' )
8(
(
*,+ 1
* 8.
? 1
2. Selection step
Multiply/ suppress samples 3 5 6 7 8 ; with high/low
% importance weights 8 , respectively, to obtain % 5 6& 7 8 ;
3
random samples
approximately distributed
% according to 2 3 5 67 8 9 : 1 7 8<; .
3. MCMC step
Apply a Markov transition kernel with invariant
& distribution given by 2 3/5 67 8 9 : 1 7 8 ; to obtain 3/5 67 8 ; ./
There are infinitely many possible choices for "+ " ,
the only condition being that its supports must include that
of !"+ " . The simplest choice is to just sample from
the prior, !"+ + , in which case the importance weight
is equal to the likelihood, !" + " . This is the
most widely used distribution, since it is simple to compute,
but it can be inefficient, since it ignores the most recent
evidence, . Intuitively, many of our samples may end up
in a region of the space that has low likelihood, and hence
receive low weight; these particles are effectively wasted.
We can show that the “optimal” proposal distribution, in
the sense of minimizing the variance of the importance
weights, takes the most recent evidence into account:
Proposition 3 The distribution that minimizes the variance of the importance weights conditional upon + and is
!" + " ' !"+ " + !"+ + " (
P" + and the associated importance weight is
! + " 0( r ! + P"+ + i + Unfortunately, computing the optimal importance sampling
distribution is often too expensive. Several deterministic
approximations to the optimal distribution have been proposed, see for example (de Freitas 1999, Doucet 1998).
Degeneracy of SIS
The following proposition shows that, for importance functions of the form (3), the variance of *+ " . can only increase (stochastically) over time. The proof of this proposition is an extension of a Kong-Liu-Wong theorem (Kong
et al. 1994, p. 285) to the case of an importance function of
the form (3).
Proposition 4 The unconditional variance (i.e. with the
observations being interpreted as random variables)
of the importance weights @+ . increases over time.
In practice, the degeneracy caused by the variance increase
can be observed by monitoring the importance weights.
Typically, what we observe is that, after a few iterations,
one of the normalized importance weights tends to 1, while
the remaining weights tend to zero.
4.2 CONVERGENCE RESULTS
To avoid the degeneracy of the sequential importance sampling simulation method, a selection (resampling) stage
may be used to eliminate samples with low importance ratios and multiply samples with high importance ratios. A
selection scheme associates to each particle + HJIJ K a numV ( D .
ber of offsprings, say D
, such that l
D
I
I
J
I
Y
Several selection schemes have been proposed in the lit
erature. These schemes satisfy # ‹ D Ž ( D HJILK , but
I
their performance varies in terms of the variance of the
particles, var D Ž . Recent theoretical results in (Crisan,
‹ I
Del Moral and Lyons 1999) indicate that the restriction
# ‹ D Ž ( D HBIJK is unnecessary to obtain convergence reI
sults (Doucet et al. 1999). Examples of these selection
schemes include multinomial sampling (Doucet 1998, Gordon et al. 1993, Pitt and Shephard 1999), residual resampling (Kitagawa 1996, Liu and Chen 1998) and stratified
sampling (Kitagawa 1996). Their computational complexity is @D .
4.1.3 MCMC step
After the selection scheme at time ’ , we obtain D particles distributed marginally approximately according to
*+ . As discussed earlier, the discrete nature of the
approximation can lead to a skewed importance weights
distribution. That is, many particles have no offspring
(
(D
 ), whereas others have a ( large number of offI
spring, the extreme case being D
D for a particular
I
value Q . In this case, there is a severe reduction in the diversity of the samples. A strategy for improving the results involves introducing MCMC steps of invariant distribution *+ " " ' on each particle (Andrieu, de Freitas and
Doucet 1999b, Gilks and Berzuini 1998, MacEachern et al.
1999). The basic idea is that, by applying a Markov transition kernel, the total variation of the current distribution
with respect to the invariant distribution can only decrease.
Note, however, that we do not require this kernel to be ergodic.
lowing theorem is a straightforward consequence of Theorem 1 in (Crisan and Doucet 2000) which is an extension
of previous results in (Crisan et al. 1999).
Theorem 5 If the importance weights G are upper
bounded and if one uses one of the selection schemes de indescribed previously, then, for all ’ /  , there exists
N
pendent of D such that for any F
2
4.1.2 Selection step
be the space of bounded, Borel measurable
Let - . The folfunctions on
. We denote
!#"
)
(
$ 5 * 7&
67 8
%$
8
+ 1
$
8 3/5 67 8 ;"24305 67 8 9<:
'*)+
'
, 8- $ 8 1 7 8 ;' 5 67 8(
where the expectation is taken w.r.t. to the randomness introduced by the PF algorithm. This results shows that, under very lose assumptions, convergence of this general particle filtering method is ensured and that the convergence
rate of the method is independent of the dimension of the
state-space. However, usually increases exponentially
with time. If additional assumptions on the dynamic system under study are made (e.g. discrete state spaces), it
is possible to get uniform convergence results ( (
for
any ’ ) for the filtering distribution !, . We do not
pursue this here.
5 EXAMPLES
We now illustrate the theory by briefly describing two applications we have worked on.
5.1 ON-LINE REGRESSION AND MODEL
SELECTION WITH NEURAL NETWORKS
.
Consider a function approximation scheme consisting of
a mixture of radial basis functions (RBFs) and a linear
regression term. The number of basis functions, , their
centers, , the coefficients (weights of the RBF centers
plus regression terms), , and the variance of the Gaussian
noise on the output, y , can all vary with time, so we treat
them as latent random variables: see Figure 1. For details,
see (Andrieu, de Freitas and Doucet 1999a).
/
0
/ .
1
.
In (Andrieu et al. 1999a), we show that it is possible to
simulate , and with a particle filter and to compute the coefficients analytically using Kalman filters.
This is possible because the output of the neural network
is linear in , and hence the system is a conditionally linear Gaussian state-space model (CLGSSM), that is it is a
linear Gaussian state-space model conditional upon the location of the bases and the hyper-parameters. This leads to
an efficient RBPF that can be combined with a reversible
jump MCMC algorithm (Green 1995) to select the number
0
0
k0
k1
k2
k3
k4
µ0
µ1
µ2
µ3
µ4
α0
α1
α2
α3
α4
σ02
σ12
σ22
σ 32
σ42
y1
y2
y3
y4
x1
x2
x3
x4
Prediction
2
1
0
−1
−2
240
250
260
270
280
290
300
310
6
k
4
2
0
0
50
100
150
200
250
300
350
400
M2(2)
M3(2)
M1(1)
M2(1)
M3(1)
L1
L2
L3
Y1
Y2
Y3
Figure 3: A Factorial HMM with 3 hidden chains. Q represents the color of grid cell Q at time ’ , represents
the robot’s location, and the current observation.
Figure 1: DBN representation of the RBF model. The
hyper-parameters have been omitted for clarity.
230
M1(2)
450
500
sensors are not perfect (they may accidentally flip bits), nor
are the motors (the robot may fail to move in the desired direction with some probability due e.g., to wheel slippage).
Consequently, it is easy for the robot to get lost. And when
the robot is lost, it does not know what part of the matrix to
update. So we are faced with a chicken-and-egg situation:
the robot needs to know where it is to learn the map, but
needs to know the map to figure out where it is.
The problem of concurrent localization and map learning for mobile robots has been widely studied. In (Murphy 2000), we adopt a Bayesian approach, in which we
maintain a belief state over both the location of the robot,
R D , and the color of each grid cell, Q R D , Q ( R D , where D is the number
of cells, and D is the number of colors. The DBN we
are
using
is
shown
in Figure 3. The state space has size
V *D . Note that we can easily handle changing environments, since the map is represented as a random variable, unlike the more common approach, which treats the
map as a fixed parameter.
The observation model is ( M C , where M is
0.4
σ2
0.2
0
0
50
100
150
200
250
300
350
400
450
Time
Figure 2: The top plot shows the one-step-ahead output
predictions [—] and the true outputs [ ] for the RBF
model. The middle and bottom plots show the true values and estimates of the model order and noise variance
respectively.
of basis functions online. For example, we generated some
data from a mixture of 2 RBFs for ’ ( R  , and
R R
(
then from a single RBF for ’
; the method


was able to track this change, as shown in Figure 2. Further
experiments on real data sets are described in (Andrieu et
al. 1999a).
5.2 ROBOT LOCALIZATION AND MAP
BUILDING
Consider a robot that can move on a discrete, twodimensional grid. Suppose the goal is to learn a map of
the environment, which, for simplicity, we can think of as
a matrix which stores the color of each grid cell, which
can be either black or white. The difficulty is that the color
a function that flips its binary argument with some fixed
probability. In other words, the robot gets to see the color
of the cell it is currently at, corrupted by noise: is a
noisy multiplexer with acting as a “gate” node. Note
that this conditional independence is not obvious from the
graph structure in Figure 3(a), which suggests that all the
nodes in each slice should be correlated by virtue of sharing
a common observed child, as in a factorial HMM (Ghahramani and Jordan 1997). The extra independence information is encoded in ’s distribution, c.f., (Boutilier, Friedman, Goldszmidt and Koller 1996).
The basic idea of the algorithm is to sample with a PF,
and marginalize out the Q nodes exactly, which can be
done efficiently since they are conditionally independent
given :
(
3
8.3 " ; A jA '8 3 ;09 1 7 8 A 1 7 8<;= + 1 3 '8 3 ;09 1 7 8 A 1 7 8<;
Some results on a simple one-dimensional grid world are
Prob. location, i.e., P(L(t)=i | y(1:t))
Prob. location, i.e., P(L(t)=i | y(1:t)), 50 particles, seed 1
1
1
1
1
0.9
0.9
2
0.8
0.7
0.6
4
0.5
5
0.7
0.6
4
0.5
5
0.4
6
0.3
0.2
7
6
0.3
0.2
7
4
6
8 10
time t
a
12
14
16
0.7
0.6
4
0.4
6
0.3
0.2
7
0.1
8
0
2
4
6
8 10
time t
12
14
16
0
2
4
6
8 10
time t
b
12
14
16
c
Figure 4: Estimated position as the robot moves from cell
1 to 8 and back. The robot “gets stuck” in cell 4 for two
steps in a row on the outgoing leg of the journey (hence the
double diagonal), but the robot does not realize this until
it reaches the end of the “corridor” at step 9, where it is
able to relocalise. (a) Exact inference. (b) RBPF with 50
particles. (c) Fully-factorised BK.
shown in Figure 4. We compared exact Bayesian inference with the RBPF method, and with the fully-factorised
version of the Boyen-Koller (BK) algorithm (Boyen and
Koller 1998), which represents the belief state as a product
of marginals:
(
3 8 A 8 3 " ; A WA 8 3 ;09 : 1 7 8 ;<= 3 8 9 : 1 7 8 ;
famous ones, there exist numerous other dynamic systems
admitting finite dimensional filters. That is, the filtering
distribution can be estimated in closed-form at any time ’
using a fixed number of sufficient statistics. These include
0.5
5
0.1
8
0
2
0.8
3
0.4
0.1
8
2
0.8
3
grid cell i
3
0.9
grid cell i
2
grid cell i
BK Prob. location, i.e., P(L(t)=i | y(1:t))
1
1
+ 1
3
8 3 ;09 : 1 7 8 ;
Dynamic models for counting observations (Smith
and Miller 1986).
Dynamic models with a time-varying unknow covariance matrix for the dynamic noise (West and Harrison
1996, Uhlig 1997).
Classes of the exponential family state space models
(Vidoni 1999).
This list is by no means exhaustive. It, however, shows that
RBPFs apply to very wide class of dynamic models. Consequently, they have a big role to play in computer vision
(where mixtures of Gaussians arise commonly), robotics,
speech and dynamic factor analysis.
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approach to state estimation in switching environments, Automatica 13: 429–434.
Andrieu, C., de Freitas, J. F. G. and Doucet, A. (1999a). Sequential Bayesian estimation and model selection applied to neural networks, Technical Report CUED/FINFENG/TR 341, Cambridge University Engineering
Department.
We see that the RBPF results are very similar to the exact results, even with only 50 particles, but that BK gets
confused because it ignores correlations between the map
R
cells. We have obtained good results
a R
.learning


map (so the state space has size ) using only 100
particles (the observation model in the 2D case is that the
robot observes the colors of all the cells in a
neighborhood centered on its current location). For a more detailed
discussion of these results, please see (Murphy 2000).
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Higher Order Statistics Workshop, Ceasarea, Israel,
pp. 130–134.
5.3 CONCLUSIONS AND EXTENSIONS
Becker, A., Bar-Yehuda, R. and Geiger, D. (1999). Random
algorithms for the loop cutset problem.
RBPFs have been applied to many problems, mostly in
the framework of conditionally linear Gaussian state-space
models and conditionally finite state-space HMMs. That is,
they have been applied to models that, conditionally upon
a set of variables (imputed by the PF algorithm), admit a
closed-form filtering distribution (Kalman filter in the continuous case and HMM filter in the discrete case). One can
also make use of the special structure of the dynamic model
under study to perform the calculations efficiently using the
junction tree algorithm. For example, if one had evolving trees, one could sample the root nodes with the PF and
compute the leaves using the junction tree algorithm. This
would result in a substantial computational gain as one only
has to sample the root nodes and apply the juction tree to
lower dimensional sub-networks.
Although the previoulsy mentioned models are the most
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Boutilier, C., Friedman, N., Goldszmidt, M. and Koller,
D. (1996). Context-specific independence in bayesian
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Boyen, X. and Koller, D. (1998). Tractable inference
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Casella, G. and Robert, C. P. (1996). Rao-Blackwellisation
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Isard, M. and Blake, A. (1996). Contour tracking by
stochastic propagation of conditional density, European Conference on Computer Vision, Cambridge,
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Crisan, D., Del Moral, P. and Lyons, T. (1999). Discrete filtering using branching and interacting particle systems, Markov Processes and Related Fields
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Kanazawa, K., Koller, D. and Russell, S. (1995). Stochastic
simulation algorithms for dynamic probabilistic networks, Proceedings of the Eleventh Conference on
Uncertainty in Artificial Intelligence, Morgan Kaufmann, pp. 346–351.
de Freitas, J. F. G. (1999). Bayesian Methods for Neural Networks, PhD thesis, Department of Engineering, Cambridge University, Cambridge, UK.
Dean, T. and Kanazawa, K. (1989). A model for reasoning about persistence and causation, Artificial Intelligence 93(1–2): 1–27.
Doucet, A. (1998). On sequential simulation-based methods for Bayesian filtering, Technical Report CUED/FINFENG/TR 310, Department of Engineering, Cambridge University.
Doucet, A., de Freitas, J. F. G. and Gordon, N. J.
(2000). Sequential Monte Carlo Methods in Practice,
Springer-Verlag.
Doucet, A., Godsill, S. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian
filtering, Statistics and Computing 10(3): 197–208.
Doucet, A., Gordon, N. J. and Krishnamurthy, V.
(1999). Particle filters for state estimation of jump
Markov linear systems, Technical Report CUED/FINFENG/TR 359, Cambridge University Engineering
Department.
Ghahramani, Z. and Jordan, M. (1997). Factorial Hidden
Markov Models, Machine Learning 29: 245–273.
Gilks, W. R. and Berzuini, C. (1998). Monte Carlo inference for dynamic Bayesian models, Unpublished.
Medical Research Council, Cambridge, UK.
Kitagawa, G. (1996). Monte Carlo filter and smoother for
non-Gaussian nonlinear state space models, Journal
of Computational and Graphical Statistics 5: 1–25.
Kong, A., Liu, J. S. and Wong, W. H. (1994). Sequential imputations and Bayesian missing data problems, Journal of the American Statistical Association
89(425): 278–288.
Liu, J. S. and Chen, R. (1998). Sequential Monte Carlo
methods for dynamic systems, Journal of the American Statistical Association 93: 1032–1044.
MacEachern, S. N., Clyde, M. and Liu, J. S. (1999).
Sequential importance sampling for nonparametric
Bayes models: the next generation, Canadian Journal of Statistics 27: 251–267.
Murphy, K. P. (2000). Bayesian map learning in dynamic
environments, in S. Solla, T. Leen and K.-R. Müller
(eds), Advances in Neural Information Processing
Systems 12, MIT Press, pp. 1015–1021.
Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann.
Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle filters, Journal of the American Statistical Association 94(446): 590–599.
Smith, R. L. and Miller, J. E. (1986). Predictive records,
Journal of the Royal Statistical Society B 36: 79–88.
Uhlig, H. (1997). Bayesian vector-autoregressions with
stochastic volatility, Econometrica.
Gordon, N. J., Salmond, D. J. and Smith, A. F. M. (1993).
Novel approach to nonlinear/non-Gaussian Bayesian
state estimation, IEE Proceedings-F 140(2): 107–
113.
Vidoni, P. (1999). Exponential family state space models
based on a conjugate latent process, Journal of the
Royal Statistical Society B 61: 213–221.
Green, P. J. (1995). Reversible jump Markov chain Monte
Carlo computation and Bayesian model determination, Biometrika 82: 711–732.
West, M. (1993). Mixture models, Monte Carlo, Bayesian
updating and dynamic models, Computing Science
and Statistics 24: 325–333.
Handschin, J. E. and Mayne, D. Q. (1969). Monte Carlo
techniques to estimate the conditional expectation in
multi-stage non-linear filtering, International Journal
of Control 9(5): 547–559.
West, M. and Harrison, J. (1996). Bayesian Forecasting
and Dynamic Linear Models, Springer-Verlag.
On Sequential Monte Carlo Sampling Methods for Bayesian
Filtering
Arnaud Doucet (corresponding author) - Simon Godsill - Christophe Andrieu
Signal Processing Group, Department of Engineering
University of Cambridge
Trumpington Street, CB2 1PZ Cambridge, UK
Email: [email protected]
ABSTRACT
In this article, we present an overview of methods for sequential simulation
from posterior distributions. These methods are of particular interest in Bayesian
filtering for discrete time dynamic models that are typically nonlinear and nonGaussian. A general importance sampling framework is developed that unifies
many of the methods which have been proposed over the last few decades in
several different scientific disciplines. Novel extensions to the existing methods
are also proposed. We show in particular how to incorporate local linearisation
methods similar to those which have previously been employed in the deterministic filtering literature; these lead to very effective importance distributions.
Furthermore we describe a method which uses Rao-Blackwellisation in order
to take advantage of the analytic structure present in some important classes
of state-space models. In a final section we develop algorithms for prediction,
smoothing and evaluation of the likelihood in dynamic models.
1
Keywords: Bayesian filtering, nonlinear non-Gaussian state space models, sequential
Monte Carlo methods, importance sampling, Rao-Blackwellised estimates
I.
Introduction
Many problems in applied statistics, statistical signal processing, time series analysis and
econometrics can be stated in a state space form as follows. A transition equation describes
the prior distribution of a hidden Markov process {x k ; k ∈ }, the so-called hidden state
process, and an observation equation describes the likelihood of the observations {y k ; k ∈ },
k being a discrete time index. Within a Bayesian framework, all relevant information about
{x0 , x1 , . . . , xk } given observations up to and including time k can be obtained from the
posterior distribution p ( x0 , x1 , . . . , xk | y0 , y1 , . . . , yk ). In many applications we are interested
in estimating recursively in time this distribution and particularly one of its marginals, the socalled filtering distribution p ( xk | y0 , y1 , . . . , yk ). Given the filtering distribution one can then
routinely proceed to filtered point estimates such as the posterior mode or mean of the state.
This problem is known as the Bayesian filtering problem or the optimal filtering problem.
Practical applications include target tracking (Gordon et al., 1993), blind deconvolution of
digital communications channels (Clapp et al., 1999)(Liu et al., 1995), estimation of stochastic
volatility (Pitt et al., 1999) and digital enhancement of speech and audio signals (Godsill et
al., 1998).
Except in a few special cases, including linear Gaussian state space models (Kalman
filter) and hidden finite-state space Markov chains, it is impossible to evaluate these distributions analytically. From the mid 1960’s, a great deal of attention has been devoted
to approximating these filtering distributions, see for example (Jazwinski, 1970). The most
popular algorithms, the extended Kalman filter and the Gaussian sum filter, rely on analytical approximations (Anderson et al., 1979). Interesting work in the automatic control field
was carried out during the 1960’s and 70’s using sequential Monte Carlo (MC) integration
2
methods, see (Akashi et al., 1975)(Handschin et. al, 1969)(Handschin 1970)(Zaritskii et al.,
1975). Possibly owing to the severe computational limitations of the time these Monte Carlo
algorithms have been largely neglected until recently. In the late 80’s, massive increases
in computational power allowed the rebirth of numerical integration methods for Bayesian
filtering (Kitagawa 1987). Current research has now focused on MC integration methods,
which have the great advantage of not being subject to the assumption of linearity or Gaussianity in the model, and relevant work includes (Müller 1992)(West, 1993)(Gordon et al.,
1993)(Kong et al., 1994)(Liu et al., 1998).
The main objective of this article is to include in a unified framework many old and
more recent algorithms proposed independently in a number of applied science areas. Both
(Liu et al., 1998) and (Doucet, 1997) (Doucet, 1998) underline the central rôle of sequential
importance sampling in Bayesian filtering. However, contrary to (Liu et al., 1998) which emphasizes the use of hybrid schemes combining elements of importance sampling with Markov
Chain Monte Carlo (MCMC), we focus here on computationally cheaper alternatives. We
describe also how it is possible to improve current existing methods via Rao-Blackwellisation
for a useful class of dynamic models. Finally, we show how to extend these methods to
compute the prediction and fixed-interval smoothing distributions as well as the likelihood.
The paper is organised as follows. In section 2, we briefly review the Bayesian filtering
problem and classical Bayesian importance sampling is proposed for its solution. We then
present a sequential version of this method which allows us to obtain a general recursive
MC filter: the sequential importance sampling (SIS) filter. Under a criterion of minimum
conditional variance of the importance weights, we obtain the optimal importance function for
this method. Unfortunately, for numerous models of applied interest the optimal importance
function leads to non-analytic importance weights, and hence we propose several suboptimal
distributions and show how to obtain as special cases many of the algorithms presented in
the literature. Firstly we consider local linearisation methods of either the state space model
3
or the optimal importance function, giving some important examples. These linearisation
methods seem to be a very promising way to proceed in problems of this type. Secondly we
consider some simple importance functions which lead to algorithms currently known in the
literature. In Section 3, a resampling scheme is used to limit practically the degeneracy of
the algorithm. In Section 4, we apply the Rao-Blackwellisation method to SIS and obtain
efficient hybrid analytical/MC filters. In Section 5, we show how to use the MC filter to
compute the prediction and fixed-interval smoothing distributions as well as the likelihood.
Finally, simulations are presented in Section 6.
II.
A.
Filtering Via Sequential Importance Sampling
Preliminaries: Filtering for the State Space Model
The state sequence {xk ; k ∈ }, xk ∈
nx ,
is assumed to be an unobserved (hidden) Markov
process with initial distribution p (x 0 ) (which we subsequently denote as p ( x 0 | x−1 ) for notational convenience) and transition distribution p ( x k | xk−1 ), where nx is the dimension of
the state vector. The observations {y k ; k ∈ }, yk ∈
ny ,
are conditionally independent
given the process {xk ; k ∈ } with distribution p ( yk | xk ) and ny is the dimension of the
observation vector. To sum up, the model is a hidden Markov (or state space) model (HMM)
described by
p ( xk | xk−1 ) for k ≥ 0
(1)
p ( yk | xk ) for k ≥ 0
(2)
We denote by x0:n
{x0 , ..., xn } and y0:n
{y0 , ..., yn }, respectively, the state sequence and
the observations up to time n. Our aim is to estimate recursively in time the distribution
p ( x0:n | y0:n ) and its associated features including p ( x n | y0:n ) and expectations of the form
I (fn ) =
Z
fn (x0:n ) p ( x0:n | y0:n ) dx0:n
4
(3)
for any p ( x0:n | y0:n )-integrable fn :
( n+1)×nx
→ . A recursive formula for p ( x0:n | y0:n ) is
given by:
p ( x0:n+1 | y0:n+1 ) = p ( x0:n | y0:n )
p ( yn+1 | xn+1 ) p ( xn+1 | xn )
p ( yn+1 | y0:n )
(4)
The denominator of this expression cannot typically be computed analytically, thus rendering
an analytic approach infeasible except in the special cases mentioned above. It will later
be assumed that samples can easily be drawn from p ( x k | xk−1 ) and that we can evaluate
p ( xk | xk−1 ) and p ( yk | xk ) pointwise.
B.
Bayesian Sequential Importance Sampling (SIS)
Since it is generally impossible to sample from the state posterior p ( x 0:n | y0:n ) directly, we
o
n
(i)
adopt an importance sampling (IS) approach. Suppose that samples x0:n ; i = 1, ..., N are
drawn independently from a normalised importance function π ( x 0:n | y0:n ) which has the
same support as the state posterior. Then an estimate Ic
N (fn ) of the posterior expectation
I (fn ) is obtained using Bayesian IS (Geweke, 1989):
∗(i)
where wn
Ic
N (fn ) =
N
X
i=1
(i)
en(i) ,
fn x0:n w
∗(i)
wn
w
en(i) = P
N
∗(j)
j=1 wn
(5)
= p ( y0:n | x0:n ) p (x0:n ) /π ( x0:n | y0:n ) is the unnormalised importance weight.
Under weak assumptions Ic
N (fn ) converges to I (fn ), see for example (Geweke, 1989). How-
ever, this method is not recursive. We now show how to obtain a sequential MC filter using
Bayesian IS.
Suppose one chooses an importance function of the form
π ( x0:n | y0:n ) = π ( x0 | y0 )
n
Y
k=1
π ( xk | x0:k−1 , y0:k )
(6)
Such an importance function allows recursive evaluation in time of the importance weights as
successive observations yk become available. We obtain directly the sequential importance
sampling filter.
5
Sequential Importance Sampling (SIS)
For times k = 0, 1, 2, ...
(i)
(i)
(i)
• For i = 1, ..., N , sample xk ∼ π xk | x0:k−1 , y0:k and x0:k
(i)
(i)
x0:k−1 , xk
.
• For i = 1, ..., N , evaluate the importance weights up to a normalising constant:
(i)
(i) (i)
p yk | xk p xk xk−1
∗(i)
∗(i)
wk = wk−1
(i) (i)
π( xk x0:k−1 , y0:k )
(7)
• For i = 1, ..., N , normalise the importance weights:
(i)
w
ek
∗(i)
wk
=P
N
∗(j)
j=1 wk
(8)
A special case of this algorithm was introduced in 1969 by (Handschin et. al, 1969)(Handschin 1970). Many of the other algorithms proposed in the literature are later shown also to
be special cases of this general (and simple) algorithm. Choice of importance function is of
course crucial and one obtains poor performance when the importance function is not well
chosen. This issue forms the topic of the following subsection.
C.
Degeneracy of the algorithm
If Bayesian IS is interpreted as a Monte Carlo sampling method rather than as a Monte Carlo
integration method, the best possible choice of importance function is of course the posterior
distribution itself, p ( x0:k | y0:k ). We would ideally like to be close to this case. However,
for importance functions of the form (6), the variance of the importance weights can only
increase (stochastically) over time.
Proposition 1 The unconditional variance of the importance weights, i.e. with the observations y0:k being interpreted as random variables, increases over time.
6
The proof of this proposition is a straightforward extension of a Kong-Liu-Wong theorem
(Kong et al., 1994) to the case of an importance function of the form (6). Thus, it is impossible
to avoid a degeneracy phenomenon. In practice, after a few iterations of the algorithm, all
but one of the normalised importance weights are very close to zero and a large computational
effort is devoted to updating trajectories whose contribution to the final estimate is almost
zero.
D.
Selection of the importance function
To limit degeneracy of the algorithm, a natural strategy consists of selecting the importance function which minimises the variance of the importance weights conditional upon the
(i)
simulated trajectory x0:k−1 and the observations y0:k .
(i)
Proposition 2 π( xk | x0:k−1 , y0:k ) = p( xk | xk−1 , yk ) is the importance function which min∗(i)
imises the variance of the importance weight w k
(i)
conditional upon x0:k−1 and y0:k .
Proof. Straightforward calculations yield
varπ
(i)
xk |x0:k−1 ,y0:k
h
∗(i)
wk
i
 
2
Z p ( yk | xk ) p xk | x(i)
k−1


(i)
dxk − p2 yk | xk−1 
= 2

(i)
p ( yk | y0:k−1 )
π xk | y0:k , x0:k−1
∗(i)
wk−1
2
(i)
(i)
This variance is zero for π xk | y0:k , x0:k−1 = p xk | yk , xk−1 .
1.
Optimal importance function
(i)
The optimal importance function p xk | xk−1 , yk was introduced by (Zaritskii et al., 1975)
then by (Akashi et al., 1977) for a particular case. More recently, this importance function has
been used in (Chen et al., 1996)(Kong et al., 1994)(Liu et al., 1995). For this distribution,
∗(i)
∗(i)
(i)
we obtain using (7) for the importance weight w k = wk−1 p yk | xk−1 . The optimal
importance function suffers from two major drawbacks. It requires the ability to sample
(i)
(i)
from p xk | xk−1 , yk and to evaluate, up to a proportionality constant, p yk | xk−1 =
7
R
(i)
p ( yk | xk ) p xk | xk−1 dxk . This integral will have no analytic form in the general case.
Nevertheless, analytic evaluation is possible for the important class of models presented
below, the Gaussian state space model with non-linear transition equation.
Example 3 Nonlinear Gaussian State Space Models. Let us consider the following model:
where f :
nx
→
nx
xk = f (xk−1 ) + vk , vk ∼ N (0, Σv )
(9)
yk = Cxk + wk , wk ∼ N (0, Σw )
(10)
is a real-valued non-linear function, C ∈
ny ×nx
is an observation
matrix, and vk and wk are mutually independent i.i.d.Gaussian sequences with Σ v > 0 and
Σw > 0, Σv and Σw being assumed known. Defining
t −1
Σ−1 = Σ−1
v + C Σw C
t −1
mk = Σ Σ−1
v f (xk−1 ) + C Σw yk
(11)
(12)
one obtains
xk | xk−1 , yk ∼ N (mk , Σ)
(13)
and
−1
1
p ( yk | xk−1 ) ∝ exp − (yk − Cf (xk−1 ))t Σv + CΣw Ct
(yk − Cf (xk−1 ))
2
(14)
For many other models, such evaluations are impossible. We now present suboptimal
methods which allow approximation of the optimal importance function. Several Monte
Carlo methods have been proposed to approximate the importance function and the associated importance weight based on importance sampling (Doucet, 1997)(Doucet, 1998) and
Markov chain Monte Carlo methods (Berzuini et al., 1998)(Liu et al., 1998). These iterative algorithms are computationally intensive and there is a lack of theoretical convergence
results. However, these methods may be useful when non-iterative schemes fail. In fact, the
8
general framework of SIS allows us to consider other importance functions built so as to approximate analytically the optimal importance function. The advantages of this alternative
approach are that it is computationally less expensive than Monte Carlo methods and that
the standard convergence results for Bayesian importance sampling are still valid. There is no
general method to build suboptimal importance functions and it is necessary to build these
on a case by case basis, dependent on the model studied. To this end, it is possible to base
these developments on previous work in suboptimal filtering (Anderson et al., 1979)(West et
al., 1997), and this is considered in the next subsection.
2.
Importance distribution obtained by local linearisation
A simple choice selects as the importance function π ( x k | xk−1 , yk ) a parametric distribution
π ( xk | θ (xk−1 , yk )), with finite-dimensional parameter θ (θ ∈ Θ ⊂
and yk , θ :
nx × ny
n
) determined by xk−1
→ Θ being a deterministic mapping. Many strategies are possible based
upon this idea. To illustrate such methods, we present here two novel schemes that result in a
Gaussian importance function whose parameters are evaluated using local linearisations, i.e.
which are dependent on the simulated trajectory i = 1, ..., N . Such an approach seems to be
a very promising way of proceeding with many models, where linearisations are readily and
cheaply available. In the auxiliary variables framework of (Pitt and Shephard, 1999), related
‘suboptimal’ importance distributions are proposed to sample efficiently from a finite mixture
distribution approximating the filtering distribution. We follow here a different approach in
which the filtering distribution is approximated directly without resort to auxiliary indicator
variables.
Local linearisation of the state space model
We propose to linearise the model locally in
a similar way to the Extended Kalman Filter. However, in our case, this linearisation is
performed with the aim of obtaining an importance function and the algorithm obtained
9
still converges asymptotically towards the required filtering distribution under the usual
assumptions for importance functions.
Example 4 Let us consider the following model
where f :
nx
xk = f (xk−1 ) + vk , vk ∼ N (0nv ×1 , Σv )
(15)
yk = g (xk ) + wk , wk ∼ N (0nw ×1 , Σw )
(16)
nx ,
→
g :
nx
→
ny
are differentiable, vk and wk are two mutually
independent i.i.d. sequences with Σ v > 0 and Σw > 0. Performing an approximation up to
first order of the observation equation (Anderson et al., 1979), we get
yk = g (xk ) + wk
∂g (xk ) (xk − f (xk−1 )) + wk
' g (f (xk−1 )) +
∂xk xk =f (xk−1 )
(17)
We have now defined a new model with a similar evolution equation to (15) but with a linear
Gaussian observation equation (17), obtained by linearising g (x k ) in f (xk−1 ). This model
is not Markovian as (17) depends on x k−1 . However, it is of the form (9)-(10) and one can
perform similar calculations to obtain a Gaussian importance function π ( x k | xk−1 , yk ) ∼
N (mk , Σk ) with mean mk and covariance Σk evaluated for each trajectory i = 1, ..., N using
the following formula:
Σ−1
k
mk
=
Σ−1
v
+
"
#t
∂g (xk ) −1 ∂g (xk ) Σ
w
∂xk xk =f (xk−1 )
∂xk xk =f (xk−1 )
"
#t
∂g
(x
)
k = Σk Σ−1
Σ−1
v f (xk−1 ) +
w ×
∂xk xk =f (xk−1 )
!!
∂g (xk ) × yk − g (f (xk−1 )) +
f (xk−1 )
∂xk xk =f (xk−1 )
The associated importance weight is evaluated using (7).
10
(18)
(19)
(20)
Local linearisation of the optimal importance function
log p ( xk | xk−1 , yk ) is twice differentiable wrt xk on
0
l (x)
00
l (x)
nx .
We assume here that l (x k )
We define:
∂l (xk ) ∂xk xk =x
∂ 2 l (xk ) ∂xk ∂xtk xk =x
(21)
(22)
Using a second order Taylor expansion in x, we get :
t
1
l (xk ) ' l (x) + l0 (x) (xk − x) + (xk − x)t l00 (x) (xk − x)
2
(23)
The point x where we perform the expansion is arbitrary (but determined by a deterministic
mapping of xk−1 and yk ). Under the additional assumption that l 00 (x) is negative definite,
which is true if l (xk ) is concave, then setting
Σ (x) = −l00 (x)−1
(24)
m (x) = Σ (x)l 0 (x)
(25)
yields
t
1
l0 (x) (xk − x) + (xk − x)t l00 (x) (xk − x)
2
1
= C − (xk − x − m (x))t Σ−1 (x) (xk − x − m (x))
2
(26)
This suggests adoption of the following importance function:
π ( xk | xk−1 , yk ) = N (m (x) + x, Σ (x))
(27)
If p ( xk | xk−1 , yk ) is unimodal, it is judicious to adopt x as the mode of p ( x k | xk−1 , yk ), thus
m (x) = 0nx ×1 . The associated importance weight is evaluated using (7).
Example 5 Linear Gaussian Dynamic/Observations according to a distribution from the
exponential family. We assume that the evolution equation satisfies:
xk = Axk−1 + vk where vk ∼ N (0nv ×1 , Σv )
11
(28)
where Σv > 0 and the observations are distributed according to a distribution from the
exponential family, i.e.
p ( yk | xk ) = exp ykt Cxk − b (Cxk ) + c (yk )
where C is a real ny × nx matrix, b :
ny
→
and c :
ny
→
(29)
. These models have
numerous applications and allow consideration of Poisson or binomial observations, see for
example (West et al., 1997). We have
l (xk ) = C + ykt Cxk − b (Cxk ) −
1
(xk − Axk−1 )t Σ−1
v (xk − Axk−1 )
2
(30)
This yields
∂ 2 b (Cxk ) l (x) = −
∂xk ∂xtk x
00
00
= −b (x) −
k =x
− Σ−1
v
Σ−1
v
(31)
but b00 (x) is the covariance matrix of yk for xk = x, thus l 00 (x) is definite negative. One
can determine the mode x = x ∗ of this distribution by applying an iterative Newton-Raphson
method initialised with x(0) = xk−1 , which satisfies at iteration j:
−1 0
l x(j)
x(j+1) = x(j) − l00 x(j)
(32)
We now present two simpler importance functions which lead to algorithms which previously appeared in the literature.
3.
Prior importance function
A simple choice uses the prior distribution of the hidden Markov model as importance function. This is the choice made by (Handschin et. al, 1969)(Handschin 1970) in their seminal
work. This is one of the methods recently proposed in (Tanizaki et al., 1998). In this case,
∗(i)
∗(i)
(i)
we have π ( xk | x0:k−1 , y0:k ) = p ( xk | xk−1 ) and wk = wk−1 p yk | xk . The method is
often inefficient in simulations as the state space is explored without any knowledge of the
12
observations. It is especially sensitive to outliers. However, it does have the advantage that
the importance weights are easily evaluated. Use of the prior importance function is closely
related to the Bootstrap filter method of (Gordon et al., 1993), see Section III..
4.
Fixed importance function
An even simpler choice fixes an importance function independently of the simulated trajectories and of the observations. In this case, we have π ( x k | x0:k−1 , y0:k ) = π (xk ) and
∗(i)
wk
∗(i)
(i)
(i) (i)
(i)
= wk−1 p yk | xk p xk xk−1 /π xk
(33)
This is the importance function adopted by (Tanizaki, 1993)(Tanizaki, 1994) who present
this method as a stochastic alternative to the numerical integration method of (Kitagawa,
1987). The results obtained are rather poor as neither the dynamic of the model nor the
observations are taken into account and leads in most cases to unbounded (unnormalised)
importance weights which will give poor results (Geweke, 1989).
III.
Resampling
As has previously been illustrated, the degeneracy of the SIS algorithm is unavoidable. The
basic idea of resampling methods is to eliminate trajectories which have small normalised
importance weights and to concentrate upon trajectories with large weights. A suitable
measure of degeneracy of the algorithm is the effective sample size N ef f introduced in (Kong
et al., 1994)(Liu, 1996) and defined as:
Nef f =
N
=
1 + varπ( ·|y0:k ) (w∗ (x0:k ))
π ( ·|y0:k )
h
N
(w∗ (x0:k ))2
i ≤N
(34)
One cannot evaluate Nef f exactly but, an estimate Nef f of Nef f is given by:
Nef f =
1
N
N
1
PN ∗(i) 2 = PN (i) 2
bk
ek
i=1 w
i=1 w
13
(35)
When Nef f is below a fixed threshold Nthres , the SIR resampling procedure is used (Rubin,
1988). Note that it is possible to implement the SIR procedure exactly in O (N ) operations
by using a classical algorithm (Ripley, 1987 p. 96) and (Carpenter et al., 1997)(Doucet,
1997)(Doucet, 1998)(Pitt et al., 1999). Other resampling procedures which reduce the MC
variation, such as stratified sampling (Carpenter et al., 1997) and residual resampling (Liu
et al., 1998), may be applied as an alternative to SIR.
An appropriate algorithm based on the SIR scheme proceeds as follows at time k.
SIS/Resampling Monte Carlo filter
1. Importance sampling
(i)
(i)
(i)
ek ∼ π( xk | x0:k−1 , y0:k ) and x
e0:k
• For i = 1, ..., N , sample x
(i)
(i)
ek
x0:k−1 , x
.
• For i = 1, ..., N , evaluate the importance weights up to a normalising constant:
(i) (i)
(i)
ek x
ek−1
ek p x
p yk | x
∗(i)
∗(i)
(36)
wk = wk−1
(i) (i)
ek x
e0:k−1 , y0:k )
π( x
• For i = 1, ..., N , normalise the importance weights:
∗(i)
w
(i)
w
ek = P k
N
∗(j)
j=1 wk
• Evaluate Nef f using (35).
(37)
2. Resampling
If Nef f ≥ Nthres
(i)
(i)
e0:k for i = 1, ..., N .
• x0:k = x
otherwise
• For i = 1, ..., N , sample an index j (i) distributed according to the discrete distribution
(l)
with N elements satisfying Pr{j (i) = l} = w
ek for l = 1, ..., N .
14
(i)
j(i)
(i)
e0:k and wk =
• For i = 1, ..., N , x0:k = x
1
N.
If Nef f ≥ Nthres , the algorithm presented in Subsection B. is thus not modified and if
Nef f < Nthres the SIR algorithm is applied and one obtains
N
1 X
b
P ( dx0:k | y0:k ) =
δ x(i) (dx0:k )
N
0:k
(38)
i=1
Resampling procedures decrease algorithmically the degeneracy problem but introduce
practical and theoretical problems. From a theoretical point of view, after one resampling
step, the simulated trajectories are no longer statistically independent and so we lose the
simple convergence results given previously. Recently, (Berzuini et al., 1998) have however
established a central limit theorem for the estimate of I (f k ) obtained when the SIR procedure
is applied at each iteration. From a practical point of view, the resampling scheme limits
the opportunity to parallelise since all the particles must be combined, although the IS
o
n
(i)
e
steps can still be realized in parallel. Moreover the trajectories x0:k , i = 1, ..., N which
(i)
have high importance weights w
ek are statistically selected many times. In (38), numerous
(i )
(i )
trajectories x0:k1 and x0:k2 are in fact equal for i1 6= i2 ∈ [1, . . . , N ]. There is thus a loss of
“diversity”. Various heuristic methods have been proposed to solve this problem (Gordon et
al., 1993)(Higuchi, 1997).
IV.
Rao-Blackwellisation for Sequential Importance Sampling
In this section we describe variance reduction methods which are designed to make the most of
any structure within the model studied. Numerous methods have been developed for reducing
the variance of MC estimates including antithetic sampling (Handschin et. al, 1969)(Handschin 1970) and control variates (Akashi et al., 1975)(Handschin 1970). We apply here the
Rao-Blackwellisation method, see (Casella et al. 1996) for a general reference on the topic.
In a sequential framework, (MacEachern et al. 1998) have applied similar ideas for Dirichlet
15
process models and (Kong et al. 1994)(Liu et al. 1998) have used Rao-Blackwellisation for
fixed parameter estimation. We focus on its application to dynamic models. We show how it
is possible to successfully apply this method to an important class of state space model and
obtain hybrid filters where a part of the calculations is realised analytically and the other
part using MC methods.
The following method is useful for cases when one can partition the state x k as x1k , x2k and
analytically marginalize one component of the partition, say x 2k . For instance, as demonstrated in example 6, if one component of the partition is a conditionally linear Gaussian
state-space model then all the integrations can be performed analytically on-line using the
Kalman filter. Let us define xj0:n
xj0 , . . . , xjn . We can rewrite the posterior expectation
I (fn ) in terms of marginal quantities:
fn x10:n , x20:n p y0:n | x10:n , x20:n p x20:n x10:n dx20:n p x10:n dx10:n
R R
I (fn ) =
p y0:n | x10:n , x20:n p x20:n x10:n dx20:n p x10:n dx10:n
R
g(x10:n )p x10:n dx10:n
= R
p y0:n | x10:n p x10:n dx10:n
R R
where
g(x10:n )
Z
fn x10:n , x20:n p y0:n | x10:n , x20:n p x20:n x10:n dx20:n
(39)
Under the assumption that, conditional upon a realisation of x 10:n , g(x10:n ) and p y0:n | x10:n
can be evaluated analytically, two estimates of I (f n ) based on IS are possible. The first
“classical” one is obtained using as importance distribution π x10:n , x20:n y0:n :
PN
2,(i)
1,(i)
∗ x1,(i) , x2,(i)
w
,
x
f
x
n
0:n
0:n
0:n
0:n
i=1
Ic
(40)
N (fn ) =
PN
2,(i)
1,(i)
∗ x
0:n , x0:n
i=1 w
2,(i) 1,(i)
2,(i)
2,(i) 1,(i)
1,(i)
where w∗ x0:n , x0:n
∝ p x0:n , x0:n y0:n /π x0:n , x0:n y0:n . The second “Rao-
Blackwellised” estimate is obtained by analytically integrating out x 20:n and using as im
R
portance distribution π x10:n y0:n = π x10:n , x20:n y0:n dx20:n . The new estimate is given
by:
c
If
N (fn ) =
1,(i)
∗ x1,(i)
w
g
x
0:n
0:n
i=1
PN
1,(i)
∗ x
0:n
i=1 w
PN
16
(41)
1,(i)
1,(i) 1,(i) where w∗ x0:n ∝ p x0:n y0:n /π x0:n y0:n . Using the decomposition of the variance,
it is straightforward to show that the variances of the importance weights obtained by RaoBlackwellisation are smaller than those obtained using a direct Monte Carlo method (40), see
for example (Doucet 1997)(Doucet 1998)(MacEachern et al. 1998). We can use this method
to estimate I (fn ) and marginal quantities such as p x10:n y0:n .
One has to be cautious when applying the MC methods developed in the previous sec-
tions to the marginal state space x1k . Indeed, even if the observations y 0:n are independent
conditional upon x10:n , x20:n , they are generally no longer independent conditional upon the
single process x10:n . The required modifications are, however, straightforward. For exam-
ple, we obtain for the optimal importance function p x1k y0:k , x10:k−1 and its associated
importance weight p yk | y0:k−1 , x10:k−1 . We now present two important applications of this
general method.
Example 6 Conditionally linear Gaussian state space model
Let us consider the following model
p x1k x1k−1
(42)
x2k = Ak x1k x2k−1 + Bk x1k vk
yk = Ck x1k x2k + Dk x1k wk
(43)
(44)
where x1k is a Markov process, vk ∼ N (0nv ×1 , Inv ) and wk ∼ N (0nw ×1 , Inw ). One wants to
t x2n x2n y0:n . It is possible
f x1n y0:n ,
x2n y0:n and
estimate p x10:n y0:n ,
to use a MC filter based on Rao-Blackwellisation. Indeed, conditional upon x 10:n , x20:n is a
linear Gaussian state space model and the integrations required by the Rao-Blackwellisation
method can be realized using the Kalman filter.
Akashi and Kumamoto (Akashi et al., 1977)(Tugnait, 1982) introduced this algorithm
under the name of RSA (Random Sampling Algorithm) in the particular case where x 1k is a
17
homogeneous scalar finite state-space Markov chain. In this case, they adopted the optimal
importance function p x1k y0:k , x10:k−1 . Indeed, it is possible to sample from this discrete
distribution and to evaluate the importance weight p yk | y0:k , x10:k−1 using the Kalman filter
(Akashi et al., 1977). Similar developments for this special case have also been proposed by
(Svetnik, 1986)(Billio et al., 1998)(Liu et al., 1998). The algorithm for blind deconvolution
proposed by ( Liu et al., 1995) is also a particular case of this method where x 2k = h is a timeinvariant channel of Gaussian prior distribution. Using the Rao-Blackwellisation method in
this framework is particularly attractive as, while x k has some continuous components, we
restrict ourselves to the exploration of a discrete state space.
Example 7 Finite State-Space HMM
Let us consider the following model
p x1k x1k−1
(45)
p x2k x1k , x2k−1
p yk | x1k , x2k
(46)
(47)
where x1k is a Markov process and x2k is a finite state-space Markov chain whose param
eters at time k depend on x1k . We want to estimate p x10:n y0:n ,
f x1n y0:n and
f x2n y0:n . It is possible to use a “Rao-Blackwellised” MC filter. Indeed, conditional
upon x10:n , x20:n is a finite state-space Markov chain of known parameters and thus the integrations required by the Rao-Blackwellisation method can be done analytically (Anderson et
al., 1979).
18
V.
Prediction, smoothing and likelihood
The estimate of the joint distribution p ( x 0:k | y0:k ) based on SIS, in practice coupled with a
resampling procedure to limit the degeneracy, is at any time k of the following form:
Pb ( dx0:k | y0:k ) =
N
X
i=1
(i)
w
ek δ x(i) (dx0:k )
(48)
0:k
We show here how it is possible to obtain based on this distribution some approximations of
the prediction and smoothing distributions as well as the likelihood.
A.
Prediction
Based on the approximation of the filtering distribution Pb ( dxk | y0:k ), we want to estimate
the p step-ahead prediction distribution, p ≥ 2 ∈ ∗ , given by:


Z
k+p
Y
p ( xk+p | y0:k ) = p ( xk | y0:k ) 
p ( xj | xj−1 ) dxk:k+p−1
(49)
j=k+1
Replacing p ( xk | y0:k ) in (49) by its approximation obtained from (48), we obtain:
N
X
i=1
(i)
w
ek
Z
k+p
Y
(i)
p xk+1 | xk
p ( xj | xj−1 ) dxk+1:k+p−1
(50)
j=k+2
(i)
To evaluate these integrals, it is sufficient to extend the trajectories x 0:k using the evolution
equation.
p step-ahead prediction
• For j = 1 to p
(i)
(i)
(i)
– For i = 1, ..., N , sample xk+j ∼ p xk+j | xk+j−1 and x0:k+j
We obtain random samples
given by
n
(i)
(i)
x0:k+j−1 , xk+j .
o
(i)
x0:k+p ; i = 1, ..., N . An estimate of Pb ( dx0:k+p | y0:k ) is
Pb ( dx0:k+p | y0:k ) =
N
X
i=1
19
(i)
w
ek δ x(i)
0:k+p
(dx0:k+p )
Thus
Pb ( dxk+p | y0:k ) =
B.
N
X
(i)
w
ek δ x(i) (dxk+p )
(51)
k+p
i=1
Fixed-Lag smoothing
We want to estimate the fixed-lag smoothing distribution p ( x k | y0:k+p ), p ∈
length of the lag.
∗
being the
At time k + p, the MC filter yields the following approximation of
p ( x0:k+p | y0:k+p ):
Pb ( dx0:k+p | y0:k+p ) =
N
X
i=1
(i)
w
ek+p δ x(i)
0:k+p
(dx0:k+p )
(52)
By marginalising, we obtain an estimate of the fixed-lag smoothing distribution:
Pb ( dxk | y0:k+p ) =
N
X
i=1
(i)
w
ek+p δ x(i) (dxk )
(53)
k
When p is high, such an approximation will generally perform poorly.
C.
Fixed-interval smoothing
Given y0:n , we want to estimate p ( xk | y0:n ) for any k = 0, ..., n. At time n, the filtering
algorithm yields the following approximation of p ( x 0:n | y0:n ) :
Pb ( dx0:n | y0:n ) =
N
X
i=1
w
en(i) δ x(i) (dx0:n )
(54)
0:n
Thus one can theoretically obtain p ( x k | y0:n ) for any k by marginalising this distribution.
Practically, this method cannot be used as soon as (n − k) is significant as the degeneracy problem requires use of a resampling algorithm. At time n, the simulated trajectories
o
n
(i)
x0:n ; i = 1, ..., N have been usually resampled many times: there are thus only a few dis-
tinct trajectories at times k for k n and the above approximation of p ( x k | y0:n ) is bad.
This problem is even more severe for the bootstrap filter where one resamples at each time
instant.
20
It is necessary to develop an alternative algorithm. We propose an original algorithm to
solve this problem. This algorithm is based on the following formula (Kitagawa, 1987):
p ( xk | y0:n ) = p ( xk | y0:k )
Z
p ( xk+1 | y0:n ) p ( xk+1 | xk )
dxk+1
p ( xk+1 | y0:k )
(55)
We seek here an approximation of the fixed-interval smoothing distribution with the following
form:
Pb ( dxk | y0:n )
i.e. Pb ( dxk | y0:n ) has the same support
n
N
X
i=1
(i)
(i)
w
e k|n δ x(i) (dxk )
(56)
k
xk ; i = 1, . . . , N
o
as the filtering distribution
n
o
(i)
Pb ( dxk | y0:k ) but the weights are different. An algorithm to obtain these weights w
e k|n ; i = 1, . . . , N
is the following.
Fixed-interval smoothing
1. Initialisation at time k = n.
(i)
(i)
• For i = 1, ..., N , w
e n|n = w
en .
2. For k = n − 1, ..., 0.
• For i = 1, ..., N , evaluate the importance weight
(i)
(j) (i)
N
w
ek p xk+1 xk
X
(i)
(j)
i
w
e k|n =
w
e k+1|n hP
(l)
(j) (l)
N
x
w
e
p
x
j=1
l=1 k
k
k+1
(57)
This algorithm is obtained by the following argument. Replacing p ( x k+1 | y0:n ) by its
approximation (56) yields
Z
p ( xk+1 | y0:n ) p ( xk+1 | xk )
dxk+1
p ( xk+1 | y0:k )
21
(i) p xk+1 xk
(i)
'
w
e k+1|n (i) p
x
y
0:k
i=1
k+1
N
X
(58)
(i) where, owing to (48), p xk+1 y0:k can be approximated by
p
(i) xk+1 y0:k
Z
=
(i) p xk+1 xk p ( xk | y0:k ) dxk
N
X
'
j=1
(59)
(j)
(i) (j)
w
ek p xk+1 xk
An approximation Pb ( dxk | y0:n ) of p ( xk | y0:n ) is thus
Pb ( dxk | y0:n )
# N
"N
X (j)
X (i)
w
e k+1|n hP
w
ek δ x(i) (dxk )
=
(j) p xk+1 xk
i
(l)
(j) (l)
N
k
x
w
e
p
x
j=1
i=1
l=1 k
k
k+1


(j)
(i)
N
N
p xk+1 xk
X
X
(i)
(j)
i  δ (i) (dxk )
=
w
ek 
w
e k+1|n hP
xk
(l)
(j) (l)
N
w
e
p
x
x
i=1
j=1
l=1 k
k+1
k
N
X
i=1
The algorithm follows.
(60)
(i)
w
e k|n δ x(i) (dxk )
k
This algorithm requires storage of the marginal distributions Pb ( dxk | y0:k ) (weights and
supports) for any k = 0, ..., n. The memory requirement is O (nN ). Its complexity is
O nN 2 , which is quite important as N 1. However this complexity is a little lower
than the one of the previous developed algorithms of (Kitagawa et al., 1996) and (Tanizaki
et al., 1998) as it does not require any new simulation step.
D.
Likelihood
In some applications, in particular for model choice (Kitagawa, 1987)(Kitagawa et al., 1996),
we may wish to estimate the likelihood of the data
p (y0:n ) =
Z
wn∗ π ( x0:n | y0:n ) dx0:n
A simple estimate of the likelihood is thus given by
pb (y0:n ) =
N
1 X (j)
wn
N
22
j=1
(61)
In practice, the introduction of resampling steps makes this approach impossible. We will
use an alternative decomposition of the likelihood:
p (y0:n ) = p (y0 )
n
Y
k=1
p ( yk | y0:k−1 )
(62)
where:
p ( yk | y0:k−1 ) =
=
Z
Z
p ( yk | xk ) p ( xk | y0:k−1 ) dxk
(63)
p ( yk | xk−1 ) p ( xk−1 | y0:k−1 ) dxk−1
(64)
Using (63), an estimate of this quantity is given by
N
X
(i)
(i)
ek w
pb ( yk | y0:k−1 ) =
p yk | x
ek−1
(65)
i=1
n
o
(i)
ek ; i = 1, . . . , N are obtained using a one-step ahead prediction based
where the samples x
on the approximation Pb ( dxk−1 | y0:k−1 ) of p ( xk−1 | y0:k−1 ). Using expression (64), it is pos
(i)
sible to avoid a MC integration if we know analytically p yk | xk−1 :
N
X
(i)
(i)
pb ( yk | y0:k−1 ) =
ek−1
p yk | xk−1 w
(66)
i=1
VI.
Simulations
In this section, we apply the methods developed previously to a linear Gaussian state space
model and to a classical nonlinear model. We make for these two models M = 100 simulations
of length n = 500 and we evaluate the empirical standard deviation for the filtering estimates
x k|k =
[ xk | y0:k ] obtained by the MC methods:
√

1/2
n
M X
X
2
1
1
xjk|l − xjk 
V AR x k|l =
n
M
j=1
k=1
where:
• xjk is the simulated state for the j th simulation, j = 1, ..., M .
23
• xjk|l
PN
(i) j,(i)
e k|l xk
i=1 w
is the MC estimate of
j,(i)
[ xk | y0:l ] for the j th test signal and xk
is the ith simulated trajectory, i = 1, ..., N , associated with the signal j. (We denote
(i)
w
e k|k
(i)
w
ek )
These calculations have been realised for N = 100, 250, 500, 1000, 2500 and 5000. The
implemented filtering algorithms are the bootstrap filter, the SIS with the prior importance
function and the SIS with the optimal or a suboptimal importance function. The fixedinterval smoothers associated with these SIS filters are then computed.
For the SIS-based algorithms, the SIR procedure has been used when Nef f < Nthres =
N/3. We state the percentage of iterations where the SIR step is used for each importance
function.
A.
Linear Gaussian model
Let us consider the following model
xk = xk−1 + vk
(67)
yk = x k + w k
(68)
where x0 ∼ N (0, 1), vk and wk are white Gaussian noises mutually independent, v k ∼
N 0, σ 2v and wk ∼ N 0, σ 2w with σ 2v = σ 2w = 1. For this model, the optimal filter is the
Kalman filter (Anderson et al., 1979).
1.
Optimal importance function
The optimal importance function is
xk | xk−1 , yk ∼ N mk , σ 2k
(69)
where
−2
σ −2
= σ −2
w + σv
k
yk
2 xk−1
mk = σ k
+ 2
σ 2v
σw
24
(70)
(71)
and the associated importance weight is equal to:
1 (yk − xk−1 )2
p ( yk | xk−1 ) ∝ exp −
2 (σ 2v + σ 2w )
2.
!
(72)
Results
For the Kalman filter, we obtain
√
V AR x k|k = 0.79. For the different MC filters, the
results are presented in Table 1 and Table 2.
With N = 500 trajectories, the estimates obtained using MC methods are similar to those
obtained by Kalman. The SIS algorithms have similar performances to the bootstrap filter
for a smaller computational cost. The most interesting algorithm is based on the optimal
importance function which limits seriously the number of resampling steps.
B.
Nonlinear series
We consider here the following nonlinear reference model (Gordon et al., 1993)(Kitagawa,
1987)(Tanizaki et al., 1998):
xk = f (xk−1 ) + vk
=
(73)
xk−1
1
xk−1 + 25
+ 8 cos (1.2k) + vk
2
1 + (xk−1 )2
yk = g (xk ) + wk
=
(74)
(xk )2
+ wk
20
where x0 ∼ N (0, 5), vk and wk are mutually independent white Gaussian noises, v k ∼
N 0, σ 2v and wk ∼ N 0, σ 2w with σ 2v = 10 and σ 2w = 1. In this case, it is not possible
to evaluate analytically p ( yk | xk−1 ) or to sample simply from p ( xk | xk−1 , yk ). We propose
to apply the method described in 2. which consists of linearising locally the observation
equation.
25
1.
Importance function obtained by local linearisation
We get
yk
∂g (xk ) ' g (f (xk−1 )) +
(xk − f (xk−1 )) + wk
∂xk xk =f (xk−1 )
f 2 (xk−1 ) f (xk−1 )
+
(xk − f (xk−1 )) + wk
20
10
f 2 (xk−1 ) f (xk−1 )
= −
+
xk + w k
20
10
=
(75)
Then we obtain the linearised importance function π ( x k | xk−1 , yk ) = N xk ; mk , (σ k )2
where
−2
(σ k )−2 = σ −2
v + σw
f 2 (xk−1 )
100
(76)
and
mk = (σ k )
2.
2
σ −2
v f
(xk−1 ) +
f
σ −2
w
(xk−1 )
10
f 2 (xk−1 )
yk +
20
(77)
Results
In this case, it is not possible to estimate the optimal filter. For the MC filters, the results
are displayed in Table 3. The average percentages of SIR steps are presented in Table 4.
This model requires simulation of more samples than the preceding one. In fact, the
variance of the dynamic noise is more important and more trajectories are necessary to
explore the space. The most interesting algorithm is the SIS with a suboptimal importance
function which greatly limits the number of resampling steps over the prior importance
function while avoiding a MC integration step needed to evaluate the optimal importance
function. This can be roughly explained by the fact that the observation noise is rather small
so that yk is highly informative and allows a limitation of the regions explored.
VII.
Conclusion
We have presented an overview of sequential simulation-based methods for Bayesian filtering
of general state-space models. We include, within the general framework of SIS, numer26
ous approaches proposed independently in the literature over the last 30 years. Several
original extensions have also been described, including the use of local linearisation techniques to yield more effective importance distributions. We have shown also how the use of
Rao-Blackwellisation allows us to make the most of any analytic structure present in some
important dynamic models and have described procedures for prediction, fixed-lag smoothing
and likelihood evaluation.
These methods are efficient but still suffer from several drawbacks. The first is the depletion of samples which inevitably occurs in all of the methods described as time proceeds.
Sample regeneration methods based upon MCMC steps are likely to improve the situation
here (MacEachern et al. 1998). A second problem is that of simulating fixed hyperparameters such as the covariance matrices and noise variances which were assumed known in our
examples. The methods described here do not allow for any regeneration of new values for
these non-dynamic parameters, and hence we can expect a very rapid impoverishment of
the sample set. Again, a combination of the present techniques with MCMC steps could be
useful here, as could Rao-Blackwellisation methods ((Liu et al. 1998) give some insight into
how this might be approached).
The technical challenges still posed by this problem, together with the wide range of
important applications and the rapidly increasing computational power, should stimulate
new and exciting developments in the field.
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31
VIII.
√
V AR x k|k
Tables
bootstrap
prior dist.
optimal dist.
N = 100
0.80
0.86
0.83
N = 250
0.81
0.81
0.80
N = 500
0.79
0.80
0.79
N = 1000
0.79
0.79
0.79
N = 2500
0.79
0.79
0.79
N = 5000
0.79
0.79
0.79
Table 1: MC filters: linear Gaussian model
32
Percentage SIR
prior dist.
optimal dist.
N = 100
40
16
N = 250
23
10
N = 500
20
8
N = 1000
15
6
N = 2500
13
5
N = 5000
11
4
Table 2: Percentage of SIR steps: linear Gaussian model
33
√
V AR x k|k
bootstrap
prior dist.
linearised dist.
N = 100
5.67
6.01
5.54
N = 250
5.32
5.65
5.46
N = 500
5.27
5.59
5.23
N = 1000
5.11
5.36
5.05
N = 2500
5.09
5.14
5.02
N = 5000
5.04
5.07
5.01
Table 3: MC filters: nonlinear time series
34
Percentage SIR
prior dist.
linearised dist.
N = 100
22.4
8.9
N = 250
19.6
7.5
N = 500
17.7
6.5
N = 1000
15.6
5.9
N = 2500
13.9
5.2
N = 5000
12.3
5.3
Table 4: Percentage of SIR steps: nonlinear time series
35
SEQUENTIAL MONTE CARLO SIMULATION OF DYNAMICAL MODELS WITH SLOWLY
VARYING PARAMETERS: AN EXTENSION
William Fong and Simon Godsill
Signal Processing Group, University of Cambridge,
Cambridge, CB2 1PZ, U.K.
[email protected] [email protected]
ABSTRACT
In this paper, we improve on the slow time-varying partial correlation (STV-PARCOR) model recently suggested
by us [2] to include any deterministic interpolator. We then
suggest a modification to the on-line filtering algorithm to
accomodate the changes. It is believed that the modification
will improve on the simulation results as it takes into account the underlying trend of the parameter evolution. The
suggested algorithm is tested with real speech data and preliminary results are shown and compared with those generated using existing approaches.
1. INTRODUCTION
Many real world data analysis problems involve sequential
estimation of filtering distribution , where is
the unobserved state of the system at time and are observations made over some time interval . In most cases, the data structures can be
very complex, typically involving elements of non-Gaussianity,
high-dimensionality and non-linearity, which may not be
solvable analytically. Sequential Monte Carlo methods, also
known as Particle Filters (PF), have been proposed to overcome these problems. Refer to [1] for an up-to-date survey
of the field. Within the particle filter framework, the filtering
distribution is approximated with an empirical distribution
formed from point masses, or particles,
Æ where Æ is the Dirac delta function and is a weight
attached to particle .
In recent years, various approaches have been developed
to apply the sequential Monte Carlo filtering strategies for
the purpose of audio signal enhancement (see, for example, [3] and citations therein). These approaches assume
a Gaussian random walk model for the system parameter
evolution at every time step, which may not give a sufficiently slow or smooth variation with time. This makes
the standard PF inefficient as it is known that the filter becomes highly degenerate for random walks with very low
variance. Recently, Fong and Godsill [2] propose a slow
time-varying partial correlation (STV-PARCOR) model to
solve this problem. In their work, the system coefficients are
considered to evolve stochastically on a block-to-block basis and all coefficients in-between are found by linear interpolator. Based on the STV-PARCOR model and under the
particle filtering framework, an algorithm for on-line joint
estimation of system parameters signal is developed.
This work serves as an extension to the work of [2]. In
this paper, we generalised the STV-PARCOR particle filter, so that any deterministic interpolator can be used. We
then describe the modified algorithm for the generation of
“delayed” state realisations. Finally, preliminary simulation
results are shown.
2. STATE-SPACE REPRESENTATION AND AUDIO
MODEL
In this section, we describe the model adopted in this paper
— the STV-PARCOR model. A lengthy time series is divided into non-overlapping blocks. If is the block size,
we define as a group of unobserved states of the system and as observations made over some blocks .
Assuming a Markovian structure for the model, the problem can then be formulated in a state-space form as follows,
State evolution density
Observation density
(1)
where and are pre-specified state evolution and
observation densities. It should be noted that the state-space
model adopted here is different from the standard one [5],
which relates the unobserved states and observations
made over a time interval . (1), however, defines the state evolution between different blocks.
For the choice of audio model, we suggest a time varying partial correlation (TV-PARCOR) model. The advantage of adopting such model is that approximate stability
can easily be enforced, provided that the PARCOR coefficients vary sufficient slowly with time [3]. The audio
signal process is then modelled as
ensure a slow and smooth evolution of the reflection coefficients [4]. In [2], we have implemented a linear interpolator, which should be considered as a special case of (5).
We believe that (5) will give a better approximation than the
simplified linear interpolator case as it takes into account the
better underlying smooth trend of the coefficient evolution.
3. SEQUENTIAL AUDIO SIGNAL AND
PARAMETER ESTIMATION
where is the TVAR coefficient at time , which is found
by transforming the PARCOR coefficient via the LevinsonDurbin recursion. is the log-excitation variance and is the time-varying model order. Refer to [3] for a detailed
description of the audio model adopted here.
The full specification of the state-space model is as follows: at any time , the state vector is partitioned as
with and being the signal state and the parameter state respectively.
In the setup of the particle filter, a proposal distribution [1, 3] similar to that of [2] has been adopted, which
takes the form:
Based on the STV-PARCOR model, [2] describes a way for
joint estimation of the signal and parameter state under the
sequential Monte Carlo filter framework. The suggested algorithm proceeds as follows:
Random samples are to be drawn from the joint filtering
which can be factorised as foldistribution lows,
´
½µ
(3)
Æ (4)
where for ,
i.e. both and are assumed to be fixed within a block.
For the PARCOR coefficients, a constrained random walk
model [3] is assumed for the block variation,
if otherwise
where . For the interpolator functions, , the Legendre polynomials, Fourier basis
function and B-splines are popular choices, all of which will
Æ (7)
as it is assumed that the proposal distribution for the parameter state being the prior (2), the importance weight will
simply take the form,
(8)
with are parameter states recently updated using the deterministic interpolator (5). The
joint filtering distribution (6) can then be approximated by
(5)
A constrained random walk model of this form will ensure approximate stability provided that the PARCOR coefficients vary sufficiently slowly. Having sampled the last
PARCOR coefficient of the block , , all the intermediate PARCOR coefficients are found by some deterministic methods using previously sampled PARCOR coefficients
,
(2)
As in [2], the block variation of the log-excitation variance
and model order take the form,
½µ
(6)
Assume there exists a particulate approximation for the marginal
parameter filtering distribution,
´ ½µ ´
Æ Æ Hence, given , signal realisations can be drawn from
For instance, if we assume a conditional Gaussian statespace model, then all the computations can be done under
the framework of the Kalman filter and smoother [3]. e.g.
for , from (8)
can be found by the prediction error decomposition [5] and
the marginal signal filtering distribution can be rewritten as:
with and (9)
are sufficient statistics found by the Kalman
smoother.
4. IMPLEMENTATION
We modify the STV-PARCOR particle filter suggested in [2]
to facilitate the generalised STV-PARCOR model. Let , assuming that the parameter realisations and the signal sufficient statistics ( of the Kalman filter over blocks to . We then
evaluate the importance weight according to (8).
! ,
Having generated the parameter set we then resample (see [1] for details) it ! times with re
placement according to . For the resampled parame
! , we run a backward sweep of
ter set the Kalman smoother and generate .
Theoretically, signal realisations ! can be generated according to (9). How
ever, as #
are going to change in the next iteration, only will be drawn. Hence, the suggested algorithm will only give “delayed” state realisations.
In addition, continuity can be ensured by taking into ac
count in the sampling of , as there is only one free
variable owing to overlapping betweem and .
5. EXPERIMENTAL RESULTS
and ) are available for ! from the
previous iteration of the filter, state random samples can be
drawn from the filtering distribution as follows:
For ! ,
Generate random sample from and ´
. For each , sample
½µ from , where
´µ
if " otherwise
Make each of the same size by appending 0 to the vector if necessary. Using these fixed
grids, #
for # being integer are found by deterministic interpolator (5). In our
simulations, we have implemented the cubic spline
with , i.e. for $ ,
%
where %
is the spline coefficient to be determined
and is the order B-spline basis function.
Given the sufficient statistics ( and )
and , we run a forward sweep
Experiments are conducted to investigate the effectiveness
of the suggested algorithm (STV-PARCOR PF) for the purpose of audio noise reduction. In particular, we would like
to verify our suggestion that the generalised STV-PARCOR
model is a better model for slow time-varying processes
(e.g. speech) then the TVAR model. Preliminary simulation results are shown and compared with those generated
using the standard extended Kalman smoother (eKS) [5].
The clean speech clips used in this experiment are:
S1: Good service should be rewarded by big tips
S2: Draw every outer line first, then fill in the interior
The experiment setup is as follows, the clean speech signal is assumed to be submerged in white Gaussian noise
(WGN) with known variance , i.e. .
The output SNR from different algorithms are recorded and
compared. Owing to the stochastic nature of the Monte
Carlo algorithm, simulation results for the STV-PARCOR
PF are found by averaging the SNR improvement over five
independent applications of the algorithm.
In our simulations, we have chosen a value of ! ,
the block size is fixed to 100 for the STV-PARCOR PF.
The hyperparameters ( , and )
adopted here are assumed to be known and fixed. In consideration of the computational cost, it is assumed that the
model order with being limited
to 20. We note that ! is extremely small for Monte
Carlo simulation but the preliminary simulation results suggest that the algorithm works pretty well in such a case. As
in other applications of the Particle Filter, simulation results
improve as ! increases.
For the eKS, a Gaussian random walk is assumed directly on the AR coefficients and the model order is fixed
to 10. This model is employed as the TV-PARCOR model
and time-varying model order is not straight forward to implement with the extended Kalman smoother. The hyperparameters adopted are adjusted so that the system parameters
will cover the same range as the generalised STV-PARCOR
model in time steps. We note that this may not be the
optimum setup for the eKS, however, this will give a fair
comparison for both algorithms.
Figure 1 and Figure 2 show the 3D histogram plots of
the first reflection coefficients ( ) at different input SNRs
(SNR ) for the word “‘reward” in S1 using the STV-PARCOR
PF. The plots are generated by grouping all PARCOR coefficients particles from five independent simulations. As
shown in the plots, the suggested algorithm gives consistent
results at different noise levels.
We then compare the performance of the suggested algorithm with the the eKS. The SNR improvements for different clips at different noise levels are summarised below:
[2] W. Fong and S. Godsill. Sequential Monte Carlo simulation of dynamical models with slowly varying parameters: Application to audio. In Proceedings of the IEEE
ICASSP, 2002. To appear.
[3] W. Fong, S. J. Godsill, A. Doucet, and M. West. Monte
Carlo smoothing with application to audio signal enhancement. IEEE Transactions on Signal Processing,
Special Issue, 50(2):438–449, February 2002.
[4] Y. Grenier. Time-dependent ARMA modeling of nonstationary signals. IEEE Transactions on Acoustics,
Speech and Signal Processing, 31(4):899–911, 1983.
[5] A. C. Harvey. Forecasting, structural time series models and the Kalman filter. Cambridge University Press,
1989.
200
180
160
Clip
S1
S1
S1
S2
S2
S2
SNR
0dB
10dB
20dB
0dB
10dB
20dB
STV-PARCOR PF
3.86dB
2.54dB
1.08dB
4.31dB
2.80dB
1.35dB
eKS
1.92dB
0.99dB
0.87dB
2.21dB
1.57dB
1.09dB
Audio outputs can be found at http://www-sigproc.eng.
cam.ac.uk/wnwf2/Eusipco2002.html. Comparing the SNR
improvements, the suggested STV-PARCOR PF consistently
outperforms the eKS, which justifies using it in practice,
even it induces a much heavier computational load when
compare with the eKS.
140
200
120
100
100
0
−0.82
3500
−0.84
80
3000
−0.86
60
2500
−0.88
2000
−0.9
40
1500
−0.92
−0.94
20
1000
−0.96
ρt
−0.98
500
t
0
Figure 1: 3D histogram plot of for the word “reward”
using the STV-PARCOR particle filter for SNR =0dB
220
6. CONCLUSION
200
We propose a generalisation to the STV-PARCOR model
recently suggested by us to include any deterministic interpolator functions, . We then describe an adaptation to
the algorithm for joint estimation for signal and parameter.
The algorithm is tested on real speech signals and compared
with other standard approaches. Encouraging results are obtained. Further simulations will be conducted to investigate
the effects of different interpolator functions and different
lags, . The results will be published in due course.
180
160
140
200
120
100
0
100
−0.82
3500
−0.84
3000
−0.86
2500
−0.88
80
60
2000
−0.9
1500
−0.92
−0.94
ρ
t
1000
−0.96
−0.98
500
40
20
t
0
7. REFERENCES
[1] A. Doucet, N. de Freitas, and N. J. Gordon, editors. Sequential Monte Carlo Methods in Practice. New York:
Springer-Verlag, 2001.
Figure 2: 3D histogram plot of for the word “reward”
using the STV-PARCOR particle filter for SNR =20dB
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in: Proceedings of the 13th International Workshop on
Principles of Diagnosis (DX02), May 2002
Hybrid Diagnosis with Unknown Behavioral Modes
Michael W. Hofbaur1 and Brian C. Williams2
Abstract. A novel capability of discrete model-based diagnosis
methods is the ability to handle unknown modes where no assumption is made about the behavior of one or several components of the
system. This paper incorporates this novel capability of model-based
diagnosis into a hybrid estimation scheme by calculating partial filters. The filters are based on causal and structural analysis of the
specified components and their interconnection within the hybrid automaton model. Incorporating unknown modes provides a robust estimation scheme that can cope, unlike other hybrid estimation and
multi-model estimation schemes, with unmodeled situations and partial information.
1
Introduction
Modern technology is increasingly leading to complex artifacts with
high demands on performance and availability. As a consequence,
fault-tolerant control and an underlying monitoring and diagnosis capability plays an important role in achieving these requirements. Monitoring and diagnosis systems that build upon the discrete
model-based reasoning paradigm[8] can cope well with complexity
in modern artifacts. As an example, the Livingstone system[22] successfully monitored and diagnosed the DS-1 space probe in flight,
a system with approximately 480 modes of operation. However, a
widespread application of discrete model-based systems is hindered
by their difficulty to reason about the continuous dynamics of an artifact in a comprehensive manner. Continuous behaviors are difficult
to capture by the pure qualitative models that are used by the reasoning engines. Nevertheless, additional reasoning in terms of the
continuous dynamics is vital for detecting functional failures, as well
as low-level incipient (i.e slowly developing) faults and subtle component degradation.
Hybrid systems theory provides a modeling paradigm that integrates both, continuous state evolution and discrete mode changes
in a comprehensive manner. Recent work in hybrid estimation[14,
16, 24, 9] attempts to overcome the shortcomings of discrete modelbased diagnosis cited above and provides schemes that integrate
model-based approaches with techniques from fault detection and
isolation (FDI)[23, 4] and multi-model adaptive filtering[13, 11, 10].
The hybrid estimation schemes, as well as their FDI and multi-model
filtering ancestors, work well whenever the underlying model(s) are
’close’ mathematical descriptions of the physical artifact. They can
fail severely whenever unforeseen situations occur. Therefore, it is
essential to provide models that capture the entire spectrum of possible behaviors/modes whenever we use the hybrid estimate for closed
loop control, for instance. Model-based diagnosis, in contrast, does
1
2
Department of Automatic Control, Graz University of Technology, A-8010
Graz, Austria, email: [email protected]
MIT Space Systems and AI Laboratories, 77 Massachusetts Ave., Rm. 37381, Cambridge, MA 02139 USA, email: [email protected]
not impose such a strong modeling assumption. Its concept of the
unknown mode allows diagnosis of systems where no assumption is
made about the behavior of one or several components of the system. In this way, it captures unspecified and unforeseen behaviors
of the system under investigation. This paper provides an approach
to incorporate the concept of an unknown mode into our hybrid estimation scheme[9]. As a result we obtain an estimation capability
that can detect unforeseen situations. Furthermore, it allows us to
continue estimation on a degraded basis. We achieve this by causal
analysis[17, 20], structural analysis[7] and decomposition of the system.
This paper starts with a brief introduction to our hybrid systems
modeling and estimation scheme. Upon this foundation, we extend
hybrid estimation to incorporate the unknown mode and demonstrate
the underlying structural analysis and decomposition task. Finally, an
experimental evaluation with computer simulated data for a Martian
live support system demonstrates the advantages of this extended hybrid estimation scheme.
2
Hybrid Systems
The hybrid automaton model used throughout this paper is based on
[9] and can be seen as a model that merges hidden Markov models
(HMM) with continuous discrete-time dynamical system models (we
present the model on the level of detail sufficient for this work and
refer the reader to the reference cited above for more detail).
2.1
Concurrent Hybrid Automata
Definition 1 A discrete-time probabilistic hybrid automaton (PHA)
A is described as a tuple hx, w, F, T, Xd , Ts i:
• x denotes the hybrid state variables of the automaton3 , composed
of x = {xd } ∪ xc . The discrete variable xd denotes the mode
of the automaton and has finite domain Xd . The continuous state
variables xc capture the dynamic evolution of the automaton. x
denotes the hybrid state of the automaton, while xc denotes the
continuous state.
• The set of I/O variables w = ud ∪ uc ∪ yc of the automaton
is composed of disjoint sets of discrete input variables ud (called
command variables), continuous input variables uc , and continuous output variables yc .
• F : Xd → FDE ∪ FAE specifies the continuous evolution of the
automaton in terms of discrete-time difference equations FDE and
algebraic equations FAE for each mode xd ∈ Xd . Ts denotes the
sampling period of the discrete-time difference equations.
3
When clear from context, we use lowercase bold symbols, such as v, to
denote a set of variables {v1 , . . . , vl }, as well as a vector [v1 , . . . , vl ]T
with components vi .
• The finite set, T , of transitions specifies the probabilistic discrete
evolution of the automaton.
Complex systems are modeled as a composition of concurrently
operating PHA that represent the individual system components. A
concurrent probabilistic hybrid automata (cPHA) specifies this composition as well as its interconnection to the outside world:
Definition 2 A concurrent probabilistic hybrid automaton (cPHA)
CA is described as a tuple hA, u, yc , vs , vo , Nx , Ny i:
• A = {A1 , A2 , . . . , Al } denotes the finite set of PHAs that represent the components Ai of the cPHA (we denote the components
of a PHA Ai by xdi , xci , udi , uci , yci , Fi , Xdi ).
• The input variables u = ud ∪ uc of the automaton consists of the
sets of discrete input variables ud = ud1 ∪ . . . ∪ udl (command
variables) and continuous input variables uc ⊆ uc1 ∪ . . . ∪ ucl .
• The output variables yc ⊆ yc1 ∪ . . . ∪ ycl specify the observed
output variables of the cPHA.
• The observation process is subject to additive, zero mean Gaussian
sensor noise. Ny : Xd → IRm×m specifies the mode dependent4
disturbance vo in terms of the covariance matrix R = diag(ri ).
• Nx specifies additive, zero mean Gaussian disturbances that act
upon the continuous state variables xc = xc1 ∪ . . . ∪ xcl . Nx :
Xd → IRn×n specifies the mode dependent disturbance vs in
terms of the covariance matrix Q.
Definition 3 The hybrid state x(k) of a cPHA at time-step k specifies the mode assignment xd,(k) of the mode variables xd =
{xd1 , . . . , xdl } and the continuous state assignment xc,(k) of the
continuous state variables xc = xc1 ∪ . . . ∪ xcl .
Interconnection among the cPHA components Ai is achieved via
shared continuous I/O variables wc ∈ uci ∪yci only. Fig. 1 illustrates
a simple example composed of 3 PHAs.
uc1
ud1
ud2
A1
yc1
wc1
A2
yc2
A3
A cPHA specifies a mode dependent discrete-time model for a
plant with command inputs ud , continuous inputs uc , continuous
outputs yc , mode xd , continuous state variables xc and additive, zero
mean Gaussian disturbances vs , vo . The discrete-time evolution of
xc and yc is described by the nonlinear system of difference equations (sampling period Ts )
yc,(k) = g(k) (xc,(k) , uc,(k) ) + vo,(k) .
(1)
The functions f(k) and g(k) are obtained by symbolically solving5
the set of equations F1 (xd1,(k) ) ∪ . . . ∪ Fl (xdl,(k) ) given the mode
xd,(k) = [xd1,(k) , . . . , xdl,(k) ]T .
4
5
F1 , F2 and F3 provide for a cPHA mode xd,(k)
[m11 , m21 , m31 ]T the equations
=
F1 (m11 ) = {uc1 = 5.0 wc1 }
F2 (m21 ) = {xc1,(k) = 0.8 xc1,(k−1) + wc1,(k−1) ,
yc1 = xc1 }
F3 (m31 ) = {xc2,(k) = xc3,(k−1) + yc1,(k−1) ,
(2)
xc3,(k) = 0.4 xc2,(k−1) + 0.5 uc1,(k−1) ,
yc2 = 2.0 xc2 + xc3 }.
This leads to the discrete-time model:
xc1,(k) = 0.8 xc1,(k−1) + 0.2 uc1,(k−1) + vs1,(k−1)
xc2,(k) = xc1,(k−1) + xc3,(k−1) + vs2,(k−1)
xc3,(k) = 0.4 xc2,(k−1) + 0.5 uc1,(k−1) + vs3,(k−1)
(3)
yc1,(k) = xc1,(k) + vo1,(k)
yc2,(k) = 2.0 xc2,(k) + xc3,(k) + vo2,(k)
2.2
Estimation of Hybrid Systems
To detect the onset of subtle failures, it is essential that a monitoring
and diagnosis system is able to accurately extract the hybrid state of
a system from a signal that may be hidden among disturbances, such
as measurement noise. This is the role of a hybrid observer. More
precisely:
Hybrid Estimation Problem: Given a cPHA CA, a sequences
of observations {yc,(0) , yc,(1) , . . . , yc,(k) } and control inputs
{u(0) , u(1) , . . . , u(k) }, estimate the most likely hybrid state
x̂(k) at time-step k.
x̂(k) := hxd,(k) , x̂c,(k) , P(k) i,
Example cPHA composed of three PHAs
xc,(k) = f(k) (xc,(k−1) , uc,(k−1) ) + vs,(k−1)
A1 = h{xd1 }, {ud1 , uc1 , wc1 }, F1 , T1 , {m11 , m12 }...i
A2 = h{xd2 , xc1 }, {ud2 , wc1 , yc1 }, F2 , T2 , {m21 , m22 }...i
A3 = h{xd3 , xc2 , xc3 }, {ud2 , uc1 , yc1 , yc2 }, F3 , T3 , {m31 }...i.
A hybrid state estimate x̂(k) consists of a continuous state estimate, together with the associated mode. We denote this by the tuple
CA
Figure 1.
Consider the illustrative cPHA in Fig. 1 with
E.g. sensors can experience different magnitudes of disturbances for different modes.
Our symbolic solver restricts the algebraic equations and nonlinear functions to ones that can be solved explicitly and utilizes a Gröbner Basis
approach[3] to derive a set of equations of form (1).
where x̂c,(k) specifies the mean and P(k) the covariance for the continuous state variables xc . The likelihood of an estimate x̂(k) is denoted by the hybrid belief-state h(k) [x̂].
We perform hybrid estimation as extended version of HMM-style
belief-state update that accounts for the influence of the continuous
dynamics upon the system’s discrete modes. A major difference between hybrid estimation and an HMM-style belief-state update, as
well as multi-model estimation, is, however, that hybrid estimation
tracks a set of trajectories, whereas standard belief-state update and
multi-model estimation aggregate trajectories which share the same
mode. This difference is reflected in the first of the following two
recursive functions which define our hybrid estimation scheme:
h(•k) [x̂i ] = PT (mi |x̂j,(k−1) , ud,(k−1) )h(k−1) [x̂j ]
(4)
h(•k) [x̂i ]PO (yc,(k) |x̂i,(k) , uc,(k) )
h(k) [x̂i ] = P
j h(•k) [x̂j ]PO (yc,(k) |x̂j,(k) , uc,(k) )
(5)
h(•k) [x̂i ] denotes an intermediate hybrid belief-state, based on transition probabilities only. Hybrid estimation determines for each
x̂j,(k−1) at the previous time-step k − 1 the possible transitions,
thus specifying candidate successor states to be tracked. Consecutive filtering provides the new hybrid state x̂i,(k) and adjusts the hybrid belief-state h(k) [x̂i ] based on the hybrid probabilistic observation function PO (yc,(k) |x̂i,(k) , uc,(k) ). The estimate x̂j,(k) with the
highest belief-state h(k) [x̂j ] = maxi (h(k) [x̂i ]) is taken as the hybrid
estimate at time-step k.
Tracking all possible trajectories of the system is almost always
intractable because the number of trajectories becomes too large after
only a few time-steps. In [9] we present an approximative anytime
anyspace algorithm that copes with the exponential growth, as well as
the large number of modes in a typical concurrent hybrid automaton
model.
Hybrid estimation and other multi-model estimation schemes have
in common that they require models that are ’close’ mathematical descriptions of the system. They can fail severely whenever unforeseen,
i.e. unmodeled, situations occur. As a consequence, we have to provide models for all operational modes as well as an exhaustive set
of models for possible failure modes. Providing all possible failure
models can be problematic even under the assumption of an exhaustive failure mode effect analysis (FMEA). For instance, consider an
incipient fault in a servo valve that causes the valve to drift off its
nominal opening value. The drift (positive, negative, slow, fast...) is
subject to the fault. It is surely difficult to provide a mathematical
model with the correct parameter values that captures all possible
drift situations. Nor is it helpful to introduce a sufficiently large set
of modes that captures possible situations of the drift fault as this
would introduce additional complexity for hybrid estimation by increasing the number of modes unnecessarily.
This requirement of hybrid mode estimation is in contrast to discrete model-based diagnosis schemes, such as GDE (e.g. [5, 6, 19]).
Model-based diagnosis deduces the possible mode of the system
based on nominal models, and few specified fault models only. The
onset of possible fault scenarios are covered by the so called unknown mode which does not impose any constraints on the system’s
variables.
The next section provides an approach that systematically incorporates the concept of the unknown mode into our hybrid estimation
scheme.
3
Estimation with Unknown Modes
The estimation scheme [9] requires a fully specified mode assignment xdi,(k) for each candidate trajectory that is tracked in the course
of hybrid estimation. Only a fully specified mode allows us to deduce
the mathematical model (1) for the overall system. This model is the
basis for the dynamic filter (e.g. extended Kalman filter) that is used
in the course of hybrid estimation.
uc1
yc1
yc2
MIMO Filter
xc1
xc2
xc3
PO
Figure 2. MIMO filter (e.g. extended Kalman filter) for the cPHA example
multi-output (MIMO) filter (see Fig. 2) for mode xdi,(k) =
[m11 , m21 , m31 ]T based on the mathematical model (3). This filter
provides the hybrid state estimate x̂i,(k) as well as the value for the
hybrid probabilistic observation function PO (yc,(k) |x̂i,(k) , uc,(k) )
for the hybrid estimator (see Appendix A for the extended Kalman
filter estimation details).
Let us assume the mode xdi,(k) = [?, m21 , m31 ]T which specifies that component 1 (A1 ) is in unknown mode. A component in unknown mode imposes no constraints (equations) among its variables
(uc1 and the internal variable wc1 , in our case). As a consequence,
we cannot deduce an overall mathematical model of the form (1) and
fail to provide the basis for the hybrid estimation scheme, the MIMO
filter for mode xdi,(k) = [?, m21 , m31 ]T .
vs1
uc1
ud1
ud2
vs2 vs3
wc1
A1
A2
yc1
vo2
yc2
A3
CA
Figure 3.
Example cPHA with explicit noise inputs
However, a close look on the PHA interconnection (Fig. 3 - the
figure extends Fig. 1 by including the implicit noise inputs, as well
as indicating the causality for the internal I/O variables) reveals that
we can still estimate component 3 by its observed output yc2 and the
observation yc1 as a substitute for the value of its input. This intuitive
approach utilizes a decomposition of the cPHA as shown in Fig. 4.
vs1
uc1
yc1
uc1
A1
vo1
yc1
A2
vs2 vs3
A3
Figure 4.
vo1
vo2
yc2
Decomposed cPHA
The decomposition allows us to treat the concurrent parts of the
system independently and calculate a filter cluster consisting of 2
independent filters. However, when calculating the individual filters
for the cluster, we have to take into account that we use the measurement of the input to the third component (yc1 ) in replacement to
its true value. This can be interpreted as having additional additive
noise at the component’s input as indicated in Fig. 4. The following
modification of the covariance matrix Q3 for the state variables of
A3 takes this into account:
Q̃3 = b3 r1 bT3 + Q3 ,
For our illustrative 3 component example introduced above
this would mean that hybrid estimation calculates a multi-input
vo1
(6)
where r1 denotes the variance of disturbance vo1 and b3 = [0, 1]T
uc1
yc1
Filter 1
Filter 2
yc2
raw model for the system given mode xd . The following decomposition performs a structural analysis of the raw model-based on
causal analysis[17, 20], structural observability analysis[7] and graph
decomposition[1].
A cPHA model does not impose a fixed causal structure that specifies directionality of automaton interconnections. Causality is implicitly specified by the set of equations. This increases the expressiveness of the modeling framework but requires us to perform a
causal analysis of the raw model (8) as a first step. The deduction of the causal dependencies is done by applying the bipartitematching based algorithm presented in [17]. The resulting directed
graph records the causal dependencies among the variables of the
system (Fig. 6 shows the graph for the the illustrative 3 PHA example). Each vertex of the graph represents one equation ei ∈ F
xc1
PO1
xc2
xc3
PO2
Filter Cluster
Figure 5.
Decomposed filter
denotes the input vector6 of A3 with respect to yc1 .
A filter cluster consisting of extended Kalman filters and the
MIMO extended Kalman filter are interchangeable as they provide
the same expected value for the continuous state (E(x̂c )) whenever
the mode of the automaton is fully specified. However, the decomposed filter has the advantage that the probabilistic observation function PO of the overall system is given by
Y
PO =
POj ,
(7)
uc1
wc1
xc1
Figure 6.
yc1
xc2
xc3
yc2
Causal graph for the cPHA example
j
where POj denotes the probabilistic observation function of the j’th
filter in the filter cluster.
This factorization of the probabilistic observation function allows
us to calculate an upper bound for PO whenever one or more components of the system are in unknown mode. We simply take the
product over the remaining filters in the cluster. This is equivalent
with considering the upper bounds of the inequalities POj ≤ 1 for
each unknown filter j. In our example with unknown component A1
this would mean:
PO ≤ PO2 ,
where PO2 denotes the observation function for the filter that estimates the continuous state of component A3 .
The following subsection provides a graph-based approach for
filer cluster deduction that grounds the informally introduced decomposition on a more versatile basis.
3.1
System Decomposition and Filter Cluster
Calculation
Starting point for the decomposition of the system for a cPHA mode
xd is the set of equations
F1 (xd1,(k) ) ∪ . . . ∪ Fl (xdl,(k) ) =: F (xd ),
In the general case, we have to calculate bj for a cPHA component Aj
and observed inputs uyc by linearization, more specifically: bj,(k) =
∂fj /∂uyc |x̂
, where fj denotes the right-hand side of
,u
cj,(k−1)
Definition 4 A causal graph of a cPHA CA at a mode xd is a directed graph
S that records the causal dependencies among the variables v ∈ i xci ∪ uci ∪ yci of CA. We denote the causal graph
by CG(CA, xd ) and sometimes omit arguments where no confusion
seems likely.
Goal of our analysis is to obtain a set of independent subsystems
that utilize observed variables as virtual inputs. Therefore, we slice
the graph at observed variable vertices with outgoing edges, insert a
new vertex to represent a virtual input and re-map the sliced outgoing edges to this vertex. Fig. 7 demonstrates this re-mapping for the
causal graph of Fig. 6. The observed variables are yc1 and yc2 . Only
the vertex with dependent variable yc1 has an outgoing edge, thus we
slice the graph at yc1 → xc2 and re-map the edge to the virtual input
uyc1 .
uc1
wc1
xc1
yc1
uyc1
xc2
xc3
yc2
(8)
where Fj (xdj,(k) ) returns the appropriate set of equations for a component Ai whenever xdj,(k) ∈ Xdj or the empty set whenever the
component is in unknown mode, i.e. xdj,(k) =?. Although we still
have to solve the set of equations to arrive at the mathematical
model of form (1) we can interpret the set of equations (8) as the
6
or an exogenous variable specification (e.g. uc1 ) and is labeled by
its dependent variable which also specifies the outgoing edge (in the
following, we will use the variable name to refer to the corresponding vertex in the graph). Vertices without incoming edges specify the
exogenous variables.
Figure 7.
Remapped causal graph for the cPHA example
cj,(k−1)
the difference equation for component Aj , uyc refers to the observed
variables that are used as inputs to the component (i.e. uyc ⊂ yc ) and
x̂cj,(k−1) as well as ucj,(k−1) represent the state estimate and the continuous input for component Aj at the previous time-step, respectively.
A dynamic filter (e.g. extended Kalman filter) can only estimate
the observable part of the model. Therefore, it is essential to perform
an observability analysis prior calculating the filter so that non observable parts of the model are excluded. We perform this analysis
on a structural basis7 .
Definition 5 We call a variable v of a cPHA CA at mode xd structurally observable (SO) whenever it is directly observed, i.e. v ∈ yc ,
or there exists at least one path in the causal graph CG(CA, xd ) that
connects the variable z to an output variable yc ∈ yc of CA.
A filter estimates the state variables xc of a dynamic system based
on observations yc and the inputs uc that act upon the state variables
xc . The required knowledge about the inputs uc indicates that the
structural observability criteria is not yet sufficient to determine the
submodel for estimation. We have to make sure, that no unknown exogenous input influences a variable. To illustrate this, consider again
the 3 PHA example with mode xd = [?, m21 , m31 ]T . Component
1 in unknown mode omits the equation that relates the variables uc1
˜ (Fig. 8), where wc1 is laand wc1 . This leads to a causal graph CG
beled as exogenous (no incoming edges). This unknown exogenous
input influences the state variable xc1 and, as a consequence, prevents us from estimating it!
Figure 8.
uc1
wc1
xc1
yc1
uyc1
xc2
xc3
yc2
Remapped causal graph for the cPHA example with unknown
component A1
We extend our structural analysis of the causal graph by the following criteria:
Definition 6 We call a variable v of a cPHA CA at mode xd structurally determined (SD) whenever it is an input variable of the automaton, i.e. v ∈ uc , or there does not exist a path in the causal
graph CG(CA, xd ) that connects an exogenous variable ue ∈
/ uc
with v.
Furthermore, it is helpful to eliminate loops in the causal graph
prior checking variables against both structural criteria. For this purpose, we calculate the strongly connected components of the causal
graph[1].
Definition 7 A strongly connected component (SCC) of the causal
graph CG is a maximal set SCC of variables in which there is a path
from any one variable in the set to another variable in the set.
Fig. 9 shows the remapped causal graph for the 3 PHA example after
grouping variables into strongly connected components.
The strong interconnection among variables in an SCC implies
that:
1. Structural observability of variables in an SCC follows directly
from structural observability of at least one variable in the SCC.
7
Throughout the paper we assume that loss of observability is caused by
a structural defect of the model. Otherwise, it is necessary to perform an
additional numerical observability test [18] as structural observability only
provides a necessary condition for observability.
uc1
uyc1
Figure 9.
wc1
xc1
xc2, xc3
yc1
yc2
Causal SCC graph for cPHA example
2. A variable in an SCC is structurally determined, if and only if all
variables in the SCC are structurally determined.
As a consequence, we can apply our structural analysis to strongly
connected components directly and operate on the SCC graph, i.e
a causal graph without loops. The analysis of a strongly connected
component with respect to structural observability and structural determination (SOD) can be outlined as follows:
function determine-SOD-of-SCC(SCC, uc , k)
when SOD-undetermined?(SCC)
if exogenous?(SCC)
then vi ← independent-var(SCC)
if vi ∈ uc then SD(SCC) ← True
else SD(SCC) ← False
else V ← uplink-SCCs(SCC)
loop for SCC i in V
do determine-SOD-of-SCC(SCC i , uc , k)
SO(SCC) ← True
SD(SCC) ← all-uplink-SCCs-are-SD?(V)
cluster-index(SCC) ← k ∪ cluster-indices(V)
SOD-determined(SCC) ← True
return Nil
Our structural analysis algorithm determines structural observability and determination (SOD) of a variable by traversing the SCC
graph backwards from the observed variables towards the inputs.
In the course of this analysis we label non-exogenous strongly connected components with an index that refers to their cluster membership. This indexing scheme allows us to cluster the variables into
non-overlapping clusters with respect to the observed variables. The
direct relation between a variable, its determining equation, and the
cPHA component that specified this equation leads to the component clusters sought. The structural analysis can be summarized as
follows:
function component-clustering(CA, xd )
returns a set of cPHA component clusters
yc ← observed-vars(CA)
˜ ← remap-causal-graph(CG(CA, xd ), yc )
CG
˜ ∪ input-vars(CA)
uc ← virtual-inputs(CG)
˜
CG SCC ← strongly-connected-component-graph(CG)
k←0
loop for SCC i in output-SCCs(CG SCC , yc )
do determine-SOD-of-SCC(SCC i , uc , k)
k←k+1
graph-clusters ← get-SOD-SSC-clusters(CG SCC )
return automaton-clusters(CA, graph-clusters)
lighting system
uc1
wc1
xc1
yc1
sod-1
sod-1
sod-1
Crew Chamber
pulse injection valves
cluster 1 { A 1 , A 2 }
Airlock
Plant Growth Chamber
flow regulator 1
CO2
CO2
tank
flow regulator 2
cluster 2 { A 3 }
uyc1
xc2, xc3
sod-2
Figure 10.
y
Example - BIO-Plex
Our application is the BIO-Plex Test Complex at NASA Johnson
Space Center, a five chamber facility for evaluating biological and
physiochemical Martian life support technologies. It is an artificial,
biosphere-type, closed environment, which must robustly provide all
the air, water, and most of the food for a crew of four without interruption. Plants are grown in plant growth chambers, where they
provide food for the crew, and convert the exhaled CO2 into O2 . In
order to maintain a closed-loop system, it is necessary to control the
resource exchange between the chambers without endangering the
crew. For the scope of this paper, we restrict our evaluation to the
sub-system dealing with CO2 control in the plant growth chamber
(PGC), shown in Fig. 11.
The system is composed of several components, such as redundant
flow regulators (FR1, FR2) that provide continuous CO2 supply, redundant pulse injection valves (PIV1, PIV2) that provide a means for
increasing the CO2 concentration rapidly, a lighting system (LS) and
the plant growth chamber (PGC), itself. The control system maintains a plant growth optimal CO2 concentration of 1200 ppm during
the day phase of the system (20 hours/day).
Hybrid estimation schemes are key to tracking system operational
modes, as well as, detecting subtle failures and performing diagnoses. For example, we simulate a failure of the second flow regulator. The regulator becomes off-line and drifts slowly towards its
positive limit. This fault situation is difficult to capture by an explicit
fault model as we do not know, in advance, whether the regulator
8
Figure 11.
Labeled and partitioned causal SCC graph for the 3 cPHA
example
Each component cluster defines the observable and determined
raw model for a subsystem of the cPHA. This raw model can be
solved symbolically and provides the nonlinear system of difference
equations (a model similar to (1), but with the additional virtual inputs) that is the basis for the corresponding filter in the filter cluster.
In this way we exclude the unobservable and/or undetermined parts
of the overall system from estimation.
Whenever a state variable xcj becomes unobservable and/or undetermined (e.g. due to a mode change) during hybrid estimation,
we hold the value for the mean at its last known estimate x̂cj and
increase its variance σj2 = pjj by a constant factor at each hybrid
estimation step. This reflects a continuously decreasing confidence
in the estimate x̂cj and allows us to restart estimation whenever the
variable becomes observable and determined again8 .
4
chamber
control
c2
sod-2
Whenever a state variable xcj is directly observed we also can utilize an
alternative approach suggested in [15] that restarts the estimator with the
observed value, thus improving the observer convergence time.
BIO-Plex plant growth chamber
drifts towards its postitive or negative limit, nor do we know the magnitude of the drift. A fault of this type, which develops slowly and
whose symptom is hidden among the noise in the system is a typical
candidate for our unknown-mode detection capability. However, we
also provide explicit failure models that describe typical situations.
For example, the PGC has 4 plant trays with one illumination bank
for each tray. A black out of one illumination bank can be interpreted
as a 25% loss in light intensity. This situation can be modeled explicitly by a dynamical model that takes this reduced light intensity into
account.
In the following we describe the outcome of a simulated experiment where the flow regulator fault with drifting symptom is injected
at time point k = 700 and an additional light fault, that harms one
of the four illumination banks, is injected at k = 900. The faults are
’repaired’ at k = 1100 and k = 1300 for the flow regulator fault and
the lighting fault, respectively. This experiment illustrates unknown
mode detection and recovery from it, nominal failure mode detection,
and the multiple fault detection capability of our approach.
ud3
uc1
ud1
A1
A5
FR1
LS
wc1
A2
FR2
ud2
A3
yc1
yc2
wc2
PIV1
A4
PIV2
Figure 12.
A6
wc3
yc3
PGC
BIO-Plex cPHA model
The simulated data is gathered from the execution of a refined subset of NASA’s JSC’s CONFIG model for the BIO-Plex system[12].
Hybrid estimation utilizes a cPHA model that consists of 6 components as shown in Fig. 12. To illustrate the complexity of the
hybrid estimation problem we should note, that the concurrent automaton has approximately 56 ≈ 15000 modes. Each mode describes the dynamic evolution of the chamber system by a third order system of difference equations. For example, the nominal operational condition for plant growth is characterized by the mode
xd = [mr2 , mr2 , mv1 , mv1 , ml2 , mp2 ], where mr2 characterizes
an partially open flow regulator, mv1 a closed pulse injection valve,
ml2 100% light on, and mp2 plant growth mode at 1200 ppm, respectively. This mode specifies the raw model:
F1 (mr2 ) = {xc1,(k) = 0.5 uc1,(k−1) , yc1 = xc1 }
F2 (mr2 ) = {xc2,(k) = 0.5 uc1,(k−1) , yc2 = xc2 }
F3 (mv1 ) = {wc2 = 0.0}
F4 (mv1 ) = {wc3 = 0.0}
F5 (ml2 ) = {wc1 = 1204.0}
F6 (mp2 ) = {xc3,(k) = xc3,(k−1) + 20.163·
The causal graph (Fig. 13) of the raw model (9) leads to the decomposition of the system as shown in Fig. 14 (our implementation
of the causal analysis and decomposition algorithms treats constant
values, such as the value 1204.0 for the photosynthetic photon flux,
as known exogenous inputs with constant value). The decomposition
of the model leads to a filter cluster with 3 extended Kalman filters one for each flow regulator and one for the remaining system (pulse
injection valves, lighting system and plant growth chamber). This
enables us to estimate the mode and continuous state of the flow regulators independent of the remaining system. As a consequence, an
unknown mode in a flow regulator does not cause any implications
on the estimation of the remaining system.
[−1.516 · 10−4 f1 (wc1,(k−1) )f2 (xc3,(k−1) )+
cluster 1 {FR1}
uc1
yc1,(k−1) + yc2,(k−1) + wc1,(k−1) + wc2,(k−1) ],
xc1
yc1
yc3 = xc3 },
cluster 2 {FR2}
(9)
xc2
where f1 and f2 denotes
2
f1 (wc1 ) := − 7.615 + 0.111 wc1 − 2.149 · 10−5 wc1
−xc3 /400.0
f2 (xc3 ) := 72.0 − 78.89 e
uyc1
cluster 3 {PIV1, PIV2, LS, PGC}
(10)
.
xc3
uyc2
1204.0
wc2
0.0
Figure 14.
approximates the CO2 gas production [g/min] due to photosynthesis according to the CO2 gas concentration and chamber
illumination[12]. This raw model defines a third order system of
discrete-time difference equations with sampling period T s = 1
[min]:
xc1,(k) = 0.5 uc1,(k−1) + vs1,(k−1)
xc2,(k) = 0.5 uc1,(k−1) + vs2,(k−1)
xc3,(k) = xc3,(k−1) + 20.163[−1.041+
1.141e−xc3,(k) /400.0 + xc1,(k−1) + xc2,(k−1) ] + vs3,(k−1)
yc1,(k) = xc1,(k) + vo1,(k)
yc2,(k) = xc2,(k) + vo2,(k)
yc2,(k) = xc3,(k) + vo3,(k) ,
yc3
wc1
xc1,(k) and xc2,(k) denote the gas flow ([g/min]) of flow regulator 1
and 2, respectively and xc3,(k) denotes the CO2 gas concentration
([ppm]) in the plant growth chamber. wc1,(k) and wc2,(k) denote the
gas flow ([g/min]) of the pulse injection valves and wc3,(k) denotes
the photosynthetic photon flux ([µ-mol/m2 s]) of the lights above the
plant trays. The nonlinear expression
−1.516 · 10−4 f1 (wc1,(k−1) )f2 (xc3,(k−1) )
yc2
wc3
Partitioned causal SCC graph of the BIO-Plex cPHA model
Fig. 15 shows the continuous input (control signal) uc1 , observed
flow rates for flow regulator 1 and 2 and the CO2 concentration for
the experiment. Both flow regulators provide half of the requested
gas injection rate up to k = 700. At this time point, the second flow
regulator starts to slowly drift towards its positive limit which it will
reach at approximately k = 800. The camber control system reacts immediately and lowers the control signal in order to keep the
CO2 concentration at the requested 1200 ppm concentration. This
transient behavior causes a slight bump in the CO2 concentration
as shown in Fig. 15-b. Our hybrid mode estimation system detects
this unmodeled fault at k = 727 and declares flow regulator 2 to be
in an unknown mode (we indicate the unknown mode by the mode
number 0 in Fig. 16). The flow regulator mode stuck-open (mr5 ) be-
(11)
Flow Regulator 2 Estimation Detail
6
wc1
0.0
wc2
wc3
uc1
xc1
5
mode number
1204.0
yc1
xc3
yc3
4
3
2
1
0
650
xc2
Figure 13.
yc2
700
Figure 16.
727
750
769
time [minutes]
800
850
Mode estimate detail for flow regulator 2
Causal graph of the BIO-Plex cPHA raw model (9)
comes more and more likely as the regulator drifts towards its open
position. Hybrid mode estimation prefers this mode as symptom ex-
1240
1
1220
0.8
CO2 concentration [ppm]
CO2 gas inflow rate [g/min]
control input
inflow rate FR2
0.6
0.4
1200
1180
1160
inflow rate FR1
0.2
1140
0
600
700 727
800
900
1000
time [minutes]
1100
1200
1300
1400
(a) Control input uc and measured CO2 input flow rates
Figure 15.
Flow Regulator 2
6
mode number
5
4
3
2
1
0
700
800
900
1000
time [minutes]
1100
1200
1300
1400
Lighting System
6
5
mode number
700 727
800
900
1000
time [minutes]
1100
1200
1300
1400
(b) CO2 level in PGC (measurement - gray/green, estimate black)
Observed data and continuous estimation of the CO2 concentration in plant growth chamber
planation from k = 769 onwards, although flow regulator 2 goes
into saturation a little bit later at k = 800.
The light fault at k = 900 is detected almost instantly at k = 904
(ml4 ). This good discrimination among the pre-specified modes
(failure and nominal) is further demonstrated at the termination
points of the faults. Repairs of the flow regulator 2 and the lighting
system are detected immediately at k = 1101 and k = 1301, respectively. Fig. 17 shows the mode estimation result for the lighting
system and flow regulator 2 over the entire experiment horizon.
600
1120
600
4
The hybrid estimator uses a cPHA description and performs decomposition and estimation, as outlined above. Decomposition is done
on-line according to the mode hypotheses that are tested in the course
of hybrid estimation. In general, it can be assumed that the the mode
in the system evolves on a lower rate than the hybrid estimation
rate, which operates on the sampling period Ts . Therefore, we cache
recent decompositions and their corresponding filters for re-use as
a compromise between a-priori calculation (space complexity) and
pure on-line deduction (time complexity).
Optimized model-based estimation schemes, such as
Livingstone[22], utilize conflicts to focus the underlying search
operation. A conflict is a (partial) mode assignment that makes a
hypothesis very unlikely. This requires a more general treatment
of unknown modes compared to the filter decomposition task
introduced above. The decompositional model-based learning
system Moriarty[21] introduced continuous variants of conflicts,
so-called dissents. We are currently reformulating these dissents for
hybrid systems and investigate their incorporation to improve the
underlying search scheme. This will lead to an overall framework
that unifies our previous work on Livingstone, Moriarty and hybrid
estimation.
3
2
REFERENCES
1
0
600
700
Figure 17.
5
800
900
1000
time [minutes]
1100
1200
1300
1400
Mode estimates for flow regulator 2 and lighting system
Implementation and Discussion
The implementation of our hybrid estimation scheme extends previous work on hybrid estimation [9] and is written in Common LISP.
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Acknowledgments
In part supported by NASA under contract NAG2-1388.
A
Extended Kalman Filter
The disturbances and imprecise knowledge about the initial state
xc,(0) make it necessary to estimate the state by its mean x̂c,(k)
and covariance matrix P(k) . We use an extended Kalman filter[2]
for this purpose, which updates its current state, like an HMM observer, in two steps. The first step uses the model to predict mean
for the state x̂c,(•k) and its covariance P(•k) , based on the previous
estimate hx̂c,(k−1) , P(k−1) i, and the control input uc,(k−1) :
x̂c,(•k)
=
A(k−1)
=
f (x̂c,(k−1) , uc,(k−1) )
∂f ∂x (12)
A(k−1) P(k−1) AT(k−1) + Q.
(14)
(13)
x̂c,(k−1) ,uc,(k−1)
P(•k)
=
This one-step ahead prediction leads to a prediction residual r(k)
with covariance matrix S(k)
r(k)
=
C(k)
=
yc,(k) − g(x̂c,(•k) , uc,(k) )
∂g ∂x (15)
(16)
x̂c,(•k) ,uc,(k)
S(k)
C(k) P(•k) CT(k) + R.
=
(17)
The second filter step calculates the Kalman filter gain K(k) , and
refines the prediction as follows:
=
P(•k) CT(k) S−1
(k)
(18)
x̂c,(k)
=
=
x̂c,(•k) + K(k) r(k)
I − K(k) C(k) P(•k) .
(19)
P(k)
K(k)
(20)
The output of the extended Kalman filter, as used in our hybrid estimation system, is a sequence of mean/covariance pairs hx̂c,(k) , P(k) i
for xc,(k) as well as the hybrid probabilistic observation function
−1
−rT
(k) S(k) r(k) /2
PO (y(k) |x̂(k) , uc,(k) ) = e
.
(21)