IEEE Signal Processing - May 2018 - 121
The proof is actually analogous to the
convergence proof for iterative average
consensus algorithms.
The iterative version of the combine
step has the inconvenience, however, that
the maximum number of iterations specified in the algorithm must be completed
before new sensor measurements {y n, r}
are available. Since the iterative procedure requires multiple internode message
passing between consecutive measurements and, ideally, for a large number of
iterations, tries to force all nodes to agree
on a merged pdf, our point of view is
that the ATC algorithm with an iterative
combine step is better classified under the
category of consensus algorithms rather
than diffusion algorithms and, accordingly, we focus on the noniterative procedure
in the remainder of this lecture note.
ATC diffusion Kalman filter
Next, we particularize the ATC diffusion filter in the special case of linear,
Gaussian state-space models. In the
sequel, let N (x|m, P) denote a multivariate Gaussian pdf with argument x
that is parameterized by a mean vector
m and a covariance matrix P.
Adapt step
Assuming the linear state-space model
in (3) and (4), it follows from (7) and
(8), after algebraic calculations that are
similar to the standard derivation of the
Kalman filter in the so-called information form [7], that, if p n -1 | n -1, r (x n -1) =
N (x n -1 | xt n -1 | n -1, r, Pn -1 | n -1, r), then, at
each node r at instant n, p n | n -1, r (x n) =
N (x n | xt n | n -1, r, Pn | n -1, r) and pu n | n, r (x n) =
N (x n | xu n | n, r, Pu n | n, r), where
xt n | n -1, r = Fn -1 xt n -1 | n -1, r
(11)
Pn | n -1, r = Fn -1 Pn -1 | n -1, r F nT -1
+ G n -1 Q n -1 G Tn -1
(12)
and
Pu n-|1n, r = P n-|1n -1, r +
/
H Tn, u R n-,1u H n, u
u ! Nr (r)
xu n | n, r = xt n | n -1, r + Pu n | n, r
(13)
/
u ! Nr (r)
1
6H Tn, u R n, u
# ^y n, u - H n, u xt n | n -1, r h@
(14)
w i t h Q n = E {U n U Tn } a n d R n, u =
E {Vn, u V Tn, u} .
Combine step
Under the asynchronous RndEx protocol
proposed in [6], each node can send
In the sequel, it follows by comonly one exchange request between conplet ing squares that, if pu n | n, u (x n) =
secutive sensor measurements. To preN (x n | xu n | n, u, Pu n | n, u) for all u ! Nr (r),
vent cyclic exchanges, after an exchange
then p n | n, r (x n) on the left-hand side of
request from node r is met by node u,
(9) is also of the form N (x n | xt n | n, r, Pn | n, r)
where
node r is barred from accepting a future
exchange request from node u; howP n-|1n, r = / a r, u Pu n-|1n, u
(15)
ever, node r may still receive exchange
u ! Nr (r)
requests from other nodes l ! u in its
P n-|1n, r xt n | n, r = / a r, u Pu n-|1n, u xu n | n, u .
neighborhood, so that a node can actualu ! Nr (r)
ly carry out more than one pdf exchange
(16)
between consecutive measurements, but
the maximum number of such exchanges
Equations (11)-(16) define an ATC
is limited to the degree of the node, i.e.,
diffusion Kalman filter, which is a modithe number of neighbors to that particufied version of the diffusion Kalman
lar node. The process then stops when
filter introduced in [1]. Compared
all the nodes in the
to the algorithm in
network have made
[1], the ATC diffuWe present, also from a
one exchange request
sion Kalman filter in
Bayesian perspective, the
or have reached their
this lecture note has
general form of the adapt- maximum number of
a modified combine
then-combine (ATC) and the allowed exchanges.
step that, as shown
random exchange (rndex)
Based on the dein [4], is compatible
scription
in the previwith the minimizadistributed diffusion filters.
ous paragraph, and,
tion, at each node r
for the purposes of this lecture note,
and each instant n, of the average KL
it suffices to say that, at the end of the
divergence in (10) in the particular case
RndEx step, which must be completed
when the underlying pdfs are Gaussbefore the sensing instant n, node r
ian. As pointed out in [4], the modified
fusion rule in (15) and (16) was already
now has in store a different posterior pdf
known in previous literature as the covap (x n - 1 |yu 0: n - 1, s), which originally came
riance intersection method; see, e.g., [9].
from a random node s that is not necessarily in the neighborhood of r. Likewise,
the
original pdf p (x n -1 |yu 0: n -1, r) from
RndEx diffusion filter
node
r ended up at a random third node
An alternative to ATC diffusion is RndEx
diffusion, inspired by the pioneering work
t, which is not necessarily equal to s.
by Kar and Moura in [2].
Data assimilation step
RndEx step
First, assume that, at instant n - 1, a
given network node r, has in store a
posterior pdf p (x n - 1 |yu 0: n - 1, r), where
yu 0: n -1, r denotes a vector of observations that is a subset of y 0: n -1, 1: R and
consists of measurements coming from
different random locations in the entire
network, including, possibly, measurements that came from nodes outside
the closed neighborhood of r. Next,
node r randomly chooses a neighboring node and requests that the chosen
neighbor exchange its stored posterior pdf (or, rather, some parametric
representation thereof) with node r's
posterior pdf.
IEEE Signal Processing Magazine
|
May 2018
|
Next, at instant n, node r updates its
new stored pdf using the locally available observations in the vector yu n, r that
collects the measurements {y n, u} for all
u ! Nr (r) . Using again the total probability theorem and Bayes law, with the
mutual independence assumptions from
the "Problem Statement" section, node r
computes at instant n
p (x n |yu 0: n -1, s) =
#0
N
6 p (x n | x n -1)
# p (x n -1 |yu 0: n -1, s)@ dx n -1
(17)
p (x n |yu n, r, yu 0: n -1, s) ? ;
%
u ! Nr (r)
p (y n, u |x n)E
# p (x n |yu 0: n -1, s) . (18)
121
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