IEEE Signal Processing - May 2018 - 122

ity of l n, thus allowing
originates at a differThe formulation of the
diffusion of informaent, random node s.
T
T T
diffusion filtering problem
yu 0: n, r = 6yu 0: n -1, s yu n, r@ .
tion. Unlike in the ATC
Finally, Figure 5
in a Bayesian framework
methodology, howillustrates the posteAs a given pdf travels through a random
as presented in this
ever, there is no comrior pdf evolution at
path of nodes {l 0, l 1, f, l n} over the netlecture note improves
bine step that merges
each node over four
work graph between instant zero and
node pdfs within local
time instants in a
instant n, then
the understanding of
neighborhoods at the
simple example with
existing linear distributed
uy 0: n, l n = 6yu 0T, l 0 yu 1T, l 1 fyu Tn, l n@T
same instant.
a linear network with
filtering algorithms.
Figure 4 presents
three nodes running
t
the
data
assimilation
the RndEx diffusion
at
node
x
l
and the estimate n | n, l n
n at instant
step at node r. Note that it is very similar to
filter algorithm. Note that the initial
n accordingly depends on measurements
posteriors {p (x 0 | yu 0,r)}, r ! {1, 2, 3} at
from multiple random locations in the netthe ATC filter's adapt step. However, the
work that are not necessarily in the vicinposterior p (x n -1 |yu 0: n -1, s) at instant n - 1
instant n = 0 are shaded according to
their corresponding nodes' colors. As n
increases, each posterior is updated with
measurements collected along a random,
~
p (xn - 1 | y0:n - 1, s)
distinct path over the network.
We redefine now

RndEx diffusion Kalman filter
In the special case of a linear, Gaussian state-space model as in (3) and (4),
p (x n |yu 0: n, r) reduces at each node r and
each instant n to a multivariate Gaussian
function specified by a mean vector xt n | n, r
and a covariance matrix Pn | n, r . During
the execution of an RndEx protocol, it
suffices then for the nodes to exchange
their respective mean vectors and covariance matrices with their neighbors.
Solving (17) and (18) when the underlying pdfs are Gaussian, it follows that,
upon receiving xt n -1 | n -1, s and Pn -1 | n -1, s
from node s at instant n - 1, node r at
instant n updates its posterior mean vector and posterior covariance matrix as

p (xn | xn - 1)

y0:n - 1, s)
p (xn | ~
p (yn, u 4 | xn)

p (yn, u 1 | xn)
p (yn, u | xn)
2
p (yn, u 3 | xn)

p (yn, u |N(r )| | xn)

~
~
∝ p (xn | yn, r , y0:n - 1, s)

Figure 4. The data assimilation step.

xt n | n -1, r = Fn -1 xt n -1 | n -1, s
n

0

1

2

3

S1
p (x0 | {y0,1, y0,2})
~
y0,1

p (x1 | ~
y1,1, ~
y0,2)
~
y0:1,1

p (x2 | ~
y2,1, ~
y0:1,3)
~
y0:2,1

p (x3 | ~
y3,1, ~
y0:2,2)
~
y0:3,1

Random
Exchanges

S2

S3

p (x0 | {y0,1, y0,2, y0,3})

p (x0 | {y0,2, y0,3})

~
y0,2

p (x1 | ~
y1,2, ~
y0,3)
~
y0:1,2

~ ,y
~
p (x2 | y
2,2 0:1,1)
~
y0:2,2

p (x3 | ~
y3,2, ~
y0:2,3)
~
y0:3,2

~
y0:1,3

~
p (x2 | ~
y2,3, y
0:1,2)
~
y0:2,3

p (x3 | ~
y3,3, ~
y0:2,1)
~
y0:3,3

+ G n -1 Q n -1 G Tn -1
and

S1

S2

S2

S3

S2

S3

S1

S2

S1
S2

S2
S3

|

May 2018

P n-|1n, r = P n-|1n -1, r +

IEEE Signal Processing Magazine

/

u ! Nr (r)

xt n | n, r = xt n | n -1, r + Pn | n, r

H Tn, u R n-,1u H n, u
(21)

/

u ! Nr (r)

1
6H nT, u R n, u

# ^y n, u - H n, u xt n | n -1, r h@ .

(22)

Equations (19)-(22) describe the RndEx
diffusion Kalman filter recursions at node
r and at instant n, which are a variation
of the gossip Kalman filter recursions
proposed in [2].

Figure 5. The RndEx protocol.
122

(20)

-

~
y0,3

p (x1 | ~
y1,3, ~
y0,1)

(19)

Pn | n -1, r = Fn -1 Pn -1 | n -1, s F nT -1

|
Table of Contents for the Digital Edition of IEEE Signal Processing - May 2018
