a distributed gaussian particle filtering using a like lead consensus and is the joint work with

the on is lou checked friends how much but there your each and

um are

so uh first let me summarise the

uh a contribution of this paper

we proposed a a a a distributed implementation of

of the gaussian particle filter that was originally introduced in

um

in a centralized for mean in the paper of co tech uh and you reach in two and

and three

so in this paper uh we we've posted the distributed implementation and in this implementation each uh

sensor computes i

global estimate based on to joint or all sensors uh likelihood function

and and

the joint likelihood function uh or its approximation is obtained uh

at each sensor in a distributed way using a the like likelihood consensus scheme which we propose to

in our previous paper two thousand ten

at the T asilomar conference

uh here we also use a a second stage of consensus algorithms

uh to reduce the complexity of the of the distributed gaussian

a particle filter

so a a is a brief comparison with some other consensus based distributed particle filters

so in this paper

in in far a lot uh two doesn't ten uh the use no approximations and so the

do estimation performance can be

better but on the other hand the communication requirements so can be much higher than it

in our case and in

uh uh two to and eight uh the use on a local like load functions

and in contrast to the use the joint likelihood function at each sensor

and

so the estimation performance of of our method is is better

um

okay so let's start with some

overview of distributed estimation wireless the sensor network

so what we consider is a wireless sensor network that's composed of capital K U

sensor nodes that joint we estimate a time varying state X and

and each of the sensors

indexed by

small K

uh obtains the measurement vector

is E

and

an example is so local aviation and

or tracking based on the sound emitted by a moving target or we can

and the cost that we would like to achieve are to forming so

it sensor nodes should obtain a global

state estimate

X head

and based on measurements of all other sensors and the network and this might be important for example in sensor

actuator or

robotic networks

and would like to use only local processing short distance communications with neighbours

and also no fusion center should be used and no routing of measurements throughout the network

okay we also wish to perform sequential estimation

uh of the time-varying state

X and

uh from the current and the past measurements of all sensors in the

in the sensor network

and and

so we consider nonlinear non gaussian state space model but with

independent additive gaussian measurement noise is

and it's uh such a system is described by the state transition pdf

and the joint likelihood function or J lf

where C N is uh

is the collection of measurements from all sensors

and

well in this case optimal bayesian estimation amounts to calculation of the posterior

uh pdf here

and sequential estimation is enabled by is uh a recursive posterior update they where we

turn the

previous post your to the current one

using the state transition pdf and and the don't by clint function

and joint like that function is important if you want to obtain

global results based on to

all sensors

measurements

okay so now let's have a look at the distributed gaussian particle filter

so it it's well known that for nonlinear non gaussian systems the optimal bayesian estimation is typically infeasible

and the computational feasible approximation is provided by a

particle filtering for

well sequential monte carlo approach

and think an example of

many one of the many particle filters is the the gaussian particle filter proposed in this paper

where the posterior is approximated by a gaussian pdf and the mean and covariance of this

gaussian approximation R

oh obtained from a set of

weighted samples or particles

and what we propose is a distributed

implementation of the

gaussian particle filter where each sensor

use

local

gaussian in particle filter to sequential track the mean and covariance of a local gaussian approximation but that the global

posterior

uh

and

in this case the the measurement update

at each sensor uses the global joint likelihood function and which ensures that global estimates are obtained

and it's end

and the J laugh is provided to to each sensor in a distributed way using the likelihood consensus

scheme that we proposed in in this paper

and some advantages are that the consensus algorithms employed by like that consensus require only local communications and operate without

putting protocols

and also no measurements or particles need to be exchange between the sensors

um so here are the i'll show some steps that each

sensor performs oh so the steps of a local

gaussian particle filter

so first couple at time and it's sends or obtains the gaussian approximation to the previous global posterior

then eight

draws

particles from this

a a gaussian approximation and it propagates so through the state this model so

basically it's samples new predicted particles from the state transition pdf

a then we need to calculate the joint likelihood function

at each sensor and to do this we used the likelihood consensus

and this step we will require communication between the

uh neighbouring sensors

and

after each sensor can update the particle weights using the

obtained trying to like lead function so

this is how it's done

so basically be we then evaluate the joint like with functions at the

but in like look function at the predicted particles so

that's why we need the joint likelihood function at each sensor

as a function of the

of the state X and four point twice evaluation

and once we have to particles and weights we can calculate again to meeting

and covariance of the of the

a gaussian approximation to the global posterior

and

the state estimate is basically equal to disk calculated the in here

so now let's have a look at how the like that consensus scheme operate

so

we we in this paper we consider the following measurement model we have here uh

measurement function H and K of X N which is

in general nonlinear it's it's a function of the

of of the sensor

index and

uh uh it it depends on the sensor and possibly also on time

and fees

additive uh

gaussian measurement noise which is

assumed to be independent from sensor to sensor

and you to this we obtain the joint like that function as a product of local likelihood functions

and therefore in the exponent of the joint likelihood

we have a sum over all sensors so this is this expression as and

a here

and for purposes of statistical inference is

S an expression completely describes the joint like lead functions will focus on a distributed calculation of of S N

and it will be

that's three to obtain this as a function of the state X N

and C N is just a collection of measurements from all sensors and it's observed and hence fixed

well

a direct calculation of of S and wood

required at each sensor knows the

measurements and also measurement functions of full other sensors in the network

but uh initial we assume that

each sensor only has its

local information so we would need to somehow

root this local information from each sensor to have every other sensor but that

so what we would like to a it so we

we choose another approach will be suitably approximate S N

by

suitably approximating the sensor measurement functions locally

and to do the approximation in such a way that we can use than consensus algorithms to compute S N

a

so

here we use a

polynomial approximation of the sensor measurement functions so which till is the polynomial approximation

uh and the

this function here P R of X and basically this is are the

the monomials all meals of of the polynomial but in principle we could use other basis functions to obtain

some more general approximation

and the the coefficients all five of this approximation there we calculate them using a least squares polynomial fitting and

as the data points for this we squares fit be use the predicted particles of the

of of the particle filter

and that's

important note that the

rocks summation so basically the

alpha coefficient of the approximation error obtained locally at it sensor so

we don't need to communicate anything to to do that

now if we have a substitute the

polynomial approximation H two the for for H in in this S expression we obtain

and approximate

S still that

uh since

H till this are

polynomials basically out of this um

overall all sensors we obtain also a polynomial but of twice the degree so

what we write this

we see her the polynomial

uh

you

coefficients the beta coefficients they contain for each sensor

all local information so it's measurement

as well as the U

alpha coefficients of the approximation of of fits a local

uh a measurement function

uh what's important is that the coefficients are independent of the state X N

and the only

way how the state and into this expression is that would these monomials or

some general basis function

and now if we exchange the order of summation here so we we get a uh

polynomial which has

coefficients T

and this coefficients here

there are obtained as a sum over all sensors

and therefore for the

these coefficients they contain information from the entire network

so we could view them

as the sufficient statistic that fully describe

is that still that

and in turn also the approximate joint likelihood function

so we see this is the approximate joint likelihood if each sensor knows these coefficients T

then it can evaluate the

joint likelihood function for more less for any any value of of the state X N

a so since this coefficients are obtained as already said

uh as a summation over all sensors state can be computed using the a distributed consensus algorithm at at each

sensor

so this is basically how would operate it check it's sensor computes locally coefficients speech of from the local available

data

and then the sum over all sensors is computed in a distributed by using consensus

and it requires only transmission of some

partial sums to the next per so we don't in to transmit measurements or or or or particles

a the communication load put therefore be much much lower

okay okay i'll just briefly mention

ah

a reduced complexity person of the distributed gaussian particle filter

a a so in in this

reduced complexity version each each of the

"'kay" set uh sensors

or

"'kay" local in particle filters

uses a reduced number of particles cheap prime

so we we use the number of particles by a factor put to the number of sensors

and we calculate a partial mean and the partial covariance variance of the global posterior but also using the joint

like with function of the using the like with sensors

and

after this partial means and covariances can be

combine by means of the second stage of consensus algorithms

and

if the second stage

use a sufficient

number of iterations then the pitch estimation performance

of the reduced complexity version will be effectively put to that of the original one so

we

reduce the computational complexity but of course we introduce some new communications so it comes at the cost of some

increasing in communications

okay now i'll show you

a target tracking sample and some simulation results

so

oh in this example the state

represents the two D position and the two D velocity of the target

and it it false according to this state transition equation

uh

and we consider or we simulate a

network of randomly deployed acoustic amplitude sensors that sounds the sound i mean that sense the sound i meet it

by to target

and

the measurement model is

the following so the

sensor measurement function is basically given here

so we have the amplitude of the

of the source divided by the distance between the

target

and the sensor

and it's in principle the sensor positions can be time varying so we could

the plight this mess the also the dynamic uh a sensor networks

a this is the setting so we deployed sensors in the field of

do mention two hundred by two very meters and

it consists of twenty five acoustic and sensors

um

and the proposed distributed gaussian particle filter and it's reduced complexity person now compared with a centralized gaussian in particle

filter

we used one thousand particles sense to approximate the measurement function we use a polynomial of degree

to

which leads to fourteen consensus algorithms that need to be executed in parallel so

basically a what in one iteration of like consensus you need to to transmit fourteen real numbers

and we compare like that consensus that use eight iterations of consensus

with a

with a case where we calculate the sums

exactly so that that could be

a that's a

as an S the asymptotic case

so infinite number of consensus iterations more less

okay okay here just as an illustration we see that the green line is the true target trajectory and the

the right one is the track one

and it's just a result

a from one of these sense but in principle all sensors obtain the same reason

okay here is to root mean square error performance of first this time

ah

the black line is the centralized case and as expected this the best one

now if you look at the distributed case the exact some calculation

that's the red plan

there is a slight performance degradation and

of course if you only use eight iterations of consensus you you get the to line which has

slightly worse performance again but even

we compare

the blue and red two

to the to the black ones with to the centralized case the the performance degradation is not so large

here it's

average average rmse which we averaged also over over the time and versus just measurement noise variance so yeah the

noise variance rises is also the

error arise is but more less the comparison between the three mats this the same as something

on the first figure

here it's the dependence of the estimation error on the number of consensus iterations and yeah of course as the

number of iterations increases the performance gets better

but what's interesting is the

when we compared to the

solid

ooh curve with the solid red once for the

strip it gaussian part before and it's reduced complexity version

for

lower number of

iterations here the the reduced used complexity version

uh

has a slightly better performance and this we could explain

more or less that

such a way that the second stage of consensus algorithms helps to diffuse for that a local information

throughout the network

okay okay so what's conclude we proposed to distributed it uh that was can particle filtering scheme that

in which each sensor around a local gaussian particle for to that computes a global state estimate that reflects the

measurements of all sensors

and to do this

the

we have to the particle weights at it's sensor using the joint like good function which we obtained in a

distributed way

i likelihood consensus

and

a think about like let can is that it requires on only local communications of

some sufficient statistics so no measurements or particles need to be communicated and

is also suitable for dynamic sensor networks

and we also propose a reduced complexity variant of the distributed option particle filter

and the simulation results indicate that the performance is good even in comparison with the centralized a in particle filter

okay so that's compose my talk thanks

i

i

i

i

yeah

should

is uh are insensitive to K to the value issue

a a take a static um a couple of lot the number of polynomials right

uh

yeah that's the order of the problem a yes i and what that this approximation

is good for K

you mean all sensors yes

uh

yeah i mean in this in this application we use the same same

type of measurement function at each sensor

so

that's what we used also the same approximation for for all sensors

but i mean in principle you could have

different measurement functions that different sensors and then you would need to

use different order of polynomials and yeah

yeah

i

and

say

a my in the same manner value

of the global one

a function

well i mean you can only guaranteed by using a

yeah

a i think you cannot guarantee these i mean it depends on the on the size of your network and

the bigger the network the more iterations you need i mean so

a

hmmm

uh yes

yes i on there are slight differences i mean depend on the number of iteration i mean you can

oh

hmmm no actually in in in the gaussian particle for to you don't need any a resampling because you construct

the gaussian posterior and then you sampled new

but yes i mean if you have insufficient number of iterations then each because each of the also operates separately

so it go

each of the nose has a

he's all its own set of particles and its own set of weights and it will there be slight difference

yeah

and

yeah

and

oh

lee

yes

that's one

i

well

yes

i

yes

uh

no no it's it's not not the case uh i mean

uh it's just what

you saying

and

as

yeah okay

yeah