Speech Transcript - DISTRIBUTED GAUSSIAN PARTICLE FILTERING USING LIKELIHOOD CONSENSUS

0:00:13	a distributed gaussian particle filtering using a like lead consensus and is the joint work with
0:00:18	the on is lou checked friends how much but there your each and
0:00:21	um are
0:00:23	so uh first let me summarise the
0:00:27	uh a contribution of this paper
0:00:29	we proposed a a a a distributed implementation of
0:00:33	of the gaussian particle filter that was originally introduced in
0:00:37	um
0:00:38	in a centralized for mean in the paper of co tech uh and you reach in two and
0:00:43	and three
0:00:44	so in this paper uh we we've posted the distributed implementation and in this implementation each uh
0:00:51	sensor computes i
0:00:54	global estimate based on to joint or all sensors uh likelihood function
0:00:59	and and
0:01:00	the joint likelihood function uh or its approximation is obtained uh
0:01:05	at each sensor in a distributed way using a the like likelihood consensus scheme which we propose to
0:01:11	in our previous paper two thousand ten
0:01:13	at the T asilomar conference
0:01:16	uh here we also use a a second stage of consensus algorithms
0:01:20	uh to reduce the complexity of the of the distributed gaussian
0:01:25	a particle filter
0:01:27	so a a is a brief comparison with some other consensus based distributed particle filters
0:01:33	so in this paper
0:01:35	in in far a lot uh two doesn't ten uh the use no approximations and so the
0:01:41	do estimation performance can be
0:01:43	better but on the other hand the communication requirements so can be much higher than it
0:01:48	in our case and in
0:01:50	uh uh two to and eight uh the use on a local like load functions
0:01:55	and in contrast to the use the joint likelihood function at each sensor
0:01:58	and
0:01:59	so the estimation performance of of our method is is better
0:02:04	um
0:02:05	okay so let's start with some
0:02:08	overview of distributed estimation wireless the sensor network
0:02:12	so what we consider is a wireless sensor network that's composed of capital K U
0:02:17	sensor nodes that joint we estimate a time varying state X and
0:02:23	and each of the sensors
0:02:24	indexed by
0:02:25	small K
0:02:27	uh obtains the measurement vector
0:02:30	is E
0:02:31	and
0:02:31	an example is so local aviation and
0:02:34	or tracking based on the sound emitted by a moving target or we can
0:02:39	and the cost that we would like to achieve are to forming so
0:02:42	it sensor nodes should obtain a global
0:02:44	state estimate
0:02:45	X head
0:02:46	and based on measurements of all other sensors and the network and this might be important for example in sensor
0:02:51	actuator or
0:02:53	robotic networks
0:02:55	and would like to use only local processing short distance communications with neighbours
0:03:00	and also no fusion center should be used and no routing of measurements throughout the network
0:03:07	okay we also wish to perform sequential estimation
0:03:09	uh of the time-varying state
0:03:12	X and
0:03:13	uh from the current and the past measurements of all sensors in the
0:03:16	in the sensor network
0:03:19	and and
0:03:20	so we consider nonlinear non gaussian state space model but with
0:03:24	independent additive gaussian measurement noise is
0:03:27	and it's uh such a system is described by the state transition pdf
0:03:31	and the joint likelihood function or J lf
0:03:34	where C N is uh
0:03:37	is the collection of measurements from all sensors
0:03:41	and
0:03:41	well in this case optimal bayesian estimation amounts to calculation of the posterior
0:03:46	uh pdf here
0:03:48	and sequential estimation is enabled by is uh a recursive posterior update they where we
0:03:53	turn the
0:03:54	previous post your to the current one
0:03:57	using the state transition pdf and and the don't by clint function
0:04:00	and joint like that function is important if you want to obtain
0:04:03	global results based on to
0:04:06	all sensors
0:04:07	measurements
0:04:09	okay so now let's have a look at the distributed gaussian particle filter
0:04:14	so it it's well known that for nonlinear non gaussian systems the optimal bayesian estimation is typically infeasible
0:04:21	and the computational feasible approximation is provided by a
0:04:25	particle filtering for
0:04:27	well sequential monte carlo approach
0:04:29	and think an example of
0:04:31	many one of the many particle filters is the the gaussian particle filter proposed in this paper
0:04:36	where the posterior is approximated by a gaussian pdf and the mean and covariance of this
0:04:42	gaussian approximation R
0:04:44	oh obtained from a set of
0:04:47	weighted samples or particles
0:04:50	and what we propose is a distributed
0:04:52	implementation of the
0:04:54	gaussian particle filter where each sensor
0:04:57	use
0:04:58	local
0:05:00	gaussian in particle filter to sequential track the mean and covariance of a local gaussian approximation but that the global
0:05:06	posterior
0:05:08	uh
0:05:09	and
0:05:10	in this case the the measurement update
0:05:12	at each sensor uses the global joint likelihood function and which ensures that global estimates are obtained
0:05:18	and it's end
0:05:20	and the J laugh is provided to to each sensor in a distributed way using the likelihood consensus
0:05:27	scheme that we proposed in in this paper
0:05:29	and some advantages are that the consensus algorithms employed by like that consensus require only local communications and operate without
0:05:38	putting protocols
0:05:40	and also no measurements or particles need to be exchange between the sensors
0:05:45	um so here are the i'll show some steps that each
0:05:49	sensor performs oh so the steps of a local
0:05:52	gaussian particle filter
0:05:54	so first couple at time and it's sends or obtains the gaussian approximation to the previous global posterior
0:06:01	then eight
0:06:02	draws
0:06:03	particles from this
0:06:04	a a gaussian approximation and it propagates so through the state this model so
0:06:09	basically it's samples new predicted particles from the state transition pdf
0:06:14	a then we need to calculate the joint likelihood function
0:06:18	at each sensor and to do this we used the likelihood consensus
0:06:21	and this step we will require communication between the
0:06:25	uh neighbouring sensors
0:06:26	and
0:06:28	after each sensor can update the particle weights using the
0:06:31	obtained trying to like lead function so
0:06:33	this is how it's done
0:06:35	so basically be we then evaluate the joint like with functions at the
0:06:39	but in like look function at the predicted particles so
0:06:42	that's why we need the joint likelihood function at each sensor
0:06:45	as a function of the
0:06:46	of the state X and four point twice evaluation
0:06:50	and once we have to particles and weights we can calculate again to meeting
0:06:54	and covariance of the of the
0:06:56	a gaussian approximation to the global posterior
0:06:59	and
0:06:59	the state estimate is basically equal to disk calculated the in here
0:07:04	so now let's have a look at how the like that consensus scheme operate
0:07:09	so
0:07:10	we we in this paper we consider the following measurement model we have here uh
0:07:15	measurement function H and K of X N which is
0:07:19	in general nonlinear it's it's a function of the
0:07:22	of of the sensor
0:07:23	index and
0:07:25	uh uh it it depends on the sensor and possibly also on time
0:07:28	and fees
0:07:29	additive uh
0:07:31	gaussian measurement noise which is
0:07:33	assumed to be independent from sensor to sensor
0:07:35	and you to this we obtain the joint like that function as a product of local likelihood functions
0:07:41	and therefore in the exponent of the joint likelihood
0:07:44	we have a sum over all sensors so this is this expression as and
0:07:48	a here
0:07:49	and for purposes of statistical inference is
0:07:52	S an expression completely describes the joint like lead functions will focus on a distributed calculation of of S N
0:07:59	and it will be
0:08:00	that's three to obtain this as a function of the state X N
0:08:03	and C N is just a collection of measurements from all sensors and it's observed and hence fixed
0:08:09	well
0:08:10	a direct calculation of of S and wood
0:08:13	required at each sensor knows the
0:08:15	measurements and also measurement functions of full other sensors in the network
0:08:19	but uh initial we assume that
0:08:22	each sensor only has its
0:08:23	local information so we would need to somehow
0:08:26	root this local information from each sensor to have every other sensor but that
0:08:30	so what we would like to a it so we
0:08:33	we choose another approach will be suitably approximate S N
0:08:37	by
0:08:38	suitably approximating the sensor measurement functions locally
0:08:41	and to do the approximation in such a way that we can use than consensus algorithms to compute S N
0:08:48	a
0:08:49	so
0:08:49	here we use a
0:08:51	polynomial approximation of the sensor measurement functions so which till is the polynomial approximation
0:08:57	uh and the
0:08:58	this function here P R of X and basically this is are the
0:09:02	the monomials all meals of of the polynomial but in principle we could use other basis functions to obtain
0:09:08	some more general approximation
0:09:11	and the the coefficients all five of this approximation there we calculate them using a least squares polynomial fitting and
0:09:18	as the data points for this we squares fit be use the predicted particles of the
0:09:23	of of the particle filter
0:09:24	and that's
0:09:25	important note that the
0:09:28	rocks summation so basically the
0:09:29	alpha coefficient of the approximation error obtained locally at it sensor so
0:09:33	we don't need to communicate anything to to do that
0:09:38	now if we have a substitute the
0:09:40	polynomial approximation H two the for for H in in this S expression we obtain
0:09:46	and approximate
0:09:48	S still that
0:09:49	uh since
0:09:51	H till this are
0:09:53	polynomials basically out of this um
0:09:55	overall all sensors we obtain also a polynomial but of twice the degree so
0:10:00	what we write this
0:10:01	we see her the polynomial
0:10:03	uh
0:10:05	you
0:10:05	coefficients the beta coefficients they contain for each sensor
0:10:09	all local information so it's measurement
0:10:12	as well as the U
0:10:13	alpha coefficients of the approximation of of fits a local
0:10:17	uh a measurement function
0:10:19	uh what's important is that the coefficients are independent of the state X N
0:10:24	and the only
0:10:25	way how the state and into this expression is that would these monomials or
0:10:29	some general basis function
0:10:31	and now if we exchange the order of summation here so we we get a uh
0:10:36	polynomial which has
0:10:38	coefficients T
0:10:40	and this coefficients here
0:10:42	there are obtained as a sum over all sensors
0:10:45	and therefore for the
0:10:46	these coefficients they contain information from the entire network
0:10:50	so we could view them
0:10:52	as the sufficient statistic that fully describe
0:10:56	is that still that
0:10:57	and in turn also the approximate joint likelihood function
0:11:00	so we see this is the approximate joint likelihood if each sensor knows these coefficients T
0:11:06	then it can evaluate the
0:11:08	joint likelihood function for more less for any any value of of the state X N
0:11:14	a so since this coefficients are obtained as already said
0:11:18	uh as a summation over all sensors state can be computed using the a distributed consensus algorithm at at each
0:11:24	sensor
0:11:25	so this is basically how would operate it check it's sensor computes locally coefficients speech of from the local available
0:11:31	data
0:11:32	and then the sum over all sensors is computed in a distributed by using consensus
0:11:38	and it requires only transmission of some
0:11:40	partial sums to the next per so we don't in to transmit measurements or or or or particles
0:11:46	a the communication load put therefore be much much lower
0:11:51	okay okay i'll just briefly mention
0:11:53	ah
0:11:54	a reduced complexity person of the distributed gaussian particle filter
0:11:59	a a so in in this
0:12:01	reduced complexity version each each of the
0:12:04	"'kay" set uh sensors
0:12:06	or
0:12:07	"'kay" local in particle filters
0:12:09	uses a reduced number of particles cheap prime
0:12:12	so we we use the number of particles by a factor put to the number of sensors
0:12:17	and we calculate a partial mean and the partial covariance variance of the global posterior but also using the joint
0:12:23	like with function of the using the like with sensors
0:12:27	and
0:12:28	after this partial means and covariances can be
0:12:31	combine by means of the second stage of consensus algorithms
0:12:36	and
0:12:36	if the second stage
0:12:38	use a sufficient
0:12:39	number of iterations then the pitch estimation performance
0:12:43	of the reduced complexity version will be effectively put to that of the original one so
0:12:48	we
0:12:49	reduce the computational complexity but of course we introduce some new communications so it comes at the cost of some
0:12:58	increasing in communications
0:13:01	okay now i'll show you
0:13:02	a target tracking sample and some simulation results
0:13:06	so
0:13:07	oh in this example the state
0:13:10	represents the two D position and the two D velocity of the target
0:13:14	and it it false according to this state transition equation
0:13:19	uh
0:13:20	and we consider or we simulate a
0:13:23	network of randomly deployed acoustic amplitude sensors that sounds the sound i mean that sense the sound i meet it
0:13:30	by to target
0:13:31	and
0:13:32	the measurement model is
0:13:33	the following so the
0:13:35	sensor measurement function is basically given here
0:13:38	so we have the amplitude of the
0:13:40	of the source divided by the distance between the
0:13:43	target
0:13:44	and the sensor
0:13:46	and it's in principle the sensor positions can be time varying so we could
0:13:51	the plight this mess the also the dynamic uh a sensor networks
0:13:56	a this is the setting so we deployed sensors in the field of
0:14:00	do mention two hundred by two very meters and
0:14:03	it consists of twenty five acoustic and sensors
0:14:07	um
0:14:08	and the proposed distributed gaussian particle filter and it's reduced complexity person now compared with a centralized gaussian in particle
0:14:16	filter
0:14:17	we used one thousand particles sense to approximate the measurement function we use a polynomial of degree
0:14:22	to
0:14:23	which leads to fourteen consensus algorithms that need to be executed in parallel so
0:14:28	basically a what in one iteration of like consensus you need to to transmit fourteen real numbers
0:14:34	and we compare like that consensus that use eight iterations of consensus
0:14:39	with a
0:14:41	with a case where we calculate the sums
0:14:44	exactly so that that could be
0:14:46	a that's a
0:14:48	as an S the asymptotic case
0:14:49	so infinite number of consensus iterations more less
0:14:54	okay okay here just as an illustration we see that the green line is the true target trajectory and the
0:14:59	the right one is the track one
0:15:02	and it's just a result
0:15:03	a from one of these sense but in principle all sensors obtain the same reason
0:15:08	okay here is to root mean square error performance of first this time
0:15:13	ah
0:15:13	the black line is the centralized case and as expected this the best one
0:15:17	now if you look at the distributed case the exact some calculation
0:15:22	that's the red plan
0:15:23	there is a slight performance degradation and
0:15:26	of course if you only use eight iterations of consensus you you get the to line which has
0:15:31	slightly worse performance again but even
0:15:34	we compare
0:15:35	the blue and red two
0:15:37	to the to the black ones with to the centralized case the the performance degradation is not so large
0:15:42	here it's
0:15:44	average average rmse which we averaged also over over the time and versus just measurement noise variance so yeah the
0:15:50	noise variance rises is also the
0:15:52	error arise is but more less the comparison between the three mats this the same as something
0:15:57	on the first figure
0:15:58	here it's the dependence of the estimation error on the number of consensus iterations and yeah of course as the
0:16:04	number of iterations increases the performance gets better
0:16:08	but what's interesting is the
0:16:11	when we compared to the
0:16:12	solid
0:16:13	ooh curve with the solid red once for the
0:16:16	strip it gaussian part before and it's reduced complexity version
0:16:19	for
0:16:20	lower number of
0:16:21	iterations here the the reduced used complexity version
0:16:25	uh
0:16:25	has a slightly better performance and this we could explain
0:16:29	more or less that
0:16:30	such a way that the second stage of consensus algorithms helps to diffuse for that a local information
0:16:36	throughout the network
0:16:38	okay okay so what's conclude we proposed to distributed it uh that was can particle filtering scheme that
0:16:45	in which each sensor around a local gaussian particle for to that computes a global state estimate that reflects the
0:16:51	measurements of all sensors
0:16:52	and to do this
0:16:54	the
0:16:55	we have to the particle weights at it's sensor using the joint like good function which we obtained in a
0:17:00	distributed way
0:17:01	i likelihood consensus
0:17:04	and
0:17:05	a think about like let can is that it requires on only local communications of
0:17:10	some sufficient statistics so no measurements or particles need to be communicated and
0:17:14	is also suitable for dynamic sensor networks
0:17:17	and we also propose a reduced complexity variant of the distributed option particle filter
0:17:23	and the simulation results indicate that the performance is good even in comparison with the centralized a in particle filter
0:17:31	okay so that's compose my talk thanks
0:17:42	i
0:17:44	i
0:17:44	i
0:17:45	i
0:17:46	yeah
0:17:47	should
0:17:48	is uh are insensitive to K to the value issue
0:17:53	a a take a static um a couple of lot the number of polynomials right
0:17:58	uh
0:17:59	yeah that's the order of the problem a yes i and what that this approximation
0:18:03	is good for K
0:18:05	you mean all sensors yes
0:18:06	uh
0:18:07	yeah i mean in this in this application we use the same same
0:18:12	type of measurement function at each sensor
0:18:14	so
0:18:16	that's what we used also the same approximation for for all sensors
0:18:19	but i mean in principle you could have
0:18:21	different measurement functions that different sensors and then you would need to
0:18:27	use different order of polynomials and yeah
0:18:34	yeah
0:18:37	i
0:18:41	and
0:18:42	say
0:18:43	a my in the same manner value
0:18:45	of the global one
0:18:46	a function
0:18:48	well i mean you can only guaranteed by using a
0:18:51	yeah
0:18:54	a i think you cannot guarantee these i mean it depends on the on the size of your network and
0:18:59	the bigger the network the more iterations you need i mean so
0:19:03	a
0:19:06	hmmm
0:19:07	uh yes
0:19:08	yes i on there are slight differences i mean depend on the number of iteration i mean you can
0:19:16	oh
0:19:19	hmmm no actually in in in the gaussian particle for to you don't need any a resampling because you construct
0:19:24	the gaussian posterior and then you sampled new
0:19:27	but yes i mean if you have insufficient number of iterations then each because each of the also operates separately
0:19:33	so it go
0:19:34	each of the nose has a
0:19:35	he's all its own set of particles and its own set of weights and it will there be slight difference
0:19:43	yeah
0:19:44	and
0:19:45	yeah
0:19:48	and
0:19:49	oh
0:19:49	lee
0:19:50	yes
0:19:51	that's one
0:19:52	i
0:19:53	well
0:19:54	yes
0:19:56	i
0:19:57	yes
0:19:58	uh
0:20:00	no no it's it's not not the case uh i mean
0:20:04	uh it's just what
0:20:06	you saying
0:20:08	and
0:20:09	as
0:20:10	yeah okay
0:20:13	yeah

DISTRIBUTED GAUSSIAN PARTICLE FILTERING USING LIKELIHOOD CONSENSUS

Distributed and Collaborative Signal Processing

Presented by: Ondrej Hlinka, Author(s): Ondrej Hlinka, Ondrej Sluciak, Franz Hlawatsch, Vienna University of Technology, Austria; Petar Djuric, Stony Brook University, United States; Markus Rupp, Vienna University of Technology, Austria