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