Non-Linear Bayesian CBRN Source Term Estimation

Transcription

1 Non-Linear Bayesian CBRN Source Term Estimation Peter Robins Hazar Assessment, Simulation an Preiction Group Dstl Porton Down, UK. Paul Thomas Hazar Assessment, Simulation an Preiction Group Dstl Porton Down, UK. Abstract - A Bayesian posterior probability ensity sampling algorithm suitable for use in estimating chemical, biological, raiological or nuclear atmospheric releases of material is propose. The estimation problem is ifferent from many other fusion problems, in that there are no state evolution equations, the forwar moel is highly non-linear an the likelihoos are non-gaussian. The algorithm is able to use store output from complex atmospheric ispersion moels for more rapi upate of the posterior from new ata without having to re-run the moels. The use of Differential Evolution Monte Carlo allows new samples to rapily iverge from egenerate sample sets. Results for inferences mae in a slightly simplifie environment of chemical releases only are presente, emonstrating that the sampling scheme performs aequately espite constraints of a short time span to calculate inferences from complex likelihoo calculations. Extensions to wier applications an improvements to the algorithm are iscusse. Keywors: Bayesian, MCMC, DE-MC, non-linear estimation, non-gaussian. 1 Introuction The threat of Chemical, Biological, Raiological an Nuclear (CBRN) attack is a frequent feature of the moern battlefiel. Face with such a threat, a Commaner nees to be able to rapily assess current CBRN hazar areas an etermine the optimum plan for force protection an maintaining operational momentum. In the future, the quality an quantity of CBRN relate information available to a Commaner will improve raically ue to the Digitization of the Battlespace initiative. In particular, evelopments in wie-area ata transmission an portable computing power will mean that large amounts of invaluable ata from the tactical groun level will be mae wiely available, along with warnings an comman ecisions. In the CBRN omain this information will be manage by the Nuclear, Biological an Chemical Battlespace Information System Application (NBC BISA). This aims to automatically collect, process an rapily isseminate CBRN information proucts to the user. Such information proucts may inclue sensor reports, observations an intelligence information as well as resultant hazar warnings an NBC s. However, since CBRN information is largely concerne with the ownwin effects of an event rather than the event itself, the incorporation of CBRN ata poses particular challenges. Detections or observations may be from ifferent locations, at ifferent times or may even relate to ifferent events. In orer to fully exploit the information containe in these raw ata an provie meaningful hazar assessments, the NBC BISA nees to intelligently combine information from multiple atasources to evelop a picture of the current CBRN threat. At the heart of the NBC BISA information fusion system will be the Source Term Estimation Moel (STEM), the Bayesian inference engine that estimates the posterior probability istribution of location, time, mass an material of a CBRN release. CBRN source term estimation is a highly non-linear Bayesian inference problem requiring high-spee inferences from slow likelihoo calculations. Likelihoo calculations require a ispersion moel which has been simplifie an approximate to take 0.1s to calculate for a simple gaseous release over flat, featureless terrain; longer for more complicate moels. The estimation problem iffers from the ata fusion more commonly use in applications such as RADAR tracking because there are no state preiction equations. In orer to simplify the inference problem, the release of material into the atmosphere is assume to be a single instantaneous event. Data associate with the release event are also elaye by several minutes. Our requirement is to estimate (with uncertainties) the location, mass, time an material of a covert release of CBRN agent within a short time from the first inication of the release. A short time frame allows potentially lifesaving ecisions to be mae about CBRN protection, evacuation or mission moifications. 2 CBRN information fusion 2.1 Hypothesis space To minimize the imensions of our inference problem, we choose to moel the simplest type of atmospheric release of a substance, namely an instantaneous point release on the groun. For many more complicate releases, a point release may be foun that results in a similar ownwin concentration istribution as long as only a single ispersing material is involve. We note however, that this simplification coul cause inconsistencies with other /05/$ IEEE

2 sources of information, e.g. line-of-sight observations of a release. A hypothesize instantaneous point release 0 can be escribe using only five parameters; mass q, location (x,y), time t an material type a. 0 ( x, ytqa,,, ) T (1) In orer to test a sampling scheme we choose a esire hypothesis space to span a 30km square battle space, 6 hours of time, 6 orers of magnitue of mass an over 100 material types. The large number of material types woul inclue harmful toxic weapons, toxic inustrial materials an harmless materials that nevertheless prouce a signal at any of the sensors we may choose to use; the so-calle interferants. In the absence of any intelligence ata to the contrary, we use Laplace s principle of inifference[1] to set our prior istribution as uniform over this hypothesis space. We choose to inclue tails on the continuous istributions outsie of the uniform omain for the continuous hypothesis components, i.e. location, mass an time; but not material. These allow the moe of the posterior to be outsie the specifie hypothesis space if the ata suggest. We can easily iagnose problems with the hypothesis space if this occurs in practice. If we were to employ a naïve gri-base posterior sampling scheme with a precision of 250m for location, one minute for time an 50% magnitue in mass; in excess of 17billion samples woul be require. Therefore we require a more intelligent sampling scheme. One of the most obvious suggestions to reuce the size of the hypothesis space woul be to confine hypotheses to the parameter values escribing a release that is etectable by the sensors. However, this has not been implemente as, in the future, the source term estimator will have to eal with other, more varie sources of information whereby the ability to etect a release is impossible to pre-calculate before the event. An obvious case is information from a human observer who has seen a release event, but until a report has been mae, the observer coul be anywhere on the battlefiel. 2.2 Available ata The ata are sensor responses to a material concentration in the atmosphere at a specific point in space (the sensor location) an time. Some sensors of this type can transmit a time series of bar reaings an material type classifications. Bar reaings are name after the rectangular lights on the front of each sensor that inicate the orer of magnitue of local concentration of isperse material. Data sampling rates can be of the orer of one secon, but isperse concentrations are likely to be highly correlate at this frequency so samples are usually taken once per minute. The time scale over which samples are truly inepenent is epenent on istance upwin from the sensors that the material was release. 2.3 Likelihoo calculation Likelihoo calculation requires output from a ispersion moel. The moel is initialize with the release parameters 0 an concentration is requeste at a series of specifie times an locations associate with ata. Dispersion is a stochastic process (see Fig. 1), Fig. 1. Material ispersion in a moelle urban environment (by Steve Walker, Dstl) so the moel output is the mean an variance of a clippe normal probability istribution for concentration, c[2]: Pc ( N, 6N) 0 { x 0} (2) 1 c Г 2 N N erf - c exp Г { x F0} N 6N 3 N where δ ( c) is the Dirac elta function, µ N an σ N are the mean an stanar eviation of the normal istribution that has been clippe. To calculate the likelihoo of a particular hypothesize release given an instance of sensor ata, the clippe normal istribution has to be multiplie by the probability of concentration given the sensor reaing: P(, ba ca, ), where b is the bar reaing, a is the h material classification carrie out by the sensor an a h is the hypothesize material. This prouct is then integrate over all possible values of c, as it is a hien variable that is never irectly observe. P( ba, ca, h) P( c µ G, σ G) c (3) 0 The etails of this calculation can be foun in [3]. 2.4 Storage of ispersion calculations For the purposes of source term estimation, in orer to calculate a new likelihoo from a new piece of ata, it woul normally be necessary to re-run the ispersion coe. This is too costly in terms of CPU effort so an alternative where ispersion moel output is store is require. Dispersion moelling is epenent on meteorological conitions, which are complex an ynamically evolving. It woul not be possible to pre-compute an store enough ispersion runs to cover all meteorological circumstances. Given that a meteorological information server may yiel

3 upate estimates as often as every half hour, only recently calculate ispersion moels are of any use to the source term estimator. 3 Sampling algorithm We now propose a posterior sampling algorithm that can make use of store ata from ispersion calculations using recent meteorological information. The algorithm shares some of the attributes of a particle filter metho[4], in that knowlege of the posterior is represente by samples an weights, however it iffers in that there are no equations for state preiction. Only likelihoo calculations are use to upate the posterior. We also use an alternative metho of aing new samples as generating a large number of new samples between the receipt of each new piece of ata woul be too computationally expensive. We efine a ata winow, T D. This sets the time bouns in which ata will be consiere. For instance, if a system is suppose to provie a reasonable inference within a short time-scale of receipt of the first inicating an attack, it may be inappropriate to consier ata referring to a time more than half an hour ago. The algorithm has two sections. The first section escribes calculations to be mae before an in between the arrival of ata. The secon section escribes calculations necessary when ata o arrive. 3.1 Ile time processing When the inference engine is waiting for new ata, the CPU time available is use to generate new samples in the hypothesis space accoring to the posterior istribution. Whenever a new release hypothesis is propose, we run a ispersion moel using the current meteorological information, with output store at each of the known sensor locations from the current time minus T D to current time plus T D at one minute intervals. This means that the likelihoo from all pieces of store ata with a time stamp of up to T D in the past, can be rapily evaluate without re-running the ispersion coe. The likelihoo is calculate instea by looking up the ispersion output parameters in the store arrays an using linear interpolation to obtain the require values. Storing output associate with times in the future ensures that any hypotheses propose now have enough ispersion information to rapily evaluate their likelihoo for any new ata arriving up to T D in the future. The un-normalize posterior probability of the propose hypothesis 0 new is calculate using the prouct of the prior istribution an the likelihoos given all the ata currently store accoring to Bayes theorem: ( 1,..., N ) ( ) ( ) i P 0 P 0 P 0, (4) N i= 1 assuming all the ata are inepenent conitional upon the ispersion moel. This posterior probability ensity is store so it can be upate from new ata in the future. We propose new hypotheses using a Differential Evolution Markov Chain (DE-MC) algorithm[5]. This algorithm is a combination of ifferential evolution genetic optimization with stanar Metropolis Markov Chain Monte Carlo (MCMC)[6]. Multiple (N=50) parallel Markov chains are upate, each using the stanar Metropolis accept/reject criterion : Accept if: uniform ( 0,1) ( 0new 1,..., N ) ( 0ol 1,..., N ) P < P otherwise reject an a another instance of 0ol to the list of samples. The population of current parallel Markov chain states i = N is use to generate jump proposals: 0, 1,2,..., i ( ) (6) inew, = iol, + γ j k + ε, i j k where ε is a narrow normally istribute multivariate an γ is a scalar inicating the size of jumps relative to the Markov chain ifferences. The inices j an k are chosen ranomly from 1,2,,N. The inex i can be iterate or chosen ranomly. We use the latter. DE-MC allows the jump proposal to aapt itself to the current estimate of the posterior, removing the responsibility of the user to provie a reasonable jump sampling istribution. The material ientifier component a, of a hypotheses is internally store as a floating point number to make use of Eq. (6) easier, but converte to an integer an constraine to the number of possible materials using moular arithmetic when it is use to select a ispersing material. We allow mass q, to become negative. This is interprete as no release an no ispersion calculation is require to calculate the likelihoo. Any prior istribution on mass has to be reflecte about zero. Upon rejection, a full copy of the hypothesis is not in fact ae to the list. This woul require unnecessary uplication of the ispersion coe storage. Instea, each hypothesis maintains a list of weights. A single weight correspons to each time the hypothesis is re-sample by rejection or the initial acceptance. These weights can be iniviually upate an accesse as if they were istinct samples. The DE-MC algorithm is much more aggressive at expaning posterior sample sets than other methos e.g. Gaussian mixtures. This property is useful when the samples become egenerate Two-step moification to the Metropolis accept/reject criterion A rule-of-thumb acceptance ratio for Metropolis MCMC is about 0.25 [7]. Using a stanar Metropolis accept/reject calculation, approximately three out of every four costly ispersion calculations woul be waste. We woul hope that the majority of the processing carrie out by the source term estimator woul inicate that the system is not uner etectable attack. In this case, the prior istributions ominate the shape of the posterior. As the prior istribution of a sample oes not require a ispersion calculation for its evaluation, it is much faster to evaluate. Hence we carry out the accept/reject stage of the sampling in two steps. We assume that the prior (5)

4 P ( 0 ) an likelihoo P( ) N i= 1 i 0 are calculate an store separately for each hypothesis. We then consier a new hypothesis if uniform ( 0,1) P < P ( 0new ) ( 0 ) Then, only if the hypothesis is accepte accoring to the prior, we carry out a ispersion calculation, calculate the likelihoo an continue to accept it if uniform ( 0,1) < N i= 1 N i= 1 P P ol ( 0 ) new ( 0 ) When the prior istribution ominates the posterior istribution, few ispersion calculations are waste. When ata inicating a release are receive, the likelihoo ominates the posterior an the efficiency of use of the ispersion calculations reverts to the 0.25 fraction mentione above. 3.2 Data processing Data processing requires several tasks. These are: upating the weights an posteriors of the samples, normalizing the weights, re-istributing the DE-MC population an removing ol hypotheses an ata. These tasks are escribe below Bayesian upate of the weights an posteriors Upon receipt of new ata, the weights of all the store hypotheses can be rapily upate by calculating the likelihoo of the ata given the store ispersion parameters. The moifie posterior probability ensity (split into prior an likelihoo components) is recore in orer for the Metropolis accept/reject step to be performe rapily in the DE-MC algorithm escribe above Re-weighting of ol hypotheses Hypotheses are re-weighte such that the maximum weight of a sample is one. If we currently have a goo sampling of the posterior, there will be many samples with weights a significant fraction of one, therefore a large total weight. The new samples ae with weight one will have a small impact on the sample approximation to the posterior. If the sampling of the posterior is poor, there will be few hypotheses with a weight close to one, an the new hypotheses ae will significantly improve the approximation to the posterior. When inferences are require, the total weight can be evaluate rapily to normalize the require marginal integrals Re-istribute the DE-MC population The initial non-zero from a chemical sensor can instantaneously reuce, by a few orers of magnitue, the with of the posterior probability istribution in the ol i i (7) (8) hypothesis space from the state it was in when influence only by the priors an null sensor s. While we hope that a number of the many store hypotheses are within the region of hypothesis space with high posterior probability ensity, the DE-MC population members, as there are far fewer, are almost certainly still wiely istribute about the hypothesis space in regions of low probability ensity. Outsie of the volume of hypothesis space influence by the non-zero sensor, the posterior is flat, as all releases that cannot reach the sensor at the time it has reporte a non-zero concentration state are equally unlikely. It can take a great eal of time for the Markov chains to ranomly fall into the likely region, so many new hypotheses coul be ae into the unlikely regions of hypothesis space, proucing a poor sample of the posterior. For this reason, after every ata upate, the DE- MC population is ranomly re-istribute amongst the existing hypotheses accoring to their weights. This way, the population (an therefore jump proposals) accurately represents the best estimate of the posterior istribution at that instant in time, as intene by the algorithm. Problems with burn-in an number of samples require for convergence are thus reuce, although not remove entirely. This re-istribution after every new piece of ata may slow the rate at which DE-MC can sprea out from a egenerate sample set. To guar against this, the acceptance ratio of the last one hunre trials is monitore. If the acceptance ratio is above 0.75, the sample set is juge to be narrow compare to the true posterior an Sample Importance Re-sampling (SIR)[4] is carrie out, reucing the relative total weight of the existing samples compare to the new ones being ae, thus allowing the algorithm to sprea outwars to the true posterior faster Removing ol hypotheses an ata Whenever the time stamp inclue with new ata suggest that the internal clock shoul be incremente, the entire list of hypotheses is searche for any that were calculate more than T D in the past. These are remove as it woul not be possible to rapily upate their posteriors as new ata arrive. At the same time, the list of store, receive ata has any s referring to a time before time T D in the past remove as new hypotheses cannot have their posteriors calculate rapily with respect to these ata (see Fig. 2.). Fig. 2 Dispersion output an ata storage in time If the sampler becomes overwhelme with the overhea of upating its list of hypotheses with ata, because there

5 are too many, an oes not manage to a new hypotheses in between processing the ata, then the hypothesis list is trimme using SIR. SIR is also use if total storage of hypotheses becomes problematic, although this has never been observe in practice as new hypotheses are not usually generate rapily enough to fill the temporary storage of a esktop PC. 3.3 Limitation of the sampling metho This algorithm, in the form reporte here is unable to eal with ata from a sensor it i not previously know about. This coul be a raio from a unit in possession of a portable chemical sensor. In orer to upate the weights of the store hypotheses, a new ispersion woul have to be calculate for each hypothesis in orer to calculate the likelihoo of the new piece of ata. This woul take too long to be practical, therefore the ata are currently ignore. Assuming such ata are relatively small in number, it is possible to store an instance of an ajoint ispersion calculation with each piece of ata an query the calculation for output corresponing to each hypothesis. This technique has been implemente for a simplistic ajoint ispersion moel, but the results are not shown here. 4 Results In orer to test the sampling algorithm escribe above it has been implemente in C++ on a esktop PC as a static library within a multi-threae -passing simulation environment that has all the necessary probabilistic algorithms an ispersion coe in orer to stimulate the sampler with realistic sensor s. The necessary sensor moels have only been constructe for chemical sensors so far. We present results of inferences mae in scenarios where the hypothesis space has been reuce from that esire to one containing eight potential chemical release materials an where releases can only be one hour in the past. Limiting the hypothesis space in time is beneficial because the length of time to calculate a realistic ispersion in a complex environment is linearly proportional to the amount of real time simulate. Therefore, hypotheses of releases many hours in the past take too long to calculate. We view the weighte samples in the five-imensional hypothesis space by a series of projections into each imension of interest. 4.1 Null reports In this section, we present projecte views of the hypothesis space in the steay state scenario where a network of sensors is yieling a time series of zero measurements. Fig. 3. shows a uniform sampling of hypotheses in the two location imensions. There is a reuction in the weights of hypotheses upwin from the sensors (Southwest from the centre in Fig. 3.). However this effect is not visible in the projecte view. Fig. 3 Store hypotheses with no sensor ata (location view) Fig. 4. shows the istribution of the hypotheses in the other three imensions (material, time an mass) which is ominate by the priors. Priors in this evaluation phase are for test purposes only. Realistic priors may iffer significantly. Fig. 4. Store hypotheses with no ata (material, time an mass views) 4.2 Non-zero ata Immeiately after the first non-zero, the samples egenerate into a single hypothesis as emonstrate by Fig. 5. This behaviour is typical, but two or three ominant samples have also been observe at this stage. The Markov chains are all re-set to this one hypothesis an we rely on the ability of the DE-MC algorithm to rapily sprea out an map the true posterior in the surrouning area of hypothesis space.

6 yiel an orer of magnitue estimate of concentration, an partly because there are not yet enough ata to exclue the larger releases upwin as we have just iscusse. The mass estimation performs better once there are enough ata to collapse the posterior istribution. The release material has been correctly ientifie (see Fig. 9.) with almost no oubt in the classification. Fig. 5. Location inference immeiately after first non-zero Five minutes after the first non-zero, the sampling algorithm has generate a useful approximation to the true posterior probability istribution (see Fig. 6. to Fig. 8.). Fig. 7 Time inference five minutes after first non-zero Fig. 8. Mass inference five minutes after first non-zero Fig. 9. Material inference five minutes after first non-zero Fig. 6. Location inference five minutes after first non-zero This istribution extens upwin of the true release as there is not enough information as yet to istinguish between the true release an a set of releases further upwin, further back in time, with larger mass. The simulate ispersing material woul have to pass entirely over the array of sensors, so the following zero bar s woul reuce the probability of some of the larger upwin releases which woul be preicte as still yieling significant concentrations somewhere in the sensor array. The simulate release time was at zero secons in this scenario. The simulate release mass was 200 (arbitrary units). There are no samples with significant probabilistic weight at this mass level, so the mass estimation is poor. This estimation error is partly because iniviual sensors only 5 Improvements to the algorithm There are many possible extensions an improvements to the algorithm we have presente. Some of these are iscusse below. 5.1 Twin hypotheses Applying a low prior on the probability that a release has occurre, forces the algorithm to create many samples that are of no use when a non-zero sensor arrives. Although the storage an posterior calculation requirements are small for each iniviual no-release hypothesis, nevertheless total overhea for maintaining many such samples can be significant. Therefore, we propose that each hypothesis shoul contain a posterior an weight for both its primary mass variable an its negative mass q. This twin hypothesis re-balances the problem, leaving us with an equal number of hypotheses

7 containing releases as those that o not, allowing us to apply a prior on the probability that a release has occurre without any technical ifficulties. 5.2 Data from other sources Other sources of information we expect to inclue in the future are: Intelligence ata, which may affect the priors. Biological sensors, which typically have more complex likelihoo calculations. A simple particle counter with an associate backgroun moel has been implemente, but the results are not reporte here. Raiological sensors, which may involve complex interaction with the ispersion moel to calculate the measurements integrate over a volume of air surrouning the sensor. Prompt effect nuclear sensors. These o not require ispersion moels to calculate the likelihoo of ata so are simple to implement. Human line-of-sight observations of the release. Once again, ispersion moels are not require an the likelihoo calculations are simple. LIDAR (Light Detection An Ranging). This is a remote measurement of concentration along a straight line through a ispersing clou. We expect the likelihoo calculation to be extremely complex. 5.3 Removal of subtle violation of Bayes theorem As ol ata are elete, their effect on oler hypotheses shoul be remove. A mismatch evelops over time regaring the amount of ata that has been use to compute the likelihoo of hypotheses. Oler hypotheses (measure in real-time from when they were calculate to now) will have ha more ata use to calculate their posterior probabilities than new ones because ol ata are constantly being elete. We currently o not notice this effect, as oler hypotheses ten to have lower weight than newer ones as a consequence of the weighting of the new hypotheses propose compare to the existing set. In aition, all our current un-normalize likelihoo ensity calculations are constructe such that they yiel values less than or equal to one. This ensures that if two hypotheses ha the same parameters, the one that ha its posterior calculate with the most ata woul have a smaller posterior probability than the hypothesis calculate using less ata. This avois pathological sampling behaviour where oler samples are preferentially selecte purely on how much ata they have use. To comply with Bayes theorem rigorously, we are require to keep account of all those hypotheses that have ha their posterior probability ensity calculate using a particular piece of ata such that, if that piece of ata is elete from the system, those hypotheses affecte by it have their posterior probability ensity ivie through by the likelihoo ensity of the piece of ata. 5.4 Heuristic scheme for multiple release hypotheses Probably the most esirable, yet most ifficult to calculate, inferences are those involving multiple iscrete releases. The probability that an enemy has attacke installations or forces with more than one iscrete release of potentially ifferent agents is significant an nees to be calculate. Within the Bayesian framework an with a suitable ispersion moel, it is trivial to calculate the posterior probability of an iniviual hypothesis containing multiple releases, with the caveat that experiments must be unertaken to etermine how sensors behave uner exposure to multiple materials. However, there are two major problems when it comes to taking samples of the posterior istribution. Firstly, the number of imensions to search an integrate over grows by five for each separate release to consier. This means a numerical explosion in the number of samples require to make inferences in such a high imensional space[8]. Seconly, any Markov Chain Monte Carlo scheme is very ifficult to efine in terms of new proposals with ifferent moel structure from those hypotheses they are erive from. Specifically, any single release hypothesis nees to be able to generate a multiple release hypothesis from itself. Where oes it put the new releases such that there is the slightest chance that the new hypothesis coul be accepte given the current ata? There is a similar problem when proposing a hypothesis with fewer releases than the current one contains. Any poorly esigne ahoc scheme will most likely prouce highly unlikely hypotheses that will be almost certainly rejecte. We will seek a scheme that is heuristic only in its jump proposal istributions for transitions between numbers of releases an attempt to remain true to the Bayesian paraigm for sample acceptance. 6 Conclusions We have presente a sampling algorithm that yiels rapi inferences in CBRN scenarios where likelihoo calculation requires output from complex moels, but the parameters to be estimate are not evolving in time. Tests were carrie out in a sub-volume of the esire test hypothesis space to ensure sufficient performance. We obtaine reasonable sampling of the posterior probability ensity an subsequent inferences given the ata receive in a short time frame. Inferences on release mass were only correct to an orer of magnitue. We also observe a strong correlation between release mass, time of release an range upwin for the source terms. We observe that the algorithm recovers rapily from sample egeneracy. A number of areas where the algorithm can be extene to wier sources of information have been iscusse as well as methos to improve accuracy an performance. We have highlighte the extension to hypotheses with multiple iscrete releases as the most esirable yet ifficult to achieve.

8 Acknowlegments We wish to thank Kevin Watkins of Dstl for his esign an coing to create a complex synthetic environment for our source term estimator. We also wish to thank Mark Hargrave for his continue avice an evelopment of Dstl s Urban Dispersion Moel (UDM). References [1] F. S. Gull. Bayesian Inuctive Inference an Maximum Entropy. Maximum-Entropy an Bayesian Methos in Science an Engineering. 1, pages Kluwer Acaemic Publishers, [2] S. W. Lewellen an R. I. Sykes. Analysis of concentration fluctuations from liar observations of atmospheric plumes. J. Clim. & Appl. Met., 25, , [3] P. Robins, V. Rapley an P. A. Thomas. A Probabilistic Chemical Sensor Moel for Data Fusion. Draft submitte to FUSION The 8th International Conference on Information Fusion. [4] B. Ristic, S. Arulampalam an N. Goron. Beyon the Kalman Filter: Particle Filters for Tracking Applications. Artech House, [5] C.T.F. Ter Braak. Genetic algorithms an Markov Chain Monte Carlo: Differential Evolution Markov Chain makes Bayesian computing easy. (last accesse 4/2/05), [6] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller an E. Teller. Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, , [7] K. M. Hanson, Tutorial on Markov Chain Monte Carlo, Workshop for Maximum Entropy an Bayesian Methos, June 9-13, 2000, Gif sur Yvette, France.