Novel approach to nonlinearnonGaussian Bayesian state estimation

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Novel approach to nonlinear/non-Gaussian Bayesian
state estimation

• N.J Gordon, D.J. Salmond and A.F.M. Smith

Presenter Tri Tran 7-2005
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Outline

• Motivations
• Recursive Bayesian estimations
• Bayesian Bootstrap filter algorithms
• Evaluations of the algorithms
• Summary

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Motivations

• In many applications of positioning, navigation
  and tracking, we want to estimate the position of
  a moving object at discrete times.
• Positioning position of an object is to be
  estimated when an navigation system is used to
  provide measurements of movement.
• Navigation besides the position also velocity,
  attitude and heading, accelerating and angular
  rates are included in the problem.
• Target tracking another objects position is to
  be estimated based on measurement of relative
  range and angles to ones own position.

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Motivations (cont.)

• The problems can be described by state space
  models where the state vector contains the
  position and derivatives of the position.
• Bayesian methods is to construct a probability
  density function (PDF) of the state based on all
  the available information.
• For linear/Gaussian models, the required PDF
  remains Gaussian at every iteration of the
  filter, Kalman filter can be used to propagate
  and update the mean and covariance of the
  distribution.
• For nonlinear/non-Gaussian models, there is no
  general analytic expression for the required PDF
  ? a new way of representing and recursively
  generating an approximation to the state PDF.
• Central idea represent the required PDF as a set
  of random samples, rather than as a function over
  state space.

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Particle Filters

• Recursive methods to estimate the dynamic state
  (a set of random samples) of a moving object in a
  discrete time nonlinear model based on posterior
  distributions.

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Recursive Bayesian estimation

• System model
• xk1 fk(xk, wk)
• where fk RnxRm?Rn the system transition function
  
• wk is white noise sequence of known PDF,
  independent of past and current states.
• Measurement model (observation equation)
• yk hk(xk, vk)
• where hk RnxRr?Rp the system transition function
  
• vk is another white noise sequence of known PDF,
  independent of past and current states.
• Assume that the initial PDF p(x1D0) of the state
  vector, the functional forms fi and hi for i1,,
  k are known.
• At time step k, available information is the set
  of measurements Dk yki1,,k
• Need to construct the PDF of the current state
  xk, given all the available information p(xkDk)

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Bayesian bootstrap filter

• A set of random samples xk-1(i) i1,,N from
  the PDF p(xk-1Dk-1).
• Propagate and update the set of random samples to
  obtain xk(i) i1,,N which approximates as
  p(xkDk)
• Prediction each sample is passed through the
  system model to obtain samples from the prior at
  time step k
• xk(i) fk-1(xk-1(i), wk-1(i))
• Update with the measurement yk, evaluate a
  normalized weight for each sample
• Thus define a discrete distribution over xk(i)
  i1,,N with probability qi associated with
  element i
• Resample N times from the discrete distribution
  to generate samples xk(i) i1,,N, so that for
  any j, Prxk(j)xk(i) qi

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Bootstrap filter

• Advantages no restrictions on the functions fk
  and hk or on the distributions of the system or
  measurement noise.
• Requirements
• (a) p(x1) is available for sampling
• (b) the likelihood p(ykxk) is a known functional
  form
• (c) p(wk) is available for sampling
• The number of random samples N depends on
• (a) the dimension of the state space
• (b) the typical overlap between the prior and
  the likelihood
• (c) the required number of time steps

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Evaluations of the bootstrap filter algorithm

• One-dimensional nonlinear example
• Consider the following nonlinear model
• Xk 0.5xk-125xk-1/(1x2k-1)8cos(1.2(k-1))wk
• Ykx2k/20vk
• Wk and vk are zero-mean Gaussian white noise with
  variances 10.0 and 1.0, respectively.
• K50, x0 0.1, N500

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Evaluations of the bootstrap filter algorithm

• Bearings-only tracking example
• Target moves within the x-y plan
• A fixed observer at the origin of the plane takes
  noisy measurements zk of the target bearing
• k24, N4000, initial state
• x1(-0.05, 0.001, 0.7, -0.055)

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Evaluations of the bootstrap filter algorithm

• Bearings-only tracking example (cont.)

• The actual co-ordinate value is always within the
  95 probability region.

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Summary

• A bootstrap filter for implementing recursive
  Bayesian filters
• The method is applied for nonlinear models and
  approximates the posterior distribution as a set
  of random samples.

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References

• N.J Gordon, D.J. Salmond and A.F.M. Smith, Novel
  approach to nonlinear/non-Gaussian Bayesian state
  estimation, in IEE Processinds-F, Vol. 140,
  1993.
• Fredrik Gustafsson, Fredrik Gunnarsson, Niclas
  Bergman, Urban Forssell, Jonas Jansson, Rickard
  karlsson, Per-Johan Nordlund, Prticle Filters
  for Positioning, Navigation and Tracking, IEEE
  Transactions on Signal Processing, vol. 50, pp.
  425- 435, Feb. 2002