Dual Extended Kalman Filter Methods

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Dual Extended Kalman Filter Methods Chapter 3
D. Joksimovic, H. van Zuylen, H. Tu
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The aim of presentation(s)

• Three parts
• I part
• General dual extended Kalman filter (algorithmic
  framework) (Dusica)
• II part
• A Probabilistic perspective of dual EKF methods
  (Henk)
• III part
• Dual EKF variance estimation and Applications
  (Hiuzhao)

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General dual extended Kalman filter

• An algorithmic framework

D. Joksimovic
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5.1 Introduction

• Extended Kalman filter (EKF) provides an
  efficient method for estimates of the
• 1) state of a discrete-time, non-linear dynamical
  system. Filter recursive procedure to optimally
  combine noisy observations with predications
  (Chapter 1)
• 2) estimating the parameters of the model (e.g.
  neural network) given clean training data of
  input and output data (Chapter 2)
• In this chapter gt dual estimation problem, both
  the states of the dynamical system and its
  parameters are estimated simultaneously (Chapter
  5)

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Discrete-time nonlinear dynamical system to be
considered

• Both states x(k) and parameters w
• Must be simultaneously estimated from only the
  observed noisy signal y
• Process noise v(k), observation noise n(k), u(k)
  inputs
• F(.) and H(.) e.g. multiplayer neural networks
  (where w are weights)

Process noise
Observed noisy signal
Observation noise
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Motivation to use dual EKF methods

• 1) the need for a model to estimate the signal
• 2) the need for good signal estimates to estimate
  the model
• Applications, for
• 1)modeling
• approximating dynamics that generated the state,
  given the only noisy observations
• 2) estimation
• All noisy data up to the current time are used to
  approximate the current value of the clean data
• 3) and prediction
• Using all available data to approximate the
  future value of clean data

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Applications on Dual EKF

• Noise reduction (speech or image enhancement)
• Prediction of financial time-series
• Predication of economical time-series
• Adaptive control
• .

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Nature of the dual EKF methods

• Heuristically, dual estimation methods work by
  alternating between
• 1) using the model to estimate the signal and
• 2) using the signal to estimate the model
• This process can be iterative of sequential

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Iterative dual EKF methods

• Repeatedly estimating the signal using the
  current model and all available data,
• And then,
• Estimating the model using the signal estimates,
• Applied to off-line applications (data are
  previously collected)
• Use large blocks of data repeatedly

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Sequential dual EKF methods

• Use each individual measurement as soon as it
  becomes available
• And update both the signal and model estimates
• Sequential approach pass over the data one point
  at a time
• These algorithms are useful in either on-line or
  off-line applications

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Literature overview of dual EKF

• First for linear models,
• Then non-linear models
• Joint KF
• Or two separate filters
• Applications on neural networks
• More information in Chapter 3

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5.2. Dual EKF- prediction error

• The basic dual EKF prediction error algorithm
  will be presented
• First, a quick review of EKF for state estimation
  (Chapter 1)
• And EKF weight estimation (Chapter 2)
• Combination of state and weight filters to form
  dual EKF algorithm

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5.2.1 EKF- State Estimation

• Linear space-state system
• Kalman filter generates the optimal estimates and
  predictions of the state x(k)
• Filter recursively updates the (posteriori) mean
  and covariance of the state by combining the
  predicted ( priory) mean and covariance with the
  current noisy measurement (y)

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5.2.1 EKF- State Estimation
A posteriori
A priori
EKF
Current noisy measurements
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5.2.1 EKF- State Estimation

• Non-linear system EKF provides approximate
  maximum-likelihood estimates
• The mean and covariance are recursively updated
• Linearization of the dynamics is necessary, and
  then Kalman filters equations are applied
• Algorithmic framework and equations are given in
  the following slides

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State-space model
Initialization k0
kk1
State estimate propagation (a priori state
estimate)
Error covariance propagation (a priori state
error estimate)
Kalman gain matrix
State estimate update (a posteriori state
estimate)
Error covariance update (a posteriori state error
estimate)
no
yes
convergence
end
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Process noise and observation noise
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Standard EKF

• Previous presented algorithm presents a standard
  EKF
• There are more accurate methods for dealing with
  non-linear dynamics (e.g. particle filters,
  second-order filters, etc)
• But, standard EKF remains most popular because of
  its simplicity (!?!)

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Another interpretation of EKF

• Optimization algorithm that recursively
  determines state x in order to minimize the cost
  function
• Cost function weighted prediction error
  estimation error components

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5.2.2. EKF Weight Estimation

• EKF used for estimating the parameters on
  non-linear modes (i.e. training neural networks)
  from clean data.
• General problem of learning using a non-linear
  function G(x(k),w)
• A training set is provided with sample pairs
  (known input and desired output) x(k), d(k)
• The error in the model ed(k)-G(x(k),w)
• Goal solving of parameter w in order to minimize
  the excepted square error

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EKF- Weight estimation

• The EKF is used then to estimate the parameters
  by writing a new state-space representation
• w(k1) w(k) r(k)
• d(k)G(x(k),w(k)) e(k)
• Where
• parameter w(k) stationary process with state
  transmission matrix
• r(k) is the process noise
• Output d(k) corresponds to a nonlinear
  observation on w(k)

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5.2.3 Dual Estimation

• When the clean data are not available, a dual
  estimation is needed.
• Basic dual KF combines both the Kalman state as
  well as weight filters
• The task is to estimate both the state and model
  from the only noisy observations.
• Two EKFs are run concurrently

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• At every time step, an EKF state filter estimates
  the state using the current model estimate w(k)
• While the EKF weight filter estimates the weights
  using the current state estimate x(k).

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A priori
A posteriori
A priori
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State-space model for dual EKF

• x(k1)F(x(k),u(k),w) v(k)
• y(k)Cx(k)n(k)
• C1 0 0
• In which the scalar observation y(k) is one of
  the states.
• Thus, we need to consider estimating the
  parameters associated with F.

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State-space model
For weight and state
1. Initialization k0
kk1
2. Weight estimate propagation and error
covariance for the weight filter (a priori weight
estimate and weight error estimate)
3. State estimate propagation and error
covariance propagation for the state filter (a
priori state and state error estimate)
4. Kalman gain matrices (state and weight)
5. State estimate update (a posteriori state
estimate) and cov.
6. Weight estimate update (a posteriori weight
estimate) and error
no
yes
convergence
end
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Dual Extended Kalman Filter Equations 1
Initialization
Initialization (for weight estimation)
Initialization (for state estimation)
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Dual Extended Kalman Filter Equations 2 Time
update equations for the weight filter
Time-update equations for the weight filter
Innovations covariance
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Dual Extended Kalman Filter Equations 3 time
update equations for the state filter
Time update equations for the state filter
Covariance of v(n)
where
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Dual Extended Kalman Filter Equations 4 Kalman
filter gain
Covariance of n(k) process noise
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Dual Extended Kalman Filter Equations 5 State
filter update
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Dual Extended Kalman Filter Equations 6 Weight
filter update
Noise covariance
Where
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Recurrent Derivative Computation

• While the dual EKF equations appear to be simple
  concatenation of the previous state and weight
  EFK,
• There is a necessary modification of the
• Associated with the weight filter

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Recurrent Derivative Computation

• The reason the signal filter, whose parameters
  are being estimated by the weight filter, has a
  recurrent architecture
• That is, x(k) is a function of x(k-1) and both
  are functions of w
• gt the linearization must be computed using
  recurrent derivatives (see Section 5.2.3)

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Example

• As an example, noisy time-series generated by
• The observation of the series y(k) contains
  measurement noise n(k)

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State-space representation
State transmission function
or
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State-space representation
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Results of the example

• Figure 5.3.a)
• Clean signal (generated by an neural network, for
  details see section 5.2.3) and (colored) noisy
  data
• Figure 5.3.b)
• Time series estimated by dual EKF (algorithm
  estimates both the clean time series and the
  neural network weights)
• For the comparison, the estimates using an EFK
  with the known neural network models is shown.

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Clean signal
Noisy data
Clean neural network signal and noisy measurements
Dual EFK estimates versus EFK estimates
Estimates with full and static derivatives
MSE profiles of EFK versus dual EFK
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Conclusions

• The dual EKF has been presented
• Dual uses two EKF run concurrently (one for state
  estimation, the other for weight estimation)
• Algorithmic framework, motivation, and an example
  are shown

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A probabilistic perspective of dual EKF methods

• H.J. van Zuylen