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Filtering a Stochastic Complex Scalar: The Prototype Test Problem
Andrew J. Majda
Department of Mathematicsand
Center for Atmosphere and Ocean Sciences,
Courant Institute for Mathematical Sciences,
New York University
Chapter 2
Filtering a Stochastic Complex Scalar: The Prototype Test Problem
2.1 Kalman Filter: one-dimensional complex variable 2.1.1 Numerical simulation on a scalar complex Ornstein-Uhlenbeck process2.2 FilteringStability2.3 ModelError 2.3.1 Mean model error. 2.3.2 ModelErrorCovariance 2.3.3 Example: Model Error through finite difference approximation 2.3.4 Information criteria for filtering with model error
Chapter 2
The Kalman filter is the optimal assuming: Model and the observation operators are both linear
Both the observation and prior forecast error uncertainties are Gaussian, unbiased, and uncorrelated. This is going to simplify the computations of Bayesian update
In particular,the observation error distribution of v at time tm+1 is a Gaussian conditional distribution
Suppose that the filter model is perfectly specified, the prior state is given by
From the probabilistic point of view, we can represent this prior estimate with a probability density
This prior distribution accounts only the earlier observations up to time tm
the posterior error covariance at time m to be computed through the Kalman filter formula
the prior mean state and the prior error covariance, computed from model dynamics via
This prior distribution accounts only the earlier observations up to time tm
while
Kalman filter
Forecast step
Correction step
2.1.1 Numerical simulation on a scalar complex Ornstein-Uhlenbeck process
Wiener process and it satisfies the following properties (see Gardiner [49])
given
we can derive equation for variance
Systematic strategies for real time filtering of turbulent signals in complex systems
Filtering a stochastic complex scalar – the prototype test problem
Filtering Stability
The Kalman filter associated with the above system is stable if we have observability (g≠0) and controllability (r≠0) (for any value of F)
Based on this we will find the equilibrium value for the covariance…
with
Filtering Stability The equation governing the asymptotic covariance can be written as
While the asymptotic Kalman gain related through the equation
The asymptotic covariance can be easily determined by solving the quadratic eq. above and keep the positive solution
Filtering Stability
Combining the above with the relation for the Kalman gain we obtain the corresponding asymptotic formula
Filtering Stability We have the posterior estimate
The condition for asymptotic stability of the above map will have the form
Cases
Stability is guaranteed since
(A & D)
Controllability is not required here
For r = 0 we will have (C)
But for r > 0 we will have > 0
Model Error We will now examine the effect of imperfect filter model parameters. Let the filter model given by
Mean model error
which is the mean deviation of the filtered solution from the truth signal
Effect of discretization error towards the mean
Measures the effect of the filter evolution
operator and its Kalman gain
Model Error We will now examine the effect of imperfect filter model parameters. Let the filter model given by
Mean model error
which is the mean deviation of the filtered solution from the truth signal
Effect of discretization error towards the mean
Measures the effect of the filter evolution
operator and its Kalman gain
Model Error Covariance model error
We directly obtain
where
Model Error through Finite - Difference Forward Euler model
Backward Euler model
Trapezoidal method
Model Error through Finite - Difference
• Forward Euler: strongly unstable – trusts heavily the observations. • Backward Euler: Fh is very small – so is the Kalman gain zero solution • Trapezoidal: Close to marginal stability region – very small noise
Information criteria for filtering with model error
We measure the information theoretic distance between:
A)the `perfect’ filtering limit mean zero Gaussian measure B)the imperfect filtering limit Mean zero Gaussian measure
The asymptotic covariance can be found by
The goal is to find an optimum system noise so that
Information criteria for filtering with model error
For implicit schemes with strong stability in stiff regimes (e.g. backward Euler or trapezoid) the above method inflates the system noise so that
The forward Euler is essentially unaffected by the information criterion
Information criteria for filtering with model error
Information criteria for filtering with model error To gain insight on the behavior of the mean model error we replace the Kalman gain by its asymptotic value
For stable schemes mean model error depends strongly on limiting Kalman gain value.
For unstable schemes the application of the information criteria will bound the mean model error:
(This comes directly from the first expression and condition D for the Kalman gain)
Rapidly converging