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