Filtering for Signals in Random Environments

Michael A. Kouritzin, Hongwei Long, and Wei Sun (2002) consider the nonlinear filtering problem in which the signal to be estimated is a (non-Markov) diffusion in a random environment that evolves in some Euclidean space.

Our motivation comes from the tracking problem of a dinghy lost at sea.  When the weather conditions are adverse, the motion of the dinghy will change dramatically due to drifting by random ocean surface wave propagation, rendering the Search and Rescue problem model inappropriate.  The dinghy itself would try to undergo the motion described by some SDE but the random wave formation would have the effect of adding random drift; hence, the signal process for the dinghy can be formally described as an Itô process whose drift term contains the gradient of W (W being a locally bounded random field in the Wiener space).  Of course, this does not really make sense as a normal SDE but is well defined through Dirichlet form or martingale problem theory.  We also allow correlation between the signal and the observation noise, which would be natural from dinghy occupant motion.

It is well known that the long-time behaviour of diffusion in a random environment is much different from that of an ordinary diffusion; note that the drift term  is not well defined, since W is a non-differentiable continuous function.  Diffusion in random medium is not a Markov process unless one fixes an instance of the random environment.  Consequently, it is impossible to construct, within the usual Itô's theory, a stochastic evolution equation for the filtering process of the diffusion in random medium based on noisy observations.  However, it can be constructed from the Dirichlet form theory in any finite dimension.

REPRESENTATIONS AND CHAOS METHODS

Although the signal itself is not Markov, it becomes Markov when a single occurrence of the random environment becomes fixed.  This results in two semigroups; one produced by the Dirichlet form on a (randomly) weighted L2-space and another on the Banach space of bounded measurable functions.  Assuming that the initial signal distribution satisfies a certain condition and we have access to the random environment, we show that there is a unique weak solution to the Zakai equation that remains in the L2-space and is expressible in terms of either semigroup. 

In the following, we remove the unrealistic assumption that we can ?x201c;see?x201d; the random environment.  We have chaos decomposition for our filtering process, namely we can represent the filtering process as a series in terms of multiple Wiener-Itô integrals, even in the case of correlated noise (i.e. when the observation noise is not independent of the signal).  We first obtain the expansion for each fixed random environment.  Then, we can use the classical filtering theory to obtain a Zakai equation for this fixed environment predictive model, from which we can express the unnormalized filtering process as the mild solution to the Zakai equation.  Further, we formulate the multiple Wiener integral representation for the unnormalized filtering process. Finally, we show that we can integrate this expansion term by term with respect to the distribution of the random environment.  We arrive at the desired unique multiple Wiener integral representation for the filtering process associated with the diffusion in the random environment.   No Zakai or Kushner-Stratonovich equation is possible.  Our representation formula in Kouritzin, Long and Sun (2002) provides the capability to simulate this filtering process on a computer.

PARTICLE APPROXIMATION

Our practical simulation experience suggests that particle and space discretization methods of implementing filters often work better than chaos methods.  Therefore, it is important to come up with a particle system approximation for the conditional distribution of the signal in a random environment given only the observations.  The difficulty that arises is that we do not want to introduce an extremely large number of particles to account for the random environment.  Instead, we try to ?x201c;learn?x201d; the environment.  In particular, we construct a functional or white noise expansion for the random environment in the Dirichlet form, placing all randomness in the coefficients (and not the bases functions).  Then, we both truncate the expansion (after a very large number of terms) and employ combined filtering-parameter estimation algorithms to ?x201c;identify?x201d; the random environment coefficients while filtering or perform functional estimation to estimate all coefficients simultaneously.  Inasmuch as we have already discussed the former approach in (c), we only discuss the later here.

Kouritzin, Sun, and Xiong (2002) investigated combined functional environment estimation and filtering.  To justify the anticipation of a solution, we simplify the problem to one dimension and constrain the original signal to be [0,2x] by reflecting at the boundaries.  These models still make sense through Dirichlet form theory and can be thought of as a formal Skorohod stochastic differential equation.  Then, we replace the W gradient with its trigometric series.  In this manner, we are also replacingby some trigometric expansion.  (Clearly, since the gradient of a Brownian motion is not a square-integrable function we cannot expect that a trigometric series for is square summable but an infinite series expansion is still valid by deweighting the coefficients and using Sobolev spaces.)  Then, motivated by the method of moments, we assume h is 1-1, take the observation noise to be independent of the signal, and form a functional estimator of the form:

which, for appropriately chosen mn, can be shown to contain consistent (as n ) estimators for all the coefficients of our trigometric coefficients for the gradient of W.  Then, for each n we define an approximate signal and particle filter, show that the approximate signal converges to actual signal as n , and use the technique of Budhiraja and Kushner (2002) to prove that as n  the long-term error of this method converges to zero.

The drawbacks of this approach are:  1) In practice, you must fix n apriori; and 2) the amount of work between observations also increases to infinity as n .  We are also trying to derive an algorithm that will sample all frequencies infinitely often with the same number of calculations between observation so that over time we have a good estimator of all frequencies and n need not be fixed. The idea is to expand W (not ) and estimate the lowest frequencies first twice, then estimate the next lowest frequencies once, repeat both once, and then sample the third lowest frequency group, etc. Suppose we label the frequency groups of approximately the same size 1,2,3, ?x2026;  Then, the algorithm would use our functional estimation procedure on the groups in the following order:  1,1,2,1,1,2,3,1,1,2,1,1,2,3,4, ... The idea here is that the coefficients for W converge to zero as the frequencies increase, so lower frequency coefficients are more important, but they cannot be estimated properly without also estimating the higher coefficients.