Gain estimation

# Gain estimation

GainEquation

Abstract type for a container representing a gain (vector field).

GainEstimationMethod

Abstract type for a method used to solve an equation of GainEquation type.

solve!(eq::GainEquation, method::GainEstimationMethod) --> eq

Solves the gain equation eq using method method.

update!(filter_state, filter_algo, obs, dt) --> filter_state

Updates the filter state by performing one forward step of the model and then assimilating the observation.

update!(eq::GainEquation)

Updates the gain equation eq such that all information contained in it is self-consistent.

update!(eq::GainEquation, ens::ParticleRepresentation)

Updates the gain equation eq by incorporating new information from the ensemble ens.

## Poisson equation

PoissonEquation(h, ensemble) ::GainEquation

Returns a Poisson equation struct representing the equation $\nabla\cdot(p\nabla \phi) = - \tilde h$, where $p$ is a probability density and $\tilde h = h-\int h p dx$. The container contains the following fields

• :h':h itself

• :positions': an i.i.d. sample fromp, represented as a matrix

• :H': the evaluation ofh at the sample points

• :mean_H': the sample average ofH'

• :potential': the evaluation of\phi at the sample points

• :gain': the evaluation ofK=\nabla \phi at the sample points

solve!(eq::PoissonEquation, method::GainEstimationMethod)

Fills the field :gain' with appropriate values. The fields:H', :mean_H', and:potential' are stored to be re-used.

update!(eq::PoissonEquation, ensemble)

Fills the fields :positions',:H', and :mean_H' according to the new samples fromensemble'.

update!(eq::PoissonEquation)

Updates fields :H', and:mean_H' to be consistent with :positions'.

## Gain estimation methods

ConstantGainApproximation()

Represents an approximation of the gain by a constant (in the Euclidean sense) vector field, given by the covariance of the observation function and x under the particle distribution.

SemigroupMethod(epsilon, delta, max_iter)

Semigroup method from Algorithm 1 in [1].

[1] Taghvaei, A., & Mehta, P. G. (2016). Gain function approximation in the feedback particle filter. In 2016 IEEE 55th Conference on Decision and Control (CDC) (pp. 5446–5452). IEEE. https://doi.org/10.1109/CDC.2016.7799105

SemigroupMethod(epsilon, delta, max_iter, lambda)

Semigroup method with regularization parameter lambda.

DifferentialRKHSMethod(epsilon, lambda)

Differential loss reproducing kernel Hilbert space (RKHS) method from [1], Section III.

Parameters

• epsilon: variance of the Gaussian kernel, Eq. 17 in [1]
• lambda: regularization parameter, Eq. 20 in [1]

Reference

[1] Radhakrishnan, A. & and Meyn, S. (2018). Feedback particle filter design using a differential-loss reproducing kernel Hilbert space. 2018 Annual American Control Conference (ACC). IEEE. https://doi.org/10.23919/ACC.2018.8431689