# Unscented Kalman Filter transformations for a Poisson state-space model

I have count data which I'm trying to model using a state-space model where

$z_t \sim Poisson(exp\{F^\prime\ x_t\})$

$x_t \sim N(G\ x_{t-1}, R)$

Where $z_t$ are the observations and $x_t$ the underlying hidden state. $F$ and $G$ are simply structure matrices, and could be considered $F=G=1$ to simplify and $R$ is a know variance matrix.

I was initially performing state estimation using a bootstrap filter, but wanted to use an Unscented Kalman Filter for comparison.

My question is, after calculating the sigma points and weights, how do I apply the model and measurement transformations?

According to the literature (considering a model in the form)

$x_t = f(x_{t-1}, k-1) + q_{t-1}$

$y_t = h(x_t, k) + r_t$

what should I use in this case for the $f(\cdot)$ and $h(\cdot)$ transformations?

• Are you familiar with the basic (linear) Kalman filter? Your notation is not completely clear, but the $x$ evolution appears to be linear Gaussian. – GeoMatt22 Sep 15 '16 at 0:12
• Hi @GeoMatt22, thanks for the comment! I've corrected the state model (added the variance). Yes, the state model is linear Gaussian. I'm familiar with the Kalman Filter, and were the $y$s also linear Gaussian I would apply the Kalman recursions. My doubt is that I need to apply a non-linear transformation to the sigma points. Given I have a link function $\eta=exp\{F^{\prime}\ x_t\}$ and a Poisson distributions, what is my "non-linear transformation"? Do I simply transform using $exp\{F^{\prime}\ m_{t-1}\}$ or sample from a Poisson using $Poisson(exp\{F^{\prime}\ m_{t-1}\}$? Many thanks! – FrankZappa Sep 15 '16 at 8:20