The Unscented Kalman Filter and Item Response Theory (Undergrad thesis)

Hi everyone,

I'm a bachelor's student of psychology and I have a question regarding my final thesis (freaking out already with the time I've got left). I'm planning to evaluate the Unscented Kalman Filter with longitudinal data using a Rasch model. However, I'm having trouble formulating a measurement equation in State Space terminology. I'll be using the following notation:


with x being the true underlying ability, f being the transition equation, u being exogenous input (which I don't have in my case), k being the process noise, z being a vector of manifest measurements (so a vector of either a 1 or a 0 for each item in my case), h being the measurement equation and w being the measurement noise.

Am I correct in assuming that I can use the right side of the following equation as h?


with beta being the same as x in the first equation and delta being the item difficulty.

I don't feel a 100% comfortable since I'm only calculating an expected value. Also, I'm unsure of how the measurement noise would then be conceptualised (for example as the difference between the expected value of h(x) and z? Is the Unscented Kalman Filter even an option with non-gaussian measurement noise?)

I'd really appreciate some help, since I've hit a dead end on my own! Thanks!
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