I am using the following equations for Kalman filter. The state vector is $x$ and observations are $z$. Both $x$ and $z$ are multivariate vectors of same length.
Forecast model: $x_{t+1} = 0.963 x_t + \mathcal{N}(2.19,3.34)$
Observation model: $z_t = -0.00542849 x_t + \mathcal{N}(0.4524,0.038952)$
These relations hold between state and observation vectors element by element (since the lengths are same).
The observations are confined within an interval [0, 1] whereas $x$ values may range from [0,80].
I am not sure if the noise values I have determined are correct. I just fit a linear regression between $z$ and $x$ and used the MSE to find the standard deviation of noise. Is that correct?
Somehow, the update step introduces a lot of noise. So much that I am in fact better off not updating at all.
How does one estimate the noise covariance matrices given a set of $(x_{t+1},x_t)$ and $(z_t,x_t)$ values? Is my methods of fitting a linear regression and using $\sqrt{MSE}$ as standard deviation of noise correct?
EDIT: I should mention here that the relationship between $z$ and $x$ seems linear element-by-element and that's why I am learning a regression model.