I am using Kalman filter for filtering noise on 2D object movement. I read a lot of examples, but no one has been explained, why noise model is distance powered by 2:
$$ s \times s = \dfrac{T^2}{2} \times \dfrac{T^2}{2} = \dfrac{T^4}{4}; $$
My process noise model is: $$ \begin{bmatrix} T^4/4 & 0 & T^3/2 & 0 \\ 0 & T^4/4 & 0 & T^3/2 \\ T^3/2 & 0 & T^2 &0 \\ 0 & T^3/2 & 0 &T^2 \end{bmatrix} $$
My state model is standard:
$$ \begin{bmatrix} x & y & \dot{x} & \dot{y} \end{bmatrix} $$
You get that process noise matrix if you discretize
$\dot{x} = w_1(t)$
$\dot{y} = w_2(t)$
where $w$ are white noise signals. The Wikipedia "Kalman Filter" page derives an example with a vehicle that is very similar.