Sorry if this is more of a debugging question, but I have been stuck on this supposedly simple NIS test for a very long while. If anyone knows any sources which cover the theory or implementation of the NIS test, I would definitely appreciate any links.
So, I am just running a very straight forward Kalman filter on a toy problem to check my understanding. I have the LTI discrete system
$x_{k+1}=F x_k+w_k$
$y_k=Hx_k+v_k$
where $w_k \sim N(0,\sigma_w^2)$ and $v_k \sim N(0,\sigma_v^2)$ are the white process and measurement noise respectively with covariances $Q$ and $R$. These are used to generate the true state $x_k$ and the obtained measurement $y_k$ (I am just using mvnrnd in MATLAB to obtain the white noise samples).
Then, I am also running the straightforward Kalman filter equations. With an initial state error covariance $\hat{P}_0$ and an initial corrected state vector $\hat{x}_0$ I then can integrate these forward using
$\tilde P_k=\hat{P}_{k-1}$
$\tilde x_k=F \hat{x}_{k-1}$
$K_k=\tilde P_k H^T (H \tilde P_k H^T +R )^{-1}$
$\hat{P}_k=(1-K_k H) \tilde P_k (1-K_k H)^T + K_k R K_k^T$
$\hat{x}_k=\tilde{x}_k+K_k(y_k-H \tilde x_k) $
Then, I perform a NIS test to ascertain its consistency following section 2.2.3 in the lecture notes here: https://www.robots.ox.ac.uk/~ian/Teaching/Estimation/LectureNotes2.pdf
To do this, I formulate the innovation covariance
$S_k=H \hat{P}_k H^T + R$
from which I formulate the normalized innovation error
$e_k=[H(x_k-\hat{x}_k)]^T S_k^{-1} [H(x_k-\hat{x}_k)]$.
Doing an $N$ trial Monte Carlo run and with the measurement having $m$ elements, I can get for every $k^{th}$ timestep the average normalized innovation error across all of the Monte Carlo runs as $\bar{e}_k$, and from the sources I saw we should have that $N \bar{e}_k$ follows a Chi squared distribution with $N*m$ degrees of freedom.
So, apparently with $\alpha=0.05$ I should have around 95% of $\bar{e}_k$ lie in the bounds $[r_1,r_2]$ obtained using the MATLAB commands
$r_1=chi2inv(alpha/2,N*m)/N$
$r_2=chi2inv(1-alpha/2,N*m)/N$
However, I always have $\bar{e}_k$ being way below the computed error bounds. I did the very similar NEES test and those pass without issue, and I double checked the lecture notes and other sources, made sure my computed Chi squared bounds matched the ones in the lecture notes, and I even tried multiple different expressions for $P$ and $S$, but so far nothing has worked for the NIS test. I was wondering is the above correct, and does anyone have any sources for the NIS test? Sorry again for the rambling debugging question.
