How will the kalman filtering model look like in the case when I just receive some data and want to filter them from noise? The data is actually an acceleration of some object. So the system must be like this:

$$x_t = A_tx_{t-1} + B_tu_t + \epsilon_t$$ $$z_t = C_tx_t + \delta_t$$ Where the $\epsilon_t$ and $\delta_t$ are the white noise. $x_t$ is a state variable. The problem is that I can't figure out what will the system look like in my case, when I receive acceleration measurements (observations - $z_t$) at each time period $\Delta t$. I think I don't need the control vector $u_t$ in my case, so the system will be: $$x_t = A_tx_{t-1} + \epsilon_t$$ $$z_t = C_tx_t + \delta_t$$ I suppose, but not sure about this kind of filter system: $$x_t = x_{t-1} + \epsilon_t$$ $$z_t = x_t + \delta_t$$ But it seems too simple. How to make the first iteration in the Kalman filtering procedure?

**EDIT 1:**
here is the kalman filtering algorithm, taken from the book: probabilistic robotics.

Kalman_filter($\mu_{t-1}$, $\Sigma_{t-1}$, $u_t$, $z_t$) $$\bar{\mu}_t = A_t\mu_{t-1} + B_tu_t$$ $$\bar{\Sigma}_t = A_t\Sigma_{t-1}A_t^T + R_t$$ $$K_t = \bar{\Sigma}_tC_t^T\left(C_t\bar{\Sigma}_tC_t^T + Q_t\right)^{-1}$$ $$\mu_t = \bar{\mu_t} + K_t\left(z_t - C_t\bar{\mu}_t\right)$$ $$\Sigma_t = \left(I - K_tC_t\right)\bar{\Sigma}_t$$ return $\mu_t$, $\Sigma_t$

The thing that I do not understand here is: Here the data that is unknown is - $\Sigma_0$, $\mu_0$ I supppose that I can choose some data by myself for that values. But one more data that is unknown for me is: $R_t$ It comes from: The state transition probability is given by $p(x_t|u_t,x_{t-1})$. And we got: $$x_t = A_t\mu_{t-1} + B_tu_t+\epsilon_t$$ as one of the equations of the Kalman filter.

We also know the normal distribution: $$p(x) = det\left(2\pi\Sigma\right)^{-1/2}exp\left(-1/2(x-\mu)^T\Sigma^{-1}(x-\mu)\right)$$ (I've already asked a question from which you can see where it comes from: question)

So we have:

$\mu_t = A_tx_{t-1} + B_tu_t$(discussed in the question, the link is above) And also $R_t$ is a covariance of the posterior state. Here we got the whole formula.

$$p(x_t|u_t, x_{t-1}) = det\left(2\pi R_t\right)^{-1/2}exp\left(-1/2(x_t-A_tx_{t-1}-B_tu_t)^TR_t^{-1}(x_t-A_tx_{t-1}-B_tu_t)\right)$$

So, how should be the $R_t$ value estimated? It depends on $t$. If I set some value by myself to $\Sigma_0$ and $\mu_0$ then what should be done with $R_t$ which appears in the Kalman filter algorithm listed above in this step: $$\bar{\Sigma}_t = A_t\Sigma_{t-1}A_t^T + R_t$$ ?

Correct me please if I am wrong: $$R_t = cov\left(x_t|x_{t-1}, u_t\right) = E\left[x_t^2|x_{t-1}, u_t \right] - \left(E\left[x_t|x_{t-1},u_t\right]\right)^2$$ $$R_t = E\left[x_t^2|x_{t-1}, u_t \right] - \left(A_tx_{t-1}+B_tu_t\right)^2$$ So how to calculate the $R_t$? Should it be also set by user? Actually $R_t$ is a covariance of the noise $\epsilon_t$ in equation: $$x_t = A_tx_{t-1} + B_tu_t + \epsilon_t$$. And it depends on $t$. The same thing about the noise covariance of $\delta_t$ in case of this equation of the Kalman filter: $$z_t = C_tx_t + \delta_t$$

**EDIT 2:**
So as I understood four parameters should be selected by the user (tuned), they are:

$Q_t$, $R_t$, $\mu_0$ and $\Sigma_0$

Am I right?

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