General questions on Kalman filter and difference

1. What are the different stages in Kalman filter?
2. Why Kalman filter is optimal?
3. What is the difference between a Kalman filter and an extended Kalman filter?
4. Why use Kalman smoother?

What are the different stages in Kalman filter?

The Kalman filter can be written as a single equation; however, it is most often conceptualized as two distinct phases: "Predict" and "Update".

Why Kalman filter is optimal?

Kalman filter is statistically optimal in a sense that it gives the minimum error covariance estimate, based on all available observation data at the present time step under the linear system.

What is the difference between a Kalman filter and an extended Kalman filter?

The Kalman filter (KF) is a method based on recursive Bayesian filtering where the noise in your system is assumed Gaussian. The Extended Kalman Filter (EKF) is an extension of the classic Kalman Filter for non-linear systems where non-linearity are approximated using the first or second order derivative.

Why use Kalman smoother?

Good reasons for Kalman smoothing are: The Kalman smoother provides very good imputations (i.e. imputed values) for missing values in your time series. The Kalman smoother provides very good estimates of the state vector in the historical period.