this
Smoother
The smoother determines the state estimates x at time indices k = 1 to k = nk that minimize the following objective:
T  -1
+ (x  - xi)  Pi (x  - xi)
1             1

nk
----              T  -1
+ >    (z  - h (x ))  R  (z  - h (x ))
----   k    k  k     k   k    k  k
k = 1

nk-1
----              T   -1
+ >    (x   - g (x ))  Q  (x    - g (x ))
----   k+1   k  k     k   k+1    k  k
k = 1
where
 Notation Meaning nk total number of time points k time index xi initial state estimate Pi initial covariance x State vector z Measurement vector h Measurement function g Transition function R Measurement variance Q Transition variance
Filter
The filter does not use data that comes after a time index for estimating the state value at that time index. To be specific, it determines the state estimate x at time index k = N by minimizing the following objective as a function of the state value at time indices k = 1 to k = N:
T  -1
+ (x  - xi)  Pi (x  - xi)
1             1

N
----              T  -1
+ >    (z  - h (x ))  R  (z  - h (x ))
----   k    k  k     k   k    k  k
k = 1

N-1
----              T   -1
+ >    (x   - g (x ))  Q  (x    - g (x ))
----   k+1   k  k     k   k+1    k  k
k = 1