The Transition function

*g*

maps a value for the
State vector
at time index *k*

to the mean value for the State vector at time index *k*+1

.
The State vector must contain enough information to make this possible.
To be specific, for time indices *k* = 1

to *k* = *nk*-1

,
*xt* = *g* (*xt* ) + *w*
*k*+1 *k* *k* *k*

where *nk*

is the number of time points,
*xt*

is the true, but unknown, value for the state vector,
and *w*

is a random variable with mean zero.
For time indices

*k* = 1

to *k* = *nk* - 1

,
the corresponding Transition variance is denoted by
/ *E*[*w*(1) *w*(1)] ... *E*[*w*(1) *w*(*nx*)] \
| *k* *k* *k* *k* |
*T* | . . . |
*Q* = *E*[ *w* *w* ] = | . . . |
*k* *k* *k* | . . . |
|*E*[*w*(*nx*) *w*(1)] ... *E*[*w*(*nx*) *w*(*nx*)] |
\ *k* *k* *k* *k* /

where *nx*

is the number of elements
in the State vector *x*

,
and *E*

denotes expected value.
The filter in

*Dt*

.
Thus the Transition function is
/ *x*(1) + *x* (3) *Dt* \
| *k* *k* |
| *x*(2) + *x* (4) *Dt* |
*g*(*x* ) = | *k* *k* |
*k* *k* | *x*(3) |
| *k* |
| *x*(4) |
\ *k* /

The transition error for this example has a standard deviation
of `1`

for the positions and of `.2`

for the velocities.
The corresponding Transition Variance is
/ 1 0 0 0 \
*R* = | 0 1 0 0 |
*k* | 0 0 .4 0 |
\ 0 0 0 .4 /