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Transition Function
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.

Transition Variance
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.

Example
The filter in EXAMPLE.KBF determines the position of a ship on the ocean given range measurements to two shore stations. The State vector for this example contains both the position and velocity of the ship. Except for random forcing, the ship is travelling at a constant velocity and the span between time indices is 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 /