Measurement Vector
For time indices k = 1 to k = nk, the corresponding Measurement vector is denoted by

           / z(1)  \
           |  k .  |
      z  = |    .  |
       k   |    .  |
           | z(nz) |
           \  k    /

where nz is the number of elements in the measurement vector.

Measurement Function
The Measurement function h maps a value for the State vector at time index k to the mean value for the Measurement vector at time index k. The State vector must contain enough information to make this possible. To be specific, for time indices k = 1 to k = nk,

      z  = h (xt ) + v
       k    k   k     k

where nk is the number of time points, xt is the true, but unknown, value for the state vector, and v is a random variable with mean zero.

Measurement Variance
For time indices k = 1 to k = nk, the corresponding Measurement variance is denoted by

                    / E[v(1) v(1)] ...  E[v(1) v(nz)] \
                    |    k    k            k    k     |
              T     |      .       .        .         |
   R = E[ v  v  ] = |      .       .        .         |
    k      k  k     |      .       .        .         |
                    |E[v(nz) v(1)] ... E[v(nz) v(nz)] |
                    \   k     k           k     k     /

where nz is the number of elements in the Measurement vector z, 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. In this example, the Measurements vector is



           / z(1) \   /range measurement to station P at time index k\

      z  = |      | = |                                              |

       k   | z(2) |   |range measurement to station S at time index k|

           \  k   /   |                                              /

Except for measurement error, the range measurements are equal to the Euclidean distance between the known shore stations and ship position. The corresponding Measurement function is

              /  _____________________________________ \
              | \/ (P(1) - x (1))^2 + (P(2) - x (2))^2 |
   h (x ) =   |             k                  k       |
    k  k      | \/ (S(1) - x (1))^2 + (S(2) - x (2))^2 |
              \             k                  k       /

Each measurement error for this example has a standard deviation of 2 and is not correlated with the other measurement. The corresponding Measurement Variance is

           / 4  0 \
      R  = |      |
       k   \ 0  4 /