ABSTRACT
The KF Algorithm 4.3 produces the BLUP of x(t) from y(1), . . ., y(t). Thus, at the end of the recursion we have calculated the vector of predictors⎡⎢⎢⎢⎢⎢⎢⎣
x(1|1) x(2|2)
.
.
. x(n|n)
⎤⎥⎥⎥⎥⎥⎥⎦ .
There are cases where this is exactly what is desired. For example, if only the current state vector at time t is of interest, then x(t|t) gives the pertinent answer and the values of random vectors x(1), . . ., x(t−1) are no longer of relevance. However, if previous values of the state vector are al-
so objects of importance then it makes sense to include the information from new response data into their predictions as well. We discussed this briefly at the end of Chapter 4. Now we will develop the idea in more detail. Because of the state-space structure, y(t+1), y(t+2),
. . ., y(n) all contain information about the state vector x(t). Consequently, by including new response in-
rithm alone. This process of modifying the BLUP x(t|t) based on y(1), . . ., y(t) to obtain a BLUP based on y(1), . . ., y(t), y(t + 1), . . ., y(r) for some r ≥ t is referred to as smoothing. The question then is how much smoothing to do or how
much new information to incorporate into the estimator x(t|t). One obvious answer would be to use the most information possible and predict the entire state vector via ⎡⎢⎢⎢⎢⎢⎢⎣
x(1|n) x(2|n)
.
