State equation

Observation equation

Y is q by 1

var(w) = Q var(v) = R

State. Summary of the past behavior of the system Markov

Advantages of approach. parsimonious estimation smoothing, forecasting/prediction missing values random effects time varying measurement error multivariate fitting ARMA by mle aggregation/disaggregation structural breaks nonlinear variants control …

r by 1 exogenous input u

Example.

Brillinger, Guckenheimer, Guttorp, Oster (1992)

Density dependent birth and death rates

Data, graphs of data

Autoregressive. VAR(m)

Var(W) =

Biomedical example with missing values

A t is either identity or all zeros in a row when variable is missing

Global warming.

drift

Actually ARMA(1,1) !

Example.

Brillinger and Kaiser “NMR”