Space time problems and applications

State space models and Kalman filter techniques have been widely used for the analysis of time series. Typically, a latent process is assessed from observations using filtering (the present), smoothing (the past) and/or prediction (the future). The model class is very broad and comprises ARIMA models, cubic spline models and structural time series models. The development of state space theory has interacted with the development of other statistical disciplines.   In the first part of the Thesis, we present the theory of state space models, including Gaussian state space models, approximative analysis of non-Gaussian models, simulation based techniques and model diagnostics.   The second part of the Thesis considers Markov random field models. These are spatial models applicable in e.g. disease mapping and in agricultural experiments. Recently, the Gaussian Markov random field models were expressed as state space models, enabling the Kalman filter machinery. Our main contribution is to extend the Markov random field models by generalising the corresponding state space model. It turns out that several non-Gaussian spatial models can be analysed by combining approximate Kalman filter techniques with importance sampling.   The third part of the Thesis contains applications of the theory. First, a univariate time series of count data is analysed. Then, a spatial model is used to compare wheat yields. Weed count data in connection with a project in precision farming is analysed using the developed methodology. Finally, a model for edge detection in digital images forms the basis of a simulation study.