The distinguishing feature of state space time series models is that observations are regarded as made up of distinct components such as trend, seasonal, regression elements and disturbance terms, each of which is modeled separately. These models for the components are put together to form a single model called a state space model which provides the basis for analysis. The book is primarily aimed at applied statisticians and
econometricians. Not much of math background is needed to go through the book,at least the first part of the book. State space time series analysis began with the path breaking paper of Kalman and early developments of the subject took place in the field of engineering. The term state space comes form engineering. Statisticians and econometricians tend to stick to the same terminology.
Part I of the book deals with linear Gaussian models and Part II deals with the extensions of it to non linear, non Gaussian world. Extensions like exponentially distributed observations, allowing nonlinearities in the model,allowing heavy-tailed densities to deal with outliers in the observations and structural shifts in the state are dealt in the second part of the book. The treatment given in the book is a simulation based approach, as excepting for a few models, there are no closed form solutions available. Instead of following the MCMC treatment, the book shows the use of importance sampling and antithetic variables in analyzing state space models. The book provides analysis from the classical treatment as well as the Bayesian treatment. In the given link at the end of this post, I have summarized Part I of the book that take up 175 out of 240 odd pages of the book. Part II of the book is all about nonlinear non Gaussian models and the use of importance sampling to estimate the filtering and smoothing estimates. In contrast to Part II of the book that is math heavy, the first part of the book is definitely manageable. In fact one can go through the entire Part I of the book by knowing just one Lemma.