Advanced time series analysis concerns relaxing the assumptions of basic time series analysis, especially those of linearity and Gaussianity.
Recommended Books

Nonlinear Time Series
Randal Douc, Eric Moulines, and David S. Stoffer
 Intext exercises
 Some code snippets
 Supporting R package
 Errata
This book is not for the faint of heart, but if you don't believe the time series you are working with satisfies the conventional assumptions, you will likely benefit from reading it. It's divided into three parts: a review of basic linear time series and some small extensions, a section on Markov models, and a section on state space and hidden Markov models. The sections don't depend very much on each other, which is nice because we think most people will be most interested in the last one. The included R package is a useful addition, but we really wish some exercise solutions were available since this book is not sufficiently popular to find them easily on the web.