## 10.1 Introduction

DFA is conceptually different than what we have been doing in the previous applications. Here we are trying to explain temporal variation in a set of $$n$$ observed time series using linear combinations of a set of $$m$$ hidden random walks, where $$m << n$$. A DFA model is a type of MARSS model with the following structure:

$$$\begin{gathered} \mathbf{y}_t = \mathbf{Z}\mathbf{x}_t+\mathbf{a}+\mathbf{v}_t \text{ where } \mathbf{v}_t \sim \text{MVN}(0,\mathbf{R}) \\ \mathbf{x}_t = \mathbf{x}_{t-1}+\mathbf{w}_t \text{ where } \mathbf{w}_t \sim \text{MVN}(0,\mathbf{Q}) \\ \end{gathered} \tag{10.1}$$$

This equation should look rather familiar as it is exactly the same form we used for estimating varying number of processes from a set of observations in Lesson II. The difference with DFA is that rather than fixing the elements within $$\mathbf{Z}$$ at 1 or 0 to indicate whether an observation does or does not correspond to a trend, we will instead estimate them as “loadings” on each of the states/processes.