Tools for time series analysis

In a typical mathematical model [9], a time series $\left(x(t_1), \ldots, x(t_n) \right)$ is generated by the flow of a smooth dynamical system on a $d$-dimensional smooth manifold $M$:

$$\mathbf{s}(t) = \phi^t(\mathbf{s}(0)).$$ (2.6)

The original $d$-dimensional states of the system cannot be observed directly. Instead, the observations consist of the possibly noisy values $x(t)$ of a one-dimensional measurement function $h$ which are related to the original states by

$$x(t) = h(\mathbf{s}(t)) + n(t)$$ (2.7)

where $n(t)$ denotes some noise process corrupting the observations. We shall for now assume that the observations are noiseless, i.e. $n(t) = 0$, and deal with the noisy case later on.