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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$:

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

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


Antti Honkela 2001-05-30