All definitions given so far concern continuous-time random processes, but they all
have a discrete time equivalent in replacing the integration by a summation. A time series
is a sequence of observations that are ordered in time, usually sampled for computational
analysis, so the times series that we are interested in is a realization of a discrete-time
random process. Yet we aim to study the properties of the random process through only
one realization, this is allowed by the fundamental property of ergodicity.
2.1.3 Ergodicity and Piecewise Stationarity
Most of the time, a process cannot be stopped and reset to its initial values, or not
enough times to get a sufficient number of realizations to perform statistical averaging.
Instead of performing averages on the number of realizations, it is more practical to
average over time. A process is called ergodic if all orders statistical averages of the
process are equal to the time averages (assuming there is an infinite number of data
points, or t -> oc). This property allows us to use one realization of a random process,
i.e., one time series, to find its statistical properties (usually just a few realizations are
available, but they contain a lot of data points). Practically, in all time series analysis
methods, the assumption of ergodicity is assumed to hold, because it would not be
possible to study the statistics of the process otherwise. It is interesting to note that
ergodicity infers stationarity, and therefore the converse is true too: non-stationarity
infers non-ergodicity. Therefore in many cases (particularly this one) the results derive
from a known wrong assumption!
Most of the times real data is non stationary but can be considered multi-modal
with stationary modes, and this is the basic assumption in segmentation problems: the
time series is assumed to be not stationary but piecewise stationary. A piecewise
stationary signal can be described in two different manners: both have in common the