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In time series analysis (or forecasting) — as conducted in statistics, signal processing, and many other fields — the innovation is the difference between the observed value of a variable at time t and the optimal forecast of that value based on information available prior to time t.
Ideally, unevenly spaced time series are analyzed in their unaltered form. However, most of the basic theory for time series analysis was developed at a time when limitations in computing resources favored an analysis of equally spaced data, since in this case efficient linear algebra routines can be used and many problems have an explicit ...
In time series analysis, the moving-average model (MA model), also known as moving-average process, is a common approach for modeling univariate time series. [1] [2] The moving-average model specifies that the output variable is cross-correlated with a non-identical to itself random-variable.
The asymptotic equipartition property holds if the process is white, in which case the time samples are i.i.d., or there exists T > 1/2W, where W is the nominal bandwidth, such that the T-spaced time samples take values in a finite set, in which case we have the discrete-time finite-valued stationary ergodic process.
This is an important technique for all types of time series analysis, especially for seasonal adjustment. [2] It seeks to construct, from an observed time series, a number of component series (that could be used to reconstruct the original by additions or multiplications) where each of these has a certain characteristic or type of behavior.
ARCH models are commonly employed in modeling financial time series that exhibit time-varying volatility and volatility clustering, i.e. periods of swings interspersed with periods of relative calm (this is, when the time series exhibits heteroskedasticity).
A bootstrap paradox, also known as an information loop, an information paradox, [6] an ontological paradox, [7] or a "predestination paradox" is a paradox of time travel that occurs when any event, such as an action, information, an object, or a person, ultimately causes itself, as a consequence of either retrocausality or time travel.
In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem), named after Herman Wold, says that every covariance-stationary time series can be written as the sum of two time series, one deterministic and one stochastic.