In
time series analysis
In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. ...
(or forecasting) — as conducted in
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
,
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
, 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''. If the forecasting method is working correctly, successive innovations are uncorrelated with each other, i.e., constitute a
white noise time series. Thus it can be said that the innovation time series is obtained from the measurement time series by a process of 'whitening', or removing the predictable component. The use of the term innovation in the sense described here is due to
Hendrik Bode and
Claude Shannon
Claude Elwood Shannon (April 30, 1916 – February 24, 2001) was an American mathematician, electrical engineer, computer scientist, cryptographer and inventor known as the "father of information theory" and the man who laid the foundations of th ...
(1950) in their discussion of the
Wiener filter problem, although the notion was already implicit in the work of
Kolmogorov.
[Mitter, S. K. (1982). Nonlinear filtering of diffusion processes a guided tour. In ''Advances in Filtering and Optimal Stochastic Control'' (pp. 256-266). Springer, Berlin, Heidelberg.]
In contrast, the residual is the difference between the observed value of a variable at time ''t'' and the optimal updated state of that value based on information available till (including) time ''t''.
See also
*
Kalman filter
In statistics and control theory, Kalman filtering (also known as linear quadratic estimation) is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unk ...
*
Filtering problem (stochastic processes)
*
Errors and residuals in statistics
*
Innovation butterfly
References
Statistical signal processing
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