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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.
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.
where is the number of data points in the time series, are the phase or frequency values, and ¯ is the average value of the time series. If used for clock stability analysis, the z t {\displaystyle z_{t}} values are the non-overlapped (or binned) averages of the original frequency or phase array for some averaging time and factor.
In order to define the notion of white noise in the theory of continuous-time signals, one must replace the concept of a random vector by a continuous-time random signal; that is, a random process that generates a function of a real-valued parameter .
Additive white Gaussian noise (AWGN) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics: Additive because it is added to any noise that might be intrinsic to the information system.
First, white noise is a generalized stochastic process with independent values at each time. [12] Hence it plays the role of a generalized system of independent coordinates, in the sense that in various contexts it has been fruitful to express more general processes occurring e.g. in engineering or mathematical finance, in terms of white noise.
In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise (also called popcorn noise or random telegraph signal).
The graphs in Figure B are a series of masking patterns, also known as masking audiograms. Each graph shows the amount of masking produced at each masker frequency shown at the top corner, 250, 500, 1000 and 2000 Hz. For example, in the first graph the masker is presented at a frequency of 250 Hz at the same time as the signal.