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# Stationarity

Module by: Nick Kingsbury. E-mail the author

Summary: This module introduces stationarity, such as strict sense stationarity (SSS) and wide sense stationarity (WSS).

Stationarity in a Random Process implies that its statistical characteristics do not change with time. Put another way, if one were to observe a stationary random process at some time tt it would be impossible to distinguish the statistical characteristics at that time from those at some other time t t .

## Strict Sense Stationarity (SSS)

Choose a Random Vector of length NN from a Random Process:

X=( X t1 X t2 X tN )T X X t1 X t2 X tN
(1)
Its NNth order cdf is
F X ( t1 ) ,     X ( tN ) x1 xN =PrX t1 x1 X tN xN F X ( t1 ) ,     X ( tN ) x1 xN X t1 x1 X tN xN
(2)
Xt X t is defined to be Strict Sense Stationary iff:
F X ( t1 ) ,     X ( tN ) x1 xN = F X ( t1 + c ) ,     X ( tN + c ) x1 xN F X ( t1 ) ,     X ( tN ) x1 xN F X ( t1 + c ) ,     X ( tN + c ) x1 xN
(3)
for all time shifts cc, all finite NN and all sets of time points t1 tN t1 tN .

## Wide Sense (Weak) Stationarity (WSS)

If we are only interested in the properties of moments up to 2nd order (mean, autocorrelation, covariance, ...), which is the case for many practical applications, a weaker form of stationarity can be useful:

Xt X t is defined to be Wide Sense Stationary (or Weakly Stationary) iff:

1. The mean value is independent of tt, for all tt
EXt=μ X t μ
(4)
2. Autocorrelation depends only upon τ= t2 t1 τ t2 t1 , for all t1 t1
EX t1 X t2 =EX t1 X t1 +τ= r X X τ X t1 X t2 X t1 X t1 τ r X X τ
(5)
Note that, since 2nd-order moments are defined in terms of 2nd-order probability distributions, strict sense stationary processes are always wide-sense stationary, but not necessarily vice versa.

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##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

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##### What are tags?

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