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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)
X⁢t 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:

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

  1. The mean value is independent of tt, for all tt
    EX⁢t=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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Definition of a lens

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?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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