2. Stationarity

Question 1

What does stationarity mean?

Answer 1

That the spatio-temporal stochastic process is “behaving the same” in all locations and times

Question 2

Complete stationarity

Answer 2

The probability of a given value U(s) does not change with s. And the joint probability of U(s) and U(s’) does not change with s and s’. etc.

Question 3

Strong stationarity

Answer 3

A synonym for complete stationarity

Question 4

Stationarity at order 1

Answer 4

Whatever the population mean in a given location at a given time may be, it is the same for all the other locations and times.
μ(x,y,z,t)=μ for all x,y,z,t

Question 5

What would you call a temporal stochastic process for which μ(x,y,z,t)=μ(t)?

Answer 5

Stationary at order 1 in 3-D space

Question 6

What would you call a temporal stochastic process for which μ(x,y,z,t)=μ(x,y,z)?

Answer 6

Stationary at order 1 in time

Question 7

What is stationarity at order 2?

Answer 7

Homogeneity of the variance

Question 8

What is stationarity at order 2 in terms of population autocovariance C(s,s’) ?

Answer 8

C(s,s’) = C(s’-s)

Question 9

What is weak stationarity? (def + 3 elements)

Answer 9

It includes stationarity at order 1 and 2 so it implies:
* E[U(s)] = μ for all s
* Var[U(s)] = σ for all s
* Corr[U(s), U(s’)] = ρ(s’-s) for any s’≠s