AR(1) Spatio-temporal Random Fields

[Nathan-Hebert]

I'm not sure I understand the purpose of the sqrt(1-rho^2) bit. What is the steady-state marginal variance? Thanks!

[ericward-noaa]

I'm not sure I understand the question -- but here's a little more info on the TMB implementation:

https://kaskr.github.io/adcomp/classdensity_1_1AR1__t.html

And even in a simpler scenario, like a 1-D time series setting, we'd also want to do the same scaling

[Nathan-Hebert]

I guess what I'm asking is, if we had only epsilon_t as the second term rather than sqrt(1-rho^2)*epsilon_t, what would happen to our fields?

[ericward-noaa]

I"ll try to illustrate this with the time series version because it's easier to see. The short version is without the adjustment, the variance blows up because it is a function of $\rho$.

In the first case, let's say we have a correlated random walk model,

$x[0] \sim N(0, \sigma)$
$x[1] = \rho * x[0] + N(0, \sigma)$
$x[2] = \rho * x[1] + N(0, \sigma)$

So Var(x[0]) is $\sigma^2$, but Var(x[1]) = $\rho^2 * \sigma^2 + \sigma^2$, and it keeps ballooning with x[2], x[3], ...

In the second case, we can do the same thing with the adjustment built in

$x[0] \sim N(0, \sigma)$
$x[1] = \rho * x[0] + N(0, sqrt(1 - \rho^2) * \sigma)$

As before, Var(x[0]) = $\sigma^2$, but now Var(x[1]) = $\rho^2 * \sigma^2 + (1 - \rho^2) * \sigma^2 = \sigma^2$, and
all other elements of x will have the same variance = $\sigma^2$.

[Nathan-Hebert]

Thank you so much! That makes perfect sense now. :)