- So state space model is a framework
for modeling a dynamic process.
Process where you have latent variables that are connected
or linked in time or space, a time series.
When you've got these latent variables that are themselves,
observed with some sort of error
and the error can be random noise,
or it can itself be a process that can change in time
or change with observer identity.
You've got these latent variables,
and they're not independent, right?
So if you have a time series,
then you know that times that are more closely
together are more likely to be similar.
So you need to have a way to model that dependent structure.
A state space framework is really powerful,
in the sense that it can take those observations,
we call them Y, it can take Y at time t
and make an estimation of the truth state.
So Y is the observation the true state called X at time t,
and the Y is at the different times
are themselves conditionally independent.
So because they depend on X, and the X is that process state
is where the dependent structure is modeled.
And so you don't have to be beholden to the assumption
that that your data have independent errors.
Because you're modeling that in the state
or process stage of the model framework.
A more specific example,
if we're thinking about a state space model in time,
a hidden Markov process is a model where the true state
at time t so it's now,
is dependent on the previous time steps
or Markov processes when you have a lag of one,
the current state and on the last state.
So a simple hidden Markov model would have a true state
dependent on the state at the previous time step
and those states would themselves be informed
by the observations Y, at t and t minus one,
but the Y is the observations,
and the observation error would themselves be independent.
For example, you may go out and count birds at time t,
and you know, you go out you do a point count of birds,
you're getting an estimate of the true population,
but it's not necessarily the true population.
And we often use a Poisson distribution to describe a count.
And so your observation model would be Y is distributed
as a Poisson with some mean,
and that mean would be equal to the X,
the true state of the population at time t.
The observed count at time t and t minus one,
would themselves be treated
as conditionally independent observations,
given the true state and so the true state at time t,
which is the mean of the Poisson,
could then itself be modeled
as a function of temperature or a region,
you could build all of the model structure
into that state variable.
And that itself would be informed
by the state at the previous time step.
States based models can also be in space.
So they can be spatial.
And in that sense, the current or the site of interest
is assumed to be the true state at site I,
is assumed to be a function of the true state
at the sites around I.
And so you have to define the adjacency matrix
which could be a grid most closely
associated with a grid point I.
But you have a lot of flexibility to define your adjacency
neighborhood and even evaluate it again within
the probabilistic framework of the state space approach.
One of the reasons this is a really powerful
and important tool for forecasting,
is that the state space approach allows us
both to separate observation error from process error,
which in itself is very important to observation error
isn't going to propagate out over time process error
is gonna increase over time,
which is obviously of importance
if you're trying to forecast out in time,
or it's gonna increase the further you get away
from whatever the last state point you actually have.
So when you have observations,
you have more information about the true state,
at that time step.
When you're trying to forecast the true state
at a future time step, or at a different site,
the further away you are, in a year
or in one neighborhood over the further away you get,
the bigger the uncertainty gets.
And that's intuitive,
but it's not actually what always happens.
If you're not partitioning uncertainty
into observation and process error.
The classic assumption of anything
we can't explain in the model
gets kinda lumped into the observation error term,
when you're forecasting out beyond the data.
If you're not pulling from that observation error
and you haven't described it accurately,
then you're likely to be overly confident in your forecasts
and more likely to miss a true future state.