- So in the last video I talked about
the simplest possible, analytically tractable
version of the forecast analysis cycle
known as the Kalman filter.
When I introduced the idea of the Kalman filter
I introduced it just in the simplest case
of a univariate model.
So a model that only has one variable,
there's only one slope,
there's only one X that we're caring about.
In many cases, for many problems,
when we're doing data assimilation we have more than one
variable that we're interested in.
So instead of having just one X,
I might have a whole vector of state variables.
These could be different state variables
at a given location,
for example: different pools in a biogeochemical model,
different stages or ages
in an age-structured population model,
different species in a community model,
so it's very common that we're trying
to make forecasts over multiple things at the same time.
Likewise, different X's could represent
the same thing, say a population,
but at different locations.
So I might be trying to forecast
different subpopulations of a single population,
or I might be trying to forecast a whole map.
When I'm doing that I now have a multivariate model.
Now, the math of the Kalman filter
can be extended to the multivariate case
through a bit of linear algebra.
We can look at what that map looks like in the slides,
but the take-home is that
the underlying results are essentially the same.
We still wanna put the analysis mean,
being a weighted average of the forecast mean,
in the data.
One of the real strengths of the multivariate
version of the Kalman filter though,
is its ability to borrow strength across observations.
So to take a simple case,
imagine I'm making a forecast that has two variables,
say X one and X two.
They might be fraction conifer on the landscape
and above-ground biomass
that we observe on the landscape.
And imagine, also, that there's some correlation
between those forecasts.
So, for example, when I predict
more conifers on the landscape,
I'm predicting more biomass on the landscape.
Now, if I then observe one of those things,
for example, I have a remote sensing observation
of the fraction conifer field survey,
or fossil-pollen data in a paleoecological sense.
I can use that new data about the fraction conifer
to update that forecast to a new analysis posterior.
Furthermore, because the fraction conifers
correlated with above-ground biomass,
I can also update the latent variable,
the above-ground biomass,
which I don't observe directly
in proportion to the strength of its correlation
with the thing that I do observe.
So if the correlation between
these two things is perfect,
no correlation curve, coefficient one,
all the information about fraction conifer
is transferred to updating above-ground biomass.
If there's no correlation between these two,
when I update, the fraction conifer has no impact
on above-ground biomass analysis.
This idea of borrowing strength across variables
works in both directions.
So if I were able to observe the fraction conifer
and the above-ground biomass,
then my post-year estimates of both
would involve both the direct constraint
coming from their direct observations,
and then an indirect constraint
due to their correlation with each other.
When we do this process of updating variables
by combining multiple sources of information
during the analysis step,
I refer to this kind of as
thinking of models as scaffolds.
So, what we're essentially doing is,
we're engaging in a data-driven exercise
of combining multiple sources of information,
but we need some way of relating
multiple sources of information to each other.
So if I have observations of fraction conifer
on the landscape and I have observations of biomass,
I need some way of relating those to each other.
And here, what I'm going to do is I'm leveraging their
covariance to do that update.
And specifically leveraging the covariance
that's coming from the models.
The model is essentially contributing
its covariance structure in the forecast
to this updating process.
This behavior we see of multiple pieces
of information being able to be synthesized,
mutually updating each other
during the multivariate analysis
is directly analogous to things
that many ecologists have seen before
in spacial statistics and in time series analysis.
So, for example, in spacial statistics,
we construct a covariance matrix
that tells us how observations
at different locations are related to each other
In a spacial statistical analysis,
we typically make the assumption
that the correlation between
information at multiple locations
is related to each other as a function of distance.
And there's some empirical distance-decay function
that we use to model how the correlation
changes the function of distance.
So in a spacial statistics model
we would fill in this covariance matrix
using this function that describes
how information decays in the function of distance.
By contrast, in data assimilation,
the math is actually completely identical.
It's the exact same math.
The only difference between
a spacial statistics analysis
and a data assimilation analysis
is how we filled in that covariance matrix.
So in a data assimilation example
we're filling in that covariance matrix
based on our process model that we use in our forecast.
So essentially we're using our hypotheses
about how ecosystems work.
Our understanding that's built
into our models and built into our forecasts.
That is what's being used to generate
this correlation structure between
different variables when we do this data assimilation,
whether they be different locations in space,
or correlations between different processes,
or different stages or, different organisms
in any sort of multivariate model.