- In this lecture I want to build on the previous lectures
talking about analytical methods for doing data simulation
and implementing the forecast analysis cycle
but now shift gears to talking about numerical methods
for assimilating data into models.
As a reminder, the framework we're thinking about
is this idea of making forecasts
and then iteratively updating those forecasts
as new information becomes available.
In the previous analytical methods,
we were using analytical methods to make that forecast
to propagate the uncertainty and the forecast
into the future.
Here I want to talk about ensemble methods
for doing data simulation.
The key distinction between this and the previous approaches
is that we're going to use the ensemble based approaches
for doing that uncertainty propagation
which we talked about in the previous lecture
on ensemble air propagation.
In a nutshell, you're going to have distributions
of your initial conditions and any other uncertainties
you need to propagate.
We'll sample from those,
you'll run your forecast model for it in the future
under different samples of these different initial
condition values, parameter values, inputs.
So you're running a whole ensemble of models
and that ensemble is capturing the uncertainty
in your prediction.
So once that is done,
you then need to do the analysis step.
For the analysis step in the general sense
we can do this using any form of basing inference
to do the analysis where we are using the model forecast
as our prior and then updating that
using the likelihood to add new information.
The thing that sets ensemble based approaches apart
from other approaches is that once we have that ensemble
and we construct the prior, we're gonna need to assume
a named probability distribution that we're fitting
that ensemble too.
So the most common form of ensemble data simulation
is the ensemble Kalman filter.
So the ensemble Kalman filter
like the other variants of the Kalman filter
assumes that that forecast takes a normal distribution.
So the essential step in the ensemble Kalman filter
is after we've used the ensemble numbers
to make the forecast
we fit the sample mean and sample variants
to that ensemble in order to put parameters on that prior
that we put in the analysis step.
Once we fit that prior distribution to the ensemble,
then that enters in to the Kalman data simulation
the same way that we did in the Kalman filter,
the extended Kalman filter
so we have a normal prior, we have a normal likelihood
we combine those things and we get a normal posterior.
Because of its simplicity, the ensemble Kalman filter
is particularly popular in ecology
and it is one that has a lot of advantages.
One of the major advantages of the ensemble Kalman filter
is its ease of implementation.
So you just need the ability to run the model
multiple times over multiple different initial conditions
and parameter settings and drivers
to propagate the uncertainty.
You do not need to analytically solve the model
for derivatives.
You don't need to restrict yourself to linear models.
In fact, the ensemble Kalman filter
and other ensemble filters
are able to deal with the nonlinearity models very easily
and unlike the extended Kalman filter,
they are able to deal with that in a way
that doesn't violate genesis and equalities
when in fact they arguably do a better job
at handling the nonlinearities in the models
than the analytical techniques.
The other advantage, like I said,
is they don't require doing any additional
mathematical analyses, additional steps.
You can use them using your existing code.
As long as that model that you're using to make a forecast
can be stopped and restart under new initial conditions
after the analysis.
You can use it with your existing model
you just need the capacity to run that model multiple times.
One of the common question with ensemble based
data simulation methods is what size is needed
for the ensemble.
So here we can rely on just basic statistical theory to say,
given amount of variability observed in the ensemble
and we want a specific level of precision
in our estimate of the posterior mean and variants,
what sample size do we need to get that?
So this is simply, you do a standard deviation
one over sample size
is standard air restorative calculation.
Rule of thumb, in ensemble based methods
that's usually in the order of tens to hundreds
because again, the goal here is just the ability
to constrain, to get a good solid estimate of the mean
and the co-variants, not to approximate
the full distribution with the ensemble.
Similar to the extended Kalman filter,
it is worth noting that the ensemble Kalman filter
does have this assumption of normality into it
but also by definition technically violates that assumption
because if you start from a normal distribution
for your initial conditions,
you transform that through a nonlinear model,
what comes out technically cannot be normal.
An important question then is how bad is that assumption?
And really the best way to check that is just by looking at
the histogram of your predictions,
so looking at the ensemble itself.
So most of us in our basic statistics are pretty comfortable
with making a histogram of our data
or making a histogram of our residuals
and looking and seeing
is this approximately normally distributed
and if so, we're fine with it, that assumption
and we can proceed on using those assumptions.
But there are a good number of places where you are likely
to violate that in ensemble filters.
So some of the ones I've encountered are for example,
where you make a forecast
and under some set of ensemble numbers ,
there are mortality or disturbance
or something where you can end up with
a bimodal distribution between
ensemble numbers where everything remained happy
and ensemble numbers where there was some event,
mortality event, disturbance event,
species going locally extinct,
that can actually create say, for example,
a bimodal distribution that would very much violate
the assumptions of normality
and is not the sort of thing that's easy to transform
to be consistent with those assumptions.
That said as I mentioned earlier there's nothing that says
we can't generalize these approaches to data simulation
and the analysis step could be handle with any choice
of distribution for the prior that we would feel comortable
representing with our ensemble.