- In this lecture, I want to make the idea of
the forecast analysis cycle a bit more concrete,
by going over what I believe to be
the simplest possible example of implementing this
that can be used for real world problems.
So in going through the simplest possible
forecast analysis problem,
I'm gonna make a lot of simplifying assumptions.
First, let's start thinking about the analysis step.
So for the analysis step, we need a prior,
which is our forecast.
So I'm gonna assume that that prior
follows a normal distribution,
with some prior mean and prior standard deviation
that captures the uncertainty in our forecast.
I'm gonna assume that we've made our forecast
and so that prior mean and variance is known.
Next, to combine that prior with data,
I need to write down a likelihood.
I'm gonna assume that likelihood is also normal
and that it follows a very simple form,
which is that the observations are just normally distributed
around the true states of the system,
with some observation error.
I'm likewise going to assume that
that observation error is known.
Which is not an unreasonable assumption,
because a lot of data products
do report their uncertainties.
If I take this normal prior from the forecast,
and a normal likelihood from the data,
this benefits from the conjugacy
between the normal prior and the normal likelihood,
and will give us a normal posterior.
Furthermore, that normal posterior
has a simple analytical solution.
If I were to express the uncertainties in my forecast
that come from the uncertainty in the data,
the observation error,
and the uncertainty in the forecast,
its prior variance.
If I express those prior variances instead
as prior precisions, which is just one over variance,
then the posterior precision is just the sum
of the precision.
So the posterior precision for the analysis step
is just the precision of the observations
plus the precision of the forecast.
And then if I wanna get back to variance,
I can just do one over that term.
Now, the mean of the analysis step
is just a weighted average
between the mean of the forecast
and the mean of the data,
each weighted by their precisions.
So I have the precision of the data
divided by the total precision,
the data precision plus the forecast precision,
giving a weight to the data.
And likewise I have the weight from the prior forecast
that comes from the uncertainty by the forecast
in terms of its precision divided by the total precision.
So, this leads to two important intuitive pieces
of understanding
that is mathematically exact for this simple problem,
but will carry through conceptually
to more complex problems.
The first is just that,
when we perform the analysis
and update the forecast with new information,
that that updated forecast will always be more precise,
because we take the prior forecast
and we add more information to it.
And so literally, we were summing the precision
of the data with the precision of the forecast.
So we're always ending up with a more precise analysis.
The second intuition is that the contribution
of information from the forecast
and the contribution of information
from the new data
are each weighted by their uncertainties.
So if I have data that is very noisy,
it's only going to have a little bit of influence
on the forecast.
By contrast, if I have data that is very precise,
but a model that is very uncertain,
then we're gonna see the analysis pulled very strongly
in the direction of that high-quality data.
This, I think has an important take-home message
for all of us,
regardless of whether we're making ecological forecast,
just as producers of ecological data,
which is that to be able to do any sort of data simulation,
to be able to integrate observations
into models and forecasts,
it is critical that we have
estimates of uncertainty on those data.
And it's also critical
that we have estimates of uncertainty on our models.
Furthermore, those estimates need to be robust.
If we have an estimate of uncertainty in the data
that is very inaccurate,
we might give very inappropriate weight to that data.
So for example, if I have some observations
and I report just some sort of calibration uncertainty
on that data,
but really there's a huge amount of sampling uncertainty,
I might give too much weight to data.
You might falsely represent its information contribution.
We've now talked through
how the simplest possible analysis step works.
Combining a normal forecast with a normal likelihood
to give us a normal analysis.
We next want to think about how the forecast step works.
We want to now make an updated forecast
from that updated analysis.
In the simplest possible case, I'm gonna assume
that the model that we wanna make a forecast with
is simply linear.
Because I'm making the assumption
that the forecast model is linear,
I can use the analytical moments approach
to uncertainty propagation
to actually analytically transform the uncertainty
in the current analysis state,
into the uncertainty in the forecast.
It's gonna end up that the forecast mean
is just what we would get
from plugging the mean of the analysis
into the forecast model.
And the analysis uncertainty is gonna have two components.
First, we're gonna take the current analysis uncertainty
and we're gonna inflate it,
or decrease it according to the slope of the model.
'Cause the simple linear model,
so as we learned when we learned
about uncertainty propagation,
the variance in the forecast
is just going to be the slope squared
times the variance in the input.
In this case, the analysis posterior,
which is the input initial condition.
So that we have a contribution
from the initial condition uncertainty
that goes into the forecast,
and then we also have a contribution
from the process error.
Which, in the simplest possible case,
we're gonna assume the process error
is also normally distributed,
and it has a process error that is known.
Which is a very strong assumption
for a lot of ecological problems.
But while it's a strong assumption,
it does result in an analytically tractable approach
to propagating the uncertainty in the forecast,
and then an analytically tractable approach
to updating that forecast
when new information becomes available.
So this very simple
analytically tractable forecast analysis cycle
that makes the simplifying assumption
that the observation errors are normal,
and that the process errors are normal,
and the model's linear,
goes by the name of the Kalman Filter
and is one of the first examples of data simulation
and will form the basis
for more complex data simulation problems
that we'll cover next.