- Statistical inference is most often engaged
in trying to separate signal that is a model
of the world and how it works from noise,
the difference in how we observe the world
and that model is often, in statistical inference,
treated as random noise.
So it classics assumptions with a linear aggression,
which is one of the most commonly used techniques
in statistics, is that uncertainty arises
from observation error.
So your error is all in that observation component.
You've got a model that describes the signal,
oftentimes, that's a mean or a line.
And if it's a mean, for instance,
then anytime your observations
are different from that mean,
you estimate that as observation errors,
so something that wasn't measured correctly.
And the general assumption there is that the error is normal
and so symmetric about zero,
and that the errors are independent.
So knowing one observation doesn't tell you anything
about the error between the model
and the next observation.
These techniques don't often handle
missing data intuitively or at all.
And the assumption of constant variance over time
is another assumption that's common
and important for progression models.
And one of the things that we're doing here is
thinking about uncertainty as more than just noise.
And so sometimes you don't know the process perfectly.
In fact, most often you don't know the process perfectly
and you certainly don't necessarily measure all of the
response variables perfectly and yet all of the error
in your model isn't necessarily
due to measurement error
or isn't necessarily error.
There are many reasons that data very data being the
observations or the response variable,
they can vary because of that observation error
because you don't measure something perfectly.
But they can also vary if they were taken
at different times or different sites.
If there's a weather difference, different slope.
There are lots of reasons that data vary.
responses vary,
we try to put as much of those into a model
and explanatory way as possible.
But there's often leftover variants
that we know has ecological relevance
and isn't just random noise.
And when that happens,
which probably happens most all of the time,
then just using the signal that we've carved
out of the data from using the model,
just using that signal to make predictions
or forecasts outside of where we have data,
whether that be in time or space is not going
to capture the true future.
Right?
So you're not going to make an precise
or an accurate forecast that has high probability
of capturing a future state if you don't describe all
of the real reasons that there's variation in data.
So another way to say it has real data,
very for a lot of reasons.
And some of those are consistent with the classic
assumptions of linear aggression,
but many of them are not.
And when they're not,
that inability to capture the many real ways
in the processes that cause data to vary,
makes forecast and even predictions
less powerful and less likely to be accurate.
Even though adding in all of that uncertainty
and acknowledging the uncertainty
as something real often makes confidence intervals
in forecast much broader.
So predictive intervals,
competence intervals will be broader,
wider uncertainty might grow
if you account for it and don't assume it's all noise.
It will certainly grow as you get further
and further away from the data points.
But you're more likely to be accurate
more likely to capture the true future state busy methods
or probabilistic models generally
or framework that allow for the flexibility
to treat all unknowns is random variables.
And so the unknown, knowns,
the things you know are unknown.
So oftentimes you might have regions that are different
and you get response variables.
And so for instance,
we're looking at for Kennedy and trees,
you know that Kennedy generally increases
as trees get bigger,
but in some regions maybe that looks a little different.
That function looks different than in other regions
and you don't necessarily need to know exactly why,
but that difference itself is not random noise.
It's not observation error,
it's due to regions.
And so there are ways to partition uncertainty
into things we don't know necessarily well enough
to fit in as fixed effects in a model.
But you can still capture the uncertainty
through higher commodifying
through random effects
and through separating process error
and observation error more coherently.