Welcome, welcome back to the second week
of Modeling Complex Systems.
I made sure to have my black cat Virgil in this shot.
Some of you saw Dawn, the white one,
over the discussions this week.
I thought the first week was great.
I was impressed by all the amazing discussions we had.
The first week was really meant to focus on models as nouns,
what are models,
what are different models we use in science
and in everyday life.
And I thought that the discussions really got to
what I believe is important when we define models,
which is the idea that there are no perfect models.
One system can be represented in different ways.
And you all did a great job of coming up
with this idea of balance
between the complexity of a model
and the purpose of a model.
So what questions are we trying to answer?
In what way are we trying to look at the system?
So for this second week,
I thought it would be fun to focus on models as verbs.
So how do we model?
Because even though there are no perfect models,
it's not the Wild West out there either.
So the idea is, I want you to feel like you can come up
with your own approaches to the problems we're gonna look.
Feels like disease modeling right now.
Definitely need new perspective.
There are no perfect way to models,
but there are people that think about the theory of modeling
and what are the different approximations, idealizations,
what is the thought process of most people
when we go from a complex reality
to a manageable computational or mathematical model?
So let me just share my screen here for a second.
So hopefully you can see my screen right now.
I thought it would be fun to go over
some of the types of models that are highlighted
in the reading for this week,
which is Models in Science, section one and two,
and at the same time to go over some examples of models,
the way we're going to use them in this class, really.
So the first type of model that is discussed in the reading
are scale models.
And these already came up in the discussions.
If we think about a maquette for an architect
or a sculpture, so creating a model of the building
or the statute that you want to build,
that would be a scale model.
You make it in a smaller size that's more manageable.
Maybe you don't use the right material right away,
but it gives you a 3D representation
that is easier to work with
than simply imagining your creation.
And in the case of this course,
we're still gonna use these scales models
often to get around computational complexity.
So we might wanna scale down a model, for example,
of the human population.
If we have this agent-based model with 7 billion nodes,
7 billion individuals, individual agents interacting,
that might be computationally extensive.
So we might want to scale down the model
for computational complexity.
And really here the process is the same,
just to make it more manageable,
allow us to highlight certain features of the model.
We might not need a full sized population.
So that would be one use of the scale model.
Most of what we're gonna do
fall in the second family of models which are idealized.
And really that's almost all mathematical models
work this way.
The idea is that we want to make simplifications
to the real system to focus on what we believe is important.
So if we're doing a forest fire model study,
the robustness of the Amazon,
then the smell of the trees might not matter.
The colors of the tree might not matter.
So we can scale that back,
make the system more idealized in a way.
So notice that idealized doesn't mean better here.
It really just means more useful to the modeler.
And that's how we're gonna allow ourselves
to do mathematical model or computer programs
that can follow reality in some way.
And all of these approximations or idealizations
fall in two families.
So let me call them 2A and 2B.
So we have Aristotelian idealizations or Galilean,
and I don't want to go too deep in the details
of the distinction between these two
or the definition of Aristotelian versus Galilean
idealizations, but I do think they're useful
for modelers to keep in mind.
So Aristotelian idealizations
are when you simply remove something.
So the previous examples I gave
removing the smell or color
of real systems,
that data is definitely useful to ignore those features
when modeling forest fire.
But we have to keep in mind
that now the model is not representative of any reality.
There is no forest where there is no smell or color
to the trees.
That might not matter, but sometime it will,
the idea that the idealizations we've made is such
that our model does not correspond to any reality.
It's just an approximation of it.
Whereas Galilean are distortions,
but we're still in the realm of the possible.
So anytime a physicist, for example,
make the approximation of frictionless movement
or ignoring air resistance,
it's convenient mathematically, but it also means
that you could test the model in a vacuum
and you should be able to predict the outcome
of some experiments.
So there is a world where those Galilean idealizations
are still possible
and the model still represents something.
In the case of what we're gonna do,
often Galilean approximations are gonna be the opposite
of a scale model or are scaled up,
which would be something
like approximating very large populations.
So all disease models, for example,
will assume an almost infinite
or literally an infinite population.
So then we know that the models we're building for disease
can't really be applied to households or small town,
but work better for cities or large homogeneous country.
So that's a Galilean approximation
where we know that the larger the population,
the better our model should fit reality.
And then that's one way of validating what we're doing.
One thing we're gonna use quite a bit
are analogical models,
which is the third family here,
and a classic example, at least for me is,
the mass spring system in physics
is used to solve almost anything.
So you can solve the sound that you would get by blowing
in a half empty beer bottle using a mass spring system,
the same way you could model to some extent
electrons moving around with a mass spring system.
So physicists use that idealized model often in analogies.
So now it's not a clear model of mechanisms,
but it still works in predicting some behavior.
We're gonna do something similar.
We're gonna have a lot of models that are gonna pop up
time and time again.
For example, we're gonna use
a lot of rich get richer approximation.
So the rich get richer mechanism will show up
if we model scientific publications.
Papers with a lot of citations are easier to find
and therefore get more citations in the future.
Similarly, wealth distribution in the US
is a clear rich gets richer process.
You gotta have money to make money.
So these mechanisms,
these general rich gets richer process,
are gonna show up often as analogies
and will allow us to translate intuition,
learn from one system to another.
And then the last family in the readings
are these phenomenological model.
I was just checking that I spelled that right.
So phenomenological models, and as you can see,
I have a harder time the more syllables there are in a word.
But here, if you've been reading the textbook,
you should be on chapter two right now.
And this is what, here like you say,
I would describe as these descriptive models.
So they don't try and get to mechanisms
behind a behavior or results,
but they just try and reproduce its phenomenology
or its behavior, the way it looks.
So you could almost say
that these models are independent of theories.
They try to get to the data
without necessarily assuming some prior knowledge
or giving you any insights into the real systems.
And we're mostly gonna steer away from that.
That's what a lot of statistical models do.
That being said, they're still an art to that.
They require a choice of description.
So there is insights that go into them.
They're not completely agnostic of reality,
but they don't necessarily tie
in what we're aiming to do with this class.
At the very end, we're still gonna do
some model fitting and selection.
So anytime you have a model and then you start fitting it
to real data, there comes a point
when you're comparing two models that are mechanistically
the same and you're just trying to tune parameters
or you're just basically building a phenomenological model
at that point.
So those categories are mo mainly useful to have in mind
just to give us a common language.
In the next video, I'll start giving you some recipes,
how do we build model from scratch
and having in mind when we're doing scale approximations
or Aristotelian idealizations
where we don't describe reality,
Galilean approximations, where we do,
just building analogies,
so that the mechanisms might not be right, but are analogous
to different systems or abandoning mechanisms altogether,
and just trying to describe phenomenology.
Those are the words we're gonna keep using
over and over again.
So this distinction from the Models in Science reading
are really useful just in terms of setting a common language
for the rest of the course.
So I'll see you in the next one
where we'll start building some recipes
for how to build models from scratch.