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Welcome back to Modeling Complex Systems.

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This will be our final
video for Module one.

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We have reached the end of our
module on Dynamical System.

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I just wanna use this video to

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talk a little bit more

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about both, the very start of the class.

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How do we model a system
as a dynamical system?

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How do we model a problem?

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And some things to keep in mind

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once we have a model and
we start analyzing it.

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What are the potential sources of error?

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So I wanna divide this
video in three parts,

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to think about modeling as a recipe.

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So we're gonna have the ingredients,

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this is how we start our model.

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The preparation, which is
really how we run our model,

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put everything together.

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And finally, how to
present our beautiful dish

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and what are the possible
sources of error.

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Because even the best models,

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if not interpreted correctly,
can be problematic.

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So the ingredients, we've
talked a lot about time,

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in these dynamical systems.

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So time could be discrete,
we have discrete models.

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It could be continuous.

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We could even think about
models that are static.

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We haven't done so yet but we will,

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as we look at statistical
models, for example.

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So first is to pick what notion of time

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best represent the problem
you're interested in.

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Then we also wanna think about space.

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Similarly, space can be
continuous or discrete.

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It could also have dimensions.

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This will all become important.

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When I say dimensions, especially,

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are we in 2D or 3D?

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Are we on a network?

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This will all become important,

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mainly as starting with the
second module on structure.

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Then the key part of this
modeling as representing a system,

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as building a representation of a system,

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is to think about what
are the important parts

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of this system.

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So in terms of epidemic
models, we knew it was

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having both susceptible
and infectious individuals.

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That was the key part, was
the separation of a system

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into boxes and following
those boxes which represent

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parts of different nature

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and asking how those different boxes

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and parts are interconnected.

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Then we wanna ask questions

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about whether the system
is closed or open.

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So when we were doing
compartmental models,

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it was about whether we
had arrows coming or going

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to the outside.

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Or whether all arrows started
in a box, went to another box,

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meaning that we had a closed system

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and some conserved quantity, often.

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Then we need to think about the rules

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and whether or not we
feel like it's important

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to have stochastic, meaning,

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rules with an element of randomness.

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So if you're flipping coins,

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the way we did a little
bit in some pseudo code,

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that was

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stochastic models.

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A set of stochastic rules,

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where we had to draw random
numbers between zero and one.

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Otherwise deterministic
rules can give you something

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like ordinary differential equations.

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Those ODEs are deterministic
by their nature.

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Tell me where I am and I'll
know exactly where I'm going.

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That's what the differential
equation is telling me.

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So that's deterministic.

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To compare, again, one good
analogy between the two,

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is that you can run the same model

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with the same initial condition twice.

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If it's a stochastic model,

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you're not gonna get the same time series,

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the same solution.

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If it's deterministic,

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your initial condition
determine the future

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and so you'll get the exact same solution,

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even if you run it more than once.

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Then there are other questions
that the system have memory.

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Are we dealing with a
Markovian, meaning, no memory.

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The present dictates the future.

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Are we dealing with non-Markovian models,

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where there might be some memory

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and we have to take that
into account somehow?

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Those were the more general questions.

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But really, when we think
about how to run a model,

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the textbook has this great

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set of principles that's
easy to remember, IOU.

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IOU a model.

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And the way I will give you that model,

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is by initializing it,
observing it and updating it.

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So initialization means
to define the variables

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of the system.

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What do you care about and
what is their initial state?

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So that could simply be a number

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of initially susceptible individuals

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and a number of officially
infectious individual.

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And that sets the initial
state of my system.

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Could be a number of predator
and a number of prey.

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Then we have to define what we observe.

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How do we monitor the system?

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What do we wanna measure,

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as the system runs, as the model runs?

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And then once we've done that,

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we can actually update the system.

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We start with our initial state.

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We apply the set of
rules that we've defined

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in the previous step

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and then measure what we wanna observe.

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Apply the set of rules again,

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measure what we wanna observe,

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apply the set of rules,
measure what we wanna observe,

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until we have enough data

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and decide that we're done.

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So this is just a good
mnemonic trick to remember

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the importance of these
three different steps.

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You do wanna play with different
way to look at the systems.

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So different things to observe.

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You wanna play with
different initial conditions.

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So the model as a set of
rules, is not always enough

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and you need to think of the other step

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of preparation as well.

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And then finally, with this presentation,

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meaning, we have a model,

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we are measuring something
and often we either wanna

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compare it to data or we just
wanna communicate the results

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of our model because we
think it's insightful

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and tells us something
about the real system

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we're interested in.

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So when we think about
presentation, really,

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what we mean is,

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what are the possible
problems at this stage?

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What are the possible sources of error

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that we need to be curious about?

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So when you present your model,

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you wanna be transparent about
everything that came before.

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What assumption and
idealization did you do?

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What did you measure and observe?

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How did you update your model?

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But there are a whole set

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and I'm putting them up
all on screen right now.

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There are a whole set of
potential sources of error,

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even once you've been transparent.

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So let's say your model, you know,

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doesn't fit well with data

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or doesn't capture the intuition
you want it to capture.

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It could be model error.

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It could be that in these
early stages of the system,

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you made a mistake, you ignored something,

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you made an assumption or an idealization,

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that you shouldn't have made.

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So any problem in the
specification of the model,

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we would call a model error.

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One very common source of
error is, is due to parameters.

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You might have the right
mechanisms in the right model

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but the bad values on these mechanisms,

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maybe the rules of the model should change

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in time and they do not.

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Or maybe you simply
didn't measure something

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like birth rate of the prey, well enough.

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So that's definitely possible.

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So often what you'll wanna do, is test

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a whole range of parameter,

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which we often call a parameter sweep

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or sensitivity analysis,

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to see how much of the
same behavior you get

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when you vary those parameters.

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And then at least you'll know
how important measuring them

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in practice will be.

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Or selecting them correctly.

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The third source of error
that I have on there,

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is measurement error.

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It is possible that the model is right

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and it is the world that is wrong, right.

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Maybe you're comparing to
data that just doesn't exactly

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capture what you wanted to capture.

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We often look at

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epidemic models by comparing
them to what we call,

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influenza-like illness.

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So we think of flu as a
disease that we wanna forecast.

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It has an important disease burden.

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And we have those well-defined flu season

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but it is very hard to diagnose.

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So the proxy that we're using,

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is these influenza-like illnesses,

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which is not just flu, it's
a bunch of other things.

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So we're comparing about
something that is super noisy.

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So the things that we're comparing against

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are not measured right.

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That could also be a
source of parameter error,

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if we're not measuring
those parameters right.

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So sometimes a model is right

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and it is the world that is wrong.

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Our fourth category here,
is discretization error.

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That is super common.

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Most complex systems are
interesting because they're

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created by a collective of parts.

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But those parts are often discrete, right.

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Discrete predators and prey,
discrete number of people.

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But most of our model are
gonna assume continuous

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mathematics, just because it
is a powerful set of tools.

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If we can use, you know,
differential equations

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over continuous variable,
that's a powerful tool to use.

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And unfortunately, reality of
our system is often discrete.

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The same goes with data.

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Our model is gonna give us

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prediction of predator or prey over time,

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in continuous time.

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But the data might just come

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from observations that are monthly.

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Those observations are gonna
have some measurement error

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but they're also gonna have
this discretization error,

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that we're only observing
them once in a while

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and putting a lot of data
together, bending them over time.

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So that's something to keep in mind.

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And finally, as we saw with chaos theory,

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there can be numerical errors.

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The model might be a little unstable,

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so it might be hard to
solve for some fixed points.

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Or it could be sensitive
to initial conditions,

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such that, even the best
algorithm won't give you

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a good prediction

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of which you should
expect from that model.

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And then it becomes
hard to compare reality

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with those noisy numerical predictions.

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So, you know,

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these are meant to just
be things to keep in mind

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as you start creating your own models.

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Ingredients, preparation,
the IOU mnemonic trick

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and the presentation.

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But we're gonna keep
applying the same recipe.

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We're gonna see other sources
of error, other ways to model.

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This is only the beginning.

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It is the end of our first module.

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But starting next week, we'll
be talking about structure.

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We'll have two modules on structure.

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The first one on Cellular Automata.

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And the second module on
structure, module three,

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will be on networks.

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So this whole step, this
next step, I should say,

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starts next week.

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I'm super excited to start

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talking a little more about structure.

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It is just, if not more
interesting than dynamics.

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And eventually we'll
put all of that together

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and we'll be creating
really, really cool models.

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So I'll see you soon.