0:11

Hello, so welcome back to our class simulation modelling of natural processes.

Â So now I'd like to start the third module of the first week,

Â which is about modeling space and time.

Â So of course,

Â you know that natural processes they take place in space and time.

Â And typically, we need to integrate this dimension in our model and

Â there are actually some difficulties about that so.

Â All the system, they evolve in time and

Â the state of the system may change in place.

Â So for instance, if you consider the atmospheric pressure or temperature,

Â it's different from one place to another.

Â I mean, if you move from one town to the next or the temperature also changes.

Â And of course, it changes during the day and for the next day.

Â So you see that everything is space and time dependent.

Â Another example is that if you are on the road, [COUGH] you can see that

Â the car is moving, hopefully and then it changes position over time.

Â So again, there's a time and space which are involved in this process.

Â 1:31

Sometime, we don't need to consider both space and time in a model.

Â So for instance, you may be just interested in global quantity

Â like how many individual of a given species I have in an area.

Â So, it's basically counting the number of individual in a population.

Â I'm not really interested in where they are, especially.

Â We can also be interested in the total amount of CO2 in the atmosphere.

Â So in some model, you can remove the spacial dimension and

Â sometime you can remove the temporal dimension if time doesn't evolve.

Â The process is steady.

Â Maybe you can get rid of the time and

Â just consider the evolution in space.

Â So I give an example, for instance, the temperature in the room,

Â you maybe interested whether it's cold near the window or warm in the middle.

Â But if you have a good heating system,

Â probably doesn't change over the day and it's something which is steady.

Â So if we focus now on the time dimension and

Â would like to see how we can describe it in a model and

Â we'll see that there are several solutions for us.

Â First, we know from physics that time is a continuous variable.

Â So time can be as detailed as you want.

Â You can have microsecond, nanosecond.

Â So every real value is potentially possible, although

Â some physicists claim that maybe at some scale we should stop and fill the gaps.

Â We'll assume that time is a continuous variable and

Â this is very difficult to describe in the computer.

Â So, only mathematical model like a differential equation that can deal with

Â the continuous time really with a real variable.

Â So most of the time when you do a model and it goes to the computer,

Â you have to change this continuous time.

Â For instance, by discretidizing it, so you basically take your time

Â into value you wanna describe and you split it in several time step and

Â then you look at your system at every of this time step.

Â So for instance, if I call delta t the time stamp can be one second,

Â one millisecond, one hour depending on the model or

Â even one year depending on the time scale we wanna capture.

Â Basically, you look at your system at time zero,

Â at time delta, and so on until you reach the end of your simulation.

Â 4:18

So you discretize the time, but basically,

Â you follow your system in a continuous way.

Â You are always able to know what your system is like.

Â But alternatively, you can have another approach to say that maybe most of

Â the time my system is not interesting and

Â like to concentrate on one thing's happening.

Â So that's the interesting moment of a system.

Â For instance, if you have a queue in a post office, you don't really

Â care to describe the evolution of the system when nothing happen.

Â If you wanna just capture the moment when your customer comes?

Â Or when one done in the booth and you have availability in between.

Â It's very static in terms of, for instance,

Â studying the time it takes for the customer to get served.

Â In that case, if you focus on the event,

Â the time t can be any value.

Â It's just the moment at which the event takes place.

Â It doesn't have to be a discretized value.

Â It is the value you get from your model with, of course,

Â the accuracy of your computer.

Â Which is I mean, this day rather good.

Â 5:36

So in that case time is not discretized, because you can potentially have any

Â value for the time for the event you're interested in.

Â But basically, you are not looking in a continuous way at your system,

Â you're just.

Â Basically, focusing on some event.

Â So basically, the history of your

Â system is broken into this event.

Â So we'll have a special chapter on this approach,

Â which is called Discrete-Event Simulation.

Â So to summarize in this graph, you see the three ways to reproduce or

Â to represent time in a model.

Â So the first is this continuous line while basically, you know everything,

Â every time within the interval between zero and capital T.

Â And again, only math can do this with this.

Â In a computer, you have to go to one of the two of the solution,

Â which is either you split in regular interval, usually smaller or

Â you just focus on the events that are interesting somewhere along the timeline.

Â 6:47

So for the space, there's also some different approaches.

Â And we'll consider, for instance,

Â the case which is known as the Eulerian approach.

Â So the idea of the Eulerian approach is that the observer is just sitting

Â somewhere in the system, of course, virtually.

Â But he just observes what happens at this specific position over time.

Â So he is basically at a fixed position acts in space and

Â determines the state of the system at this point.

Â It could be, for instance, the atmospheric pressure.

Â At this specific position at given time, but

Â it could be also in the traffic example.

Â Simply the number of cars that have crossed some line on the road that

Â an observer can just count.

Â He can see every minute how many cars is so passing in front of him.

Â 7:58

So space again, supposed to be continuous from what we know from physics.

Â And again, only mathematical model like a partial and

Â differential equation can describe that in a proper way.

Â Otherwise, in computer model,

Â we have to discretize the space into cells or little chunks and

Â that usually makes a mesh, which covers the space you wanna describe.

Â 8:31

and the idea in that case is to take the point of view of the object in this space.

Â And basically, the observer sits on one of these object and moves with it.

Â Then he can give the position of this object over time and the position,

Â again is given with as much accuracy as your computer can give it.

Â So then the space is not anymore discretized.

Â So for instance, when you wanna describe the movement of the Moon,

Â you will give its trajectory.

Â So its position to the accuracy that you can over time and

Â you are not just in spacing order.

Â I've seen the moon at this position and now it's gone somewhere else, so you just

Â give trajectory rather than Eulerian description of whether the Moon is down.

Â 9:20

So in a traffic model, of course, you can give the position of the cars over time,

Â which is different approach than just looking at a specific position on

Â the road and saying whether there's a car or

Â not and what's the average density of car disposition.

Â So as I told you,

Â the Lagrangian approach is taking the viewpoint of the moving object.

Â And if you want to represent these two ways of modeling

Â the space, this feature can illustrate that.

Â So on the left, you have part of the space,

Â which is a square here, which is divided in little cells.

Â So, I've discretized the space in the mesh.

Â And for each of these cells,

Â I give a color describing the state of this position.

Â So it could be, for instance, the temperature.

Â So when it's white, that means that it's cold.

Â When it's black, it's maybe warm.

Â And when it's gray, it's in between.

Â So then that's the way you would represent a phenomena in space

Â according to the value of quantity at each of the discretized position,

Â but the glycogen point of view is traded under right.

Â So here, you see particles, for instance, particle can be anywhere

Â in the domain and you can described them as their coordinates,

Â Cartesian coordinates with all the accuracy you need.

Â And of course, this can change all the time.

Â 11:10

So for instance, if you have a system of person interacting like,

Â for instance, you take a social network or

Â whatever, you may have very interesting phenomena.

Â But it's not so much whether the people are physically close to each other,

Â it's rather whether they interact or not.

Â They can interact with the phone or with any other way.

Â So I would say, the spatial relation is not really distance in the real space,

Â but more whether there's an interaction or not an interaction.

Â So what really matters is really the link that relates the component of your system.

Â 11:52

As I said, this is the case in social economical

Â model whether you interact or not.

Â So if you would take the example of an economical system,

Â you would say that two agents are Interacting.

Â If they exchange information, they exchange money or goods or whatever.

Â And that's what makes the fact that they are close to each

Â other not in terms of distance, but in terms of action and

Â the right way to represent this in a model is to use a graph and

Â a graph means that you put a link between component or agent that are the relation.

Â And of course, this graph can be also evolving in time.

Â So it can be dynamical in saying that we can create new links,

Â meaning that you create new relation or you remove old relations or

Â links are destroyed, for instance.

Â So as an example, I show here a graph.

Â And in this context, people like to talk about complex network rather than graphs.

Â But mathematically speaking, that's a graph, but they can be very big and

Â they have very interesting properties.

Â So this example is just an example of a community where you

Â have this circle that represents persons and

Â the color represents their opinion and the size of the circle representing

Â the person is just a number of people they're connected to.

Â So if you're a big circle, you mean that this person is connected to a lot

Â of these person and so we could study how opinion maybe evolved in such a structure.

Â And on the right, you have time evolution according to our model,

Â which again, I have no time to discuss here.

Â But you see that in that case, initially you have a lot of green person and

Â some blue person.

Â But as time goes on,

Â you end up with having everybody convinced that blue is the right.

Â 13:57

So there are a lot of important problem that we can discuss on complex network.

Â We won't have time in this specific class to do it.

Â So I mean, it's just an important Information I give you that sometimes

Â the spatial structure or the spatial relation is represented as a graph.

Â It's a field, which is extremely important now.

Â You find a lot of article and books on this.

Â And what's interesting is the graph topology, it creates a rich structure.

Â Which of course, has an impact on the dynamic of what happened on this graph.

Â There are many interesting property of this extended space I would say,

Â which is degree distribution.

Â For instance, the clustering coefficient, centrality measure and so on so.

Â Again, also that would deserve a full chapter that I'm just

Â making very short, so that you know that this is also a way to

Â represent interaction between component in a model.

Â So, I thank you for attention.

Â This is the end of this module on Modeling Space and Time.

Â Thank you.

Â [MUSIC]

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