851-0585-04l – modeling and simulating social …... modeling and simulating social systems with...

2012-11-12 © ETH Zürich |

851-0585-04L – Modeling and Simulating Social Systems with MATLAB

Lecture 8 – Continuous Simulations

© ETH Zürich |

Chair of Sociology, in particular of

Modeling and Simulation

Karsten Donnay and Stefano Balietti

2012-11-12 K. Donnay & S. Balietti / [email protected] [email protected] 2

Schedule of the course 24.09. 01.10. 08.10. 15.10. 22.10. 29.10. 05.11. 12.11. 19.11. 26.11. 03.12. 10.12. 17.12.

Introduction to MATLAB

Introduction to social-science modeling and simulations

Working on projects (seminar thesis)

Handing in seminar thesis and giving a presentation


Schedule of the course 24.09. 01.10. 08.10. 15.10. 22.10. 29.10. 05.11. 12.11. 19.11. 26.11. 03.12. 10.12. 17.12.

Introduction to MATLAB

Working on projects (seminar thesis)

Handing in seminar thesis and giving a presentation

Dynamical Systems (no-space) Cellular Automata (grid)

Networks (graphs)

Continuous Space (…)

Different ways of Representing space

2012-11-12 K. Donnay & S. Balietti / [email protected] [email protected]

Goals of Lecture 8: students will 1.  Consolidate knowledge acquired during lecture 7, through

brief repetition of the main points. 2.  Receive an essential, but commensurate introduction

about Stochastic Processes. 3.  Discover the Micro-Macro link in continuous modeling,

bridging Random Walk and Diffusion Equation. 4.  Understand the basics of Motion Dynamics and learn

how to simulate processes like Brownian Motion in continuous space.

5.  Have a closer glance into two examples of Pedestrian Simulation, confronting grids with continuous space.

4


Repetition - Generators  Generating random, realistic graphs?

1.  Probabilistic generators Erdos-Renyi: start with n nodes, connect with probability p

2.  Degree-based generators Assign degrees to nodes; add edges, so that they match the original degree distribution

3.  Process-based generators preferential attachment (Barabasi), attach new node:

5


Repetition  We also saw a demonstration with Gephi

and some efficient network generating MATLAB functions

 USE THEM and existing implementation from previous terms in your work (and cite them properly!)

6


All began with the Drunkard’s Walk   The drunkard takes a series of steps away from

the lamp post, each with a random angle.

7

(Figure Resource: Sethna JP, 2006)


Stochastic Process   A stochastic process:

(Xt)t ∈ I is a family of random variables indexed on an interval I , where I ⊂ R.

8

  Intuitively, a stochastic process is an infinite- dimensional random vector

0 0.3 0.4 0.7 0.6 … 1 2 3 4 5


Stochastic Process   In reality, we observe just a single realization of such

process, among all the possible ones.   E.g. Stock Market Boom vs Stock Market Slump

9


White Noise (WN)   White Noise is a stochastic process whose t-th

element has the following properties:   E(xt) = 0   E(xt

2) = σ2   γk= 0 for |k| > 0 (autocorrelation coeff)

  That means that the white noise is succession of uncorrelated random variables with zero mean and constant variance.

10


Random Walk (RW)   Random Walk is a stochastic process whose t-

th element is determined as follows:

Xt = µ + Xt −1 + ϵt

  µ is the drift, constantly leading the process towards its sign

  ϵt represents the random increments.

11


Random Walk (RW)

12

function rw = rw(steps,drift) rw = zeros(steps,1); for i=2:steps rw(i) = drift + rw(i-1) + randn(1,1); end end


RW: on the random increments ϵt   RW1. ϵt = IID (0, σ2 ). Independent and Identically

Distributed.   classical hypothesis -> computational simplification

  RW2. ϵt are still independent, but no more identically distributed.   Different density functions across times are possible

  RW3. ϵt are simply uncorrelated and basically White Noise. This is the weakest RW form.   Admits non linear correlations among past values.

13

Campbell et al 1997


Random Walk and Diffusion Equation

14

  Consider a general, uncorrelated random walk where at

each time step the particle’s position changes by

a step :

  The probability distribution for each step is , which

has zero mean and standard deviation .


From Random Walk to Diffusion Equation   For the particle to go from at time to at time ,

the step . Such event occurs with probability

times the probability density . Therefore,

we have

15


The Micro-Macro Link   Using Taylor expansion, we have

16

The following is the diffusion equation


The Micro-Macro Link: Take Home Msg   Diffusion:

  Constant drift of the ensemble with the characteristic

constant D given by the properties of RW

  A simple and controllable process such as RW applied to

single agents can give rise to seemingly organized

macroscopic behavior of the ensemble.

17

€

∂ρ∂t

=α 2

2Δt∂2ρ∂x 2

Diffusion Constant D

Differential Term for Diffusion


The Taylor Series   The Taylor series is an approximation of a

function as an infinite sum of terms calculated from the values of its derivatives at a single point.

  F(x) can thus be rewritten as:

 

 

18

  a = single point


The Taylor Series in Matlab

19

>> syms x >> f = exp(x); >> taylor(f)

ans =

x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1

  What’s the Taylor Series for f(x) = ex ?



20

>> syms x >> f = exp(x); >> taylor(f)

ans =

x^5/120 + x^4/24 + x^3/6 + x^2/2 + x + 1

  What’s the Taylor Series for f(x) = ex ?

And this is good because we reduced the complexity of the function from exponential to polinomial!



21

  Play with taylortool to get familiar Taylor Series


The Continuity Assumption

22

Resources: Google


From discrete to continuous space

  Cellular automaton (CA)

  Continuous simulation


Modelling motion in continuous space

  Agents are characterized by:

  position -  X (m) -  Y (m)

  velocity -  Vx (m/s) -  Vy (m/s)

x

y

Vx

Vy



  Next position after 1

second is given by:

  x(t+1) = x(t) + Vx(t)

  y(t+1) = y(t) + Vy(t)

x

y

Vx

Vy

Time t Time t+1



  Time step dt is often

different from 1:

  x(t+dt) = x(t) + dt Vx(t)

  y(t+dt) = y(t) + dt Vy(t)

x

y

Vx

Vy

Time t Time t+dt dt<1

Time t+dt dt>1



  The velocity is also

updated in time.

x

y



  The velocity changes as a result

of inter-individual interactions

  Repulsion

  Attraction

  Examples: atoms, planets,

pedestrians, animals swarms…



  The changes of the velocity

vector is defined by an

acceleration.



  The velocity changes as a

result of interactions with the

environment

  Repulsion

  Attraction

  Examples: bacteria, ants, pedestrian trails …



  The velocity changes as a

result of a random process:

  With or without bias

  Example: exploration behaviour in many species of animals



  How to define a random move?

  Choose a random angle in a normal distribution

  Rotate the velocity by this angle

  In Matlab:

  newAngle = randn()

Mean=0

Variance=1



  In Matlab:

  meanVal=2;

  v=0.1;

  newAngle = meanVal + v*randn()

  How to define a random move?

  Choose a random angle in a normal distribution

  Rotate the velocity by this angle



  How to define a BIASED random

move?

  Unbalance the distribution toward positive or negative values

  In Matlab:

  meanVal=2;

  v=0.1;

  newAngle = meanVal + v*(offset + randn())


Programming a Continuous Simulator

X = [1 1 3 ]; Y = [2 1 1]; Vx = [0 0 0.1]; Vy = [1 1 0.5];

for t=1:dt:T for p=1:N [Vx(p) Vy(p)] = update( …) ; X(p) = X(p) + dt*Vx(p) ; Y(p) = Y(p) + dt*Vy(p) ; end

end

  Initialization: Four elements are required to define the state of individuals

  The velocity is updated first, and then the position as a function of the velocity and the time step

2012-11-12 K. Donnay & S. Balietti / [email protected] [email protected] 36 36

Brownian Motion %Brownian Motion!t = 100;n = 1000;dt=t/n;!x = zeros(1,n);!y = zeros(1,n);!dx = zeros(1,n);!dy = zeros(1,n);!x(1) = 0;y(1) = 0;!hold on;!k = 1;!for j=2:n! x(j)= x(j-1) +…! sqrt(dt)*randn;! y(j)= y(j-1) +…! sqrt(dt)*randn;! plot([x(j-1),x(j)],…! [y(j-1),y(j)]);!! k = k + 1;!end!


Pedestrian Simulation

  CA model

  Many-particle model


CA Model (A. Kirchner, A. Schadschneider, Phys. A, 2002)


Many-particle Model (Helbing, Nature, 2000)

Social Force Model   Desired velocity

  Repulsive force


  Easier

  Implementation dependant

  More Limited

  One Element per Cell

  Computationally more

intensive

  (more) Complex

  Elegant

  Extremely Powerful

  Agents Float in Space

  Computationally faster

(with analytical derivates)

40

Grids Vs Continuum Space


  Easier

  Implementation dependant

  More Limited

  One Element per Cell

  Computationally more

intensive

  (more) Complex

  Elegant

  Extremely Powerful

  Agents Float in Space

  Computationally faster

(with analytical derivates)

41

Grids Vs Continuum Space

You can write a good paper with both approaches if you have a good idea


  Train Boarding Simulation (Graf & Krebs, Fall 2010)

42

Examples from Previous Semesters


  Love Parade Disaster (Guthrie & Fougner, Fall 2010)

43



  Airplane Evacuation (Bühler & Heer, Spring 2011)

44



  Flooding Evacuation (Crameri & Thielmann, Fall 2011)

45



  Desert Ant Navigation (Megaro & Rudel, Fall 2011)

46



Projects   Today, there are no exercises.  You can work on your projects and we will

supervise you.

  If you have preliminary results we are happy to discuss them with you…

47


References   http://en.wikipedia.org/wiki/Taylor_expansion

  http://en.wikipedia.org/wiki/Random_walk

  Toolbox for different random walks:

http://www.mathworks.com/matlabcentral/fileexchange/22003

  Nature 407, 487-490 (2000) Simulating dynamical features of

escape panic. Dirk Helbing, Illés Farkas and Tamás Vicsek

  The Econometrics of Financial Markets. 1997. John Y. Campbell,

Andrew W. Lo, A. Craig MacKinsley. Princeton University Press.

48


GitHub Links   Train Boarding Simulation (Graf & Krebs, Fall 2010)

https://github.com/msssm/Train-Boarding

  Love Parade Disaster (Guthrie & Fougner, Fall 2010) https://github.com/msssm/Love_Parade_2010

  Airplane Evacuation (Bühler & Heer, Spring 2011) https://github.com/msssm/Airplane_Evacuation_2011_FS

  Flooding Evacuation (Crameri & Thielmann, Fall 2011) https://github.com/mthielma/SOCIAL_MODELING

  Desert Ant Navigation (Megaro & Rudel, Fall 2011) https://github.com/erudel/desert_ant_behavior_gordonteam

49

851-0585-04l – modeling and simulating social …... modeling and simulating social systems with...

Documents