modeling and simulating social systems with matlab · 2015-09-21 · 2015-09-21 modeling and...

2015-09-21 ETH Zürich

Modeling and Simulating Social Systems with MATLAB

Lecture 1B– Introduction to MATLAB

Computational Social Science

Olivia Woolley, Stefano Balietti, Dirk Helbing

2015-09-21 Modeling and Simulating Social Systems with MATLAB 2

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MATLAB environment


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What is MATLAB? §  MATLAB derives its name from matrix laboratory

§  Interpreted language §  No compilation like in C++ or Java §  The results of the commands are immediately

displayed

§  Procedural/imperative programming

§  Matrix/array/vector programming

§  (Object-oriented programming)


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Overview - What is MATLAB? §  MATLAB derives its name from matrix laboratory

§  Scalars §  Vectors §  Matrices

x11


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x11 x12 x13

x11

x12

x13


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x11 x12 x13

x21 x22 x23


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§  Scalars §  Vectors §  Matrices §  Multi-dimensional

x111 x121 x131

x211 x221 x231


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Pocket calculator §  MATLAB can be used as a pocket calculator:

>> 1+2+3

ans=

6

>> (1+2)/3

ans=

1


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Variables and operators §  Variable assignment is made with ‘=’

§  Variable names are case sensitive: §  Num, num, NUM are all different variables

>> num=10

num =

10

§  The semicolon ‘;’ cancels the validation display >> B=5;

>> C=10*B

C =

50


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Variables and operators §  Basic operators:

§  + - * / : addition subtraction multiplication division §  ^ : Exponentiation §  sqrt() : Square root §  % comment >> a=2; % First term

>> b=5; % Second term

>> c=9; % Third term

>> R=a*(sqrt(c) + b^2);

>> R

R =

56


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Data structures: Vectors §  Vectors are used to store a set of scalars

§  Vectors are defined by using square bracket [ ]

>> x=[0 2 4 10]

x =

0 2 4 10


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Data structures: Defining vectors

>> x=0:2:6

x =

0 2 4 6

§  Vectors can be used to generate a regular list of scalars by means of colon ‘:’ §  n1:k:n2 generate a vector of values going from n1 to n2

with step k

§  The default value of k is 1 >> x=2:5

x =

2 3 4 5


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Data structures: Accessing vectors

>> x=1:0.5:3;

>> x(2)

ans =

1.5

§  Access to the values contained in a vector §  x(i) return the ith element of vector x

§  x(i) is a scalar and can be assigned a new value >> x=1:5;

>> x(3)=10;

>> x

x =

1 2 10 4 5


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Data structures: Size of vectors

>> x=1:0.5:3;

>> s=length(x)

s =

5

§  Vectors operations §  The command length(x) return the size of the vector x

§  x(i) return an error if i>length(x) >> x=1:0.5:3;

>> x(6) ??? Index exceeds matrix dimensions.


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Data structures: Increase size of vectors

>> x=1:5;

>> x(6)=10;

>> x

x =

1 2 3 4 5 10

§  Vectors operations §  Vector sizes can be dynamically increased by

assigning a new value, outside the vector:


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Data structures: Increase size of vectors

>> x=1:5;

>> x(6)=10;

>> x

x =

1 2 3 4 5 10

§  Vectors operations §  Vector sizes can be dynamically increased by

assigning a new value, outside the vector:

Important: the first element of a vector has index 1


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Data structures: Sub-vectors

§  Vectors operations §  Subvectors can be addressed by using a colon §  x(i:j) return the sub vector of x starting from the ith

element to the jth one

>> x=1:0.2:2;

>> y=x(2:4);

>> y

y =

1.2 1.4 1.6


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Data structures: Matrices

§  Matrices are two dimensional vectors §  Can be defined by using semicolon into square

brackets [ ] >> x=[0 2 4 ; 1 3 5 ; 8 8 8]

x = 0 2 4 1 3 5 8 8 8

>> x=[1:4 ; 5:8 ; 1:2:7]

x = 1 2 3 4 5 6 7 8 1 3 5 7


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>> x=[0 2 4 ; 1 3 5 ; 8 8 8]

x = 0 2 4 1 3 5 8 8 8

>> y=x(2,3)

y =

5

§  Accessing the elements of a matrix §  x(i,j) return the value located at ith line and jth column §  i and j can be replaced by a colon ‘:’ to access the

entire line or column


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>> x=[0 2 4 ; 1 3 5 ; 8 8 8]

x = 0 2 4 1 3 5 8 8 8

>> y=x(2,:)

y =

1 3 5

§  Access to the values contained in a matrix §  x(i,j) return the value located at ith line and jth column §  i and j can be replaced by a colon ‘:’ to access the



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>> x=[0 2 4 ; 1 3 5 ; 8 8 8]

x = 0 2 4 1 3 5 8 8 8

>> y=x(:,3)

y =

4 5 8

§  Access to the values contained in a matrix §  x(i,j) return the value located at ith line and jth column §  i and j can be replaced by a colon ‘:’ to access the



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Matrices operations: Transpose §  Transpose matrix

§  Switches lines and columns §  transpose(x) or simply x’ >> x=[1:3 ; 4:6]

x = 1 2 3 4 5 6 >> transpose(x) x = 1 4 2 5 3 6 >> x’ x = 1 4 2 5 3 6


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Matrix operations §  Inter-matrices operations

§  C=A+B : returns C with C(i,j) = A(i,j)+B(i,j) §  C=A-B : returns C with C(i,j) = A(i,j)-B(i,j)

A and B must have the same size, unless one of them is a scalar

>> A=[1 2;3 4] ; B=[2 2;1 1];

>> C=A+B

C = 3 4 4 5

>> C=A-B

C = -1 0 2 3


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Matrix operations: Multiplication §  Inter-matrices operations

§  C=A*B is a matrix product. Returns C with C(i,j) = ∑ (k=1 to N) A(i,k)*B(k,j) N is the number of columns of A which must equal the

number of rows of B


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Element-wise multiplication §  Inter-matrices operations

§  C=A.*B returns C with C(i,j) = A(i,j)*B(i,j) A and B must have the same size, unless one of them is a scalar

>> A=[2 2 2;4 4 4];

>> B=[2 2 2;1 1 1];

>> C=A.*B C = 4 4 4 4 4 4


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Element-wise division §  Inter-matrices operations

§  C=A./B returns C with C(i,j) = A(i,j)/B(i,j) A and B must have the same size, unless one of them is a scalar

>> A=[2 2 2;4 4 4];

>> B=[2 2 2;1 1 1];

>> C=A./B C = 1 1 1 4 4 4


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Matrices operations: Division §  Inter-matrices operations

§  x=A\b returns the solution of the linear equation A*x=b A is a n-by-n matrix and b is a column vector of size n

>> A=[3 2 -1; 2 -2 4; -1 0.5 -1];

>> b=[1;-2;0];

>> x=A\b

x = 1 -2 -2

3x + 2y � z = 12x� 2y + 4z = �2

�x +12y � z = 0 Attention:

/ (slash) and \ (back slash) are different operators


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Matrices: Creating

§  Matrices can also created by these commands: rand(n, m) a matrix of size n x m, containing random numbers [0,1] zeros(n, m), ones(n, m) a matrix containing 0 or 1 for all elements


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Matrices Dimensions

>> A = zeros(3,4);

>> size(A)

ans =

3 4

>> size(A,1)

ans =

3

>> size(A,2)

ans =

4

§  size() returns info about a matrix’s dimensions.


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The for loop

§  Vectors are often processed with loops in order to access and process each value, one after the other: §  Syntax : for i=x

…. end

§  With -  i the name of the running variable -  x a vector containing the sequence of values assigned to i


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The for loop

>> for i=1:3

i^2

end

i =

1

i =

4

i =

9

§  MATLAB waits for the keyword end before computing the result.


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The for loop

>> for i=1:3

y(i)=i^2;

end

>> y

y =

1 4 9

§  MATLAB waits for the keyword end before computing the result.


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Conditional statements: if

§  The keyword if is used to test a condition §  Syntax :

if (condition) ..sequence of commands.. end

§  The condition is a Boolean operation §  The sequence of commands is executed if the tested

condition is true


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Logical operators

§  Logical operators §  < , > : less than, greater than §  == : equal to §  && : and §  || : or §  ~ : not ( ~true is false) §  (1 stands for true, 0 stands for false)


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Conditional statements: Example

>> threshold=5;

>> x=4.5;

>> if (x<threshold)

diff = threshold - x;

end

>> diff

diff =

0.5

§  An example:


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Conditional statements: else §  The keyword else is optional

§  Syntax : if (condition)

..sequence of commands n°1..

else

..sequence of commands n°2..

end

>> if (x<threshold)

diff = threshold - x ;

else

diff = x – threshold;

end


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Scripts and functions §  External files used to store and save sequences

of commands.

§  Scripts: §  Simple sequence of commands §  Global variables

§  Functions: §  Dedicated to a particular task §  Inputs and outputs §  Local variables


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Scripts and functions §  Scripts :

§  Create .m file, e.g. sumVector.m. §  Type commands in the file. §  Type the file name, .e.g sumVector, in the command

window.

%sum of 4 values in x x=[1 3 5 7]; R=x(1)+x(2)+x(3)+x(4); R

sumVector.m

>> sumVector

R = 16


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Scripts and functions §  Make sure that the file is in your working

directory!


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Scripts and functions §  Functions :

§  Create .m file, e.g. absoluteVal.m §  Declare inputs and outputs in the first line of the file,

function [out1, out2, …] = functionName (in1, in2, …) e.g. function [R] = absoluteVal(x)

§  Use the function in the command window functionName(in1, in2, …) e.g. absoluteVal(x)


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Scripts and functions

function [R] = absoluteVal(x) % Compute the absolute value of x if (x<0) R = -x ; else R = x ; end

absoluteVal.m

>> A=absoluteVal(-5);

>> A

A =

5


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Scripts and functions

function [R] = absoluteVal(x) % Compute the absolute value of x if (x<0) R = -x ; else R = x ; end

absoluteVal.m

>> A=absoluteVal(-5);

>> A

A =

5


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Exercise 1

§  Compute: a) b)

c) §  Slides/exercises:

http://www.coss.ethz.ch/education/matlab.html

25510718

×

+ ∑=

100

0ii

∑=

−10

5

2 )(i

ii


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Exercise 2

§  Solve for x:

452232223321432

4321

4321

4321

4321

=++−

=−−−

=+−+

=+−−

xxxxxxxxxxxxxxxx


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Exercise 3 §  Fibonacci sequence: write a function which

computes the Fibonacci sequence of a given number n and returns the result in a vector.

§  The Fibonacci sequence F(n) is given by :


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References §  http://www.mathworks.ch/products/matlab/

index.html

§  http://www.mathworks.ch/matlabcentral/index.html

§  https://ides.ethz.ch/

modeling and simulating social systems with matlab · 2015-09-21 · 2015-09-21 modeling and...

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