1. Modelling and Simulating Social Systems with MATLAB
Modelling and Simulating Social Systems with MATLAB
Lesson 2 – Statistical plotting and animation
A. Johansson & W. Yu
© ETH Zürich |

2. Lesson 2 - Contents Repetition Statistical Plotting Animation
Lesson 2 - Contents
Repetition
Statistical Plotting
Animation

3.
Repetition
Creating a scalar:
>> a=10
a = 10

4. Repetition Creating a row vector: >> v=[1 2 3 4 5] v = 1 2 3 4 5
Repetition
Creating a row vector:
>> v=[ ]
v =

5. Repetition Creating a row vector, 2nd method: >> v=1:5 v =
Repetition
Creating a row vector, 2nd method:
>> v=1:5
v =

6. Repetition Creating a column vector: >> v=[1; 2; 3; 4; 5] v =
Repetition
Creating a column vector:
>> v=[1; 2; 3; 4; 5]
v =

7. Repetition Creating a column vector, 2nd method:
Repetition
Creating a column vector, 2nd method:
>> v=[ ]’
v =

8. Repetition Creating a matrix: >> A=[1 2 3 4; 5 6 7 8] A =
Repetition
Creating a matrix:
>> A=[ ; ]
A =

9. Repetition Creating a matrix, 2nd method: >> A=[1:4; 5:8] A =
Repetition
Creating a matrix, 2nd method:
>> A=[1:4; 5:8]
A =

10.
Repetition
Accessing a scalar:
>> a
a = 10

11. Repetition Accessing an element in a vector: v(index) >> v(2)
Repetition
Accessing an element in a vector:
v(index) >> v(2)
ans = 2

12. Repetition Accessing an element in a matrix:
Repetition
Accessing an element in a matrix:
M(rowIndex, columnIndex) >> A(2, 1)
ans = 5

13. Repetition - operators
Repetition - operators
Scalar operators: Basic: +, -, *, / Exponentialisation: ^ Square root: sqrt()

14. Repetition - operators
Repetition - operators
Matrix operators: Basic: +, -, *
Element-wise operators: Multiplication: .* Division: ./ Exponentialisation: .^
Solving Ax=b: x = A\b

15. Repetition – for loop Computation can be automized with for loops:
Repetition – for loop
Computation can be automized with for loops:
>> y=0; for x=1:4 y = y + x^2 + x; end
>> y
y = 40

16. Repetition – if case Conditional computation can be made with if:
Repetition – if case
Conditional computation can be made with if:
>> val=-4; if (val>0 ) absVal = val; else absVal = -val; end

17. Statistics Function
Statistics in MATLAB can be performed with the following commands:
Mean value: mean(x) Median value: median(x) Min/max values: min(x), max(x) Standard deviation: std(x) Variance: var(x) Covariance: cov(x) Correlation coefficient: corrcoef(x)

18. Plotting Function
Plotting in MATLAB can be performed with the following commands: Plot vector x vs. vector y: Linear scale: plot(x, y) Logarithmic scale: loglog(x, y)
Plot histogram of x: hist(x)

19. Details of plot
An additional parameter can be provided to plot() to define how the curve will look like: plot(x, y, ‘key’) Where key is a string which can contain: Color codes: ‘r’, ‘g’, ‘b’, ‘k’, ‘y’, … Line codes: ‘-’, ‘--’ (solid, dashed, etc.) Marker codes: ‘*’, ‘.’, ‘s’, ’c’ Examples: plot(x, y, ‘r--’) plot(x, y, ‘g*’) * * * *

20. Where can I get help?
>>help functionname
>>helpwin

21. Plotting Tips
To make the plots look nicer, the following commands can be used: Set label on x axis: xlabel(“text”) Set label on y axis: ylabel(“text”) Set title: title(“text”)

22. Plotting Tips Two additional useful commands
Plotting Tips
Two additional useful commands
hold on|off
grid on|off
>> x=[-5:0.1:5];a=2;b=3;
>> y1=exp(-x.^2);
>> y2=a*exp(-x.^2);
>> y3=exp(-(x.^2)/b); 22 22

23. Plotting Tips >> plot(x,y1); >> hold on
Plotting Tips
>> plot(x,y1);
>> hold on
>> plot(x,y2,’r’);
>> plot(x,y3,’g’); 23 23

24. Plotting Tips >> plot(x,y1); >> hold on
Plotting Tips
>> plot(x,y1);
>> hold on
>> plot(x,y2,’r’);
>> plot(x,y3,’g’);
>> grid on 24 24

25. File I/O Saving a plot >> hgsave(‘fileName.fig’) 10.06.2018 25

26. Tips for I/O Some commands in Linux are also applicable in Matlab.
rmdir,mkdir
delete,edit
ls,cd
Tab

27. Animation A simple example of computer animation.
r = 10;k = 1;pi = 3.14;
hold on;
for t = 0 : 0.01 : 2*pi
x = r*cos(t);y = r*sin(t);
plot(x,y,'o','MarkerSize',20);
axis([-10,10,-10,10]); %adjust the range of axis
M(k) = getframe;%get the current frame
k = k+1;
end%for

28. Replay a Movie
movie(M,N) plays the movie in matrix M for N times.

29. Save Movie as avi File
movie2avi(M,’demo.avi’) saves the movie in matrix M as ‘demo.avi’.

30. 3 Dimensions Plots Plot a surface surf(X,Y,Z) with: x2 x1 x3 y1 y2 y3
3 Dimensions Plots
Plot a surface
surf(X,Y,Z) with:
X a vector containing the x-axis
Y a vector containing the y-axis
Z a matrix with Z(i,j) is the corresponding value for Y(i) and X(j)
Number of lines and columns in Z must respectively equal length(Y) and length(X). x2 x1 x3 y1 y2 y3
z11
z12
z13
z21
z22
z23
z31
z32
z33 30 30

31. 3 Dimensions Plots Plot a surface >> X=-2:0.1:2;
3 Dimensions Plots
Plot a surface
>> X=-2:0.1:2;
>> Y=0:0.1:1;
>> Z = Y(2)*exp(-X.^2)
Z =
>> for i=1:length(Y)
Z(i,:)= Y(i)*exp(-X.^2);
end 31 31

32. 3 Dimensions Plots Plot a surface >> surf(X,Y,Z) 10.06.2018 32

33. 3 Dimensions Plots Plot a surface >> surf(X,Y,Z) 10.06.2018 33

34. 3 Dimensions Plots Plot a surface >> surf(X,Y,Z) 10.06.2018 34

35. 3 Dimensions Plots Plot a surface
3 Dimensions Plots
Plot a surface
>> contourf(X,Y,Z) >> contour(X,Y,Z) 35 35

36. Datasets
The datasets for statistical plotting can be found at the course web page you will find the files: countries.m cities.m

37. Datasets
Download the files countries.m and cities.m and save them in the working directory of MATLAB.

38. Datasets - countries
This dataset countries.m contains a matrix A with the following specification:
Rows: Different countries
Column 1: Population
Column 2: Annual growth (%)
Column 3: Percentage of youth
Column 4: Life expectancy (years)
Column 5: Mortality

39. Datasets - countries
Most often, we want to access complete columns in the matrix. This can be done by A(:, index) For example if you are interested in the life- expectancy column, it is recommended to do: >> life = x(:,4); and then the vector life can be used to access the vector containing all life expectancies.

40. Datasets - countries
The sort() function can be used to sort all items of a vector in inclining order. >> life = A(:, 4); >> plot(life)

41. Datasets - countries
The sort() function can be used to sort all items of a vector in inclining order. >> life = A(:, 4); >> lifeS = sort(life); >> plot(lifeS)

42. Datasets - countries
The histogram hist() is useful for getting the distribution of the values of a vector. >> life = A(:, 4); >> hist(life)

43. Datasets - countries
Alternatively, a second parameter specifies the number of bars: >> life = A(:, 4); >> hist(life, 30)

44.
Exercise 1
Statistics: Generate a vector of N random numbers with randn(N,1) Calculate the mean and standard deviation. Do the mean and standard deviation converge to certain values, for an increasing N? Optional: Display the histogram with hist(randn(N,1)) and compare with the histogram of rand()

45.
Exercise 2
Demographics: From the countries.m dataset, find out why there is such a large difference between the mean and the median population of all countries. Hint: Use hist(x, n) Also sort() can be useful.

46.
Exercise 3
Demographics: From the countries.m dataset, see which columns have strongest correlation. Can you reason why these columns have stronger correlations? Hint: Use corrcoef() to find the correlation between columns.

47.
Exercise 4 - optional
Zipf’s law: Zipf’s law says that the rank, x, of cities (1: largest, 2: 2nd largest, 3: 3rd largest, ...) and the size, y, of cities (population) has a power-law relation: y ~ xb Test if Zipf’s law holds for the three cases in the cities.m file. Try to estimate the b parameter. Hint: Use log() and plot() (or loglog())