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6.094 Introduction to MATLAB®
January (IAP) 2009

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6.094
Introduction to Programming in MATLAB®

Lecture 4: Advanced Methods

Sourav Dey
Danilo Šćepanović
Ankit Patel
Patrick Ho

IAP 2009

Outline

(1) Probability and Statistics
(2) Data Structures
(3) Images and Animation
(4) Debugging
(5) Symbolic Math
(6) Other Toolboxes

Statistics

• Whenever analyzing data, you have to compute statistics
» scores = 100*rand(1,100);

• Built-in functions
mean, median, mode

• To group data into a histogram
» hist(scores,5:10:95);
makes a histogram with bins centered at 5, 15, 25…95
» N=histc(scores,0:10:100);
returns the number of occurrences between the specified
bin edges 0 to <10, 10 to <20…90 to <100.

Random Numbers

• Many probabilistic processes rely on random numbers

• MATLAB contains the common distributions built in
» rand
draws from the uniform distribution from 0 to 1
» randn
draws from the standard normal distribution (Gaussian)
» random
can give random numbers from many more distributions
see doc random for help
the docs also list other specific functions
• You can also seed the random number generators
» rand(‘state’,0)

Changing Mean and Variance

• We can alter the given distributions
» y=rand(1,100)*10+5;
gives 100 uniformly distributed numbers between 5 and 15
» y=floor(rand(1,100)*10+6);
gives 100 uniformly distributed integers between 10 and
15. floor or ceil is better to use here than round
400

350

» y=randn(1,1000)
300

250

200

» y2=y*5+8 150

100

increases std to 5 and makes the mean 8 50

0
-25 -20 -15 -10 -5 0 5 10 15 20 25

90

80

70

60

50

40

30

20

10

0
-25 -20 -15 -10 -5 0 5 10 15 20 25

Exercise: Probability
• We will simulate Brownian motion in 1 dimension. Call the script
‘brown’
• Make a 10,000 element vector of zeros
• Write a loop to keep track of the particle’s position at each time
• Start at 0. To get the new position, pick a random number, and if
it’s <0.5, go left; if it’s >0.5, go right. Store each new position in
the kth position in the vector
• Plot a 50 bin histogram of the positions.

Exercise: Probability
• We will simulate Brownian motion in 1 dimension. Call the script
‘brown’
• Make a 10,000 element vector of zeros
• Write a loop to keep track of the particle’s position at each time
• Start at 0. To get the new position, pick a random number, and if
it’s <0.5, go left; if it’s >0.5, go right. Store each new position in
the kth position in the vector
• Plot a 50 bin histogram of the positions.

» x=zeros(10000,1);
» for n=2:10000
» if rand<0.5
» x(n)=x(n-1)-1;
» else
» x(n)=x(n-1)+1;
» end
» end
» figure;
» hist(x,50);

Advanced Data Structures

• We have used 2D matrices
Can have n-dimensions
Every element must be the same type (ex. integers,
doubles, characters…)
Matrices are space-efficient and convenient for calculation

• Sometimes, more complex data structures are more
appropriate
Cell array: it's like an array, but elements don't have to be
the same type
Structs: can bundle variable names and values into one
structure
– Like object oriented programming in MATLAB

Cells: organization

• A cell is just like a matrix, but each field can contain
anything (even other matrices):

3x3 Matrix 3x3 Cell Array
2
J o h n
1.2 -3 5.5 4
32
-2.4 15 -10
M a r y 27 1
7.8 -1.1 4 18

L e o []

• One cell can contain people's names, ages, and the ages of
their children
• To do the same with matrices, you would need 3 variables
and padding

Cells: initialization

• To initialize a cell, specify the size
» a=cell(3,10);
a will be a cell with 3 rows and 10 columns

• or do it manually, with curly braces {}
» c={'hello world',[1 5 6 2],rand(3,2)};
c is a cell with 1 row and 3 columns

• Each element of a cell can be anything

• To access a cell element, use curly braces {}
» a{1,1}=[1 3 4 -10];
» a{2,1}='hello world 2';
» a{1,2}=c{3};

Structs

• Structs allow you to name and bundle relevant variables
Like C-structs, which are objects with fields

• To initialize an empty struct:
» s=struct([]);
size(s) will be 1x1
initialization is optional but is recommended when using large
structs

• To add fields
» s.name = 'Jack Bauer';
» s.scores = [95 98 67];
» s.year = 'G3';
Fields can be anything: matrix, cell, even struct
Useful for keeping variables together

• For more information, see doc struct

Struct Arrays

• To initialize a struct array, give field, values pairs
» ppl=struct('name',{'John','Mary','Leo'},...
'age',{32,27,18},'childAge',{[2;4],1,[]});
size(s2)=1x3
every cell must have the same size
» person=ppl(2);
person is now a struct with fields name, age, children
the values of the fields are the second index into each cell
» person.name
returns 'Mary' ppl ppl(1) ppl(2) ppl(3)

name: 'John' 'Mary' 'Leo'
age: 32 27 18
childAge: [2;4] 1 []

Structs: access

• To access 1x1 struct fields, give name of the field
» stu=s.name;
» scor=s.scores;
1x1 structs are useful when passing many variables to a
function. put them all in a struct, and pass the struct

• To access nx1 struct arrays, use indices
» person=ppl(2);
person is a struct with name, age, and child age
» personName=ppl(2).name;
personName is 'Mary'
» a=[ppl.age];
a is a 1x3 vector of the ages

Exercise: Cells

• Write a script called sentGen
• Make a 3x2 cell, and put people’s names into the first
column, and adjectives into the second column
• Pick two random integers (values 1 to 3)
• Display a sentence of the form ‘[name] is [adjective].’
• Run the script a few times

Exercise: Cells

• Write a script called sentGen
• Make a 3x2 cell, and put people’s names into the first
column, and adjectives into the second column
• Pick two random integers (values 1 to 3)
• Display a sentence of the form ‘[name] is [adjective].’
• Run the script a few times

» c=cell(3,2);
» c{1,1}=‘John’;c{2,1}=‘Mary-Sue’;c{3,1}=‘Gomer’;
» c{1,2}=‘smart’;c{2,2}=‘blonde’;c{3,2}=‘hot’
» r1=ceil(rand*3);r2=ceil(rand*3);
» disp([ c{r1,1}, ’ is ‘, c{r2,2}, ’.’ ]);

Importing/Exporting Images

• Images can be imported into matlab
» im=imread('myPic.jpg');

• MATLAB supports almost all image formats
jpeg, tiff, gif, bmp, png, hdf, pcx, xwd, ico, cur, ras, pbm,
pgm, ppm
see help imread for a full list and details

• To write an image, give an rgb matrix or indices and
colormap
» imwrite(mat,jet(256),'test.jpg','jpg');
see help imwrite for more options

Animations

• MATLAB makes it easy to capture movie frames and play
them back automatically

• The most common movie formats are:
avi
animated gif

• Avi
good when you have ‘natural’ frames with lots of colors and
few clearly defined edges
• Animated gif
Good for making movies of plots or text where only a few
colors exist (limited to 256) and there are well-defined
lines

Making Animations

• Plot frame by frame, and pause in between
» close all
» for t=1:30
» imagesc(rand(200));
» colormap(gray);
» pause(.5);
» end

Saving Animations as Movies

• A movie is a series of captured frames
» close all
» for n=1:30
» imagesc(rand(200));
» colormap(gray);
» M(n)=getframe;
» end
• To play a movie in a figure window
» movie(M,2,30);
Loops the movie 2 times at 30 frames per second
• To save as an .avi file on your hard drive
» movie2avi(M,'testMovie.avi','FPS',30);
• See book appendix or docs for more information

Handles

• Every graphics object has a handle
» h=plot(1:10,rand(1,10));
gets the handle for the plotted line
» h2=gca;
gets the handle for the current axis
» h3=gcf;
gets the handle for the current figure
• To see the current property values, use get
» get(h);
» yVals=get(h,'YData');
• To change the properties, use set
» set(h2,'FontName','Arial','XScale','log');
» set(h,'LineWidth',1.5,'Marker','*');
• Everything you see in a figure is completely customizable
through handles

display

• When debugging functions, use disp to print messages
» disp('starting loop')
» disp('loop is over')
disp prints the given string to the command window

• It's also helpful to show variable values
» disp(strcat(['loop iteration ',num2str(n)]));
strcat concatenates the given strings
Sometimes it's easier to just remove some semicolons

Debugging
• To use the debugger, set breakpoints
Click on – next to line numbers in MATLAB files
Each red dot that appears is a breakpoint
Run the program
The program pauses when it reaches a breakpoint
Use the command window to probe variables
Use the debugging buttons to control debugger

Clear breakpoint Stop execution; exit

Step to next

Two breakpoints

Where the program is now
Courtesy of The MathWorks, Inc. Used with permission.

Exercise: Debugging

• Use the debugger to fix the errors in the following code:


Performance Measures

• It can be useful to know how long your code takes to run
To predict how long a loop will take
To pinpoint inefficient code

• You can time operations using tic/toc:
» tic
» CommandBlock1
» a=toc;
» CommandBlock2
» b=toc;
tic resets the timer
Each toc returns the current value in seconds
Can have multiple tocs per tic

Performance Measures

• For more complicated programs, use the profiler
» profile on
Turns on the profiler. Follow this with function calls
» profile viewer
Displays gui with stats on how long each subfunction took


What are Toolboxes?

• Toolboxes contain functions specific to a particular field
for example: signal processing, statistics, optimization

• It's generally more efficient to use MATLAB's toolboxes
rather than redefining the functions yourself
saves coding/debugging time
some functions are compiled, so they run faster
HOWEVER there may be mistakes in MATLAB’s functions
and there may also be surprises

• MATLAB on Athena contains all the toolboxes

• Here are a few particularly useful ones for EECS…

Symbolic Toolbox

• Don’t do nasty calculations by hand!
• Symbolics vs. Numerics

Advantages Disadvantages

Symbolic •Analytical solutions •Sometimes can't be
•Lets you intuit solved
things about •Can be overly
solution form complicated
Numeric •Always get a •Hard to extract a
solution deeper understanding
•Can make solutions •Num. methods
accurate sometimes fail
•Easy to code •Can take a while to
compute

Symbolic Variables

• Symbolic variables are a type, like double or char

• To make symbolic variables, use sym
» a=sym('1/3');
» b=sym('4/5');
fractions remain as fractions
» c=sym('c','positive');
can add tags to narrow down scope
see help sym for a list of tags

• Or use syms
» syms x y real
shorthand for x=sym('x','real'); y=sym('y','real');

Symbolic Expressions

• Multiply, add, divide expressions
» d=a*b
does 1/3*4/5=4/15;

» expand((a-c)^2);
multiplies out

» factor(ans)
factors the expression

Cleaning up Symbolic Statements

» pretty(ans)
makes it look nicer

» collect(3*x+4*y-1/3*x^2-x+3/2*y)
collects terms

» simplify(cos(x)^2+sin(x)^2)
simplifies expressions

» subs(‘c^2’,c,5) ans=
25
Replaces variables with numbers
or expressions

» subs(‘c^2’,c,x/7) ans=
1/49*x^2

More Symbolic Operations

• We can do symbolics with matrices too
» mat=sym('[a b;c d]');

» mat2=mat*[1 3;4 -2];
compute the product
» d=det(mat)
compute the determinant
» i=inv(mat)
find the inverse

• You can access symbolic matrix elements as before
» i(1,2)

Exercise: Symbolics

• The equation of a circle of radius r centered at (a,b) is
given by: (x-a)^2 + (y-b)^2 = r^2.
• Expand this equation into the form Ax^2 + Bx+Cxy + Dy +
Ey^2 = F and find the expression for the coefficients in
terms of a,b, and r.

Exercise: Symbolics

• The equation of a circle of radius r centered at (a,b) is
given by: (x-a)^2 + (y-b)^2 = r^2.
• Expand this equation into the form Ax^2 + Bx+Cxy + Dy +
Ey^2 = F and find the expression for the coefficients in
terms of a,b, and r.

» syms a b r x y
» pretty(expand((x-a).^2 + (y-b).^2))

Signal Processing Toolbox

• MATLAB is often used for signal processing (fft)
• What you can do:
filter design
statistical signal processing
Laplace transforms

• Related Toolboxes
Communications
Wavelets
RF
Image Processing

Control System Toolbox

• The control systems toolbox contains functions helpful for
analyzing systems with feedback

• Simulation of LTI system function
• Discrete time or continuous time
• You will be exposed to it in 6.003
• Can easily study step response, etc. modal analysis.

• Related toolboxes:
System Identification
Robust Control – modern control theory
Model Predictive Control

Statistics Toolbox

• For hardcore statistics and data-analysis
Principal component analysis
Independent component analysis
Tests of significance (chi squared, t-tests…)

• Related Toolboxes
Spline – for fitting
Bioinformatics
Neural Networks

Optimization Toolbox

• For more hardcore optimization problems – that occur in
OR, business, engineering
linear programming
interior point methods
quadratic methods

SIMULINK

• Interactive graphical environment
• Block diagram based MATLAB add-on environment
• Design, simulate, implement, and test control, signal
processing, communications, and other time-varying
systems


Central File Exchange

• The website – the MATLAB Central File Exchange!!
• Lots of people's code is there
• Tested and rated – use it to expand MATLAB's functionality

• http://www.mathworks.com/matlabcentral/

MATLAB Final Exam

• Brownian Motion stop-animation – integrating loops,
randomization, visualization

• Make a function brown2d(numPts), where numPts is the
number of points that will be doing Brownian motion
• Plot the position in (x,y) space of each point (start initially
at 0,0). Set the x and y limits so they’re consistent.
• After each timestep, move each x and y coordinate by
randn*.1
• Pause by 0.001 between frames
• Turn on the DoubleBuffer property to remove flicker
» set(gcf,’DoubleBuffer’,’on’);

• Ask us for help if needed!

End of Lecture 4

(1) Data Structures
(2) Symbolics
(3) Probability
(4) Toolboxes

THE END

Monte-Carlo Simulation

• A simple way to model complex stochastic systems
• Use random numbers to control state changes

20 60 20
A B
47
2 2
60 47 E 47 60
2
2
47
D C
20 60 20

• This system represents a complex reaction
• The numbers by the arrows show the propensity of the
system to go from one state to another
• If you start with 1 molecule of A, how does the system
behave with time?

Example: Monte-Carlo

• This MATLAB file will track the behavior of the molecule


Example: Monte-Carlo

• We can run the code 1000 times to simulate 1000 molecules
» s=zeros(200,5);
» for n=1:1000
» st=MC(200);
» for state=0:4
» s(:,state+1)= s(:,state+1)+(st==state);
B D

» end 450

400
400

350

» end 350

300
300

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200
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150
150
A
100
1000 100

900 50
50

800
0 0
0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200
700 C E
350 1000
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0
0 20 40 60 80 100 120 140 160 180 200
100 300

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0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200

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