%  INTRODUCTION TO USING MATLAB  
%    for MATLAB 7 
%    with TOOLBOXES
%     SYMBOLIC MATH
%     STATISTICS
%   modified and updated in May, 2008
%   by Carol Horvitz

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% WORKING IN MATH COMPUTER LAB
%(note to instructor: increase fontsize in command window
% choose: FILE from Menu bar, Preferences, Fonts, try setting to Tahoma 15)

% GET files of interest onto the local computer or your portable data medium:
%  
%   1) make a folder for today's session on the desktop/or on memory
%      stick or CD
%   2) use web browser to go to the course website:
%        bio.miami.edu/horvitz/bil235/brasil08
%   3)click on m-files folder
%   4)right click on filename of interest and choose "save link as"
%   5)save file to folder on local computer or on memory stick or CD
%   6)IF you have put it on the desktop for the session, be sure to save to
%     your own portable data medium before leaving the lab

% NOW open MATLAB
%   1) click on MATLAB
%   2) in menu bar at top, use the button to the right of the window to navigate
%      to directory where your files are
%     ****IMPORTANT: YOU DO NEED TO HAVE THAT DIRECTORY 
%        SHOWING IN THE WINDOW
%        TO USE FILES IN IT AND SAVE STUFF TO IT.******
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% MATLAB lessons:
%
%  First: GETTING OUT!!!! 
% 
%  'ctrl-C'    breaks an ongoing operation
%   >> quit      closes program
%
%  Second: GETTING HELP!!!!
% 
%   command line help
%   >> help dsolve
%   >> help plot

%   menu bar help
%     click on Help
%     click on MATLAB Help
%
%  Now for the rest of the tutorial
%
%   1) working directory and windows
%       command, editor, workspace,
%       command history, current directory
%   2) working on the command line in the command window
%   3) variables, expressions, statements
%      a)types of variables (all are matrices)
%       i) numeric
%       ii) strings
%       iii) symbolic
%      b)expressions may contain operators
%   4) matrices
%   5) numbers and arithmetic expressions
%   6) matrix vs array operations
%   7) vector and matrix manipulation
%   8) do loops, relational and logical operators
%   9) working with files
%      a) m-files: scripts
%      b) m-files: functions
%                   i) user-defined
%                  ii) built-in   
%      c) mat-files: store values of variables
%  10) simple graphics
%  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%MATH ECOLOGY LESSONS
%  (in later labs this will be filled in!)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%HOW TO USE m-SCRIPTS provided in the course:
%  1) FOLLOW THE DIRECTIONS IN THE COMMENTS. 
%    (some syntax in MATLAB has changed with different versions,
%     I have been updating the code; some lines of code will 
%     differ from the resource material in the text books)
%  2A) RUN THE COMMANDS INTERACTIVELY WITH THE "F9" COMMAND.
%  or
%  2B) TYPE THE COMMANDS INTO THE COMMAND WINDOW TO PRACTICE COMPOSING
%      YOUR OWN SCRIPT
%  3) ANSWER THE Questions BY FILLING IN THE ANSWERS IN COMMENT MODE.
%  4) TRY OTHER STUFF AND ADD YOUR OWN COMMANDS AND COMMENTS.
%  5) ADD YOUR NAME AND DATE TO THE TOP OF THE SCRIPT and save your customized
%     version of it.
%  6) PRINT OUT THE REVISED SCRIPT THAT HAS YOUR COMMENTS IN IT FOR YOUR NOTEBOOK.
%  7) Also print out FIGURES you have created for your notebook.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%original source of examples in this script%%%%%%%%%%%%%%%%%%%%
%    main reference for this tutorial is 
% Roughgarden, 1998, appendix, but this is updated to MATLAB 7 SYNTAX
% Cavers, I. 1998, http://www.cs.ubc.ca.spider/cavers/MatlabGuide/


%   1) working directory and windows
%IMPORTANT!!!!!!!!!
%     pick a working directory
%     point MATLAB to it (the window in the menu bar at top shows where you are) 

what  %What files are in it?
who   %What variables are in the workspace?
whos  %What are the sizes and types of variables?
why   %What is the meaning of your existence?
date

%   2) working on the command line in the command window

%use up and down arrows to cycle through previous commands
%use the 'return','enter'  to carry out the command: 
%        translates your code into machine language and does the job
%        and prints the result

%   3) variables, expressions, statements
%  variable name starts with a letter, can have 19 characters, including
%  underscore
%  
x=2
y=2+3
z_1=x+y
z_2=y-x
z_3=x*y; %semicolon suppresses the printing of the result
         % but the job has been done, check it:
whos
z_3

%      a)types of variables (all are matrices)
%       i) numeric
%       ii) strings
%       iii) symbolic
%      b)expressions may contain operators


%character variables are denoted with single quotes
students=['Antonio Aguiar Neto']

%declare symbolic variables for manipulation of symbols
syms f g
m =(f+g)^2
m2 = expand(m)
f=4
eval(m)

whos

%you may now want to save all these variables with their current values, so
%you can go for a coffee break and return and not have to re-calculate
%them... here is where a special kind of file comes in a "mat-file"
save resultsofmywork_5mai08    

what
%a mat-file has all the variables in it that were defined and active at the
%time you issued the command save; it is a binary file, not a text file.

%simulate a fresh start after your break by using the 
clear
%command to remove all previously defined variables 
who

%to get back your pre-break results:
load resultsofmywork_5mai08
who

clear all; %ok, let's go on...
close all;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%highlight and run each set separately...
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   4) matrices
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

A=[ 1 2 4.5;8/2.0 6 5]
A(2,3)

rowthing=[1 2 3]
colthing=[1; 2; 3;]

%transpose
[1 2 3]'

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%  some of Matlab's preset matrices
%   
 ones(7)
 zeros(7)
 eye(7)
 rand(7)

% you can also have non square matrices of many of these
 ones(3,2)
 zeros(3,2)
 rand(3,2)
 
 %note rand generate random numbers between 0 and 1 
 % from a uniform distribution
 
 %note the semicolon at the end of the line suppresses printing the result
 unis=rand(1,100);
 figure(1)
 hist(unis)
 
 % for normally distributed random numbers
 
 norms=randn(1,100);
 figure(2)
 hist(norms)

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % before going on close graphics windows and clear variables
 close all
 clear all
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%other convenient premade matrices and the use of the colon
% refer to elements and manipulate matrices
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
 %integer steps
 a=1:5
 
 %any steps you choose
 abyhalfs=1:0.5:5
 
 abytenths=1:0.1:5
 
 abytwos=1:2:5
 
 % some ways to extract particular elements from a matrix
 
 a=[1 2 3; 4 5 6; 7 8 9]
 
 bbysubscripts=[a(1,2:3); a(2,2:3)]
 
 bytriangle=triu(a)
  
 help triu
 
 abovediag=triu(a,1)
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 % some ways to rearrange elements in a matrix
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 %the transpose is very useful
 transa=a'
 
%what else can you do?

%HTML help
% for example...check out the symbolic math stuff
%  go to Help on the frame around the command window
%    symbolic math...compatibility... obs functions

% check out the reference list of matlab commands
%  functions,  by category, mathematics...choose a topic
%   e.g. "arrays and matrices", "operations and manipulations", "flipud"

%cool we can flip the matrix around

updowna=flipud(a)
leftrighta=fliplr(a)
 

% last let's work one example of referring to a elements 
%  in a  matrix of symbolic math entities

% note that in my example I use the NEW syms declaration command
%   not available when Roughgarden wrote the book... so
%    my syntax is much less cumbersome... than 
%       what is on Roughgarden, p. 434: mine has fewer quotes and brackets 
syms w x y z 
a=[w x; y z]

%change the 2,2 element to z-squared
a(2,2)=z^2


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   5) numbers and arithmetic expressions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;

nums=[4 
-189
.032
-2.21E-23
0.9772E18
0.00001]

format short
format short e
format long
format long e
format +
format

%arithmetic operators
2+3
2-3
2*3
2^3
2/3
2\3

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   6) matrix vs array operations
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%element-by-element operations
% precede one of the opertators by a period (.)
clear all;
close all;
A=[1 2 3; 4 5 6]; B=[2 2 2; 3 3 5]; C=A.*B
D=A.^2

%matrix multiplication 
%NOT commutative
% number of rows of matrix on the left 
% must equal
% number of columns of matrix on the right

A*B    % A has two rows, B has three columns, this won't work
A*(B') % this will, B' has two columns

syms a11 a12 a13 b11 b21 b31
%note the indices are (row,col) as below
a=[a11 a12 a13]
b=[b11; b21; b31;]

c1=a*b  %defined
c2=b*a  %also defined, but distinct from above

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   7) vector and matrix manipulation (see above)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   8) do loops, relational and logical operators
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%"Do Loops" 
clear all;
close all;

for i=1:3
    %do something for each value of i..
    % for example
    isq=i^2
    isqrt=i^(1/2)
end

%i does not have to be an integer
for i=1:0.5:3
    %do something for each value of i..
    % for example
    isq=i^2
    isqrt=i^(1/2)
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%intro for USE WITH "If..." and other logical issues

x=[-2 2 0 -1 1 0 -3 3 0]

%understand some ways matlab chooses elements logically
%  in logic  1= it is true, 0 = it is false
%identifies the elements of x that are equal to 0 
x==0

%identifies the elements of x that are not equal to 0
x~=0

%identifies the elements of x that are >  0
x>0

%identifies the elements of x that are <  0
x<0

%we will return to how to use these later in the course

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%   9) working with files
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      9a) m-files: scripts(such as this one)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 1) open a new window in the m-editor (or any text editor)
% 2) write your commands in the window
% 3) save the script using the menu bar FILE drop down menu in the _editor_ window
%  type name of script into window given to you by the drop down choice "save-as"
                                myfirstscript

% 4) invoke the whole set of commands from the command window
myfirstscript %don't type the ".m"
% OR
% 5) open the file in the editor window and run it interactively by highlighting
%    certain lines and using F9 to execute them
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      9b) m-files: FUNCTIONS user-defined
%     
% something you want to do over and over...instead of having to repeat the
% long script each time...write a special m-file called a 
%     function
%you give it inputs, called  "arguments"
% if gives you results, called "outputs"
% it does the details and intermediate steps in secret
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% example the pythagorean theorem
% given the a and b sides of a triangle, find the c side
% name it "hypot"...
% "a" and "b" will be the arguments
% "c" is the output

function c=hypot(a,b)
c=sqrt(a^2+b^2);

%save this as a function:
%  open a new window for an m file 
%  copy this into into and save it as "hypot"

% so all you have to do on the command line is 
%    name the function and 
%     give the input argument

%try it
hypot(1,2)
hypot(4,5)
hypot(7.2,5.6)


%"global" is a statement used to "de-classify" the secrets: the variables calculated and
% defined and used in functions  remain the same in the
% command space and in the function space..  we'll see its use later

%here is a more fun one to try
% from Higham and Higham 2005, p. 15

function yprime = lorenzde(t,y)
%LORENZDE  lorenz equations
yprime= [10*(y(2)-y(1))
               28*y(1)-y(2)-y(1)*y(3) 
               y(1)*y(2)-8*y(3)/3]
           
%copy this to its own user-defined function m-file, name the file 'lorenzde'.
% open a blank m-file in the editor
% copy these lines to it
% save it as "lorenzde.m"

%then use this function together 
% with the built-in function "ode45.m" as follows
help ode45

tspan=[0 50];                            % solve for  times 0 to 50, inclusive
yzero=[0;1;0];                           % Specifies Initial conditions

[t,y]=ode45(@lorenzde,tspan,yzero)  %calls the function, sets time span, initial conditions
plot(y(:,1),y(:,3))
xlabel('y_1')
ylabel('y_2')
title('Lorenz equations, phase plane plot', 'FontSize', 16)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    9b) m-files: FUNCTIONS:  built-in, CREATED BY MATLAB   
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% the function 'exp'
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% exp is _not_   the number e in matlab.
% it is the exponential function... e to the x
%thus it is e raised to the power in the parentheses
%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;

help exp
exp(1)

%so this makes the natural log funtion work as follows
%  recall that log to the base e 
% answers the question: to what power must we raise e to get the number
log(exp(1))

% review some other logs
log10(100)

log10(1000)

log2(4)

log2(16)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%spread sheet functions
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%treats rows as observations, columns as variables
%e.g.: dots on frog legs...each row is a frog, 
%          the first column is left leg
%          the second column is right leg
clear all;
close all;
dotsonlegs=[5 6; 4 5; 7 8;8 12]
meandots=mean(dotsonlegs)
sumdots=sum(dotsonlegs)
mindots=min(dotsonlegs)
maxdots=max(dotsonlegs)
stddots=std(dotsonlegs)

% hmm is this the with the n or the n-1 weighting?
help std
%oh...so now try the other weighting
stddotsnweighting=std(dotsonlegs,1)

%what about relationships between dots on right and left legs
corrleftright=corrcoef(dotsonlegs(:,1),dotsonlegs(:,2))
covleftright=cov(dotsonlegs(:,1),dotsonlegs(:,2))
%note the variances are on the diagonal
vardots=var(dotsonlegs)

% a few other things:
maxdotsbothlegs=max(max(dotsonlegs))
cumsumdots=cumsum(dotsonlegs)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Linear algebra functions
%  the most important thing for us will be the function that finds the eigenvalues
%  and eigenvectors of a matrix: no small feat...
% routines at the core of this were the public domain fortran programs called
% EISPAK
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all;
close all;
a=[.2 0   0  23  25
   .1 .1  0  0   0
    0  0 .5  0   0
    0 0  .2  .7  0
    0 0 0    .2 .3]

%there are two ways to to call this function
vals=eig(a)

[vecs,vals]=eig(a)

%we will return to this in detail later in the course


% Plotting basics (Roughgarden, p. 438)
clear all
close all
a=1:10
b=2:2:20

figure(1)
plot(a)

figure(2)
bar(a)

figure(3)
hist(a)

figure(4)
plot(a,b)

figure(5)
semilogy(b)

%now add some stuff to figures
%first close the old ones 
close all

figure(1)
plot(a)
xlabel('vector index')
ylabel('a')
title ('plot a vector as the single argument')

figure(2)
bar(a)
xlabel('vector index')
ylabel('a')
title('bar graph for a vector')

figure(3)
hist(a)
xlabel('bin index')
ylabel('frequency')
title('histogram')

figure(4)
plot(a,b)
xlabel('a')
ylabel('b')
title('plot y vs. x')


figure(5)
semilogy(b)
xlabel('vector index')
ylabel('log10(b)')
title('plot a vector on a log scale')


%there is a lot more you can do...we will learn some of it but for now:
close all
clear all

%but before we go...there is also "ezplot" to see the form of a function
ezplot('x^2')

ezplot('exp(x)')

ezplot('log(x)')


% "differential equations..." (Roughgarden, p. 440)
% we will do this two ways
%  symbolically
%  numerically

%why?  sometimes we know something about the rate of change
% and we want to know what the pattern across time will look like...
% for example
% we may know that "populations grow at a rate proportional to their size"
%    the equation that says this is
%                    Dn/dt = r * n
% how can we  see what happens over time given a certain starting point?

%symbolically this problem is:
dsolve('Dn=r*n', 'n(0)=a', 't')


%numerically this problem can be addressed by writing a function 
% we can call it  "expgrow"

function ndot=expgrow(t,n)
global r; %declares the r as a non-secret, global variable to be used in  function and command space
ndot=r*n; 

%open an new mfile
%copy these lines there
% save it as a user-defined function "expgrow.m"

%NOW, use it in conjuction with built-in function 'ode23'
help ode23

clear all
global r;
r=0.5;

% try it for  time series that goes from 0 to 10 and initial value=2
[t,n]=ode23('expgrow',[0,10],2)
plot(t,n)
xlabel('time')
ylabel('population size')


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Think of a biological problem 
%
% Write your own function file
% Write your own m-file script that calls the function
% Solve the problem
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%