MATLAB

=About MATLAB= MATLAB is a numerical computing environment and programming language. Created by The MathWorks, MATLAB allows easy matrix manipulation, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs in other languages. Although it specializes in numerical computing, an optional toolbox interfaces with the Maple symbolic engine, making it a full computer algebra system. It is used by more than one million people in industry and academia and runs on most modern operating systems, including Windows, Mac OS, Linux and Unix. The current version is MATLAB 7.1 Service Pack 3. It is available for commercial use for approximately US$2000 and US$100 for an academic license with a limited set of Toolboxes. — from Wikipedia:MATLAB

For Bioinformatics and Systems Biology, there are useful toolboxes available:
 * SBMLToolbox v2.0.2, by the SBML Team download from sourceforge
 * SimBiology 1, from Mathworks
 * Bioinformatics Toolbox v2.2, from Mathworks

=Elementary Tutorial= The following is an unstructured quick bare bones introduction to some of MATLAB's important commands/functions. MATLAB is reasonably easy to use and intuitive to pick up, so go ahead and experiment and learn!

GENERAL COMMANDS

 * clc: clears the command window
 * clear: removes all variables from the workspace
 * clear all: removes all variables, globals, functions and MEX links
 * ls: Returns listing of the current directory (works on Windows as well)
 * !ls: Use ! to run shell commands both in windows and linux. In windows, !explorer . maybe a useful command, to browse the current directory
 * pwd: Shows the present working directory (in Windows too)
 * help : gives a brief documentation
 * doc : gives the complete documentation
 * lookfor : looks for the string in the first comment line of the HELP text in all M-files (MATLAB functions/scripts) found on MATLABPATH.
 * whos: It lists all the variables in the current workspace, together with information about their size, bytes, class, etc.
 * whos -file : lists the variables in the specified .MAT file.
 * type : 'cats' or echoes the file to the screen

ESSENTIAL FUNCTIONS
>> 1:7:100   ans = 1    8    15    22    29    36    43    50    57    64    71    78    85    92    99    >>       >> f=inline('sin(x.^2+y.^2+x.*y)') f = Inline function: f(x,y) = sin(x.^2+y.^2+x.*y)
 * size: returns the size of the matrix
 * length: returns the maximum dimension of a matrix
 * A':transposes a matrix
 * 1:7:100 : Generates numbers upto 100 beginning from 1 in steps of 7.
 * linspace: linspace(a,b,n) generates n linearly spaced points between a and b
 * logspace: generates logarithmically spaced points
 * : : : (colon) is a wildcard (similar to * in shell). e.g. A(:,4:end). This returns all rows of A and columns from 4 onwards. end is a keyword that tells MATLAB to take elements untill the last column is reached.
 * Entering a vector: x= [1 2 3 4 5]
 * Entering a matrix: I_4= [1 0 0 0; 0 1 0 0 ; 0 0 1 0; 0 0 0 1]</tt> Instead of a semi-colon, line-breaks (Enter) will also work.
 * A([1 3 5 7],[2 4 6 8]): any submatrix can be addressed; it can even be assigned a value. For example A([1 3 5 7],[2 4 6 8])=0</tt> sets those elements to zero!
 * prod(size(x)): a simple construct for calculating the number of elements in x.
 * find(A): returns the indices of the non-zero elements in a vector. For matrices, it is better to give two outputs: [i,j]=find(A)</tt>.
 * sqrt(i): complex variables are handled seamlessly. i and j are both square roots of -1. Avoid i,j in loops, therefore.
 * inline: Can create functions like $$f(x,y)=\sin(x^2+y^2+xy)$$. e.g.:
 * sparse: Use whenever memory problems are possible due to the usage of large matrices. MATLAB intelligently applies the corresponding algorithms. Typically, just saying A=sparse(A)</tt> may cause your code to run 10 times faster, if your matrix is large and sparse.

SPECIAL MATRICES
Note that a single argument usually creates a square matrix for the above functions.
 * ones(m,n[,p,..]): Generates a matrix of the corresponding size with all elements 1.
 * zeros(m,n): Generates a matrix filled with zeroes; useful for initialisation some times.
 * rand(n): gives an n-by-n matrix with random entries, chosen from a uniform distribution on the interval (0.0,1.0).
 * eye(n): eye(n)</tt> is the n-by-n identity matrix.
 * eye(m,n): is an m-by-n matrix with 1's on the diagonal and zeros elsewhere.
 * diag: Can be used to extract the diagonal of a matrix or even create diagonal matrices.

INPUT/OUTPUT RELATED
>> x=1 x = 1   >> x=1; >>    >> x    x = 1   >> disp(x) 1   >>
 * x=1 vs x=1;: ; suppresses the output
 * disp: Suppresses some of the extra lines...
 * sprintf: similar to sprintf</tt> of C/++.
 * fprintf: similar to fprintf</tt> of C/C++. Use either this or sprintf to print on stdout (command window).
 * load: loads files. Simplest file is one with equal number of entries in each row separated by space or tab. MATLAB will crib if number of entries in any row are different from others.
 * importdata: Useful for loading text files and other files you would usually load using 'File>Import...'
 * xlsread: Can read Microsoft Excel files directly
 * save: Without filename, all variables are saved to 'matlab.mat</tt>'. With filename, all are saved to the corresponding file. You can even specify what variables to save, even with a regular expression!
 * format long: Displays many digits of precision (default format shows only 4 decimals)
 * format rat: Displays output as rational numbers!
 * format: Resets format

MATRIX FUNCTIONS
MATLAB has a host of matrix-related functions. lookfor should come handy for searches.
 * inv: Inverse of a matrix.
 * inv vs \: Inverse of a matrix is expensive to compute and to be avoided. MATLAB supports the beautiful slash operator: Just as for $$ax=b$$, $$x=b/a$$, even for matrix equations (simulataneous equations), i.e. when $$\mathbf{A}\mathbf{x}=\mathbf{b}$$ MATLAB allows, x=A\b (try to see it is a b 'over' A!)
 * *: Matrix multiplication
 * .*: Element-wise multiplication
 * . : . does element-wise operations.
 * For example: A^5</tt> vs A.^5</tt>: A^5</tt> multiplies A with itself 5 times (matrix product; possible only when A is square) whereas A.^5</tt> raises every element of A to the 5th power.

$$e^\mathbf{A}=\mathbf{I}+\mathbf{A}+\frac{\mathbf{A^2}}{2!}+\frac{\mathbf{A^3}}{3!}+\dots \infty$$ This is of great use while solving systems of differential equations.
 * norm: Various norms maybe calculated for a matrix.
 * expm(A) vs exp(A): exp(A) does $$e^{a_{ij}}\ \forall \ a_{ij} \in \mathbf{A}$$, whereas expm is defined as:
 * svd: Singular Value Decomposition
 * eig: Eigenvalue/Eigenvector finding
 * chol: Cholesky Decomposition

POLYNOMIAL FUNCTIONS

 * polyval(P,X): returns the value of a polynomial P evaluated at X. P is a vector of length N+1 whose elements are the coefficients of the polynomial in descending powers. $$ Y = P(1)*X^N + P(2)*X^{N-1} + ... + P(N)*X + P(N+1)$$. If X is a matrix or vector, the polynomial is evaluated at all points in X.
 * compan: Can be used for solving polynomial equations. Beautiful function! The eigenvalues of the companion matrix of a polynomial are the roots of the polynomial!
 * Example: Calculate the cube roots of unity, i.e. the roots of the equation $$x^3-1=0$$:

>> eig(compan([1 0 0 -1])) ans = -0.5000 + 0.8660i -0.5000 - 0.8660i 1.0000   >>

ADVANCED

 * cell
 * symbolic

FIGURE RELATED FUNCTIONS

 * figure:Opens a new figure window
 * clf:Clears the current figure
 * close:Closes the curent figure window
 * close all:Closes all figure windows
 * close all hidden:Closes all figure windows including those hidden (clustergrams are strangely 'hidden')
 * close (1:gcf) : a workaround for closing all including hidden I guess...
 * hold on:Holds the current plot on the figure window
 * waitforbuttonpress:Waits on the plot during a script/function, till a button is pressed, before further execution
 * subplot: Create a grid of plots in a single figure.
 * plot: Versatile plot function.
 * ezplot: Can plot anything you imagine! Just try ezplot('sin(x*y)')!
 * surf: Plots a surface
 * mesh: Plots a mesh (similar to surf but ~no solid)
 * title: Titles a figure
 * xlabel: Add X-Axis title
 * ylabel: Add Y-Axis title
 * colormap: Change the colours of plots/images
 * spy: Sparsity plot of a matrix — puts blue dots wherever entries are non-zero. Shows number of non-zeroes as well.

PROGRAMMING

 * function name and file name should be the same
 * % comments
 * help and comments
 * help myfun will return the first block of comments in myfun.m
 * (see the following example as well)

% my function to calculate max of two numbers %    function a=mymax(x,y) if (x>y) a=x; else a=y' end return %optional: whatever a is at the end of the %function file, that value will be returned.
 * return is implicit. For example
 * programs vs scripts
 * Scripts don't have a function keyword. They're just 'scripts', which run a series of commands. The variables are all in memory after the execution of the script


 * if (x~=1)
 * This is a significant difference from C/C++: instead of x!=1, use x~=1.


 * elseif
 * end - ENDs any blocks - for, if, elseif, while etc (roughly equivalent of } in C++; there is no { if (x~=1) itself implies — loosely— if(x!=1){ of C++))
 * for k=1:100 (not i!!)
 * All addressing is from 1:n
 * handling arguments
 * Multiple arguments can be specified in the function definition

function [mean,median,mode,variance]=statistics(x)
 * for loops is bad programming... there should be vectorisable alternatives!!

INTEGRATING ODEs
Matlab has a number of excellent tools for numerically integrating systems of ordinary differential equations. Here is a brief HOWTO:
 * Step 1) Convert your system of high order, non-autonomous ODEs into autonomous first order ODEs. For example, if the equation is d2y/dt2 + a*dx/dt + b*x + c = f(t), then create auxillary variables y1 = t, y2 = x, y3 = dx/dt. Then the time derivatives of these auxillary variables will be dy1/dt = 1, dy2/dt = dx/dt = y3, dy3/dt = d2x/dt2 = -a*y3 - b*y2 - c + f(y1), which is a system of first order ODEs.


 * Step 2) Create a Matlab M-file with your equations. If we continue with the example, the file should look like:

function dY = MyMFile(t,Y) a=1;b=2;c=3; f = @(t) sin(t); dY(1) = 1; dY(2) = Y(3); dY(3) = -a*Y(3) - b*Y(2) - c + f(Y(1)); dY=dY';

Save this file with name 'MyMFile.m'. Here, we've also used an "anonymous function" called 'f'. It's a fast way of creating a function with one or more parameters.


 * Step 3) Call a Matlab ODE solver function using the M-file we just created. For example,

[T,Y] = ode45(@MyMFile, [TStart TEnd], Yo);

This will use the 'ode45' solver to integrate the ODEs in the MyMFile M-file from a beginning time of TStart to an ending time of TEnd with an initial condition Yo. So ...

Yo = [0;-1;0];

Remember that the length of Yo should be equal to the number of ODEs in your system. You can also use [TStart:TStep:TEnd] to force Matlab to output the solution at TStart, TStart+TStep, and so on until TEnd. Otherwise, the solver will dynamically choose an optimal time step and output the solution at these times.

Matlab comes with at least 7 ODE solvers: ode45, ode23, ode113, ode15s, ode23s, ode23t, and ode23tb. Each uses a different numerical method to numerically integrate the system of ODEs. The order of the method is given in the name. The 'ode23' solver is a 2nd-3rd order method with global error proportional to &Delta;t2 or &Delta;t3, depending on the system of ODEs. If the last letter of the solver is 's', then the numerical method is particularly good for "stiff" systems. A stiff system of ODEs has at least two very different timescales (fast and slow). Stiff solvers are more efficient at solving stiff systems of ODEs. The best practice is to try the 'ode45' solver first and (if it's super slow) then switch to the 'ode23s' or 'ode15s' one. The syntax for each ODE solver is very similar (if not exact).


 * Step 4) Plot the result!

plot(T,Y)


 * There are many available options to make integrating large systems of ODEs more efficient, such as providing the explicit Jacobian or describing the band structure of the Jacobian. Matlab also supports the execution of events. If you're solving a partial differential equation with finite differences, Matlab can also solve the resulting M(t,y)*dy/dt = f(t,y) problem, where M is the "mass" matrix.

BIOINFORMATICS TOOLBOX

 * clustergram: useful for clustering microarray data etc. Has a lot of options.

Symbolic Toolbox
Store an equation in GrowthEq using t as an independent variable syms t; GrowthEq = K / (1 + exp(-r* (t - l )));

Differentiate with respect to t GrowthRate = diff(GrowthEq,t);

plot the function with t = [0..50] ezplot(GrowthEq,[0 50]);

SBMLToolBox

 * Rougly a libSBML for MATLAB! More on http://sbml.org/software/sbmltoolbox/

=Troubleshooting=

Running Matlab remotely on OS X

 * X11 Forwarding in Matlab on Mac OS X
 * Using X11 display with Matlab 7.4 on OS X
 * make sure X11 forwarding is enabled in /etc/sshd_config</tt> on the remote machine (X11Forwarding yes</tt>)

=External Links=
 * Loren on the Art of MATLAB
 * MATLAB WikiBook