MATH-680: Mathematics of genome analysis Introduction to MATLAB

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1 MATH-680: Mathematics of genome analysis Introduction to MATLAB Jean-Luc Bouchot, Simon Foucart January 14, Introduction As for the case of R, MATLAB is another example of interpreted programming language. Therefore the code is not compiled but translated on-the-fly by the interpreter. Note that MATLAB being very expensive, many open source copy have appeared around on the web. The most curious of you might want to google for Octave or Scilab. The first one is more or less a copy of MATLAB with a pretty much similar syntaxe while the second one is somewhat different (and the community working on it is much smaller). 2 Data types Dealing with different data in MATLAB is somewhat easier than with R. Strings and in general characters will barely be used except as an argument of a function (if for instance, you choose to use another method than the usual one). Most of the data you will be working with will be numeric. However MATLAB differentiates between floating point (real numbers, quotient etc...) and integers. Operations on integer are (almost) always much faster then on floating points numbers. Matrices are particularly powerful in MATLAB (whence the name: MATrix LABoratory) and you should try to use matrix representation whenever you can. Matrices in MATLAB are not necessarily two dimensional and can be of any dimension (similar to arrays in R). They are ordered in a column wise manner and are also considered as a vector. Therefore, if M denotes an m n matrix (a matrix with m rows and n columns), if you try to access the element on the first row and the i th column (1 i n) you can equivalently write: myvalue = M(1,i); myvalue = M( (i-1)*m + 1); If you want an element on the first column located on the i th row you can write myscdvalue = M(i,1); myscdvalue = M( (1-1)*m + i); % actually, M(i) is sufficient Exercise 1. Assume M is a matrix as above, and you are given two integers i and j such that 1 i m and 1 j n. How would you access the element M(i, j) using the vector notation of the matrix? A drawback of matrices is that all columns have to have the same size (or all rows have to have the same length, depending on how to look at the matrix). To overcome this issue one can use cells instead of matrices. They are used with curly braces { } instead of braces. The advantage of using cells is that it allows to store vectors of data of different sizes DNAsequences{1} = ATTCGTA ; DNAsequences{2} = AATGCCTATTGCA ; 1

2 There is no way to deal with such an example using only matrices in MATLAB. However when it comes to k-mers representations, the number of k-mers (4 k ) is already known and hence, an individual can be represented by a fixed size vector and all specimens will have the same size. Finally a last aspect for using different data is by defining what is called structures. A structure is an object where different fields can be specified. It can be understood as a list in the R language. Each field of a structure has a particular key or name and can be assigned anything (a matrix, a cell, a vector or even another structure). astudent.name = John Doe ; astudent.birthyear = 1990 ; astudent.address = 123 Foobar avenue ; In this example the variable is astudent while the different fields represent different characteristics of that variable. The boolean type is difficult to use in MATLAB. Even if a true and a false keywords exist, a number 0 will be considered as a boolean false whenever asked and any non zero number is considered as being a boolean true. Try copy pasting the following statements in your MATLAB window to convince yourself. if( 0.5 ) disp( 0.5 is true ) % appears only if the condition in IF is true end if( 0 ) disp( 0 is true ) % appears only if the condition in IF is true end As numeric variables or values can be sometimes used in conditions, some care should be taken in some places. 3 Basic operators Understanding the different operators in MATLAB is rather straigthforward. The = sign is used for assignment (a=5 affects the value 5 to a variable a) while the double one == is used for comparison (a==b) and returns a numeric (used as boolean in this case) 1 if the statement is true (a indeed equals b) and 0 (a boolean false) otherwise. More general comparisons work similarly using <=, <, > and >=. The negation is done using the tilda sign ~: a = 2; b = 1; myfirstbool = (a >= b) % should display 1 (a boolean true!) mysecondbool = (~(a < b) ) myfirstbool == mysecondbool The last statement compares the two boolean variables and should always return a 1 given any numbers a and b. As you might have noticed so far, MATLAB s syntax is quite different from the one of R. For instance, we have used the semicolon ; at the end of some lines and not at others. This is not random. As default MATLAB displays every result it computes. In order to avoid this annoying effect, the semicolon added at the end of a line tells MATLAB to stay quiet. Another major difference is dealing with comments. While R uses the # command to tell the interpreter to ignore everything afterwards, MATLAB uses %. Hence a=5 %and this will be ignored is equivalent to a=5. Remember to always comment your code. What is clear for you might not be that clear for someone else, even future you. A last difference with R is when asking for help. MATLAB uses the help command: help help for help about the help function. 2

3 Arithmetic and logical operators As for any other programming languages, MATLAB comes with a certain amount of built-in functions. Here are the few you really need to get used to. More advanced functions (such as singular value decomposition) can be studied later for more complex topics. Operations are used in a classical way: a=4; b=5; product = a*b sum = a+b quotient = a/b difference = a-b Note that the choice of name for a variable can be important in some cases. For instance in the previous example, the choice of sum on the fourth line is not smart. As we will see later sum() is also the name of a MATLAB function and this instruction will overwrite the definition of the function making it unavailable for the rest of the runtime (or until you clear the workspace). Note that some variables are also predefined in MATLAB and should not be overwritten (unless you are sure not to need them later), for instance: i, j as two representations of the same complex unit (the square root of 1). MATLAB will not tell you if you try to overwrite an already existing variable; so be careful! Variable creation and assignment When working only with single valued variable (any number in a usual sense or string) you do not need to take any care and any myvariabe=acertainvalue will work perfectly fine. However creating vectors and matrices can be done in many different ways. The first way is by defining directly the vector you will work with. For this, you need to use the brackets [] and write the values inside: myfirstvector = [ ]; This instruction creates a row vector containing all numbers from 1 to 4. Another (somewhat clearer) way to separate the different values is by using a coma. Hence the previous formulation is equivalent to [1,2,3,4] and to [1, 2, 3, 4]. Creating a column vector (i.e. oriented vertically) can be achieved using semicolon ; instead of periods or by transposing (using the command) a row vector. myfirstcolvector = [1; 2; 3; 4] myfirstcolvector2 = myfirstvector yield two exact same vectors (try it to convince yourself!). While it might not appear clear why MATLAB distinguishes between row vectors and column vectors, we will see on the example of the scalar product how these considerations make sense. In a more general sense, the brackets are used to concatenate or stack variables together. For instance if we need to add the number 5 at the end of the vector ( end means to the right for the row vector and to the bottom for the column one) we can reuse the previous one and just concatenate it with the value you want to add myextendedrowvector = [myfirstvector, 5] myextendedcolvector = [myfirstcolvector; 5] Note that in the first case we have used the coma separator, meaning that we add to the right, on the same row as the rest, while in the second case, we have used the semicolon, which means add to the bottom, in the same column. Exercise 2. Now we can see why the column and row vectors are different. Try for instance to add a value (say 6) to the right of column vector or to the bottom of the row vector. What is happening? What is the problem? Whenever you wish to concatenate variables or values together, you need to make sure the dimensionality are consistent. If you use the concatenation to the right, you want to make sure the number of rows are the same and when stacking together, the number of columns should be the same. 3

4 Exercise 3. Create a two row vectors of length 5 of you choice. Using the concatenation as above create 1. A vector of size A table of size 2 5. (Here you have created you first matrix!) 3. A vector of size (two equivalent ways) 4. A table of size Two tables of size 5 5 and 2 2 ( hint: try multiplying matrices together. Be careful about dimension conditions) Exercise 4. Try adding the value 0 at the beginning of the previous column and row vectors A last possibility to create a row vector is by using the sequence operator: :. It works as follows, myvector=firstidx:increment:lastidx. It creates a row vector with its leftmost value starting at firstidx and increasing (or decreasing for negative steps) step by step by increment until the maximum value lastidx is reached. If the increment is omitted, it will be replaced with 1 hence firstidx:1:lastidx and firstidx:lastidx are equivalent. otherrowvector = 1:4 othercolvector = (1:1:4) The symbol : is particularly important in MATLAB as it allows to select a complete row or vector, as we will see later. Creating a matrix can be done the exact same way by first filling the first row, then stack another row, and so on mymatrix = [1, 2, 3, 4; 5, 6, 7, 8] creates the following matrix: mymatrix = ( It is common to separate with a new line whenever we add a new row to the matrix (but this is just for the beauty of the code) mymatrix = [1, 2, 3, 4;... 5, 6, 7, 8] The dots at the end of the line are not mandatory (in this case) but it helps keeping track that the statement is not finished yet. Once again, you can deal with concatenation and stacks as for vectors and we suggest you try practicing by yourself. Some operations can be extended to the case of matrices. Hence, the following statements make sense C = A+B % A and B have same sizes D = A-B % A and B have same sizes E = A*B % nb of columns of A == nb or rows of B. What size does E have? F = A/B % B is invertible, dimensions are coherent G = A^2 % A is square On top of that, the component wise operations (for instance exponential, quotient and multiplication) can be done by adding a dot in front of the operator: A.*B is a matrix C of the same size as A and B whose components are the component wise products of each component. In a more formal way, say A and B are m n matrices, then C is a m n matrix too such that 1 i m, 1 j n, C(i, j) = A(i, j) B(i, j) ) 4

5 4 Working in a MATLAB environment Data creation can be done in a smarter way than the ones presented in the previous section. Unfortunately adding rows to a matrix during runtime is time consuming, and it is always better, if known, to preallocate the memory for the whole matrix beforehand. For this you can create different matrices using built-in functions m = 5; n = 10; % define the matrix size amatrix = zeros(m,n); % creates a 5x10 matrix filled with 0s anothermatrix = ones(m,n); % creates a 5x10 matrix filled with 1s rdmmatrix = rand(m,n); % creates a random matrix with values iid, uniform distribution rdmmatrix2 = randn(m,n); % same as above but normally distributed Once the memory for the matrix has been allocated you can always access (and modify as you wish) using the mymatrix(i,j) notation. Another useful notation for accessing and modifying matrix elements is the : symbol. It is interpreted as take everything along this direction. For instance M(:,i) is interpreted as all the components along the first direction and at the i th position in the second, or in other words, the i th column of the matrix M. You can also select subsets of the matrix by using vectors of indices inside the brackets M(1:3:12,2:2:16) will select only some predefined values of the matrix. Exercise 5. Create a diagonal 5 5 matrix D with the number 1 to 5 on the diagonal: D = now try to use the function diag() for the same purpose. (Remember to use the help: help diag) Dimensional information It is often very useful to recover information about the size of the data you are manipulating. For instance the length() function returns largest dimension of the matrix passed as argument. The size() command returns a vector containing the different lengths in each dimension. The values can be stored independently in separated variables. You can also give a second parameter to precisely recover the size of the given dimension m = 10; n = 20; % size parameters A = randn(m,n); % create a random matrix mysize = size(a) % returns a vector [10, 20] [l1, l2] = size(a) % affects the length of the matrix along dim 1 to l1 [l1,l2,l3] = size(a) % third output will be assigned a value of 1 scddimension = size(a,2) % recovers only the length of the scd dimension A last function used sometimes is numel() which tells you the number of elements in a certain variable. This is sometimes useful to check whether an assignment occurred as expected or if something wrong happened. In general, for an m n q array X, we have numel(x) = m n q. Vector information (Not the best name we can give though...) Many pieces of information can be recovered (almost) instantaneously using basic functions. Most of the function will work on a vector basis. For instance, the function sum(v) computes the sum of all elements of the vector v. However if the argument is a matrix, all the operations (in this case summing up) will be done along the first dimension (the sum function applied to a matrix returns a row vector where each element is the sum of all the element of the associated column. 5

6 amatrix = [1, 2, 3; 4, 5 6]; firstsum = sum(amatrix); firstsum2 = sum(amatrix,1); firstsum == firstsum2 % should return a vector of 1s isequal(firstsum,firstsum2) % compares the two vectors GLOBALLY -> important function! sum(amatrix ) == sum(amatrix,2) % should be 1 too As you can see from the examples, the second parameter tells along which direction to sum up. In some cases (for instance when one needs to have cumulative probability distribution) it is interesting to keep track of the cumulated sums. See the function cumsum() for that, which has a similar behavior as the normal sum regarding its parameters. Exercise 6. Compute the sum of all elements of a 2 dimensional matrix. The product command prod() has the exact same behavior as the sum. The functions min() and max() can be used in a similar way: max(x,dim). Exercise 7. Find the range of a matrix M using the min() and max() functions An interesting behaviour of the min and max functions is that it allows to recover the position of the max or min in the vector. A = randn(5,10); % Random matrix [minvalues, indices] = min(a) % Recover min values and position along each columns Exercise 8. Recover the maximum value and its position in a two dimensional array. (this is a hard task for novice programmers) Statistical information can be computed out of vectors using the var() and std() functions. As before, it works along the first dimension and a second argument can be passed in order to change the direction of computation. Together with the mean() function you should be able to get most of the basic statistics out a vector. Exercise Compute numel() without using this function 2. Compute length() without using this function Hint: Remember the the dim() function can be very useful sometimes. These two computations should work for any kind of data (matrix, vectors or even much higher dimensional data) Exercise 10. Assume you are given two column vectors x and y of the same size. Compute the scalar product of these two vectors without using loops (we don t know how to implement loops anyway). Remember, the scalar product is defined as the real number α = x i y i. One method uses only one line of code and another one uses two. Once we will see how to use loops, we can compare the efficiency of these methods compared to the one using loops for large vectors. Useful functions are already implemented in MATLAB. And therefore you can use without any problem exp() as the natural exponential function. If applied on the matrix, it corresponds to the coordinate wise exponential function. sqrt() computes the component-wise square root of a matrix. Note that if you are looking for the square root of a square matrix X (i.e. a matrix A such that A A = X), you want to use the function sqrtm(). Exercise 11. The square root and the square functions can be considered as inverse of each other. As an effect applying first the square root then the power two function should return the original number. As an example, try the following code a = 5; recovereda = (sqrt(a))^2 Does it give the expected result? Now compute the difference a-expecteda. What is the result? Is it normal? Why? Try the same process with the following matrix: A=[1,2,3;4,5,6]. Does it work? Why? Can you correct the code to get the expected solution? 6

7 Integer functions are function returning integers depending on the floating points. floor() returns the largest integer smaller than or equal to a given parameter, ceil() returns the smallest integer greater than or equal to a given parameter and round() returns the closest integer to a number, regardless of it being bigger or smaller. Exercise How would you compute the modulo (i.e. the remainder of the integer division)? 2. Pick a random integer in the range [0, 255] and check whether it is odd or even. 5 Scripts and functions As for the case of R it is often very useful to separate certain sets of instructions to another file and to call that file in the interpreter. Once again, calling a script is be equivalent to a copy-paste of the instructions in the file while calling a function you have created allows you to modify the variables locally and get the computed results in some other variables. In MATLAB the filenames end with the.m extension. Assume you want to normalise a certain vector x to have 0 mean value and unit variance and that this vector x will change along time. You can write the following separate donormalisation.m file: mu = mean(x); xnormalised = x - mu; sigma = var(xnormalised); xnormalised = x/sigma; Now in the flow of instructions, you could have: % x is already assigned as output of a function of some initilisations donormalisation; % Check the content of your workspace x = rand(1,123); donormalisation; % The content of the workspace has not changed, but their values have. For this kind of task the use of a separate function would be much smarter, as it avoids to overwrite variables and make it easier to remember names. Indeed, if you do not remember how you called the vector in the donormalisation.m file, you might end up initialising a vector named y which will not be affected by the script. To overcome this problem, modify this file by adding the line function [xnormalised, mu, sigma] = donormalisation(x) By doing this, you tell MATLAB that donormalisation is a function of the variable x (internally to that function) and that [xnormalised, mu, sigma]are the three outputs of this function. You do not have to use all of them if you do not want to: y = rand(1,150); % assign a random vector donormalisation(y); % does not do anything normalisedvector = donormalisation(y); % stores the normalised vector in this new variable [yn, avgy, deviationy] = donormalisation(y); % keep track of the interesting values of y In case you have optional arguments to pass, you need to use the nargin statement to check the number of argument or the isempty() function together with a if...then...else...end conditional block, as we will see in the next section. Loops and conditions are useful in MATLAB. They allow to repeat processes of to process a certain way under certain conditions. However, it is important to always vectorise your calculus as much as you can, as we will see in the next exercise. As for R we have access to two kinds of loops: the for loop and the while loop. As always for such cases, while is used whenever the stopping criteria can not be calculated beforehand: for instance while(convergenceisnotdone) or while(dbnotempty). Note that in the case of the database not empty case, we could actually recover the size of that database using size() and pass it into a for loop as follows: 7

8 % load a database represented as a matrix X [nbindividuals, lengthindividuals] = size(x); for oneindex=1:nbindividuals % do something to X(oneIndex,:) end % loop until nbindividuals are done You can also stipulate much more complicated sets to iterate on by simply writting for oneidx=acertainset where this certain set as already been defined (as a row or column vector). Note how you end your loop with the end statement. This is also the case for the while loop. It will also be the case to end a conditional block. Exercise 13. Create a myslowscalarproduct(x,y) function implementing the scalar product using a for loop. (you will basically iterate along the dimension of the vector and add each contribution to a auxiliary variable). Once this is done, make sure you get the correct result on some easy examples. Then copy-paste the following lines of code in your interpreter and conclude. vectorlength = ; % A big number; play with it! firstvector = rand(vectorlength,1); scdvector = rand(vectorlength,1); % second random number tic % this is a function to set the timer scalarproduct1 = y *x toc % end of the timer tic scalarproduct2 = sum(x.*y) toc tic scalarproduct3 = myslowscalarproduct(x,y) toc As a last tool for MATLAB programming we want to introduce the if... else... statement. This allows the user/programmer to distinguish between different use cases which can not be predicted beforehand. The best way to understand how it works is by observing the following example arandomnumber = rand(1); if( arandomnumber < 1/3 ) disp( The random number is pretty small ) else if( arandomnumber > 2/3 ) disp( The random number is pretty high ) else disp( The random number is close to 0.5 ) end; By running these few line a certain number of times, we can see that depending on the value of the random number (modify the code to display it if you wish) the interpreter will display different sentences. 8