cycle6_week2.m

% These examples are adapted from the YAGTOM (yet another guide to MATLAB)
% section "Matrix Operations"

% Matrices are key in matlab as we will use them to store most types of
% data. For example an image can be represented as a 2d matrix and a time
% course of a signal can be represeted by a 1d matrix.

% Our goals are to introduce commands which generate matrices with different 
% properties (eg ones, zeroes, randn, rand, etc)
% and those that report properties of the matrices (ie, number of elements
% or the mean value, standard deviation).  We also learn to isolate specific 
% elements of a matrix. This is known as indexing.

% http://ubcmatlabguide.github.io/html/matrixOperations.html

% First a review from week 1: the semi colon

% If you have got a handle on ; skip to line 61

% Recall "Bonus: run the same command again with a semi colon symbol ; at the end
% of the command.  What does the semi colon do?"

% Explanation: I wanted to point out the semi colon early on as a lot of people 
% find its use confusing when beginning to use Matlab. Common questions arise: 
% "Do I need to put a semi colon here??"

% The role of a semi colon when following a command is to prevent matlab from 
% displaying the result of a command in the command window. Recall the ones 
% command which gives a matrix filled with ones. Evaluate:

a=ones(2,2)

% In this case the variable a is assigned a 2 row x 2 column matrix
% filled with ones. It should now be in your workspace. And notice the 
% ones are printed in the command window.

% Evaluate: 

clear a
a=ones(2,2);

% The command clear a deletes a from the workspace. The next line creates a and 
% assigns it ones again. You will see it in the worksapce, but if you look 
% in the command window, the values have not been displayed.

% The semi colon suppresses output in the command window when assigning
% values to variables.

% In general evaluating commands without a semi colon is helpful when you
% are checking your work and need to see a particular variable's value or a
% single element or a few elements of a matrix. For example you might need
% to check the value in one of the elements of a. Evaluate:

a(1,2)

% However, if you are writing a script to automate your data analysis and
% values will be assigned to variable a 100's of 1000's of times, you probably want
% the semi colon as displaying them all to the screen is not helpful. (NB.: Another
% role for semi colons is to separate rows of numbers when manually
% entering matrices. We will see this below.)

% Ok, enough about ; ... on to creating matrices.

% As we saw last time as well, variables can be created directly by
% assigning them values using the = operator. This can be a single
% numerical value, for example evaluate:

a=1

% Or a matrix, for example: (NB: Here ; is used to separate rows of the
% matrix elements. It's not often we manually type numbers in this way,
% but it is useful for examples.)

a=[1 2 3; 4 5 6 ; 7 8 9]

% An important shorthand exists in matlab for generating matrices which
% contain lists of numbers which can be generated by a simple rule.

% A typical command looks like this. Evaluate:

a=1:10

% This assigns a a list of numbers starting at 1 and ending with 10.
% Incrementing by 1. Plus 1 is the default increment, but you can specify
% it. A typical command looks like:

a=1:.2:5

% which means start at 1 and go to 5 in steps of 0.2, the increment appears
% between the start and end, separated by the colons. Negative increments
% are possible as well.

% 1. Write a command to assign a list of numbers starting at 4 ending at -7
% decrementing by 1 each time.

% matlab also has many built in functions which create matrices for you.
% We introduced zeros and ones last time. Try these:

a=zeros(3,4)
b=ones(4,3)

% There is also randn which fills the matrix with random, normally distributed 
% numbers. We will use this in many examples during the tutorials.

a=randn(2,5)

% and its counterpart rand which generates random numbers distributed
% uniformly between 0 and 1.

b=rand(5,2)

% 2. Use randn to create a matrix called dat of size 1000 columns in a single row.
% We will use this matrix in later examples.

dat = randn(1,10000);

% matlab has many useful built in functions to give you properties of your
% matrices or their contents.

% For example if we needed to know the number of elements we'd stored in
% dat we can use numel:

npts=numel(dat)

% in this command dat is given as the input argument between the
% parenthesis to the function numel.  numel calculates the number of
% elements and the result is assigned to the variable npts by the =

% size is another very useful function.  

% 3. Give dat as the input argument of size in an analogous fashion to
% the command for numel above. Assign the result to a variable called s.
% make sure to leave the ; out so you can see the result directly in the
% command window.

% If you get stuck try:

doc size

% to see examples in the documentation

% As you can see the result (check the workspace) s is itself a matrix.  
% In this instance it has 2 elements the first should be equal to 1 
% (the number of rows) and the second to 1000 (the number of columns of dat).
% It generalizes to more than 2 dimensions as well.

% 4. Try size on a multi-dimensional matrix.  Use the command randn in an
% analogous fashion to the example above to create a matrix with 3 dimensions.
% Make the first dimension have 3 possible values the second 4 and the
% third 5. Assign this to a variable called test3d. Run size on test3d.
% Assign the result to s again overwriting the previous result.

% Matlab also has built in functions to calculate for you the mean and
% standard deviation of values stored in your matrices.

% 5. Use the built-in matlab functions mean and std to calculate the standard 
% deviation and mean of the values you stored in dat above. Store the
% results in the variables dat_std and dat_avg respectively.

% Check doc mean and doc std for help

% Bonus1: Evaluate the following command to generate a figure of a histogram of the
% values stored in dat. This will give you a graphical representation
% of the distribution of the random numbers. We will spend more time on
% figures and plots in week 5.

figure;hist(dat)

% Do the values you get for dat_std and dat_avg make sense based on the 
% properties of samples drawn randomly from a standard normal distribution?

% Bonus2: Write a command to generate 1000 normally distributed random numbers
% with a mean of 5 and standard deviaiton of 2.  Store the result in the
% variable named dat2. 
% Use an analogous command to the figure command above to view a histogram 
% of the values in dat2.
% Use commands mean and std to check that dat2 has the correct properties.
% Hint: Check the following website to find more information on means and
% and standard deviations: http://www.stat.yale.edu/Courses/1997-98/101/rvmnvar.htm

% Matrices can be combined by concatenation.  For example if we create two
% matrices by evaluating these commands:

a=[1 2 3]
b=4:10

% but we want the list of numbers stored in a and b in a single variable,
% we could use the command:

c=[a b]

% Alternatively we can use the built-in matlab function cat. It is useful
% in particular for multidimensional matrices.

c=cat(2,a,b)

% Cat requires 3 input arguments. The first is the dimension in which we
% are combining in this case columns (2). The other two arguments are the 2
% matrices we are combining.

% We will now move on to indexing. Indexing is very important as it is how
% we specify subsets of the elements of our matrices to work with. These
% may correspond to a region of interest in an image or a specific window
% in time in a time course of a signal.

% For example if we wanted to see the first element of our normally
% distributed random numbers stored in dat we could use

dat(1)

% The matrix's variable name is followed by the index we want (in this case
% 1) in parenthesis.

% We can also use a range of numbers using the shorthand we discussed
% above. For instance if we wanted to see the first 10 elements

dat(1:10)

% or elements 15 through 25

dat(15:25)

% Notice that these subsets of dat that we obtain by specifying the
% indices are themselves matrices. The results are stored in the default
% variable ans that we discussed in week 1.

% We can assign them to a new variable name using the equals sign if we
% need to use them later.

% 6. Imagine that dat actually represents a time course of a signal
% (EEG voltage or Ca2+ fluorescence for example) in response to a stimulus.  
% Imagine that the time course is divided into a baseline and a response.
% The baseline consists of the first 250 time points. Write a command to
% isolate the baseline by indexing and storing the result in a new variable
% called base_line.

% Bonus1: Use mean to compute the mean of the baseline time points. Is it
% close to the value you might expect given the properties of the values
% generated by randn?