对于fmri的设计矩阵构造的一个很直观的解释-by 西南大学xulei教授

简介:

本程序意在解释这样几个问题:完整版代码在本文的最后。

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1 .实验的设计如何转换成设计矩阵?
 
2 .设计矩阵的每列表示一个刺激条件,如何确定它们?
 
3 .如何根据设计矩阵和每个体素的信号求得该体素对刺激的敏感性?

 

程序详解:

1.构造hrf  

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hrf_small = [  0   4   2   - 1   0  ];
figure( 1 );
clf;                      
plot( 0 : 4 ,hrf_small, 'o-' );                            
grid on;                  
xlabel( 'Time (in units of TRs, 4s long each)' );
ylabel( 'fMRI signal' );
title( 'This is what an HRF would look like if you measure once every 4s' )

  

 

2.构造刺激序列,并与hrf做卷积:

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word_stim_time_series =  [  0  1  0  0  0  0  ];
object_stim_time_series= [  0  0  0  1  0  0  ];
 
predicted_signal_that_word_would_evoke = conv(word_stim_time_series,hrf_small);
predicted_signal_that_object_would_evoke = conv(object_stim_time_series,hrf_small);
 
figure( 2 );
clf;           
subplot( 3 , 1 , 1 );
                
         
hold on;       
 
h1=stem(word_stim_time_series, 'b' );
h2=stem(object_stim_time_series, 'r' );
               
hold off;
grid on;
legend([h1( 1 ) h2( 1 )], 'Word stim onset time' , 'Object stim onset time' );         
axis([ 1  9  0  1.2 ]); 
ylabel( 'Stimulus present / absent' );
subplot( 3 , 1 , 2 );
plot(predicted_signal_that_word_would_evoke, 'b*-' );                                    
grid on;
legend( 'Word-sensitive voxel would give this fMRI signal' );
axis([ 1  10  - 1.5  7 ]);
ylabel( 'fMRI signal' );
 
subplot( 3 , 1 , 3 );
plot(predicted_signal_that_object_would_evoke, 'r^-' );
  
grid on;
legend( 'Object-sensitive voxel would give this fMRI signal' );
axis([ 1  10  - 1.5  7 ]);
xlabel( 'Time (measured in TRs, i.e. one time-point every 4secs)' );
ylabel( 'fMRI signal' );

 

3.利用两个刺激构造设计矩阵,并绘图

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predicted_word_response_column_vec = predicted_signal_that_word_would_evoke';
 
predicted_object_response_column_vec = predicted_signal_that_object_would_evoke';
 
%%% Now let's look at the actual vectors in the Matlab workspace window
 
predicted_word_response_column_vec  % Because there is no semi-colon after  this ,
                                     % it will display in workspace window
predicted_object_response_column_vec
 
 
%%%%%% Now we can join these two column vectors together
%%%%%% to make the design matrix. We simply put the two columns side-by-side.
%%%%%% In Matlab, you make  new  matrices and vectors by
%%%%%% putting the contents inside [ square brackets ]
%%%%%% Note that to join them together in  this  way, they must be
%%%%%% the same length as each other.
%%%%%%
%%%%%% Because the names of my variables are so  long  and verbose,
%%%%%% the command below spills over onto two lines. In Matlab,
%%%%%% we can split a command over two lines by putting three dots ...
 
design_matrix = ...   % The three dots here mean  "continued on the next line"
  [ predicted_word_response_column_vec  predicted_object_response_column_vec ];
 
design_matrix      % No semi-colon, so it displays in window
 
%%%%%% Translation guide:
%%%%%% In equations, the design matrix is almost always called X
%%%%%% Note that  this  is a capital  "X" .
%%%%%%
%%%%%% X = design_matrix;
%%%%%%
%%%%%% Capitals are typically used  for  matrices, and small- case  is
%%%%%% used  for  vectors.
%%%%%% The only difference between a vector and a matrix is that
%%%%%% a vector is just a bunch of numbers in a row (a row-vector)
%%%%%% or a bunch of numbers in a column (a column-vector),
%%%%%% whereas a matrix is bunch of vectors stacked up next to each
%%%%%% other to make a rectangular grid, with rows *and* columns of numbers.
 
%%%%%% Now let's view a grayscale plot of the design matrix,
%%%%%% in the way that an fMRI-analysis  package , such as SPM, would show it.
%%%%%% To  do  this , we use the Matlab command  "imagesc" .
%%%%%% This takes each number in the design matrix and represents
%%%%%% it as a colour, with the colour depending on how big the number is.
%%%%%% In  this  case , we'll be using a gray colour-scale, so low numbers
%%%%%% will be shown as darker grays, and high numbers are lighter grays.
%%%%%% The  "sc"  part at the end of  "imagesc"  stands  for  "scale" , which
%%%%%% means that Matlab scales the mapping of numbers onto colours so
%%%%%% that the lowest number gets shown as black, and the highest as white.
%%%%%%
%%%%%% For examples of how to use the imagesc command to make
%%%%%% pictures of brain-slices, see the companion program
%%%%%% showing_brain_images_tutorial.m
 
figure( 3 );
clf;                      % Clear the figure
imagesc(design_matrix);   %  'imagesc'  maps the numbers to colors,
                           % normalising so that the max goes to white
                           % and the min goes to black
 
colormap gray;            % Show everything in gray-scale
colorbar;                 % Shows how the numbers lie on the colour scale
                           % Note that the highest number in the design matrix,
                           % which is  4 , is shown as white, and the lowest, - 1 ,
                           % gets shown as black.
title( 'Gray-scale view of design matrix' );
xlabel( 'Each column represents one stimulus condition' );
ylabel( 'Each row represents one point in time, one row per TR (every 4secs)' );

  

 

4.设计矩阵和敏感度矩阵相乘,这里假设某一个体素仅仅对某一种刺激有反应,而对另外的刺激没有反应。

 

%% Now suppose we have a voxel which responds only to words, not to objects.
%% We can calculate how it would be predicted to respond 
%% to our word+object display as follows:
%%
%% Predicted response from word-sensitive voxel = 
%% 1 * Response which word-presentation would evoke 
%% + 0 * Response which object-presentation would evoke 
%%
%% Note that this is how the voxel would be predicted to respond
%% if there were no noise whatsoever in the system.
%% Clearly a real fMRI signal would never be this clean.
%%
%% Now, let's make a "sensitivity vector" for this voxel, 
%% in which each entry will say how sensitive that voxel is to
%% the corresponding stimulus condition.
%%
%% This voxel is sensitive to words, which are our *first* stimulus-type.
%% And we made the predicted word response into the first column of 
%% the design matrix.
%% So, the sensitivity of this voxel to words will be the first element
%% in the sensitivity-vector.
%% 
%% Similarly, the sensitivity of this voxel to the second stimulus-type,
%% which are objects, will be the second element in the sensitivity vector.
%%
%% So, the sensitivity vector for a voxel with 
%% sensitivity = 1 to the first stimulus-type, which are words
%% and sensitivity = 0 to the second stimulus-type, which are objects
%%
%% will be [ 1 0 ]
%%
%% I know this seems trivial !!
%% Things will get more interesting in a minute...

sensitivity_vec = [ 1 0 ]'; % The dash makes this a column vector

sensitivity_vec % No semi-colon, so it displays in window

%% Translation guide:
%% In equations, the numbers in the sensitivity-vector are typically
%% called "beta-values", or sometimes "beta-coefficients" or "beta-weights".
%% The columns of the design matrix are called "regressors" and
%% the value that is assigned to each regressor is the beta-value.
%%
%% Note that in the example above, we are pretending that we already *know*
%% how sensitive our voxel is to the various stimuli, but in the real world
%% we don't know this. We're trying to figure out what stimuli our voxel
%% is sensitive to, using the fMRI data that we collect in the scanner.
%% This will be described more below.
%% In math-speak, that means that we are trying to *estimate* the betas.
%% When people want to distinguish between the true beta-value
%% (which we don't know) and the estimated beta-value that we figure out
%% from our data, then they call the true one beta and 
%% the estimated one "beta hat" (beta with a circumflex sign on top of it: ^

%% [ End of that part of the translation guide, back to the main theme... ]
%% So, we can now express our predicted voxel response in terms
%% of entries in the sensitivity vector multiplied by 
%% columns in the design matrix:
%%
%% Predicted response from word-sensitive voxel = 
%% 1 * Response which word-presentation would evoke 
%% + 0 * Response which object-presentation would evoke 
%%
%% And because of the way we made our sensitivity vector and design matrix,
%% this can be re-written as:
%%
%% Predicted response from word-sensitive voxel = 
%% (First element in sensitivity vector) * (First column in design matrix)
%% + (Second element in sensitivity vector) * (Second column in design matrix)
%%
%% Here's an important bit:
%% The process above, of going through the elements in a vector, 
%% multiplying each element by the corresponding column in a matrix,
%% and then adding up the results of the multiplication, 
%% is precisely what matrix multiplication does.
%%
%% In Matlab, everything is by default assumed to be a matrix,
%% (or a vector --- you can think of a vector as simply a matrix that only
%% has one row or column in it), and every multiplication is
%% by default assumed to be a matrix multiplication.
%% So, to matrix-multiply our design matrix by our sensitivity-vector,
%% we just use the standard "multiply by" sign, which is *

predicted_word_selective_voxel_response = design_matrix * sensitivity_vec;

predicted_word_selective_voxel_response 
% Let's display this vector in the command window,
% by entering it without a semi-colon after it.

%% When we multiply the design matrix by the sensitivity vector,
%% we make the i-th row of the result by taking the i-th row
%% of the matrix, rotating it 90 degrees, multiplying it element-by-element
%% with the sensitivity vector, and then adding that all up.
%%
%% Since the sensitivity vector is in this case [ 1 0 ],
%% multiplying each matrix row by it element-by-element means that
%% we end up getting 1* the first element in each row, and 0* the second
%% element in each row.
%%
%% So, by the time we have gone through all the rows, we have 
%% 1* the first column of the design matrix, plus 0* the second column,
%% which is what we wanted.

%% Let's plot all this

figure(4);
clf; % Clear the figure
subplot(2,1,1); % This is just to make the plots line up prettily
hold on; % "Hold" is one way of putting more than one plot on a figure

h1=plot(predicted_word_response_column_vec,'b*-');
h2=plot(predicted_object_response_column_vec,'r^-');

hold off;
grid on;
legend([h1 h2],'Word-response column vector','Object-response column vector');
axis([1 10 -1.5 7]);
xlabel('Time (measured in TRs, i.e. one time-point every 4secs)');
ylabel('fMRI signal');

subplot(2,1,2);
plot(predicted_word_selective_voxel_response,'ms-'); % Magenta squares
grid on;
legend('Word-selective voxel-response: 1*word-response + 0*object-response');
axis([1 10 -1.5 7]);
xlabel('Time (measured in TRs, i.e. one time-point every 4secs)');
ylabel('fMRI signal');

 

5.假设某一个体素对两种刺激都会产生反应,则它的beta矩阵应当是[1,1]:

 

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%%% Now let's  try  a voxel which responds equally to both words and objects
%%% So, it's sensitivity vector will be [  1  1  ]
%%%
%%% This means that its response will be
%%%  1 * the first column of the design matrix, plus  1 * the second column
%%% i.e.
%%%  1 * the response which the word stimulus evokes  +
%%%  1 * the response which the object stimulus evokes
 
sensitivity_vec = [  1  1  ]';    % The dash makes  this  a column vector
 
predicted_unselective_voxel_response = design_matrix * sensitivity_vec;
 
predicted_unselective_voxel_response     % Display in Matlab command window
 
%% Let's plot all  this
 
figure( 5 );
clf;            % Clear the figure
subplot( 2 , 1 , 1 ); % This is just to make the plots line up prettily
hold on;        %  "Hold"  is one way of putting more than one plot on a figure
 
h1=plot(predicted_word_response_column_vec, 'b*-' );
h2=plot(predicted_object_response_column_vec, 'r^-' );
 
hold off;
grid on;
legend([h1 h2], 'Word-response column vector' , 'Object-response column vector' );
axis([ 1  10  - 1.5  7 ]);
xlabel( 'Time (measured in TRs, i.e. one time-point every 4secs)' );
ylabel( 'fMRI signal' );
 
subplot( 2 , 1 , 2 );
plot(predicted_unselective_voxel_response, 'ms-' );  % Magenta squares
grid on;
legend( 'Unselective voxel-response: 1*word-response + 1*object-response' );
axis([ 1  10  - 1.5  7 ]);
xlabel( 'Time (measured in TRs, i.e. one time-point every 4secs)' );
ylabel( 'fMRI signal' );

  

 

6.假设某一个体素对两种刺激都会产生反应,但是它的beta矩阵是[1,2],即对第二种刺激反应更强烈:

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%%% Ok, I hope  this  isn 't overkill: let' try  a voxel which gives a normal
%%% response to words, but which gives a response to objects which is
%%% *twice* as strong.
%%% So, it's sensitivity vector will be [  1  2  ]
%%%
%%% This means that its response will be
%%%  1 * the first column of the design matrix, plus  2 * the second column
%%% i.e.
%%%  1 * the response which the word stimulus evokes  +
%%%  2 * the response which the object stimulus evokes
 
sensitivity_vec = [  1  2  ]';   % The dash makes  this  a column vector
 
predicted_object_preferring_voxel_response = design_matrix * sensitivity_vec;
 
predicted_object_preferring_voxel_response    % Display in Matlab command window
 
%% Let's plot all  this
 
figure( 6 );
clf;            % Clear the figure
subplot( 2 , 1 , 1 ); % This is just to make the plots line up prettily
hold on;      
 
h1=plot(predicted_word_response_column_vec, 'b*-' );
h2=plot(predicted_object_response_column_vec, 'r^-' );
 
hold off;
grid on;
legend([h1 h2], 'Word-response column vector' , 'Object-response column vector' );
axis([ 1  10  - 2  10 ]);
xlabel( 'Time (measured in TRs, i.e. one time-point every 4secs)' );
ylabel( 'fMRI signal' );
 
subplot( 2 , 1 , 2 );
plot(predicted_object_preferring_voxel_response, 'ms-' );  % Magenta squares
grid on;
legend( 'Object-preferring voxel-response: 1*word-response + 2*object-response' );
axis([ 1  10  - 2  10 ]);
xlabel( 'Time (measured in TRs, i.e. one time-point every 4secs)' );
ylabel( 'fMRI signal' );

 

7.我们现在模拟出一个真实测量得到的生理信号体素激活值:

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measured_voxel_data = [  1   - 1   12   8   - 1   5   - 3  1   - 2   - 1  ]';
 
% This is is what often gets called  "y" .
% This measured signal is probably some kind of mixture of
% a response to the word stimulus and a response to the object stimulus,
% with random noise thrown on top.
 
% Let's plot it
 
figure( 7 );
clf;                % Clear the figure
plot(measured_voxel_data, 'o-' );    
                     % Plot HRF against time, with one time-point every TR seconds.
                     % A line with circles on it
grid on;
xlabel( 'Time (in units of TRs, 4s long each)' );
ylabel( 'fMRI signal' );
title( 'Measured voxel data' );

 

8.进行数据拟合,矩阵求逆,求伪逆,然后绘图plot,进行比对:

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%%% What is the estimated sensitivity vector of  this  voxel ?
%
% Well, we make the pseudo-inverse of the design matrix, and multiply
% it by the vector of measured voxel data:
 
estimated_voxel_sensitivity = pinv(design_matrix) * measured_voxel_data;
 
%%% This estimated_voxel_sensitivity is what gets called beta-hat in the math.
 
%%% Let's display  this  in the workspace, by typing it without a semi-colon
 
estimated_voxel_sensitivity
 
%%% This makes the following show up in the Matlab command window:
%
% estimated_voxel_sensitivity =
%
%     3.2965
%     1.0565
%
%%% So, the estimate is that  this  voxel is around  3  times more sensitive to
%%% words than it is to objects
 
%%% Now, let's make a plot of what the predicted response would be of
%%% a voxel that has a sensitivity matrix which is *exactly* our estimate,
%%% and compare it to the voxel response which we measured.
%%% They won't be exactly the same, because of the noise in the signal.
 
predicted_voxel_output = design_matrix * estimated_voxel_sensitivity;
 
%%% This predicted overall voxel output is just the
%%% predicted response to the word, plus the predicted response to the object.
%%% As we saw in hrf_tutorial.m, the idea that we can calculate the overall
%%% response simply by adding up these two separate responses is what it
%%% means to say that we are assuming that the system is LINEAR.
%%%
%%% If we want to look at the predicted responses to the separate stimulus
%%% types, we can calculate them by separately multiplying the
%%% corresponding column of the design matrix by the corresponding element
%%% of the estimated sensitivity vector.
 
predicted_response_to_word = predicted_word_response_column_vec * ...
                              estimated_voxel_sensitivity( 1 );
                              
predicted_response_to_object = predicted_object_response_column_vec * ...
                                estimated_voxel_sensitivity( 2 );
 
%%%%% Let's plot all  this
 
figure( 8 );
clf;            % Clear the figure
 
subplot( 3 , 1 , 1 ); % This is just to make the plots line up prettily
hold on;       
 
h1=stem(word_stim_time_series, 'b' );
h2=stem(object_stim_time_series, 'r' ); % Word onset in blue, object onset in red
 
hold off;
grid on;
legend([h1( 1 ) h2( 1 )], 'Word stim onset time' , 'Object stim onset time' );
axis([ 1  10  0  1.2 ]); % This just sets the display graph axis size
ylabel( 'Stimulus present / absent' );
 
subplot( 3 , 1 , 2 );
hold on;
h1=plot(predicted_response_to_word, 'b*-' );
h2=plot(predicted_response_to_object, 'ro-' );
h3=plot(predicted_voxel_output, 'ms:' , 'linewidth' , 2 );
                 %%%  'ms:'  means plot in the colour magenta (m),
                 %%% with squares as the markers (s), using a dotted line (:).
                 %%% Then we make the width of the line broader, linewidth= 2 ,
                 %%% so that it shows up better.
                 %%% Note that the predicted_voxel_output is simply the sum of
                 %%% predicted_response_to_word and predicted_response_to_object
grid on;
legend([h1 h2 h3], 'Predicted response to word' , ...
               'Predicted response to object' , 'Predicted total voxel response' );
                
axis([ 1  10  - 3  14 ]);
ylabel( 'fMRI signal' );
 
subplot( 3 , 1 , 3 );
hold on;
h1=plot(measured_voxel_data, 'g^-' );
h2=plot(predicted_voxel_output, 'ms:' , 'linewidth' , 2 );
hold off;
grid on;
legend([h1 h2], 'Measured voxel response' , 'Predicted voxel response' );
axis([ 1  10  - 3  14 ]);
xlabel( 'Time (measured in TRs, i.e. one time-point every 4secs)' );
ylabel( 'fMRI signal' );

  

 

 

 

 

 

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%%%% Tutorial on the basic structure of an fMRI design matrix, using Matlab
%%%% Written by Rajeev Raizada, July  23 2002 .
%%%%
%%%% This file follows up on a preceding one: hrf_tutorial.m
%%%%
%%%% Neither file assumes any prior knowledge of linear algebra
%%%%
%%%% Please mail any comments or suggestions to: raizada at cornell dot edu
%%%%
%%%% Probably the best way to look at  this  program is to read through it
%%%% line by line, and paste each line into the Matlab command window
%%%% in turn. That way, you can see what effect each individual command has.
%%%%
%%%% Alternatively, you can run the program directly by typing
%%%%
%%%%   design_matrix_tutorial
%%%%
%%%% into your Matlab command window.
%%%% Do not type  ".m"  at the end
%%%% If you run the program all at once, all the Figure windows
%%%% will get made at once and will be sitting on top of each other.
%%%% You can move them around to see the ones that are hidden beneath.
%%%%
%%%% Note that  this  tutorial only shows the method where the
%%%% design matrix assumes a specific shape to the HRF.
%%%% It is also possible to estimate the HRF without making
%%%% any assumptions about its shape. This is called using the
%%%% Finite Impulse Response method, or FIR.
%%%% This involves using a slightly more complicated design-matrix
%%%% than the one we make below.
%%%%
%%%% First, let's make a pretend mini-hrf, just to show examples.
%%%% This is similar in shape to the HRFs that we looked at in
%%%% the program hrf_tutorial.m, but it doesn't have as many time-points.
%%%% One reason to use a shortened HRF like  this  is just to save typing!
%%%% But in fact,  this  is approximately what a real HRF would look like
%%%%  if  you only measured from it once every four seconds.
%%%% In fMRI, the time it takes to make a whole-brain measurement is called
%%%% the TR (Time  for  Repetition, although people say  "Repetition Time" ).
%%%% So,  this  HRF is similar to what we'd measure
%%%%  if  our scanner had a TR of  4  seconds. These days, fast scanners
%%%% can usually manage to get a whole-brain full of data in only 2s.
 
hrf_small = [  0   4   2   - 1   0  ];
 
%%%% Plot it
 
figure( 1 );
clf;                        % Clear the figure
plot( 0 : 4 ,hrf_small, 'o-' );   % Plot HRF against a time-vector [ 0 , 1 , 2 , 3 , 4 ]
                             'o-'  means  "use a line with circles on it"
                             % Type  "help plot"  in the Matlab command window
                             % to get a list of all the line-styles and markers
                             % that you can use. There are lots of them!
                             
grid on;                    % Overlay a dotted-line grid on top of the plot
xlabel( 'Time (in units of TRs, 4s long each)' );
ylabel( 'fMRI signal' );
title( 'This is what an HRF would look like if you measure once every 4s' )
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Just as we did in hrf_tutorial.m, now we're going to make
% a time-series of  1 's and 0' s representing the times when stimuli
% are shown. These time-series will be convolved with the HRF,
% in order to see what kinds of fMRI signals would be evoked in voxels
% that respond to the stimuli. These predicted responses will form
% the columns of our design matrix, as is shown in more detail below.
%
% Just  for  purposes of illustration, we're going to imagine that
% one of our stimuli is flashing up a word on the screen, and that
% the other is flashing up a picture of an object.
%
% These stimulus onsets will probably produce more complex patterns
% of neural firing than the sudden flash of light that we talked about
% in HST_hrf_tutorial.m, but we're going to ignore that complication
for  now. We'll simply suppose that each stimulus instantly kicks off
% its own standard-shaped HRF.
% This is what's typically done in event-related fMRI, and it turns
% out that it usually works pretty well.
 
%%%%%%%%%%%%%%% Now suppose we present a word at time t= 2
 
word_stim_time_series =  [  0  1  0  0  0  0  ];
 
%%%%%%%%%%%%%%% And let's present a picture of an object at time t= 4
 
object_stim_time_series= [  0  0  0  1  0  0  ];
 
%%%% Let's convolve these with our mini-HRF to see what kind of fMRI
%%%% signals they would evoke in voxels which respond to words or pictures
 
predicted_signal_that_word_would_evoke = conv(word_stim_time_series,hrf_small);
 
predicted_signal_that_object_would_evoke = conv(object_stim_time_series,hrf_small);
 
%%% Let's plot all  this
 
figure( 2 );
clf;            % Clear the figure
subplot( 3 , 1 , 1 ); % This is just to make the plots line up prettily.
                 % The first number is how many rows of subplots we have:  3
                 % The second number is how many columns:  1
                 % The third number is which subplot to draw in: the first one.
                 % So, we end up with three plots stacked on top of each other,
                 % and we draw in the first one (which is the uppermost subplot)
         
hold on;        %  "Hold"  is one way of putting more than one plot on a figure
 
h1=stem(word_stim_time_series, 'b' );
                 % Stem makes a nice looking plot with lines and circles
h2=stem(object_stim_time_series, 'r' ); % Word onset in blue, object onset in red
                 % The  "h1="  and  "h2="  bits are called  "handles" .
                 % They are pointers to the plots that we are making,
                 % which are the stem plots in  this  case .
                 % Making handles like  this  is useful  for  manipulating
                 % pretty much any aspect of the plot afterwards.
                 % In  this  instance, we use them to put a legend on the plot.
                 % That's done by the  "legend"  command, a couple of lines below.
                 % There's no need to worry about these handles at  this  stage,
                 % I just wanted to explain what those mysterious-looking h's
                 % were doing there. Usually you can make a nice-looking legend
                 % without worrying about handles, but it turns out that
                 for  stem plots they help to make the legend look better.
hold off;
grid on;
legend([h1( 1 ) h2( 1 )], 'Word stim onset time' , 'Object stim onset time' );
                 % We use the h1 and h2 handles here.
                 % This helps us to get the right symbols displayed in the legend,
                 % in  this  case , blue and red circles.
                 
axis([ 1  9  0  1.2 ]);  % This just sets the display graph axis size
                     % The first two numbers are the x-axis range:  1  to  9
                     % The last two numbers are the y-axis range:  0  to  1.2
ylabel( 'Stimulus present / absent' );
 
subplot( 3 , 1 , 2 );
plot(predicted_signal_that_word_would_evoke, 'b*-' ); 
                                     'b*-'  means blue stars on a solid line
grid on;
legend( 'Word-sensitive voxel would give this fMRI signal' );
axis([ 1  10  - 1.5  7 ]);
ylabel( 'fMRI signal' );
 
subplot( 3 , 1 , 3 );
plot(predicted_signal_that_object_would_evoke, 'r^-' );
                                     'r^-'  means red triangles
                                     % pointing up, lying on a solid line.
grid on;
legend( 'Object-sensitive voxel would give this fMRI signal' );
axis([ 1  10  - 1.5  7 ]);
xlabel( 'Time (measured in TRs, i.e. one time-point every 4secs)' );
ylabel( 'fMRI signal' );
 
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%
%%%%%% What the design matrix has in it
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%
%%%%%% Here's the key part.
%%%%%% The design matrix is built up out of these predicted responses.
%%%%%%
%%%%%% Each column of the design matrix is the predicted fMRI signal
%%%%%% that a voxel would give,  if  it were perfectly and exclusively
%%%%%% sensitive to a particular stimulus-condition.
%%%%%%
%%%%%% In our  case , the first column of the design matrix
%%%%%% would be the vector  "predicted_signal_that_word_would_evoke"
%%%%%% that we made just above, and the second column would be
%%%%%% the vector  "predicted_signal_that_object_would_evoke"
%%%%%%
%%%%%% So, the most important part of the design matrix
%%%%%% is simply these two vectors side-by-side.
%%%%%%
%%%%%% A real design matrix would have some other columns in it too,
%%%%%% which have other types of predicted fMRI signals in them,
%%%%%% e.g. what the signal would look like  if  the scanner's output
%%%%%% were slowly drifting in time.
%%%%%% But those other columns don't deal with the signal that the stimuli
%%%%%% would be predicted to evoke in the brain, and so we can ignore them
%%%%%%  for  now. (Later in the HST583 course, Doug will talk more about
%%%%%% how you might model slow-scanner drift etc.)
%%%%%%
%%%%%% It's the *columns* of the design matrix that get built up out of
%%%%%% these predicted responses to the different stimulus types,
%%%%%% but the actual vectors that we made above are row vectors,
%%%%%% i.e. just a bunch of numbers in a row.
%%%%%% So, to match the format of the design matrix, we need to turn
%%%%%% these into column vectors, by transposing them (i.e. flipping them).
%%%%%% We  do  this  by putting a dash/apostrophe at the end of the vector
 
predicted_word_response_column_vec = predicted_signal_that_word_would_evoke';
 
predicted_object_response_column_vec = predicted_signal_that_object_would_evoke';
 
%%% Now let's look at the actual vectors in the Matlab workspace window
 
predicted_word_response_column_vec  % Because there is no semi-colon after  this ,
                                     % it will display in workspace window
predicted_object_response_column_vec
 
 
%%%%%% Now we can join these two column vectors together
%%%%%% to make the design matrix. We simply put the two columns side-by-side.
%%%%%% In Matlab, you make  new  matrices and vectors by
%%%%%% putting the contents inside [ square brackets ]
%%%%%% Note that to join them together in  this  way, they must be
%%%%%% the same length as each other.
%%%%%%
%%%%%% Because the names of my variables are so  long  and verbose,
%%%%%% the command below spills over onto two lines. In Matlab,
%%%%%% we can split a command over two lines by putting three dots ...
 
design_matrix = ...   % The three dots here mean  "continued on the next line"
  [ predicted_word_response_column_vec  predicted_object_response_column_vec ];
 
design_matrix      % No semi-colon, so it displays in window
 
%%%%%% Translation guide:
%%%%%% In equations, the design matrix is almost always called X
%%%%%% Note that  this  is a capital  "X" .
%%%%%%
%%%%%% X = design_matrix;
%%%%%%
%%%%%% Capitals are typically used  for  matrices, and small- case  is
%%%%%% used  for  vectors.
%%%%%% The only difference between a vector and a matrix is that
%%%%%% a vector is just a bunch of numbers in a row (a row-vector)
%%%%%% or a bunch of numbers in a column (a column-vector),
%%%%%% whereas a matrix is bunch of vectors stacked up next to each
%%%%%% other to make a rectangular grid, with rows *and* columns of numbers.
 
%%%%%% Now let's view a grayscale plot of the design matrix,
%%%%%% in the way that an fMRI-analysis  package , such as SPM, would show it.
%%%%%% To  do  this , we use the Matlab command  "imagesc" .
%%%%%% This takes each number in the design matrix and represents
%%%%%% it as a colour, with the colour depending on how big the number is.
%%%%%% In  this  case , we'll be using a gray colour-scale, so low numbers
%%%%%% will be shown as darker grays, and high numbers are lighter grays.
%%%%%% The  "sc"  part at the end of  "imagesc"  stands  for  "scale" , which
%%%%%% means that Matlab scales the mapping of numbers onto colours so
%%%%%% that the lowest number gets shown as black, and the highest as white.
%%%%%%
%%%%%% For examples of how to use the imagesc command to make
%%%%%% pictures of brain-slices, see the companion program
%%%%%% showing_brain_images_tutorial.m
 
figure( 3 );
clf;                      % Clear the figure
imagesc(design_matrix);   %  'imagesc'  maps the numbers to colors,
                           % normalising so that the max goes to white
                           % and the min goes to black
 
colormap gray;            % Show everything in gray-scale
colorbar;                 % Shows how the numbers lie on the colour scale
                           % Note that the highest number in the design matrix,
                           % which is  4 , is shown as white, and the lowest, - 1 ,
                           % gets shown as black.
title( 'Gray-scale view of design matrix' );
xlabel( 'Each column represents one stimulus condition' );
ylabel( 'Each row represents one point in time, one row per TR (every 4secs)' );
 
%% Now suppose we have a voxel which responds only to words, not to objects.
%% We can calculate how it would be predicted to respond
%% to our word+object display as follows:
%%
%% Predicted response from word-sensitive voxel =
%%       1  * Response which word-presentation would evoke
%%    +  0  * Response which object-presentation would evoke
%%
%% Note that  this  is how the voxel would be predicted to respond
%%  if  there were no noise whatsoever in the system.
%% Clearly a real fMRI signal would never be  this  clean.
%%
%% Now, let's make a  "sensitivity vector"  for  this  voxel,
%% in which each entry will say how sensitive that voxel is to
%% the corresponding stimulus condition.
%%
%% This voxel is sensitive to words, which are our *first* stimulus-type.
%% And we made the predicted word response into the first column of
%% the design matrix.
%% So, the sensitivity of  this  voxel to words will be the first element
%% in the sensitivity-vector.
%%
%% Similarly, the sensitivity of  this  voxel to the second stimulus-type,
%% which are objects, will be the second element in the sensitivity vector.
%%
%% So, the sensitivity vector  for  a voxel with
%%      sensitivity =  1    to the first stimulus-type, which are words
%% and  sensitivity =  0    to the second stimulus-type, which are objects
%%
%% will be [   1   0   ]
%%
%% I know  this  seems trivial !!
%% Things will get more interesting in a minute...
 
sensitivity_vec = [  1  0  ]';  % The dash makes  this  a column vector
 
sensitivity_vec              % No semi-colon, so it displays in window
 
%% Translation guide:
%% In equations, the numbers in the sensitivity-vector are typically
%% called  "beta-values" , or sometimes  "beta-coefficients"  or  "beta-weights" .
%% The columns of the design matrix are called  "regressors"  and
%% the value that is assigned to each regressor is the beta-value.
%%
%% Note that in the example above, we are pretending that we already *know*
%% how sensitive our voxel is to the various stimuli, but in the real world
%% we don 't know this. We' re trying to figure out what stimuli our voxel
%% is sensitive to, using the fMRI data that we collect in the scanner.
%% This will be described more below.
%% In math-speak, that means that we are trying to *estimate* the betas.
%% When people want to distinguish between the  true  beta-value
%% (which we don't know) and the estimated beta-value that we figure out
%% from our data, then they call the  true  one beta and
%% the estimated one  "beta hat"  (beta with a circumflex sign on top of it:  ^
 
%% [ End of that part of the translation guide, back to the main theme... ]
%% So, we can now express our predicted voxel response in terms
%% of entries in the sensitivity vector multiplied by
%% columns in the design matrix:
%%
%% Predicted response from word-sensitive voxel =
%%       1