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？ |
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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 itfigure(1);clf; % Clear the figureplot(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 plotxlabel(‘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=2word_stim_time_series = [ 0
1 0 0 0 0
]; %%%%%%%%%%%%%%% And let‘s present a picture of an object at time t=4object_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 picturespredicted_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 thisfigure(2);clf; % Clear the figuresubplot(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 figureh1=stem(word_stim_time_series,‘b‘); % Stem makes a nice looking plot with lines and circlesh2=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 1.2ylabel(‘Stimulus present / absent‘);subplot(3,1,2);plot(predicted_signal_that_word_would_evoke,‘b*-‘); % ‘b*-‘
means blue stars on a solid linegrid 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 vectorpredicted_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 windowpredicted_word_response_column_vec % Because there is no semi-colon after this, % it will display in workspace windowpredicted_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 figureimagesc(design_matrix); % ‘imagesc‘
maps the numbers to colors, % normalising so that the max goes to white % and the min goes to blackcolormap gray; % Show everything in gray-scalecolorbar; % 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 vectorsensitivity_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 thisfigure(4);clf; % Clear the figuresubplot(2,1,1); % This is just to make the plots line up prettilyhold on; % "Hold"
is one way of putting more than one plot on a figureh1=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 squaresgrid 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‘);%%% 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 evokessensitivity_vec = [ 1
1 ]‘; % The dash makes this
a column vectorpredicted_unselective_voxel_response = design_matrix * sensitivity_vec;predicted_unselective_voxel_response % Display in Matlab command window %% Let‘s plot all thisfigure(5);clf; % Clear the figuresubplot(2,1,1); % This is just to make the plots line up prettilyhold on; % "Hold"
is one way of putting more than one plot on a figureh1=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 squaresgrid 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‘);%%% Ok, I hope this
isn‘t overkill: let‘s 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 evokessensitivity_vec = [ 1
2 ]‘; % The dash makes this
a column vectorpredicted_object_preferring_voxel_response = design_matrix * sensitivity_vec;predicted_object_preferring_voxel_response % Display in Matlab command window %% Let‘s plot all thisfigure(6);clf; % Clear the figuresubplot(2,1,1); % This is just to make the plots line up prettilyhold 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 squaresgrid 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‘);%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% So, to recap:%% Voxel response = Design matrix * sensitivity vector%% Each column of the design matrix is the response to a particular stimulus.% Each row of it is a moment in time, with one row per MRI image-acquisition.% So, reading down a column (through the rows), gives the response through% time to a particular stimulus.%% Each element in the sensitivity vector is a measure of how much that voxel% responds to the stimulus in the corresponding column of the design matrix.%% When we multiply the design matrix by the sensitivity vector, this
produces a % result which takes each column, which is the responses that each stimulus-type% would evoke, then multiplies that column by how sensitive that voxel is to % that particular stimulus, and then adds together the results of all % those multiplications.%% But so far we‘ve only been talking about an imaginary situation% in which we already *know* which stimuli our voxel is sensitive to, and we use% that knowledge to calculate how the voxel ought to respond.% That‘s why we have been talking about *predicted* voxel responses so far.%% Here‘s the key bit: in fMRI, we have exactly the reverse situation:% we *measure* how the voxels respond, and we want to figure out which % stimuli they must therefore have been sensitive to.%% ie. Voxel response = Design matrix * sensitivity vector %% ^ ^ ^% | | |% We measure this
We build this
We want to find this
out% with the scanner from stimulus This is what the analysis works out% onset times%%% So, we measure a voxel‘s response, and we know that it should be% equal to (Design matrix * sensitivity vector)% It won‘t be exactly equal to that, because the signal is noisy.% We‘ll ignore the noise for now, but we‘ll come back to it soon below.%% What we need to do
is to unpack the result of this
multiplication,% so that we can take (Design matrix * sensitivity vector)% and pull out the part that we don‘t already know and that we want,% namely the sensitivity vector.%% To do
that, we need the concept of a MATRIX INVERSE.%% If multiplying by a matrix, M, does one thing,% then multiplying by its inverse, inv(M), does the opposite.%% From above, we know the value of Design matrix * sensitivity vector%% (its value is the voxel response), but what we need to find out% is just the sensitivity vector on its own.%% So, we can achieve this
by multiplying by the inverse of the design matrix%% inv(design matrix) * design matrix * sensitivity vector% % = sensitivity vector%%% But since design matrix * sensitivity vector = voxel response,%% the above is the same as:%% inv(design matrix) * voxel response%% Given that we *know* the design matrix (we built it), we just need% to calculate its inverse, multiply it by the voxel response, and% then we will get that voxel‘s sensitivity vector.%% sensitivity vector = inv(design matrix) * voxel response% % And since the voxel‘s sensitivity vector is just a list of the% responses which it gives to each of the stimuli which we presented,% it therefore tells us which stimuli make that voxel light up.% % And that is what we wanted to find out!%% This is pretty much what any fMRI-analysis package
does, % although they often organise the results a bit differently. % The "sensitivity vector"
above is a list of numbers for
a single voxel: % each number describes how closely the BOLD signal time-course from that % voxel matches to the corresponding column of the design matrix.%% In an fMRI-analysis package, instead of getting a separate% "sensitivity vector"
for each voxel, you may instead get % a "sensitivity image"
for each design matrix column,% where each image is a brain-full of sensitivity values.% Since these sensitivity values are called "betas", the% brains-full of beta-values are called "beta-images".% The value in a given voxel is the measure of how closely% that voxel‘s BOLD time-course matches to the% corresponding column of the design matrix.% In SPM, for
example, beta_001.img is a brain-full of numbers% saying how sensitive each voxel is to the 1st column in the % design matrix. So, the beta-images are made up of the same% numbers as we are calculating here for
the "sensitivity vector",% it‘s just that they‘re grouped into brain-sized images,% rather than given one voxel at a time.%% Now, it turns out that what I just told you about inverses% isn‘t really true. We don‘t multiply by the inverse of the design matrix.% We multiply by something that is basically the same, only slightly% more complicated, called the "pseudo-inverse".% In Matlab, the pseudo-inverse of X is written pinv(X).%% If you really want to know, pinv(X) = inv(X‘*X)*X‘%% This isn‘t a really important difference.% The key point is to see that trying to figure out a voxel‘s sensitivity% vector is the problem of trying to work out which vector would have to be% multiplied by the design matrix, in order to give the voxel response vector% which we measured with the scanner. %% So, the equation for
figuring out a voxel‘s sensitivity is:% % Voxel response = Design matrix * sensitivity vector%% which means that we can calculate the sensitivity vector like this:%% Sensitivity vector = pinv(design matrix) * voxel response%% We mentioned above that there‘s noise in the signal.% It turns out that% With the noise included, the equation is:%% Voxel response = Design matrix * sensitivity vector + noise%% ... where noise means % "anything in the measured signal that our design matrix can‘t explain".% % This is a problem, because with the noise, it‘s no longer true% that the measured voxel response is exactly equal to the% design matrix multiplied by the sensitivity vector.% Luckily, it turns out that this
doesn‘t stop us from being able% to *estimate* a sensitivity vector, even though the noise prevents us% from being able to calculate exactly what the voxel‘s sensitivities are.% It turns out that we can still use the pseudo-inverse of the design matrix, % and that this
gives us the best estimate of the sensitivity vector that% we could get, despite the noise.%% So, although the noise prevents us from calculating the "true"
sensitivity% vector, it doesn‘t stop us from getting a good estimate:%% estimated sensitivity vector = pinv(design matrix) * voxel response%% Translation guide:% The fMRI signal that we measure from the scanner, which% we call "voxel response"
or "measured_voxel_data"
here, is % usually called "y"
in equations.%% As before, the design matrix is called X, and the % voxel sensitivities are called beta-values.% To show that a beta-value is estimated, rather than being the real but% unknown sensitivity of the voxel, a hat sign gets put on it: beta-hat%% So, instead of the equation that we write below:% estimated_voxel_sensitivity = pinv(design_matrix) * measured_voxel_data;%% .. you‘ll see an equation that looks like this:%% beta = inv(X‘*X)*X‘
* y;%% or, with the hat-sign to show that beta is just an estimate:%% beta_hat = inv(X‘*X)*X‘
* y;% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Ok, let‘s try
that with an example.%% Suppose we measure this
data: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 itfigure(7);clf; % Clear the figureplot(measured_voxel_data,‘o-‘); % Plot HRF against time, with one time-point every TR seconds. % A line with circles on itgrid on;xlabel(‘Time (in units of TRs, 4s long each)‘);ylabel(‘fMRI signal‘);title(‘Measured voxel data‘);%%% 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-colonestimated_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 thisfigure(8);clf; % Clear the figuresubplot(3,1,1); % This is just to make the plots line up prettilyhold on; h1=stem(word_stim_time_series,‘b‘); h2=stem(object_stim_time_series,‘r‘); % Word onset in blue, object onset in redhold 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 sizeylabel(‘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_objectgrid 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‘);%%%%%% From Fig.8, we can see that the voxel-sensitivities that we estimated%%%%%% give a predicted overall voxel response which matches reasonably%%%%%% closely to the actual measured voxel data.%%%%%%%%%%%% But the match isn‘t perfect.%%%%%% That‘s because the MRI signal has noise in it.%%%%%% By "noise", we basically mean, "any changes in the MRI signal that%%%%%% our design matrix can‘t explain".%%%%%%%%%%%% All that our design matrix talks about is the predicted response%%%%%% to the word stimulus and the predicted response to the object stimulus.%%%%%% These predicted responses are made from HRFs, and so they change%%%%%% on a slow, HRF kind of time-scale, i.e. over several seconds.%%%%%%%%%%%% So, if
there are either much more rapid changes in the fMRI signal,%%%%%% or much slower changes, then the design matrix won‘t be able to%%%%%% account for
them.%%%%%%%%%%%% In a real design matrix, there would be extra columns that would%%%%%% try
to account for
any slower changes that there might be, %%%%%% e.g. slow drifts in the signal that the scanner is giving out.%%%%%%%%%%%% Sometimes it‘s also possible to explain away very rapid changes.%%%%%% For example, if
we put columns in the design matrix that describe%%%%%% how much the subject‘s head moved, then it might turn out %%%%%% that some of the rapid MRI signal changes correlate closely with %%%%%% the amount of head-movement. This is what people are referring to %%%%%% when they talk about "putting in motion as a regressor".%%%%%%%%%%%% But there‘s always some noise that we simply can‘t get rid of.%%%%%% If there‘s not much left-over noise, then we can be fairly %%%%%% confident that the voxel-sensitivity vector that we calculated above%%%%%% is a good estimate.%%%%%% And if
there‘s a lot of left over noise, then we probably won‘t%%%%%% be very confident.%%%%%%%%%%%% That‘s the basis of the statistical tests that%%%%%% any fMRI-analysis package
starts to apply after it has%%%%%% used the design matrix to estimate how sensitive each voxel is%%%%%% to the various stimulus-types that we presented.%%%%%%%%%%%% However, those statistical tests are a topic for
a different talk.%%%%%%%%%%%% A couple of good websites to check out, which also have%%%%%% accompanying Matlab code, are these ones by Matthew Brett:%%%%%% http://www.mrc-cbu.cam.ac.uk/Imaging/spmstats.html%%%%%% http://www.mrc-cbu.cam.ac.uk/Imaging/statstalk.m%%%%%%%%%%%% and also several programs by Russ Poldrack, which are listed here:%%%%%% http://www.nmr.mgh.harvard.edu/~poldrack/spm/tutorials/ |
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,布布扣,bubuko.com
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
原文：http://www.cnblogs.com/haore147/p/3633598.html