Deconvolution Analysis of FMRI Time Series Data - Waisman ...

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Program 3dDeconvolve Command Line for Example 1.4.1.13dDeconvolve n-nodata n-nlast 59 n-polort 0 n-num stimts 1 n-stim file 1 Block.1D n-stim label 1 ''Block'' n-stim maxlag 1 4The -nodata command is used to specify that there is no input data (i.e., no measurementdata). This indicates that the program is only to evaluate the experimental design.There is one input stimulus function (-num stimts 1), which is specied by the contents ofle Block.1D (-stim le 1 Block.1D). The maximum duration for the estimated IRF is 4TR (-stim maxlag14). The program creates the experimental design matrix X from timelagged versions of the input stimulus functions, as explained in Section 1.2.3.Try executing the above script. The program will attempt to calculate the (X t X) ;1matrix. In this particular case, the program is unable to calculate the inverse matrixtherefore, the program prints the following error message:3dDeconvolve Error:Improper X matrix (cannot invert X'X)This error message results from multicollinearity in the experimental design. See Section1.2.4 for a discussion of multicollinearity andhow to avoid it. Can you explain why thisdesign suers from multicollinearity? Since the stimulus function has a period of 8 TR,does it surprise you that the problem is not solvable for a maximum time lag of only 4 TR?If you cannot explain the result, add the -xout command option to the above script, andrepeat the execution. This time, the program prints out the experimental design matrixX. The X matrix has 6 columns: column #0 contains all 1's columns #1 through #5contain time-lagged versions of the stimulus function (corresponding to 0-4 TR). What isthe result if you add column #1 and column #5 (one row at a time)?If you can write a linear combination (with some coecients non-zero) of the columnsof X that is equal to a column of zeros, then the columns of X are said to be linearlydependent. In the current example, we have:1 C 0 ; 1 C 1 ; 1 C 5 = 0where C i = ith column of X:Now, try executing the above script, but reduce maxlag to 3 (i.e., use the command-stim maxlag 1 3 in place of the -stim maxlag 1 4 command). Does this resolve the multicollinearityproblem? You should see the following printout on the screen:26
Program 3dDeconvolve Screen Output for Example 1.4.1.1Program:3dDeconvolveAuthor:B. Douglas WardInitial Release: 02 Sept 1998Latest Revision: 27 July 2000(X'X) inverse matrix:0.0820 -0.0656 0.0000 -0.0000 -0.0656-0.0656 0.1382 -0.0714 -0.0000 0.06670.0000 -0.0714 0.1429 -0.0714 -0.0000-0.0000 -0.0000 -0.0714 0.1429 -0.0714-0.0656 0.0667 0.0000 -0.0714 0.1382Stimulus: Blockh[0] norm. std. dev. = 0.3717h[1] norm. std. dev. = 0.3780h[2] norm. std. dev. = 0.3780h[3] norm. std. dev. = 0.3717If the program is able to evaluate the (X t X) ;1matrix, this matrix is printed to thescreen. As mentioned in Section 1.2.7, the variance-covariance matrix for the regressioncoecients is given by:s 2 (b) =MSE ; X t X ;1where MSE = sample variance. Since there is no input measurement data, MSE isunknown. However, if MSE is normalized to a value of 1, we see that (X t X) ;1is the(normalized) variance-covariance matrix. By taking the square roots of the correspondingdiagonal elements, we obtain the (normalized) standard deviations of the estimatedimpulse response coecients. These values are also printed to the screen (labeled norm.std.dev.). Note that the above procedure can be repeated using dierent stimulus functions.This allows a comparative analysis of the IRF coecient estimation accuracy for dierentexperimental designs, under the assumption that MSE will be constant for a given voxel.We have seen that, using the above experimental design, it is not possible to estimatethe system IRF for a maxlag of 4 however, it is possible to estimate the IRF for a maxlagof 3 (or less). Question: If it is required that the \On" period and the \O" period beequal, can you devise a block type design that allows estimation of the IRF for a maxlagof 4 (or greater)? Verify your answer using 3dDeconvolve with the -nodata option.Example 1.4.1.2 Evaluation of a Random DesignIn place of the above \block" type design, we willnowinvestigate the use of a \random"design. Toss a coin 60 times for each coin toss, record a 1if\heads"occurs, and recorda 0 if \tails" occurs. The record of the coin tosses might look something like this:27
Page 1: Deconvolution Analysis of FMRI Time
Page 7 and 8: Since the system is linear, the ope
Page 9: the above equation can be written:Z
Page 13 and 14: where s(h k ) is the square root of
Page 15 and 16: Here, SSE(F )isthe error sum of squ
Page 17 and 18: concatenated runs, this is not the
Page 19: This equation relates the output y
Page 22 and 23: value is pnum = 1 (corresponding to
Page 24 and 25: model parameters, and P columns, ea
Page 28 and 29: f(t) = f 0 0 1 1 1 0 1 1 0 0 1 0 1
Page 30 and 31: If we model the measured response b
Page 32 and 33: Note that, since there is only one
Page 34 and 35: Full Model:MSE = 0.9618R^2 = 0.9835
Page 36 and 37: that are not of inherent interest,
Page 38 and 39: Program 3dDeconvolve Command Line f
Page 40 and 41: Now, model the measured data as a c
Page 42 and 43: -stim label 1 Random -stim label 2
Page 44 and 45: Note: This example is for illustrat
Page 46 and 47: The output is the same as for Examp
Page 48 and 49: After program 3dDeconvolve has nish
Page 50 and 51: Program 3dDeconvolve Command Line f
Page 52 and 53: Program:3dDeconvolveAuthor:B. Dougl
Page 54 and 55: cat y.1D y.1D > ycat.1DBe sure to r
Page 56 and 57: Recall that in Example 1.4.4.6, we
Page 58 and 59: and the noisy measurement data:wp(t
Page 60 and 61: The model that we will use for the
Page 62 and 63: Program 3dConvolve Command Line for
Page 64 and 65: 2 Program plug deconvolve2.1 Purpos
Page 66 and 67: voxels indicated by the crosshairs,
Page 68 and 69: Deconvolution Plugin Screen Output
Page 70 and 71: The following Random Permutation an
Page 72 and 73: Shuffled array:0 5 6 1 3 5 0 2 1 1
Page 74 and 75: Program RSFgen Command Line for Exa
Page 76 and 77:
Example 3.3.3.1 Initial Random Bloc
Page 78 and 79:
Compare the output for this example
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Markov chain time series:0 0 1 5 0
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(X'X) inverse matrix::065 ;:000 ;:0
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Program 3dDeconvolve Screen Output
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then the standard deviation for the
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4 Program 3dConvolve4.1 PurposeAs t
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-input1DThe -input1D command specie
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-stim minlag k m-stim maxlag k n-st
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parameters b 0 and b 1 , and impuls
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Program 3dConvolve Command Line for
Page 98 and 99:
Example 4.4.6 Single Stimulus No No
Page 100:
5 References[1 ]F.Stremler, Fourier
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