fMRI: Biological Basis and Experiment Design Lecture 22: GLM 101 Which linear equations? Design matrix Solution assuming HIRF, single voxel  +==+

Linear algebra A 1 x 1 + A 2 x 2 + A 3 x 3 + A 4 x 4 = y Ax = y is the same as where A =x = A 1 A 2 A 3 A 4 x1x2x3x4x1x2x3x4

Linear algebra A 1,1 x 1 + A 2,1 x 2 + A 3,1 x 3 + A 4,1 x 4 = y 1 A 1,2 x 1 + A 2,2 x 2 + A 3,2 x 3 + A 4,2 x 4 = y 2 A 1,3 x 1 + A 2,3 x 2 + A 3,3 x 3 + A 4,3 x 4 = y 3 A 1,m x 1 + A 2,m x 2 + A 3,m x 3 + A 4,m x 4 = y m A 1,1 A 2,1 A 3,1 A 4,1 A 1,2 A 2,2 A 3,2 A 4,2 A 1,3 A 2,3 A 3,3 A 4,3 A 1,m A 2,m A 3,m A 4,m A = x = x1x2x3x4x1x2x3x4 y = y1y2y3ymy1y2y3ym stimulus 1 at t=2 response to stimulus 2 time stimulus type

A 1,1 A 2,1 A 3,1 A 4,1 A 1,2 A 2,2 A 3,2 A 4,2 A 1,3 A 2,3 A 3,3 A 4,3 A 1,m A 2,m A 3,m A 4,m Linear model for BOLD in a voxel A = x = x1x2x3x4x1x2x3x4 y = y1y2y3ymy1y2y3ym Ax = y Design matrix, [m x n] - m time-points - n stimulus types Data [m x 1] - response through time Responses [n x 1] - for each stimulus, a scalar (single number) representing how well that voxel responds to that stimulus

Design matrix stim 1stim 2stim 3 stim 1 stim 2 stim 3 Matrix form for GLM

Design matrix: assuming shape of HIRF stim 1stim 2stim 3stim 1stim 2stim 3  = time

Design matrix: modeling data A x  = stim 1 stim 2 stim 3 BOLD

Solving linear model for BOLD in a voxel Ax = y A 1,1 A 2,1 A 3,1 A 4,1 A 1,2 A 2,2 A 3,2 A 4,2 A 1,3 A 2,3 A 3,3 A 4,3 A 1,m A 2,m A 3,m A 4,m A = x = x1x2x3x4x1x2x3x4 y = y1y2y3ymy1y2y3ym Design matrix, [m x n] - m time-points - n stimulus types Data [m x 1] - response through time Responses [n x 1] - for each stimulus, a scalar (single number) representing how well that voxel responds to that stimulus measured known... the answer

Solving linear model for BOLD in a voxel Ax = y A T Ax = A T y (A T A) -1 (A T A)x = (A T A) -1 A T y x = (A T A) -1 A T y

Solving linear model for BOLD in a voxel x = (A T A) -1 A T y x = (A T A) -1 A T y =

Linear model for BOLD in a voxel, with noise A(x +  ) = y +  where A = design matrix [nTimepoints x nStimTypes ], x = concatenated responses [nStimTypes x 1], y = true response [nTimepoints x 1]  = noise in data [nTimepoints x 1]  = error in estimating response [nStimTypes x 1] Solution: x est = x +  = (A T A) -1 A T (y +  ), so  = (A T A) -1 A T 

A(x +  ) = y +  x est = x +  = (A T A) -1 A T (y +  ) A*x est y +  x est = x =

Example: block design with linear trend Design matrix:Solution: x = [ 1 ; 0.2]; x est = [ 1.01 ; 0.24 ]

Example: block design with linear trend Design matrix:Solution: x = [ -0.5 ; 0.2]; x est = [ ; 0.25 ]