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Published byCharles Dalton Modified over 9 years ago
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General Linear Model
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Y1Y2...YJY1Y2...YJ = X 11 … X 1l … X 1L X 21 … X 2l … X 2L. X J1 … X Jl … X JL β1β2...βLβ1β2...βL + ε1ε2...εJε1ε2...εJ Y = X * β + ε Observed data Design Matrix ParametersResiduals/Error time points time points regressors time points
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Design Matrix time rest On Off Off On Conditions Use ‘dummy codes’ to label different levels of an experimental factor (eg. On = 1, Off = 0). task 00000000000000 11111111111111
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Design Matrix Covariates Parametric variation of a single variable (eg. Task difficulty = 1-6) or measured values of a variable (eg. Movement). 544231631652544231631652
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Design Matrix Constant Variable Models the baseline activity (eg. Always = 1) 11111111...11111111...
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Design Matrix The design matrix should include everything that might explain the data. Regressors Time
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General Linear Model Y1Y2...YJY1Y2...YJ = X 11 … X 1l … X 1L X 21 … X 2l … X 2L. X J1 … X Jl … X JL β1β2...βLβ1β2...βL + ε1ε2...εJε1ε2...εJ Y = X * β + ε Observed data Design Matrix ParametersResiduals/Error time points time points regressors time points
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Error Independent and identically distributed iid
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Ordinary Least Squares Residual sum of square: The sum of the square difference between actual value and fitted value. e
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Ordinary Least Squares e N t t e 1 2 = minimum
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Ordinary Least Squares x1β1x1β1 x2β2x2β2 y e XβXβ Y = Xβ+e e = Y-Xβ X T e=0 => X T (Y-Xβ)=0 => X T Y-X T Xβ=0 => X T Xβ=X T Y => β=(X T X) -1 X T Y
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fMRI 12 Y = X * β + ε Observed dataDesign MatrixParametersResiduals/Error
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Problems with the model
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The Convolution Model = Impulses HRF Expected BOLD
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Convolve stimulus function with a canonical hemodynamic response function (HRF): HRF Original Convolved HRF
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Physiological Problems
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Noise Low-frequency noise Solution: High pass filtering
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blue = data black = mean + low-frequency drift green = predicted response, taking into account low-frequency drift red = predicted response, NOT taking into account low-frequency drift discrete cosine transform (DCT) set
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Assumptions of GLM using OLS All About Error
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Unbiasedness Expected value of beta = beta
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Normality
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Sphericity
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Homoscedasticity
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not
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Independence
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Autoregressive Model y = Xβ + e over time e t = ae t-1 + ε autocovariance function a should = 0
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Thanks to… Dr. Guillaume Flandin
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References http://www.fil.ion.ucl.ac.uk/spm/doc/books/hbf2/pdfs/Ch7.pdf http://www.fil.ion.ucl.ac.uk/spm/course/slides10- vancouver/02_General_Linear_Model.pdf http://www.fil.ion.ucl.ac.uk/spm/course/slides10- vancouver/02_General_Linear_Model.pdf Previous MfD presentations
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