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General Linear Model

General Linear Model. General Linear Model. regressors. β 1 β 2 . . . β L. ε 1 ε 2 . . . ε J. Y 1 Y 2 . . . Y J. X 11 … X 1 l … X 1L X 2 1 … X 2 l … X 2 L . . . X J1 … X J l … X JL. =. +. time points. time points. time points. regressors.

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General Linear Model

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  1. General Linear Model

  2. General Linear Model regressors β1 β2 . . . βL ε1 ε2 . . . εJ Y1 Y2 . . . YJ X11 … X1l … X1L X21… X2l… X2L . . . XJ1 … XJl… XJL = + time points time points time points regressors Y = X *β + ε Design Matrix Observed data Parameters Residuals/Error

  3. Design Matrix 0 0 0 0 0 0 0 rest task Conditions On Off Off On 1 1 1 1 1 1 1 Use ‘dummy codes’ to label different levels of an experimental factor (eg. On = 1, Off = 0). time

  4. Design Matrix 5 4 4 2 3 1 6 3 1 6 5 2 Covariates Parametric variation of a single variable (eg. Task difficulty = 1-6) or measured values of a variable (eg. Movement).

  5. Design Matrix 1 1 1 1 1 1 1 1 . . . Constant Variable Models the baseline activity (eg. Always = 1)

  6. Design Matrix Time Regressors The design matrix should include everything that might explain the data.

  7. General Linear Model regressors β1 β2 . . . βL ε1 ε2 . . . εJ Y1 Y2 . . . YJ X11 … X1l … X1L X21… X2l… X2L . . . XJ1 … XJl… XJL = + time points time points time points regressors Y = X *β + ε Design Matrix Observed data Parameters Residuals/Error

  8. Error • Independent and identically distributed iid

  9. Ordinary Least Squares Residual sum of square: The sum of the square difference between actual value and fitted value. e

  10. Ordinary Least Squares N å 2 e = minimum t = t 1 e

  11. Ordinary Least Squares Y = Xβ+e e = Y-Xβ XTe=0 => XT(Y-Xβ)=0 => XTY-XTXβ=0 => XTXβ=XTY => β=(XTX)-1XTY y e Xβ x1β1 x2β2

  12. fMRI Y = X *β + ε Observed data Design Matrix Parameters Residuals/Error

  13. Problems with the model

  14. The Convolution Model Expected BOLD HRF Impulses  =

  15. Convolve stimulus function with a canonical hemodynamic response function (HRF): OriginalConvolvedHRF  HRF

  16. Physiological Problems

  17. Noise Low-frequency noise Solution: High pass filtering

  18. 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

  19. Assumptions of GLM using OLS All About Error

  20. Unbiasedness Expected value of beta = beta

  21. Normality

  22. Sphericity

  23. Homoscedasticity

  24. not

  25. Independence

  26. Autoregressive Model y = Xβ + e overtime et= aet-1 + ε autocovariance function a should = 0

  27. Thanks to… • Dr. Guillaume Flandin

  28. 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 • Previous MfD presentations

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