1. General Linear Model

2. 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

3. 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

4. Design Matrix Covariates Parametric variation of a single variable (eg. Task difficulty = 1-6) or measured values of a variable (eg. Movement). 544231631652544231631652

5. Design Matrix Constant Variable Models the baseline activity (eg. Always = 1) 11111111...11111111...

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

7. 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

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 e   N t t e 1 2 = minimum

11. 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

12. fMRI 12 Y = X * β + ε Observed dataDesign MatrixParametersResiduals/Error

13. Problems with the model

14. The Convolution Model  = Impulses HRF Expected BOLD

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

16. Physiological Problems

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

20. 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

21. Assumptions of GLM using OLS All About Error

22. Unbiasedness Expected value of beta = beta

23. Normality

24. Sphericity

25. Homoscedasticity

26. not

28. Independence

29. Autoregressive Model y = Xβ + e over time e t = ae t-1 + ε autocovariance function a should = 0

30. Thanks to… Dr. Guillaume Flandin

31. 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