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General Linear Model Beatriz Calvo Davina Bristow

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Overview Summary of regression Matrix formulation of multiple regression Introduce GLM Parameter Estimation Residual sum of squares GLM and fMRI fMRI model Linear Time Series Design Matrix Parameter estimation Summary

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Summary of Regression Linear regression models the linear relationship between a single dependent variable, Y, and a single independent variable, X, using the equation: Y = βX + c + ε The regression coefficient, β, reflects how much of an effect X has on Y ε is the error term and is assumed to be independently, identically, and normally distributed (mean 0 and variance σ 2 )

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Summary of Regression Multiple regression is used to determine the effect of a number of independent variables, X 1, X 2, X 3 etc, on a single dependent variable, Y The different X variables are combined in a linear way and each has its own regression coefficient: Y = β 1 X 1 + β 2 X 2 +…..+ β L X L + ε The β parameters reflect the independent contribution of each independent variable, X, to the value of the dependent variable, Y. i.e. the amount of variance in Y that is accounted for by each X variable after all the other X variables have been accounted for

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Matrix formulation Multiplying matrices reminder: a b c d e f GHIGHI =x G x a + H x b + I x c G x d + H x e + I x f

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Matrix Formulation Write out equation for each observation of variable Y from 1 to J: Y 1 = X 11 β 1 +…+X 1l β l +…+ X 1L β L + ε 1 Y j = X j1 β 1 +…+X jl β l +…+ X jL β L + ε j Y J = X J1 β 1 +…+X Jl β l +…+ X JL β L + ε J Y1YjYJY1YjYJ = X 11 … X 1l … X 1L X j1 … X 1l … X 1L X 11 … X 1l … X 1L Can turn these simultaneous equations into matrix form to get a single equation: β1βjβJβ1βjβJ + ε1εjεJε1εjεJ Y = X x β + ε Observed data Design MatrixParametersResiduals/Error

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General Linear Model This is simply an extension of multiple regression Or alternatively Multiple Regression is just a simple form of the General Linear Model Multiple Regression only looks at ONE dependent (Y) variable Whereas, GLM allows you to analyse several dependent, Y, variables in a linear combination i.e. multiple regression is a GLM with only one Y variable ANOVA, t-test, F-test, etc. are also forms of the GLM

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GLM - continued.. In the GLM the vector Y, of J observations of a single Y variable, becomes a MATRIX, of J observations of N different Y variables An fMRI experiment could be modelled with matrix Y of the BOLD signal at N voxels for J scans However SPM takes a univariate approach, i.e. each voxel is represented by a column vector of J fMRI signal measurements, and it processed through a GLM separately (this is why you then need to correct for multiple comparisons)

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GLM and fMRI How does the GLM apply to fMRI experiments? Y = X. β + ε Observed data: SPM uses a mass univariate approach – that is each voxel is treated as a separate column vector of data. Y is the BOLD signal at various time points at a single voxel Design matrix: Several components which explain the observed data, i.e. the BOLD time series for the voxel Timing info: onset vectors, O m j, and duration vectors, D m j HRF, h m, describes shape of the expected BOLD response over time Other regressors, e.g. realignment parameters Parameters: Define the contribution of each component of the design matrix to the value of Y Estimated so as to minimise the error, ε, i.e. least sums of squares Error: Difference between the observed data, Y, and that predicted by the model, Xβ. Not assumed to be spherical in fMRI

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Parameter estimation In linear regression the parameter β is estimated so that the best prediction of Y can be obtained from X i.e. sums of squares of difference between predicted values and observed data, (i.e. the residuals, ε) is minimised Remember last week’s talk & graph! The method of estimating parameters in GLM is essentially the same, i.e. minimising sums of squares (ordinary least squares), it just looks more complicated

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Last week’s graph ε y = βx + c ε = residual error = y i, true value = ỹ, predicted value

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Residual Sums of Squares Take a set of parameter estimates, β Put these into the GLM equation to obtain estimates of Y from X, i.e. fitted values, Y: Y = X x β The residual errors, e, are the difference between the fitted and actual values: e = Y - Y = Y - Xβ Residual sums of squares is: S = Σ j J e j 2 When written out in full this gives: S = Σ j J (Y j - X j1 β 1 -…- X jL β L ) 2

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Minimising S If you plot the sum of squares value for different parameter, β, estimates you get a curve S is minimised when the gradient of this curve is zero Gradient = 0 min S S = Σ( Y - Xβ) 2 e = Y - Xβ S = Σ j J e j 2

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Minimising S cont. so to calculate the values of β which gives you the least sums of squares you must find the partial derivative of S = Σ j J (Y j - X j1 β 1 -…- X jL β L ) 2 Which is ∂S/∂β = 2Σ(-X jl )(Y j – X j1 β 1 -…- X jL β L ) and solve this for ∂S/∂β = 0 In matrix form of the residual sum of squares is S = e T e this is equivalent to Σ j J e j 2 (remember how we multiply matrices) e = Y - X β therefore S = (Y - X β ) T (Y - X β )

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Minimising S cont. Need to find the derivative and solve for ∂S/∂β = 0 The derivative of this equation can be rearranged to give X T Y = (X T X)β when the gradient of the curve = 0, i.e. S is minimised This can be rearranged to give: β = X T Y(X T X) -1 But a solution can only be found, if (X T X) is invertible because you need to divide by it, which in matrix terms is the same as multiplying by the inverse!

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GLM and fMRI How does the GLM apply to fMRI experiments? Y = X. β + ε Observed data: SPM uses a mass univariate approach – that is each voxel is treated as a separate column vector of data. Y is the BOLD signal at various time points at a single voxel Design matrix: Several components which explain the observed data, i.e. the BOLD time series for the voxel Timing info: onset vectors, O m j, and duration vectors, D m j HRF, h m, describes shape of the expected BOLD response over time Other regressors, e.g. realignment parameters Parameters: Define the contribution of each component of the design matrix to the value of Y Estimated so as to minimise the error, ε, i.e. least sums of squares Error: Difference between the observed data, Y, and that predicted by the model, Xβ. Not assumed to be spherical in fMRI

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fMRI models Completed the experiment, after preprocessing, the data are ready for STATS. STATS: (estimate parameters, β, inference) indicating evidence against the Ho of no effect at each voxel are computed->an image of this statistic is produce This statistical image is assessed (other talk will explain that)

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Example: 1 subject. 1 session Moving finger vs rest 7 cycles of rest and moving Time series of BOLD response in one voxel Time seconds Responses at voxel (x, y, z) Question: Is there any change in the BOLD response between moving and rest? Each epoch 6 scans Whole brain acquisition data

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TIME SERIES: consist on the sequential measures of fMRI data signal intensities over the period of the experiment The same temporal model is used at each voxel Mass-univariated model and perform the same analysis at each voxel Therefore, we can describe the complete temporal model for fMRI data by looking at how it works for the data from a voxel. Single Voxel Time Series Time Linear Time Series Model

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Y: My data/ observations Single Voxel Time Series My Data Time Y1YsYNY1YsYN Time series of N observations Y 1,…,Y s,…,Y n. N= scan number Acquired at one voxel at times t s, where S=1:N

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Model specification The overall aim of regressor generation is to come up with a design matrix that models the expected fMRI response at any voxel as a linear combinations of columns. Design matrix – formed of several components which explain the observed data. Two things SPM need to know to construct the design matrix: Specify regressors Basis functions that explain my data

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Model specification … Specify regressors X Timing information consists of onset vectors O m j and duration vectors D m Other regressors e.g. movement parameters Include as many regressors as you consider necessary to best explain what’s going on. Basis functions that explain my data (HRF) Expected shape of the BOLD response due to stimulus presentation

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GLM and fMRI data Model the observed time series at each voxel as a linear combination of explanatory functions, plus an error term Y s = β 1 X 1 (tS) + …+ β l X l (tS) + …+ β L X L (tS) + ε s Here, each column of the design matrix X contains the values of one of the continuous regressors evaluated at each time point t s of fMRI time series That is, the columns of the design matrix are the discrete regressors

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Consider the equation for all time points, to give a set of equations Y1YsYNY1YsYN β1βlβLβ1βlβL εNεsεNεNεsεN + = X 1 (t1) X l (t1) X L (t1) X 1 (tS) X l (tS) X L (tS) X 1 (tN) X l (tN) X L (tN) Y 1 = β 1 X 1 (t1) + …+ β l X l (t1) + …+ β L X L (t1) + ε 1 Y s = β 1 X 1 (tS) + …+ β l X l (tS) + …+ β L X L (tS) + ε s Y N = β 1 X 1 (tN) + …+ β l X l (tN) + …+ β L X L (tN) + ε N Y = X β + ε In matrix notation: GLM and fMRI data … In matrix form:

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Getting the design matrix Regressors ε Errors are normally and independently and identical distributed Intensity Time = β1 β1 β2β2 + + Observations y = x 1 x 2

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Getting the design matrix … Regressors Intensity Time = β1 β1 β2β2 + + Observations y = β 1 x 1 + β 2 x 2 + ε Error

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Design matrix Regressors = β2 β2 β1β1 + Observations Y = X β + ε Error x

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Design matrix Regressors β1 β1 β2β2 ObservationsError Y = X β + ε N N N l L l l L N: nuber of scans P: number of regressors Y = X β + ε Model is specified by: design matrix Assumptions about ε Y1YsYNY1YsYN X 1 (t1) X l (t1) X L (t1) X 1 (tS) X l (tS) X L (tS) X 1 (tN) X l (tN) X L (tN) β1βlβLβ1βlβL εNεsεNεNεsεN

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Parametric estimation = β1 β1 β2β2 + + Y X ε Estimate parameters β The error is minimal when Least squares Parameter estimates β = X T Y(X T X) -1 ε = Y - Y = Y - Xβ S = Σ t J ε t 2 (Get this by putting into matrix form and finding derivative)

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Summary The General Linear Model allows you to find the parameters, β, which provide the best fit with your data, Y The optimal parameters estimates, β, are found by minimising the Sums of Squares differences between your predicted model and the observed data The design matrix in SPM contains the information about the factors, X, which may explain the observed data Once we have obtained the βs at each voxel we can use these to do various statistical tests but that is another talk….

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THE END Thank you to Lucy, Daniel and Will and to Stephan for his chapter and slides about GLM and to Adam for last year’s presentation Links:

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