General Linear Model Beatriz Calvo

Yüklə 458 b.

tarix	25.06.2018
ölçüsü	458 b.
	#51850

General Linear Model

Beatriz Calvo
Davina Bristow

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

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 σ2)

Summary of Regression

Multiple regression is used to determine the effect of a number of independent variables, X1, X2, X3 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 = β1X1 + β2X2 +…..+ βLXL + 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

Matrix formulation

Multiplying matrices reminder:

Matrix Formulation

Write out equation for each observation of variable Y from 1 to J:
Y1 = X11β1 +…+X1lβl +…+ X1LβL + ε1
Yj = Xj1β1 +…+Xjlβl +…+ XjLβL + εj
YJ = XJ1β1 +…+XJlβl +…+ XJLβL + εJ

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

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)

GLM and fMRI

How does the GLM apply to fMRI experiments?

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

Last week’s graph

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 = ΣjJej2
When written out in full this gives:
S = ΣjJ(Yj - Xj1β1 -…- XjLβL)2

Minimising S

If you plot the sum of squares value for different parameter, β, estimates you get a curve

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 = ΣjJ(Yj - Xj1β1 -…- XjLβL)2
Which is
∂S/∂β = 2Σ(-Xjl)(Yj – Xj1β1-…- XjLβL)
and solve this for ∂S/∂β = 0
In matrix form of the residual sum of squares is
S = eTe
this is equivalent to ΣjJej2
(remember how we multiply matrices)
e = Y - X β therefore S = (Y - X β )T(Y - X β )

Minimising S cont.

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

GLM and fMRI

How does the GLM apply to fMRI experiments?

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)

Example: 1 subject. 1 session

Moving finger vs rest
7 cycles of rest and moving

Linear Time Series Model

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.

Y: My data/ observations

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

Model specification …

Specify regressors X

Timing information consists of onset vectors Omj and duration vectors Dm
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

GLM and fMRI data

Model the observed time series at each voxel as a linear combination of explanatory functions, plus an error term
Ys= β1 X1(tS)+ …+ βl Xl(tS)+ …+ βL XL(tS) + εs
Here, each column of the design matrix X contains the values of one of the continuous regressors evaluated at each time point ts of fMRI time series
That is, the columns of the design matrix are the discrete regressors

GLM and fMRI data …

Consider the equation for all time points, to give a set of equations

Getting the design matrix

Getting the design matrix …

Design matrix

Parametric estimation

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….

THE END

Yüklə 458 b.

Dostları ilə paylaş:

General Linear Model Beatriz Calvo

General Linear Model

Beatriz Calvo

Davina Bristow

Overview

Summary of regression

Matrix formulation of multiple regression

Introduce GLM

Parameter Estimation

GLM and fMRI

fMRI model

Summary

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 σ2)

Summary of Regression

Multiple regression is used to determine the effect of a number of independent variables, X1, X2, X3 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 = β1X1 + β2X2 +…..+ βLXL + 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

Matrix formulation

Multiplying matrices reminder:

Matrix Formulation

Write out equation for each observation of variable Y from 1 to J:

Y1 = X11β1 +…+X1lβl +…+ X1LβL + ε1

Yj = Xj1β1 +…+Xjlβl +…+ XjLβL + εj

YJ = XJ1β1 +…+XJlβl +…+ XJLβL + εJ

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

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)

GLM and fMRI

How does the GLM apply to fMRI experiments?

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

Last week’s graph

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 = ΣjJej2

When written out in full this gives:

S = ΣjJ(Yj - Xj1β1 -…- XjLβL)2

Minimising S

If you plot the sum of squares value for different parameter, β, estimates you get a curve

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 = ΣjJ(Yj - Xj1β1 -…- XjLβL)2

Which is

∂S/∂β = 2Σ(-Xjl)(Yj – Xj1β1-…- XjLβL)

and solve this for ∂S/∂β = 0

In matrix form of the residual sum of squares is

S = eTe

this is equivalent to ΣjJej2

(remember how we multiply matrices)

e = Y - X β therefore S = (Y - X β )T(Y - X β )

Minimising S cont.

Need to find the derivative and solve for ∂S/∂β = 0

The derivative of this equation can be rearranged to give

XTY = (XTX)β

when the gradient of the curve = 0, i.e. S is minimised

This can be rearranged to give:

β = XTY(XTX)-1

But a solution can only be found, if (XTX) is invertible because you need to divide by it, which in matrix terms is the same as multiplying by the inverse!