Linear Algebra, Matrices (and why they matter to (f)MRI!) Methods for Dummies FIL October 2008 Nick Henriquez & Nick Wright Theory & Application Theory & Application

Sources and further information   Previous FIL slides! YES we copied some....   SPM course   Web Guides – – – – (Formal Modelling in Cognitive Science course) – –

fMRI and Linear Algebra (fMRI) measures signal changes in the brain that are due to changing neural activity. brainneural Increases in neural activity cause changes in the MR signal via T2* changes; this mechanism is referred to as the BOLD (blood-oxygen-level dependent) effect.blood-oxygen-level dependent To estimate the MAGNITUDE of signal changes we need to measure MR INTENSITY levels for each “volumetric pixel”=VOXEL at various TIME POINTS. In SPM each VOXEL is observed/considered independently over time Any relationship between EXPERIMENT and VOXEL CHANGE is established using standard statistics Calculation and notation require Linear Algebra and MATRICES

Defining scalars, vectors and matrices   Scalar: Variable described by a single number – e.g. Image intensity (pixel value)  Vector: Variable described by magnitude and direction – e.g. pixel value+(relative) location  Matrix: Rectangular array of vectors defined by number of rows and columns – e.g. Intensities of several voxels or same voxel at different times Square (3 x 3)Rectangular (3 x 2) d r c : r th row, c th column 3 2 (Roman Catholic) V E V N Volumetric pixel (VOXEL) intensity is expressed as a VECTOR. The size (=> MAGNITUDE) is determined by its direction. V= V N V E

Matrices in Matlab   Vector formation: [1 2 3]   Matrix formation: X = [1 2 3; 4 5 6; 7 8 9] = Subscripting – each element of a matrix can be addressed with a pair of numbers; row first, column second (Roman Catholic) e.g. X(2,3) = 6 X(3, :) = X( [2 3], 2) = ‘;’ is used to signal end of a row ‘:’ is used to signify all rows or columns “Special” matrix commands: zeros(3,1) = ones(2) = magic(3) more to come…

Addition (matrix of same size) –Commutative: A+B=B+A –Associative: (A+B)+C=A+(B+C) Subtraction (consider as the addition of a negative matrix) Matrix addition

Matrix multiplication a 11 a 12 a 13 b 11 b 12 a 21 a 22 a 23 X b 21 b 22 a 31 a 32 a 33 b 31 b 32 a 41 a 42 a 43 Scalar multiplication: Multiplication of vectors/matrices: b 11 b 12 a 11 a 12 a 13 b 21 b 22 x a 21 a 22 a 23 b 31 b 32 a 31 a 32 a 33 a 41 a 42 a 43 Matrix multiplication rule: “When A is a mxn matrix & B is a kxl matrix, AB is only viable if n=k. The result will be an mxl matrix” nl m k

Sum over product of respective rows and columns For larger matrices, following method might be helpful: Multiplication method m l m l X= = = Define output matrix Sum over c rc Matlab does all this for you! Simply type: C = A * B N.B. If you want to do element-wise multiplication, use: A.* B rc

Defining the identity matrix   Is there a matrix which plays a similar role as the number 1 in number multiplication? Consider the nxn matrix: A square nxn matrix A has one A I n = I n A = A An nxm matrix A has two!! I n A = A & A I m = A X010= Worked example A In = A for a 3x3 matrix: In Matlab: eye(r, c) produces an r x c identity matrix

Inverse matrices   Definition. A matrix A is nonsingular or invertible if there exists a matrix B such that: worked example:  Common notation for the inverse of a matrix A is A -1  If A is an invertible matrix, then (A T ) -1 = (A -1 ) T  The inverse matrix A -1 is unique when it exists.  If A is invertible, A -1 is also invertible  A is the inverse matrix of A -1.11X = = In Matlab: A -1 = inv(A) Matrix division: A/B = AB -1

Determinants In Matlab: det(A) = det(A)  A matrix A has an inverse matrix A -1 if and only if det(A)≠0 (see next slide) The determinant of a 2×2 matrix is the product of the elements on the main diagonal minus the product of the elements off the main diagonal.

Calculation of inversion using determinants ax 1 = 1-cx 2 => x 1 = (1-cx 2 ) Etc. thus Note: det(A)≠0 Or you can just type inv(A) ! bx 1 +dx 2 =0 => b(1-cx 2 )/a = -dx 2

Transposition column → row row → column M rc = M cr In Matlab: A T = A’ T

Outer product = matrix Inner product = scalar Two vectors: Outer and inner products of vectors (1xn)(nx1)  (1X1) (nx1)(1xn)  (nXn)

Applications

Recap!   Why matrices? Lots of data and calculation   Why algebra? Allows you to find unknowns.   To do matrix algebra you need to use the matrix manipulations you’ve just learnt, e.g. – –Addition and subtraction – –Multiplicatione.g. Inner / outer – –“Division”e.g. Inverse – –“Powers”e.g. Transpose

Scalar and vector algebra   Algebra with scalars (“Normal school algebra”): e.g. y = x  – –Solve for  :  = y / xor  = y x -1   But our experiments get more data, which we want to represent vectors or matrices. We still want to do algebra with matrices. e.g. Y = X  – –Solve for  :  = X -1 Y   Examples of using matrix algebra to solve equations: – –Simultaneous equations with 2 unknowns and 2 equations – –Many equations and unknowns – real world GLM and fMRI

Example: Using matrices to solve simultaneous equations   A pair of simultaneous equations:   In matrix form   We want to rearrange to find the unknowns 0.4p + 0.2q = 4 0.6p + 0.8q = 11

  Need to find the inverse of A   Multiply both sides by A -1 of our first matrix by A -1   Get the answer! A = A -1 = = A -1 A = A -1 = = So… p = 5 and q = 10

Analysing data from one voxel   Getting the data as a vector of intensities Intensity at time 1 Intensity at time 2 Intensity at time 3 Time

  +  =  + YX data vector (Voxel) design matrix parameters error vector × = One voxel: The GLM Our aim: Solve equation for β – tells us how much BOLD signal is explained by X

Some matrix algebra   Initially can think of this as a system of simultaneous equations.   But, there are more equations (y = x  ) than unknowns (  )   So … clever maths e.g. Y = X  X T Y = X T X  (X T X) -1 X T Y =   = (XT X)-1 XT Y ^

  +  =  + YX data vector (Voxel) design matrix parameters error vector × = One voxel: The GLM Our aim: Solve equation for β – tells us how much BOLD signal is explained by X

Take home messages 1. 1.Matrices are at the core of SPM – it is how the data and design “matrix” are built and manipulated You need to be able to manipulate matrices to do matrix algebra and find unknowns – the basic results for the experiment! 3. 3.Simple example: use matrix algebra to solve 2 simultaneous equations with 2 unknowns Use matrix algebra to solve bigger problems, e.g. the GLM.

Thanks Justin. The End…