Linear Algebra and Matrices Methods for Dummies FIL

Linear Algebra and Matrices Methods for Dummies FIL November 2011 Narges Bazargani and Sarah Jensen

ONLINE SOURCES Web Guides – http: //mathworld. wolfram. com/Linear. Algebra. html – http: //www. maths. surrey. ac. uk/explore/emmaspages/option 1. html – http: //www. inf. ed. ac. uk/teaching/courses/fmcs 1/ Online introduction: - http: //www. khanacademy. org/video/introduction-tomatrices? playlist=Linear+Algebra

What Is MATLAB? - And why learn about matrices? MATLAB = MATrix LABoratory Typical uses include: • Math and computation • Algorithm development • Modelling, simulation, and prototyping • Data analysis, exploration, and visualization • Scientific and engineering graphics • Application development, including Graphical User Interface building

Everything in MATLAB is a matrix ! Zero-dimentional matrix A Scalar - a single number is really a 1 x 1 matrix in Matlab! 1 dimentional matrix A vector is a 1 xn matrix with 1 row 4 [1 2 3] n m A matrix is an mxn matrix Even a picture is a matrix! 2 7 4 3 8 9

Building matrices I MATLAB with [ ]: A = [2 7 4] 2 7 4 3 8 9 2 A = [2; 7; 4] 7 4 ; separates the different rows A = [2 7 4; 3 8 9] : separates collums

Matrix formation in MATLAB X = [1 2 3; 4 5 6; 7 8 9] = Submatrices in MATLAB Subscripting – each element of a matrix can be addressed with a pair of numbers; row first, column second X(2, 3) = 6 X(3, : ) = ( 7 8 9 ) X( [2 3], 2) =

Matrix addition and subtraction NB Only matrices of the same size can be added and substracted Addition Subtraction

Matrix Multiplication I Different kinds of multiplication I MATLAB Scalar multiplication Matlab does all this for you!: 3 * A

Matrix multiplication II Sum product of respective rows and columns n Matrix multiplication rule: m A x B is only viable if n=k. Matlab does all this for you!: C = A * B l a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a 42 a 43 b 11 b 12 b 21 b 22 b 31 b 32 x b 11 b 12 b 21 b 22 b 31 b 32 a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a 42 a 43 X k

Elementwise multiplication Matrix multiplication rule: Matrixes need the exact same ‘m’ Ma and ‘n’ Matlab does all this for you!: A. * B a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 a 41 a 42 a 43 b 11 b 12 b 21 b 22 b 31 b 32 x X b 11 b 12 b 21 b 22 b 31 b 32 a 11 a 12 a 21 a 22 a 31 a 32

Transposition – reorganising matrices column → row → column In Matlab: AT = A’

Identity matrices Tool to solve equation This identity matrix Is a matrix which plays a similar role as the number 1 in number multiplication Worked example A In = A for a 3 x 3 matrix: 1 2 3 4 5 6 7 8 9 X 1 0 0 0 1 = 1+0+0 0+2+0 0+0+3 4+0+0 0+5+0 0+0+6 7+0+0 0+8+0 0+0+9 In Matlab: eye(r, c) produces an r x c identity matrix

Inverse matrices Definition: Matrix A is invertible if there exists a matrix B such that: 1 1 -1 2 X 2 3 -1 3 1 3 = 2+1 3 3 -1 + 1 3 3 -2+ 2 3 3 1+2 3 3 = 1 0 0 1 Notation for the inverse of a matrix A is A-1 If A is invertible, A-1 is also invertible A is the inverse matrix of A-1. • In Matlab: A-1 = inv(A)

Determinants Determinant is a function: A Matrix A has an inverse matrix (A-1)if and only if det(A) ≠ 0 • In Matlab: det(A) = det(A)

With more than 1 equation and more than 1 unknown Can use solution equation to solve from the single For example In matrix form AX = B

Need to find determinant of matrix A (because X =A-1 B) From earlier (2 x -2) – (3 x 1) = -4 – 3 = -7 So determinant is -7 To find A-1: if B is So

scalars, vectors and matrices in SPM • Scalar: Variable described by a single number – e. g. intensity of each voxel in MRI scan • Vector: Physics vector is Variable described by magnitude and direction – Here we talk about column of numbers e. g. voxel intensity at a different times or different voxels at the same time Matrix: Rectangular array of vectors defined by number of rows and columns. x 11 x 12 ………x 1 n. . xn 1……………xnn

Vectorial Space and Matrix Rank Vectorial space: is a space that contains vectors and all the those that can be obtained by multiplying vectors by a real number then adding them (linear combination). In other words, because each column of the matrix can be represented by a vector, the ensemble of n vector-column defines a vectorial space for a matrix. Rank of a matrix: corresponds to the number of vectors that are linearly independents from each other. So, if there is a linear relationship between the lines or columns of a matrix, then the matrix will be rank-deficient (and the determinant will be zero). For example, in the graph below there is a linear relationship between X 1 and X 2, and the determinent is zero. And the Vectorial space defined will has only 1 dimension.

Eigenvalues et eigenvectors Eigenvalues are multipliers. They are numbers that represent how much linear transformation or stretching has taken place. An eigenvalue of a square matrix is a scalar that is represented by the Greek letter λ (lambda). Eigenvectors of a square matrix are non-zero vectors, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix. All eigenvalues and eigenvectors satisfy the equation Ax = λx for a given square matrix A. i. e. Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A. One can represent eigenvectors of A as a set of orthogonal vectors representing different dimensions of the original matrix A. (Important in Principal Component Analysis, PCA)

Matrix Representations of Neural Connections • Can create a mathematical model of the connections in a neural system • Connections are the excitatory or inhibitory Excitatory Connection Input Neuron Output Neuron Inhibitory Connection Input Neuron Output Neuron

Matrix Representations of Neural Connections Excitatory = Makes it easier for the post synaptic cell to fire #2 -1 #1 +1 #3 Inhibitory = Makes it harder for the post synaptic cell to fire We can translate this information into a set of vectors (1 row matrices) Input vector = (1 1) relates to activity (#1 #2) Weight vector = (1 -1) relates to connection weight (#1 #2) Activity of Neuron 3 Input x weight Cancels out! But it is more complicated than this!

How are matrices relevant to f. MRI data? Basics of MR Physics Angular Momentum: Neutrons, protons and electrons spin about their axis. The spinning of the nuclear particles produces angular momentum. Certain nuclei exhibit magnetic properties. A proton has mass, a positive charge, and spins, it produces a small magnetic field. This small magnetic field is referred to as the magnetic moment that is a vector quantity with magnitude and direction and is oriented in the same direction as the angular momentum. Under normal circumstances these magnetic moments have no fixed orientation (so no overall magnetic field). However, when exposed to an external magnetic field (B 0), nuclei begin to align. To detect net magnetisation signal a second magnetic field is introduced (B 1) which is applied perpendicular to B 0, and it has to be at the resonant frequency.

How are matrices relevant to f. MRI data? Y = X. β + ε Observed = Predictors * Parameters + Error BOLD = Design Matrix * Betas + Error Time Y is a matrix of BOLD signals Each column represents a single voxel sampled at successive time points. Each voxel is considered as independent observation So, we analysis of individual voxels over time, not groups over space Y Intensity

s er er ve ror ct or m et ra pa da ve ta ct or d m esig at n ri x Response variable A single voxel sampled at successive time points. Each voxel is considered as independent observation. Solve equation for β – tells us how much of the BOLD signal is explained by X Explanatory variables These are assumed to be measured without error. May be continuous, indicating levels of an experimental factor. a m N of scans b 3 b 4 = b 5 + b 6 b 7 b 8 b 9 Y Observed = X Predictors ´ b + e

Pseudoinverse In SPM, design matrices are NOT square matrices (more lines than columns, especially for f. MRI). So, there is not a unique solution, i. e. there is more than one solution possible. SPM will use a mathematical trick called the pseudoinverse, which is an approximation, where the solution is constrained to be the one where the values that are minimum.

How are matrices relevant to f. MRI? Image time-series Realignment Spatial filter Design matrix Smoothing General Linear Model Statistical Parametric Map Statistical Inference Normalisation Anatomical reference Parameter estimates RFT p <0. 05

In Practice Estimate MAGNITUDE of signal changes and MR INTENSITY levels for each voxel at various time points Relationship between experiment and voxel changes are established Calculation require linear algebra and matrices manipulations SPM builds up data as a matrix. Manipulation of matrices enables unknown values to be calculated.

Thank you! Questions?