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Linear Algebra and Matrices

Linear Algebra and Matrices

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Linear Algebra and Matrices

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  1. Linear Algebra and Matrices Methods for Dummies 20th October, 2010 Melaine Boly Christian Lambert

  2. Overview Definitions-Scalars, vectors and matrices Vector and matrix calculations Identity, inverse matrices & determinants Eigenvectors & dot products Relevance to SPM and terminology Linear Algebra & Matrices, MfD 2010

  3. Part IMatrix Basics Linear Algebra & Matrices, MfD 2010

  4. Scalar e.g. Intensity of each voxel in an MRI scan • A quantity (variable), described by a single real number Linear Algebra & Matrices, MfD 2010

  5. Vector EXAMPLE: VECTOR= i.e. A column of numbers Not a physics vector (magnitude, direction) Linear Algebra & Matrices, MfD 2010

  6. Matrices Rectangular display of vectors in rows and columns Can inform about the same vector intensity at different times or different voxels at the same time Vector is just a n x 1 matrix Linear Algebra & Matrices, MfD 2010

  7. Matrices Matrix locations/size defined as rows x columns (R x C) d i j : ith row, jth column Square (3 x 3) Rectangular (3 x 2) 3 dimensional (3 x 3 x 5) Linear Algebra & Matrices, MfD 2010

  8. Matrices in MATLAB Description Type into MATLAB Meaning Matrix(X) X=[1 4 7;2 5 8;3 6 9] ;=end of a row Reference matrix values (X(row,column)) Note the : refers to all of row or column and , is the divider between rows and columns 3rd row X(3, :) 2nd Element of 3rd column X(2,3) 8 Elements 2&3 of column 2 X( [1 2], 2) Special types of matrix zeros(3,1) All zeros size 3x1 All ones size 2x2 ones(2,2) Linear Algebra & Matrices, MfD 2010

  9. Transposition column row row column Linear Algebra & Matrices, MfD 2010

  10. Matrix Calculations Addition Commutative: A+B=B+A Associative: (A+B)+C=A+(B+C) Subtraction - By adding a negative matrix Linear Algebra & Matrices, MfD 2010

  11. Scalar multiplication Scalar * matrix = scalar multiplication Linear Algebra & Matrices, MfD 2010

  12. Matrix Multiplication Simply put, can ONLY perform A*B IF: Number of columns in A = Number of rows in B n l A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 B13 B14 B15 B16 B17 B18 k m x = m x l matrix • Hint: • If you see this message in MATLAB: • ??? Error using ==> mtimes • Inner matrix dimensions must agree -Then columns in A is not equal to rows in B “When A is a mxnmatrix & B is a kxl matrix, AB is only possible if n=k. The result will be an mxlmatrix” Linear Algebra & Matrices, MfD 2010

  13. Matrix multiplication Matlab does all this for you! Simply type: C = A * B • Hints: • You can work out the size of the output (2x2). In MATLAB, if you pre-allocate a matrix this size (e.g. C=zeros(2,2)) then the calculation is quicker Multiplication method: Sum over product of respective rows and columns Linear Algebra & Matrices, MfD 2010

  14. Matrix multiplication • Matrix multiplication is NOT commutative i.e the order matters! • AB≠BA • Matrix multiplication IS associative • A(BC)=(AB)C • Matrix multiplication IS distributive • A(B+C)=AB+AC • (A+B)C=AC+BC Linear Algebra & Matrices, MfD 2010

  15. Identity matrix A special matrix which plays a similar role as the number 1 in number multiplication? For any nxnmatrix A, we have A In = InA = A For any nxm matrix A, we have InA = A, and A Im = A (so 2 possible matrices) If the answers always A, why use an identity matrix? Can’t divide matrices, therefore to solve may problems have to use the inverse. The identity is important in these types of calculations. Linear Algebra & Matrices, MfD 2010

  16. Identity matrix Worked example A I3 = A for a 3x3 matrix: • In Matlab:eye(r, c) produces an r x c identity matrix Linear Algebra & Matrices, MfD 2010

  17. Part IIMore Advanced Matrix Techniques Linear Algebra & Matrices, MfD 2010

  18. Vector components& orthonormal base y axis b x axis a • A given vector (a b) can be summarized by its components, but only in a particular base (set of axes; the vector itself can be independent from the choice of this particular base). example a and b are the components of in the given base (axes chosen for expression of the coordinates in vector space) Orthonormal base: set of vectors chosen to express the components of the others, perpendicular to each other and all with norm (length) = 1 Linear Algebra & Matrices, MfD 2010

  19. Linear combination & dimensionality Vectorial space: space defined by different vectors (for example for dimensions…). The vectorial space defined by some vectors is a space that contains them and all the vectors that can be obtained by multiplying these vectors by a real number then adding them (linear combination). A matrix A (mn) can itself be decomposed in as many vectors as its number of columns (or lines). When decomposed, one can represent each column of the matrix by a vector. The ensemble of n vector-column defines a vectorial space proper to matrix A. Similarly, A can be viewed as a matricial representation of this ensemble of vectors, expressing their components in a given base. Linear Algebra & Matrices, MfD 2010

  20. Linear dependency and rank If one can find a linear relationship between the lines or columns of a matrix, then the rank of the matrix (number of dimensions of its vectorial space) will not be equal to its number of column/lines – the matrix will be said to be rank-deficient. Example When representing the vectors, we see that x1 and x2 are superimposed. If we look better, we see that we can express one by a linear combination of the other: x2 = 2 x1. The rank of the matrix will be 1. In parallel, the vectorial space defined will has only one dimension. Linear Algebra & Matrices, MfD 2010

  21. Linear dependency and rank • The rank of a matrixcorresponds to the dimensionality of the vectorial space defined by this matrix. It corresponds to the number of vectors defined by the matrix that are linearly independents from each other. • Linealy independent vectors are vectors defining each one one more dimension in space, compared to the space defined by the other vectors. They cannot be expressed by a linear combination of the others. • Note.Linearly independent vectors are not • necessarily orthogonal (perpendicular). • Example: take 3 linearly independent vectors • x1, x2 et x3. • Vectors x1 and x2 define a plane (x,y) • And vector x3 has an additional non-zero • component in the z axis. • But x3 is not perpendicular to x1 or x2. Linear Algebra & Matrices, MfD 2010

  22. Eigenvalues et eigenvectors One can represent the vectors from matrix X (eigenvectors of A) as a set of orthogonal vectors (perpendicular), and thus representing the different dimensions of the original matrix A. The amplitude of the matrix A in these different dimensions will be given by the eigenvalues corresponding to the different eigenvectors of A (the vectors composing X). Note: if a matrix is rank-deficient, at least one of its eigenvalues is zero. In Principal Component Analysis (PCA), the matrix is decomposed into eigenvectors and eigenvalues AND the matrix is rotatedto a new coordinate system such that the greatest variance by any projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. For A’: u1, u2 = eigenvectors k1, k2 = eigenvalues Linear Algebra & Matrices, MfD 2010

  23. Vector Products Two vectors: Inner product = scalar Outer product = matrix • Outer product XYT is a matrix (nx1) (1xn) Inner product XTY is a scalar (1xn) (nx1) Linear Algebra & Matrices, MfD 2010

  24. Scalar product of vectors Calculate the scalar product of two vectors is equivqlent to make the projection of one vector on the other one. One can indeed show that x1x2 = x1 . x2 . cos where is the angle that separates two vectors when they have both the same origin. x1x2 =   .  . cos In parallel, if two vectors are orthogonal, their scalar product is zero: the projection of one onto the other will be zero. Linear Algebra & Matrices, MfD 2010

  25. Determinants Le déterminant d’une matrice est un nombre scalaire représentant certaines propriétés intrinsèques de cette matrice. Il est noté detA ou |A|. Sa définition est un détour indispensable avant d’aborder l’opération correspondant à la division de matrices, avec le calcul de l’inverse. Linear Algebra & Matrices, MfD 2010

  26. Determinants For a matrix 11: For a matrix 22: For a matrix 33: a11 a12 a13 a21 a22 a23=a11a22a33+a12a23a31+a13a21a32–a11a23a32–a12a21a33 –a13a22a31 a31 a32 a33 = a11(a22a33 –a23a32)–a12(a21a33–a23a31)+a13(a21a32–a22a31) The determinant of a matrix can be calculate bymultiplying eachelement of one of its lines by the determinant of a sub-matrix formed by the elements that stay when one suppress the line and column containing this element. One give to the obtained product the sign (-1)i+j. Linear Algebra & Matrices, MfD 2010

  27. Determinants • Determinants can only be found for square matrices. • For a 2x2 matrix A, det(A) = ad-bc. Lets have at closer look at that: [ ] a b c d = ad - bc det(A) = • In Matlab:det(A) = det(A) The determinant gives an idea of the ’volume’ occupied by the matrix in vector space A matrix A has an inverse matrix A-1 if and only if det(A)≠0. Linear Algebra & Matrices, MfD 2010

  28. Determinants The determinant of a matrix is zero if and only if there exist a linear relationship between the lines or the columns of the matrix – if the matrix is rank-deficient. In parallel, one can define the rank of a matrix A as the size of the largest square sub-matrix of A that has a non-zero determionant. Here x1 and x2 are superimposed in space, because one can be expressed by a linear combination of the other: x2 = 2 x1. The determinant of the matrix X will thus be zero. The largest square sub-matrix with a non-zero determinant will be a matrix of 1x1 => the rank of the matrix is 1. Linear Algebra & Matrices, MfD 2010

  29. Determinants • In a vectorial space of n dimensions, there will be no more than n linearly independent vectors. • If 3 vectors (21) x’1, x’2, x’3 are represented by a matrix X’: Graphically, we have: Here x3 can be expressed by a linear combination of x1 and x2. The determinant of the matrix X’ will thus be zero. The largest square sub-matrix with a non-zero determinant will be a matrix of 2x2 => the rank of the matrix is 2. Linear Algebra & Matrices, MfD 2010

  30. Determinants The notions of determinant, of the rank of a matrix and of linear dependency are closely linked. Take a set of vectors x1, x2,…,xn, all with the same number of elements: these vectors are linearly dependent if one can find a set of scalars c1, c2,…,cn non equal to zero such as: c1 x1+ c2 x2+…+ cn xn= 0 A set of vectors are linearly dependent if one of then can be expressed as a linear combination of the others. They define in space a smaller number of dimensions than the total number of vectors in the set. The resulting matrix will be rank-deficient and the determinant will be zero. Similarly, if all the elements of a line or column are zero, the determinant of the matrix will be zero. If a matrix present two rows or columns that are equal, its determinant will also be zero Linear Algebra & Matrices, MfD 2010

  31. Matrix inverse • Definition. A matrix A is callednonsingularorinvertibleifthereexists a matrix Bsuchthat: • Notation. A common notation for the inverse of a matrix A is A-1. So: • The inverse matrix is unique when it exists. So if A is invertible, then A-1 is also invertible and then (AT)-1 = (A-1)T • In Matlab:A-1 = inv(A) • Matrix division: A/B=A*B-1 Linear Algebra & Matrices, MfD 2010

  32. Matrix inverse • For a XxX square matrix: • The inverse matrix is: • E.g.: 2x2 matrix For a matrix to be invertible, its determinant has to be non-zero (it has to be square and of full rank). A matrix that is not invertible is said to be singular. Reciprocally, a matrix that is invertible is said to be non-singular. Linear Algebra & Matrices, MfD 2010

  33. Pseudoinverse In SPM, design matrices are not square (more lines than columns, especially for fMRI). The system is said to be overdetermined – 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 used in overdetermined systems, where the solution is constrained to be the one where the  values that are minimum. Linear Algebra & Matrices, MfD 2010

  34. Part IIIHow are matrices relevant to fMRI data? Linear Algebra & Matrices, MfD 2010

  35. Image time-series Statistical Parametric Map Design matrix Spatial filter Realignment Smoothing General Linear Model StatisticalInference RFT Normalisation p <0.05 Anatomicalreference Parameter estimates Linear Algebra & Matrices, MfD 2010

  36. Model specification Parameter estimation Hypothesis Statistic Voxel-wise time series analysis Time Time BOLD signal single voxel time series SPM Linear Algebra & Matrices, MfD 2010

  37. How are matrices relevant to fMRI data? GLM equation design matrix parameters data vector error vector a m b3 b4 b5 b6 b7 b8 b9 N of scans = + Y X b e ´ = + Linear Algebra & Matrices, MfD 2010

  38. Time Intensity How are matrices relevant to fMRI data? Preprocessing ... data vector Y • Response variable • e.g BOLD signal at a particular voxel • A single voxel sampled at successive time points. • Each voxel is considered as independent observation. Time Y Y = X . β + ε Linear Algebra & Matrices, MfD 2010

  39. How are matrices relevant to fMRI data? design matrix parameters Explanatory variables • These are assumed to be measured without error. • May be continuous; • May be dummy, indicating levels of an experimental factor. a m b3 b4 b5 b6 b7 b8 b9 Solve equation for β – tells us how much of the BOLD signal is explained by X X b ´ Y = X . β + ε Linear Algebra & Matrices, MfD 2010

  40. In Practice • Estimate MAGNITUDE of signal changes • MR INTENSITY levels for each voxel at various time points • Relationship between experiment and voxel changes are established • Calculation and notation require linear algebra and matrices manipulations Linear Algebra & Matrices, MfD 2010

  41. Summary • SPM builds up data as a matrix. • Manipulation of matrices enables unknown values to be calculated. Y= X . β+ ε Observed = Predictors * Parameters + Error BOLD = Design Matrix * Betas + Error Linear Algebra & Matrices, MfD 2010

  42. References • SPM course http://www.fil.ion.ucl.ac.uk/spm/course/ • Web Guides http://mathworld.wolfram.com/LinearAlgebra.html http://www.maths.surrey.ac.uk/explore/emmaspages/option1.html http://www.inf.ed.ac.uk/teaching/courses/fmcs1/ (Formal Modelling in Cognitive Science course) • http://www.wikipedia.org • Previous MfD slides Linear Algebra & Matrices, MfD 2010

  43. ANY QUESTIONS ?