linear algebra and matrices methods for dummies fil november 2011
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Linear Algebra and Matrices Methods for Dummies FIL November 2011 Narges Bazargani and Sarah Jensen. ONLINE SOURCES 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/ - PowerPoint PPT PresentationTRANSCRIPT
Linear Algebra and Matrices Linear Algebra and Matrices
Methods for DummiesMethods for DummiesFILFIL
November 2011November 2011
Narges Bazargani and Sarah JensenNarges Bazargani and Sarah Jensen
Linear Algebra and Matrices Linear Algebra and Matrices
Methods for DummiesMethods for DummiesFILFIL
November 2011November 2011
Narges Bazargani and Sarah JensenNarges Bazargani and Sarah Jensen
ONLINE SOURCES
Web Guides
– http://mathworld.wolfram.com/LinearAlgebra.htmlhttp://mathworld.wolfram.com/LinearAlgebra.html
– http://www.maths.surrey.ac.uk/explore/emmaspages/option1.htmlhttp://www.maths.surrey.ac.uk/explore/emmaspages/option1.html
– http://www.inf.ed.ac.uk/teaching/courses/fmcs1/http://www.inf.ed.ac.uk/teaching/courses/fmcs1/
Online introduction::- http://www.khanacademy.org/video/introduction-to-matrices?- http://www.khanacademy.org/video/introduction-to-matrices?playlist=Linear+Algebraplaylist=Linear+Algebra
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
What Is MATLAB?- And why learn about matrices?
Everything in MATLAB is a matrix !
Zero-dimentional matrixA Scalar - a single number is really a 1 x 1 matrix in Matlab!
1 dimentional matrixA vector is a 1xn matrix with 1 row [1 2 3]
A matrix is an mxn matrix
Even a picture is a matrix!
4
2 7 4
3 8 9
m
n
2 7 4
2
7
4
2 7 4
3 8 9
Building matrices I MATLAB with [ ]:
A = [2 7 4]
A = [2; 7; 4]
A = [2 7 4; 3 8 9]
; separates the different rows; separates the different rows
: separates collums: separates collums
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) =
X = [1 2 3; 4 5 6; 7 8 9] =
Matrix formation in MATLAB
Submatrices in MATLAB
NB Only matrices of the same size can be added and substractedNB Only matrices of the same size can be added and substracted
AdditionAddition
SubtractionSubtraction
Matrix addition and subtraction
Scalar multiplication
Matlab does all this for you!: 3 * AMatlab does all this for you!: 3 * A
Matrix Multiplication I
Different kinds of multiplication I MATLAB
Matrix multiplication IISum product of respective rows and columnsSum product of respective rows and columns
Matlab does all this for you!: C = A * BMatlab does all this for you!: C = A * B
Matrix multiplication rule:Matrix multiplication rule:
A x B is only viable if n=k.
aa1111 aa1212 aa1313 bb1111 bb1212
aa2121 aa2222 aa2323 XX bb2121 bb2222
aa3131 aa3232 aa3333 bb3131 bb3232
aa4141 aa4242 aa4343
bb1111 bb1212 aa1111 aa1212 aa1313
bb2121 bb2222 xx aa2121 aa2222 aa2323
bb3131 bb3232 aa3131 aa3232 aa3333
aa4141 aa4242 aa4343
n l
mk
Elementwise multiplication
Matlab does all this for you!: A .* BMatlab does all this for you!: A .* B
aa1111 aa1212 aa1313 bb1111 bb1212
aa2121 aa2222 aa2323 XX bb2121 bb2222
aa3131 aa3232 aa3333 bb3131 bb3232
aa4141 aa4242 aa4343
bb1111 bb1212 aa1111 aa1212
bb2121 bb2222 xx aa2121 aa2222
bb3131 bb3232 aa3131 aa3232
Matrix multiplication rule:Matrix multiplication rule:
MaMatrixes need the exact same ‘m’ and ‘n’
column → row row → column
In Matlab: AT = A’ In Matlab: AT = A’
Transposition – reorganising matrices
Identity matrices
Tool to solve equationThis identity matrix Is a matrix which plays a similar role as the number 1 in number multiplication
11 22 33 11 00 00 1+0+01+0+0 0+2+00+2+0 0+0+30+0+3
44 55 66 XX 00 11 00 == 4+0+04+0+0 0+5+00+5+0 0+0+60+0+6
77 88 99 00 00 11 7+0+07+0+0 0+8+00+8+0 0+0+90+0+9
Worked exampleA In = A
for a 3x3 matrix:
100
010
001
In Matlab: eye(r, c) produces an r x c identity matrixIn Matlab: eye(r, c) produces an r x c identity matrix
Definition: Matrix A is invertible if there exists a matrix B such that:
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.
11 11 XX2 2 33
-1-1 33
==22 + + 11 3 33 3
-1-1 + + 11 3 33 3 == 11 00
-1-1 22 1 1 33
1 1 33
-2-2+ + 22 3 3
33
11 + + 22 3 3 3 3 00 11
• In Matlab: A-1 = inv(A)• In Matlab: A-1 = inv(A)
Inverse matrices
DeterminantsDeterminantsDeterminantsDeterminants
• In Matlab: det(A) = det(A)• In Matlab: det(A) = det(A)
Determinant is a function: Determinant is a function:
A Matrix A has an inverse matrix (A Matrix A has an inverse matrix (AA-1-1)if and only )if and only if if det(A) ≠0det(A) ≠0
Can use solution from the single Can use solution from the single equation to solve equation to solve
For exampleFor example
In matrix formIn matrix form
bax 1
12
532
21
21
xx
xx
AX = B
1
5
21
32
2
1
x
x
With more than 1 equation and more
than 1 unknown
21
32
bcaddc
baA )det(
21
32
7
1
21
32
)7(
11A
4
1
if B isif B is
1
2
7
14
7
1
4
1
21
32
7
1
XSoSo
Need to find determinant of matrix A (because X =A-1B)
From earlier
(2 x -2) – (3 x 1) = -4 – 3 = -7So determinant is -7
To find A-1:
scalars, vectors and matrices in scalars, vectors and matrices in SPM SPM
scalars, vectors and matrices in scalars, vectors and matrices in SPM SPM
e
n
v
vv
•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
•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
MatrixMatrix: : Rectangular array of vectors defined by number of rows and columns. Rectangular array of vectors defined by number of rows and columns.
xn
x
x
2
1
x11 x12 ………x1n..xn1……………xnn
Vectorial Space and Matrix RankVectorial Space and Matrix RankVectorial Space and Matrix RankVectorial 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 X1 and X2, and the determinent is zero. And the Vectorial space defined will has only 1 dimension.
2x
1x
y
x
4
21
2
Eigenvalues et eigenvectorsEigenvalues et eigenvectorsEigenvalues et eigenvectorsEigenvalues 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)
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
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 Inhibitory Connection
Input Neuron Output Neuron Input Neuron Output Neuron
Matrix Representations of Neural Matrix Representations of Neural ConnectionsConnections
Matrix Representations of Neural Matrix Representations of Neural ConnectionsConnections
#1 #3
#2
-1
+1
Excitatory = Makes it easier for the Excitatory = Makes it easier for the post synaptic cell to firepost synaptic cell to fire
Inhibitory = Makes it harder for the Inhibitory = Makes it harder for the post synaptic cell to firepost synaptic cell to fire
Excitatory = Makes it easier for the Excitatory = Makes it easier for the post synaptic cell to firepost synaptic cell to fire
Inhibitory = Makes it harder for the Inhibitory = Makes it harder for the post synaptic cell to firepost synaptic cell to fire
We can translate this information into a set of vectors (1 row matrices)We can translate this information into a set of vectors (1 row matrices)
Input vector = (1 1) Input vector = (1 1) relates to activity (#1 #2)relates to activity (#1 #2) Weight vector = (1 -1)Weight vector = (1 -1) relates to connection weight (#1 #2relates to connection weight (#1 #2))
0)1()1()11()11(1
1.11
Activity of Neuron 3Activity of Neuron 3Input x weightInput x weight
Cancels out! But it is more complicated than this!
How are matrices relevant to fMRI data?How are matrices relevant to fMRI data?Basics of MR PhysicsBasics of MR Physics
Angular MomentumAngular Momentum: Neutrons, protons and electrons spin about their axis. : Neutrons, protons and electrons spin about their axis. The spinning of the nuclear particles produces angular momentum.The spinning of the nuclear particles produces angular momentum.
Certain nuclei exhibit Certain nuclei exhibit magnetic propertiesmagnetic properties. A proton has mass, a positive . A proton has mass, a positive charge, and spins, it produces a small magnetic field. This small magnetic 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 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 magnitude and direction and is oriented in the same direction as the angular momentum. momentum.
Under normal circumstances these magnetic moments have no fixed Under normal circumstances these magnetic moments have no fixed orientation (so no overall magnetic field). However, when exposed to orientation (so no overall magnetic field). However, when exposed to an an external magnetic field (external magnetic field (BB00), nuclei begin to align. To detect net ), nuclei begin to align. To detect net
magnetisation signal a second magnetic field is introduced magnetisation signal a second magnetic field is introduced ((BB11) which is ) which is
applied applied perpendicularperpendicular to B to B00, and it has to be at the resonant frequency., and it has to be at the resonant frequency.
How are matrices relevant to fMRI data?How are matrices relevant to fMRI data?Basics of MR PhysicsBasics of MR Physics
Angular MomentumAngular Momentum: Neutrons, protons and electrons spin about their axis. : Neutrons, protons and electrons spin about their axis. The spinning of the nuclear particles produces angular momentum.The spinning of the nuclear particles produces angular momentum.
Certain nuclei exhibit Certain nuclei exhibit magnetic propertiesmagnetic properties. A proton has mass, a positive . A proton has mass, a positive charge, and spins, it produces a small magnetic field. This small magnetic 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 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 magnitude and direction and is oriented in the same direction as the angular momentum. momentum.
Under normal circumstances these magnetic moments have no fixed Under normal circumstances these magnetic moments have no fixed orientation (so no overall magnetic field). However, when exposed to orientation (so no overall magnetic field). However, when exposed to an an external magnetic field (external magnetic field (BB00), nuclei begin to align. To detect net ), nuclei begin to align. To detect net
magnetisation signal a second magnetic field is introduced magnetisation signal a second magnetic field is introduced ((BB11) which is ) which is
applied applied perpendicularperpendicular to B to B00, and it has to be at the resonant frequency., and it has to be at the resonant frequency.
How are matrices relevant to fMRI How are matrices relevant to fMRI data?data?
How are matrices relevant to fMRI How are matrices relevant to fMRI data?data?
YY = X= X . . ββ + ε+ ε
Observed = Predictors * Parameters + ErrorObserved = Predictors * Parameters + Error
BOLD = Design Matrix * Betas + ErrorBOLD = Design Matrix * Betas + Error
YY = X= X . . ββ + ε+ ε
Observed = Predictors * Parameters + ErrorObserved = Predictors * Parameters + Error
BOLD = Design Matrix * Betas + ErrorBOLD = Design Matrix * Betas + Error
Y is a matrix of BOLD signalsY is a matrix of BOLD signals Each column represents a single Each column represents a single
voxel sampled at successive time voxel sampled at successive time points.points.
Each voxel is considered as Each voxel is considered as independent observationindependent observation
So, we analysis of individual voxels So, we analysis of individual voxels over time, not groups over spaceover time, not groups over space
Tim
e
Intensity
Y
Response variable A single voxel sampled at successive time points. Each voxel is considered as independent observation.
Response variable A single voxel sampled at successive time points. Each voxel is considered as independent observation.
=
+
= +Y X
data
vecto
rdes
ign
mat
rixpar
amete
rs
erro
r
vecto
r
Explanatory variablesThese are assumed to be measured without error.May be continuous, indicating levels of an experimental factor.
Observed Predictors
Solve equation for β – tells us how much of the BOLD signal is explained by X
Solve equation for β – tells us how much of the BOLD signal is explained by X
PseudoinversePseudoinverse
In SPM, design matrices are NOT square matrices (more lines than columns, especially for fMRI).
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.
NormalisationNormalisation
Statistical Parametric MapStatistical Parametric MapImage time-seriesImage time-series
Parameter estimatesParameter estimates
General Linear ModelGeneral Linear ModelRealignmentRealignment SmoothingSmoothing
Design matrix
AnatomicalAnatomicalreferencereference
Spatial filterSpatial filter
StatisticalStatisticalInferenceInference
RFTRFT
p <0.05p <0.05
How are matrices relevant to fMRI?How are matrices relevant to fMRI?How are matrices relevant to fMRI?How are matrices relevant to fMRI?
In PracticeIn PracticeIn PracticeIn Practice Estimate Estimate MAGNITUDEMAGNITUDE of signal changes and of signal changes and
MR INTENSITY MR INTENSITY levels for each voxel at various levels for each voxel at various time pointstime points
Relationship between experiment and voxel Relationship between experiment and voxel changes are established changes are established
Calculation require linear algebra and matrices Calculation require linear algebra and matrices manipulationsmanipulations
SPM builds up data as a matrix.SPM builds up data as a matrix. Manipulation of matrices enables unknown Manipulation of matrices enables unknown
values to be calculated.values to be calculated.
Thank you!Thank you!Thank you!Thank you!
Questions?Questions?