Multivariate Approaches to Analyze fMRI Data
Yuanxin Hu
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Nature of fMRI data
1. Multivariate
2. Subspaces / high dimensions/directions
a) Space: region of brain with similar temporal behavior
b) Time course
c) Space & Time course
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Principle Component Analysis (PCA)
Goal
To find linear combinations of the original variables reflecting the structural dependence of data. The strategy
Is to create a new set of orthogonal variables that contain the same information as the original set, and the previous
Orthogonal axe occupies the majority of sample variance, and determine the direction of dimension of the dataset.
Steps
1. Find independent components;
X1, X2, ---------, Xp
~ multivariate distribution ( µ, Σ)
1ST component = a1t X (with maximal sample variance: a1t Sa1, and a1t a1 = 1);
2nd component = a2t X (a2t a2 = 1, and a1t a2 =0: it indicates that its coefficient vector is orthogonal to the coefficient
vector of 1st component)
--------Kth component = akt X (akt aK = 1, ak-1t aK = 0);
2. Transform components into coordinates.
Serial components will be transformed into a new set of coordinates given values in appropriate eigenvectors
Consequences
1. Sample variance comparison among components:
1st > > 2nd > > 3rd >> -------------- > ; so the 1st component has the principle axis of the p-dimensional scatter cloud;
2. The coefficient vector of sub sequential component is orthogonal to its previous one.
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Canonical Correlation Analysis (CCA)
A way to quantify correlation between sets of variables.
Pairs of canonical variables:
Radom variables
canonical variables: 1st:
2nd
.
.
.
kth
X
Y
a1T X
a2T X
.
.
.
akT X
b1T X
b2T X
.
.
.
bkT X
Cor (aiT X, biT Y), and its coefficients:
(ΣxyΣyy -1Σyx-CiΣxx)*ai = 0
(ΣyxΣxx -1 Σxy-CiΣyy)*bi = 0
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Independent Component Analysis (ICA)
(Originates from “Cocktail-Party Problem”)
In the Cocktail-Party-Problem, you are attending a party with simultaneous
conversations of hundreds guests. Same amount microphones located at
different places in the room, are simultaneously recording the conversations.
Each microphone recording can be considered as a linear mixture of individual
'independent' conversations.
Key of ICA
Non-normality
Each microphone signal (X) can be modeled as linear superpositions of
the recorded source signals (linear mixture by unknown matrix A).
Recover original source signals by finding a matrix (W)
ICA in studying fMRI data
Sensor1
Sensor2
Sensor3
Difference between PCA and ICA
Jung TP, et al 2001, Proceedings of the IEEE, 89(7);
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Preprocessing for ICA
1. Centering:
The most basic and necessary preprocessing is to center x, by subtracting its mean
vector m = E{x} to make x a zero-mean variable. It will simplify ICA algorithms;
2. Whitening
Linearly transform observed vector X to make the components uncorrelated,
and their covariance matrix of ˜x equals the identity matrix: E{˜x˜xT } = I.
3.Data reduction
Remove noise signal to decrease data dimension, and make the data meet biological
sense.
Hyvärinen A, et al, 2000, Neural Networks, 13(4-5):411-430
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
ICA Multivariate Analyses
1. Spatial ICA
2. Temporal ICA
voxels
t1
n
Time course
1n
Spatially independent time course associated the tth component image
1n
t1
The tth component image
Time course
1t
voxels
n1
associated temporally independent image
Temporally independent time course
n1
nt
In theory, once the independent components are identified, the statistical
test can be further investigated, for example: the distribution of probability
of all voxels and correlation of activation of different regions upon stimuli,
and so on. However, the nature of the procedure makes us not that
confident.
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Validation of ICA Results
Reasons of validation
1.
Different algorithms can yield different components, which will
contribute different interpretations for same data;
2.Algorithms always have stochastic elements, as a result, different
runs of same algorithms can contribute different results.
Validation of ICA Results
Strategies of validation
1.Fixed-point based:
Normalize differential entropy/negentropy, and maximize negentropy
to find directions of maximal non-normality of the data;
2.Bootstrap:
The validation is to find out whether the statistical test is
reproducible or consistent. To avoid the variation caused by
stochastic element from algorithms operation, the analysis can start
at different initial value, which can be accomplished via Bootstrap.
In practice, researchers can repeat running same operation, and find
the tight cluster of point, which will be real independent component;
if the clusters are wildly scattered, which should not be selected,
because they are not real independent components. This can be
judged by Cluster Quality Index, higher is better.
Selection of Clusters
Himberg J, et al, 2004 NeuroImage, 22(3):1214–1222
Himberg J, et al, 2004 NeuroImage, 22(3):1214–1222
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Methods of Group ICA
1. Averaging Across Subjects
2. Calhound’s model: (Temporal basis, Subject-wise concatenation)
Combination of data from individual subjects. The data is large, so data reduction is
essential: Clean individual data; transform original data into Talairach coordinates.; and
then concatenate all individuals’ data together for analysis
3. Svensѐn model: (Spatial basis, row-wise concatenation)
Data reduction by masking air out sir voxels, decrease about 50% data dimension, so
there is no need to transform the data into Talairach coordinates.
1) Calhoun VD, et al (2001): NeuroImage 14(5):1080-1088.
2) Beckmann CF, et al. (2005): NeuroImage 25(1):294-311.
3) Calhoun VD, et al (2001): Hum.Brain Map. 14(3):140-151.
4) Esposito F, Neuroimage. 25(1):193-205.
5) Schmithorst VJ, et al (2004): J.Magn Reson.Imaging 19(3):365-368.
6) Svensen M, et al. (2002): NeuroImage 16:551-563.
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Accuracy Comparison of Three Methods by Simulation
MSE = mean-squared error between original and estimated sources
Average CC = average cross-correlation value between original and estimated associated time courses
Accuracy Comparison of Three Methods by Simulation
(+ data from 5 subjects)
MSE = mean-squared error between original and estimated sources
Average CC = average cross-correlation value between original and estimated associated time courses
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering
2) whitening
3) data reduction
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results
3.Group ICA
a) models/methods
b) comparison of methods
c) modifications of classical group ICA
Modification of Basic ICA Apporaches
1. Spatiotemporal ICA
either sICA or tICA are dual dimension, which is meaningless for scientific basis;
2. Skew-ICA
Real images are surrounded by homogeneous background, which will cause skewed
distribution. To solve this, the method uses more realistically long tail instead of heavy
tail to represent the distribution.
GLM
PCA
tICA
sICA
stICA
Skewed-ICA
Correlation Between the Four Time Courses Extracted by Each Method
Method
GLM
PCA
tICA
sICA
stICA
Skew-sICA
Skew-stICA
Source 1
0.87
0.76
0.87
0.48
0.87
0.83
0.87
Source 2
0.93
0.55
0.90
0.75
0.89
0.90
0.92
Source 3
0.95
0.77
0.91
0.40
0.91
0.88
0.94
Source 4
0.84
0.81
0.72
0.71
0.72
0.81
0.83
Mean
0.90
0.72
0.85
0.59
0.85
0.86
0.89
Stone JV, 2002, NeuroImage 15: 407-421
Outlines
1.Summary of principles of three approaches
a) Principle Component Analysis (PCA)
b) Canonical Correlation Analysis (CCA)
c) Independent Component Analysis (ICA)
2.Procedure to analyze fMRI data by ICA approach
a) data preprocessing
1) centering (simplify calculation)
2) whitening (linearly transformation, to ensure components are uncorrelated)
3) data reduction (remove noise signal, keep biologically related information only)
b) strategies of ICA
1) temporal ICA
2) spatial ICA
c) validation of ICA results (point-fixed method, bootstrap to indentify real independent
clusters )
3.Group ICA
a) models/methods (Averaging, row-wise group ICA, subject-wise group ICA)
b) comparison of methods (the subject-wise group ICA is more accurate)
c) modifications of classical group ICA (skewed ICA is more consistent)