Analytical Techniques

Analytical Techniques

• Data Driven• Principal Component Analysis (PCA)• Independent Component Analysis (ICA)• Fuzzy Clustering

• Others• Structural equation modeling

• Hypothesis Driven

Matrix Notation of fMRI Data1 voxel

t=1

t=2

t=3

t=4. . .

Data Matrix

Voxels

time XSlice 1

BOLD signal

Calculating level of Significance

GX

fMRI Data =

significance: ~ t statistic i/i

+

total variability = Variability explained by the model

+ noise

SPM Nomenclature for Design Matrix

G (interesting)

Design matrix G

G1 Gc

H (non-interesting)

Indicator variable

Covariate

E.g. dose of drug

H1

subject

HcGlobal activity

Linear trends

Some General Linear Model (GLM) Assumptions:

• Design matrix known without error

• the ’s follow a Gaussian distribution

• the design matrix is the same everywhere in the brain

• the residuals are well modeled by Gaussian noise

• each voxel is independent of the others

• the voxels are temporally aligned

• each time point is independent of the others (time courses of voxels are white)

Hypothesis

Test voxel

Global Signal

Inclusion of Global Signal in Regression

< 5 degrees difference between Global Signal & Hypothesis !

1 2-2

-1

0

1

2

3

4

Regression Coefficients

< 0!!!Globalsignal

Hypothesis

Inclusion of Global Covariate in Regression:Effect of non orthogonality

1

2X1

X’1

db1

db2

1

2

“Reference Function, R”

db2

X1

X’1

b = (GTG)-1GTX

Consider an fMRI experiment with only 3 time points

Consider an fMRI experiment with only 3 time points

Analysis of Brain Systems

reference function R1

R2

Although R1 and R2 both somewhat correlated with the reference

function, they are uncorrelated with each other

ref

R1

R2

Corr(R1, ref)

Corr(R2, ref)

Correlation viewed as

a projection

Principal Component Analysis (PCA)

Voxel 1Voxel 2

Vox

el 3

PC1

Voxel 1 Voxel 2

Voxel 3

Eigenimage + time course

t

Independent Component Analysis (ICA)

Without knowing position of microphones or what any person is saying, can you isolate each of the voices?

Independent Component Analysis (ICA)

Assumption: each sound from speaker unrelated to others (independent)

Some ICA assumptions

• Position of microphones and speakers is constant (mixing matrix constant)

• Sources Ergodic

• The propagation of the signal from the source to the microphone is instantaneous

• Sources sum linearly

• Number of microphones equals the number of speakers

• In Bell-Sejnowski algorithm, the non-linearity approximates the cdf of the sources

g(C) :

Independent Component Analysis (ICA)

Independent Sources(individuals’ speech)

time

Mixing

matrix= Data

S?M X=

Goal of ICA: given Data (X), can we recover the sources (S), without knowing M?

Independent Components

time

=Data

X =W

Weight matrix

C

Goal of ICA: Find W, so that Kullback-Leibler divergence between f1(C) and f2(S) is minimized ?g(C) y ,0

)(

W

yHg(C) :

‘InfoMax’ algorithm: Iteratively estimate W, so that:

Key point: maximizing H(y) implies that rows of C are maximally independent

Measured Signal

Task Non task-relatedactivations

(e.g. Arousal)

PulsationsMachine Noise

Independent Component Analysis (ICA)

Assumption: spatial pattern from sources of variability unrelated (independent)

The fMRI data at each time point is considered a mixture of activations from each component map

n

COMPONENTMAPS

MEASURED fMRI SIGNAL

‘mixing matrix’,

M

#1

#2

t = 1

t = 2

t = n

Mixing

time

Selected Components:Consistentlytask-related

Transientlytask-related

Quasi-periodic Slowly-varying Slow headmovement

Abrupt headmovement

ActivatedSuppressed

PCA (2nd order) 4th order ICA (all orders)

Comparison of Three Linear Models

r = 0.46 r = 0.85 r = 0.92

Increasing spatial independence between components

Are Two Maps Independent?0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ...0.1, 1.2, 1.3, -1.9, ... -0.1, 4.2 ...

?

A BStatistically

Independent

Decorrelated

Higher-order

statistics

Identical2nd-orderstatistics

0i

qi

pi BA

ICA (all orders)

Comon’s 4th order

0i

ii BAPCA (2nd

order)

0.4, 1.2, 4.3, -6.9, ... -2.1, 0.2 ...

A component map specified by voxel values

Histogram of voxel values for component map

0

z > 1

associated time course

Derived Independent Components

Component map after thresholding

ICA Component

0 10 20 30 40 50 60

RestSelf-paced movement Movie

Unexpected Frontal-cerebellar activation detected with ICA

A Transiently task-related (TTR) component (active during first two trials)

Martin J. McKeown, CNL, Salk Institute, martin@salk.edu

Single trial fMRI

ICA component time course Aligned ICA component spatial distribution

(a)

(b)

Trial 1

Single trial fMRI

All p < 10-20

(c)

(d)

(e)

19-sec

PRESS Statistic:

Assessing Statistical Models

Data

Eliminate 1 time point

=

+ -iG ^

Reference function

fMRI (X) Data

Voxel #

time = +

G

How well does G-i match data?

• Gives some idea of the influence of the ith time point

Hybrid Techniques

Data Driven

Hypothesis

Driven

Exp Exp Exp Exp

Con Con ConCon

0 10 20 30 40 50 60

HYBICA: L arm pronation/supination

hypothesis

Hybrid activation

S1

Use of HYBICA for Memory Load Hypothesis testing

Maintenance

Use of HYBICA for Memory Load Hypothesis testing

S2

Use of HYBICA for Memory Load Hypothesis testing