Analytical Techniques Hypothesis Driven Data Driven Principal Component

Analytical Techniques • Hypothesis Driven • Data Driven • Principal Component Analysis (PCA) • Independent Component Analysis (ICA) • Fuzzy Clustering • Others • Structural equation modeling

Matrix Notation of f. MRI Data 1 voxel BOLD signal t=1 t=2 t=3 t=4. . . Voxels time Slice 1 Data Matrix X

Calculating level of Significance significance: ~ t statistic i/ i G X f. MRI Data total variability + = = Variability explained by the model + noise

SPM Nomenclature for Design Matrix G (interesting) Covariate Indicator variable G 1 Gc H (non-interesting) H 1 Hc Global activity Linear trends E. g. dose of drug subject Design matrix G

Some General Linear Model (GLM) Assumptions: • Design matrix known without error • the design matrix is the same everywhere in the brain • the ’s follow a Gaussian distribution • the residuals are well modeled by Gaussian noise • the voxels are temporally aligned • each time point is independent of the others (time courses of voxels are white) • each voxel is independent of the others

Inclusion of Global Signal in Regression Global Signal Hypothesis Regression Coefficients Hypothesis Test voxel Global signal < 0!!! < 5 degrees difference between Global Signal & Hypothesis !

Inclusion of Global Covariate in Regression: Effect of non orthogonality 2 db 2 X 1 X’ 1 db 2 1 b = (GTG)-1 GTX 2 X 1 X’ 1 1 “Reference Function, R”

Consider an f. MRI experiment with only 3 time points

Consider an f. MRI experiment with only 3 time points

Analysis of Brain Systems reference function R 1 Correlation viewed as a projection R 2 Although R 1 and R 2 both somewhat correlated with the reference function, they are uncorrelated with each other Corr(R 2, ref) ref Corr(R 1, ref) R 1

Principal Component Analysis (PCA) Voxel 1 Voxel 3 PC 1 Vox el 2 1 l e x Vo Voxel 2 t Voxel 3 Eigenimage + time course

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) ? M S Mixing matrix Independent Sources (individuals’ speech) = X = Data time Goal of ICA: given Data (X), can we recover the sources (S), without knowing M? W X = Weight matrix Data = C Independent Components time ‘Info. Max’ algorithm: Iteratively estimate W, so that: Goal of ICA: Find W, so that Kullback-Leibler divergence between f 1(C) and f 2(S) is minimized ? g(C) : Key point: maximizing H(y) implies that rows of C are maximally independent

Independent Component Analysis (ICA) Non task-related activations (e. g. Arousal) Task Machine Noise Measured Signal Pulsations Assumption: spatial pattern from sources of variability unrelated (independent)

The f. MRI data at each time point is considered a mixture of activations from each component map COMPONENT MAPS Mixing #1 ‘mixing matrix’, S t=1 S t=2 M S n S t=n time #2 MEASURED f. MRI SIGNAL

Selected Components: Consistently task-related Transiently task-related Abrupt head movement Quasi-periodic Slowly-varying Slow head movement Activated Suppressed

Comparison of Three Linear Models PCA (2 nd order) r = 0. 46 4 th order ICA (all orders) r = 0. 85 Increasing spatial independence between components r = 0. 92

Are Two Maps Independent? 0. 1, 1. 2, 1. 3, -1. 9, . . . 0. 4, 1. 2, 4. 3, -6. 9, . . . -0. 1, 4. 2. . . -2. 1, 0. 2. . . ? A Statistically Independent Identical 2 nd-order statistics B å Ai B = 0 p q i i Higherorder statistics Decorrelated ICA (all orders) Comon’s 4 th order å Ai Bi = 0 i PCA (2 nd order)

Derived Independent Components ICA Component Histogram of voxel values for component map A component map specified by voxel values 0. 4, 1. 2, 4. 3, -6. 9, . . . -2. 1, 0. 2. . . z>1 0 associated time course Component map after thresholding

Unexpected Frontal-cerebellar activation detected with ICA Self-paced movement Rest Movie

A Transiently task-related (TTR) component (active during first two trials) Martin J. Mc. Keown, CNL, Salk Institute, martin@salk. edu

Single trial f. MRI ICA component time course (a) (b) Trial 1 Aligned ICA component spatial distribution

Single trial f. MRI (c) (d) (e) 19 -sec All p < 10 -20

Assessing Statistical Models PRESS Statistic: G Eliminate 1 time point Data ^ -i = How well does G -i match data? • Gives some idea of the influence of the ith time point +

Hybrid Techniques Hypothesis Data Driven Con Exp Exp Con

HYBICA: L arm pronation/supination hypothesis Hybrid activation

Use of HYBICA for Memory Load Hypothesis testing S 1

Use of HYBICA for Memory Load Hypothesis testing Maintenance

Use of HYBICA for Memory Load Hypothesis testing S 2