Deconvolution of Overlapping Time Series: an fMRI Perspective - - PowerPoint PPT Presentation

Deconvolution of Overlapping Time Series: an fMRI Perspective

Indrayana Rustandi indra+@cs.cmu.edu

Before we begin

• Feel free to interrupt with comments or
• questions. If you are lost, let me know.
• Comments on how this talk in general or parts of

it can be improved are appreciated.

Overlapping Time Series

5 10 15 20 25 20 40 60 80 100 120 5 10 15 20 25 20 40 60 80 100 120

+ =

5 10 15 20 25 −200 −100 100 200 300

Simple linear regression review

• Known/observed:
• Unknown (want to estimate): β0, β1

Yi = β0 + β1xi + i xi, Yi i ∼ N(0, σ2)

Simple linear regression, least-squares solution

Squared error: Minimizing the squared error:

∂E ∂β0 = 0 ∂E ∂β1 = 0

Least-squares estimators:

E =

n

• i=1

(yi − (β0 + β1xi))2 ˆ β0 = ¯ yn − ˆ β1¯ xn ˆ β1 = n

i=1 xiyi − n¯

xn¯ yn n

i=1 x2 i − n¯

x2

n

Simple linear regression, maximum-likelihood solution

L ∝ 1 σ exp

• − 1

2σ2

n

• i=1

(yi − (β0 + β1xi))2

• ∝ −n log σ −

1 2σ2

n

• i=1

(yi − (β0 + β1xi))2

The maximum-likelihood estimators are the least-squares estimators.

Ordinary least-squares

y = Xh + u

• Known/observed: output y, design matrix X
• W

ant to estimate h

u ∼ N(0, σ2I)

More about the design matrix X

• Row: time point in the time series
• Column: parameter of the time series
• Binary elements (0 and 1)

    1 1 1 1 1 1    

Example:

Ordinary least-squares, least-squares solution

Squared error:

E = (y − Xh)T(y − Xh) = yTy − 2yTXh + hTXTXh

Minimizing the squared error:

∂E ∂h = 0 XTXh = XTy

Least-squares estimator:

ˆ h = (XTX)−1XTy

Ordinary least-squares, maximum-likelihood solution

L = 1 (2πσ2)n/2 exp

• −1

2 (y − Xh)T(y − Xh) σ2

• = −n

2 log 2π − n 2 log σ2 − 1 2 (y − Xh)T(y − Xh) σ2

The maximum-likelihood estimators are the least-squares estimators.

Generalized least-squares

• Otherwise similar to OLS

y = Xh + u u ∼ N(0, σ2Σ)

Generalized least-squares, continued

Σ Σ−1 = LTL

Because u is Gaussian, is positive definite, there exists a nonsingular matrix L such that T ransform u using L:

˜ u = Lu E(˜ u) = 0 V(˜ u) = σ2I

Hence, we can apply OLS to solve

Ly = LXh + Lu

Generalized least-squares, continued

Let

˜ y ≡ Ly ˜ X ≡ LX ˜ u ≡ Lu

Solve

˜ y = ˜ Xh + ˜ u

using OLS GLS estimator:

ˆ h = (˜ XT ˜ X)−1 ˜ X˜ y = (XTΣ−1X)−1XTΣ−1y

Can also be verified that this is the maximum- likelihood estimator (see [Hamilton 1994])

Estimating the covariance matrix

ut = ρut−1 + t

In fMRI time series, the covariance matrix is unknown. The AR(1) model closely matches data in fMRI time series [Purdon, W eisskoff 1998]:

|ρ| < 1 t ∼ N(0, σ2) Σ =       1 ρ ρ2 · · · ρn−1 ρ 1 ρ · · · ρn−2 . . . . . . . . . · · · . . . ρn−1 ρn−2 ρn−3 . . . 1      

Estimating the covariance matrix, continued

L =       

• 1 − ρ2

· · · −ρ 1 · · · −ρ 1 · · · . . . . . . . . . · · · . . . . . . · · · −ρ 1       

Use L to transform the model with

ˆ ρ = n

i=2 ˆ

ui ˆ ui−1

• i=1 ˆ

ui

2

ˆ u = y − Xˆ hOLS

Summary of algorithm

• Do an OLS regression on y = Xh + u to obtain

residuals .

• Calculate from .
• Use to create L. T

ransform the original regression model y = Xh + u using L.

• Do an OLS regression on the transformed model,

giving .

ˆ u ˆ ρ ˆ u ˆ h ˆ ρ

Parameters in fMRI experiments

• Controllable: can be controlled by the

experiments

• Uncontrollable

Controllable parameters

• sampling rate (TR)
• interstimulus interval (ISI)
• number of stimuli
• stimulus duration

Uncontrollable parameters

• duration of the hemodynamic response function
• signal-to-noise ratio

Synthetic data

• Look at the distance of the estimate to the true

function, using mean-squared error of the sampled points as the metric.

• V

ary the controllable parameters, except for stimulus duration.

Synthetic data, continued

T rue response function: Birn impulse [Birn et.al. 2002]

h(t) = At8.60 exp

• −

t 0.547

• UseA = 1.0

Defaults: sampling = 1, ISI = 9, number of stimuli = 10, sigma = 100 (signal-to-noise ratio = 1.10) Run 100 times to get the distribution.

OLS, sampling

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 x 10

4

sampling MSE 0.5 1 1.5 2 2.5 3 3.5 500 1000 1500 2000 2500 sampling MSE

OLS, sampling jitter vs. non-jitter

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 x 10

4

sampling MSE

OLS, ISI

2 4 6 8 10 12 14 16 0.5 1 1.5 2 2.5 3 x 10

4

ISI MSE

OLS, number of stimuli

20 40 60 80 100 120 2000 4000 6000 8000 10000 12000 14000 no of events MSE 20 40 60 80 100 120 500 1000 1500 2000 no of events MSE

OLS, number of stimuli jitter vs. non-jitter

20 40 60 80 100 120 2000 4000 6000 8000 10000 12000 14000 no of events MSE

GLS, sampling

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 x 10

4

sampling MSE 0.5 1 1.5 2 2.5 3 3.5 500 1000 1500 2000 2500 3000 sampling MSE

GLS, sampling jitter vs. non-jitter

0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 x 10

4

sampling MSE

GLS, ISI

2 4 6 8 10 12 14 16 0.5 1 1.5 2 2.5 3 x 10

4

ISI MSE

GLS, number of stimuli

20 40 60 80 100 120 2000 4000 6000 8000 10000 12000 14000 16000 no of events MSE 20 40 60 80 100 120 500 1000 1500 2000 no of events MSE

GLS, number of stimuli jitter vs. non-jitter

20 40 60 80 100 120 2000 4000 6000 8000 10000 12000 14000 16000 no of events MSE

Brainlex

5 10 15 −0.5 0.5 5 10 15 −0.5 0.5 5 10 15 −0.5 0.5 5 10 15 −0.5 0.5 5 10 15 −0.5 0.5 5 10 15 −0.5 0.5

Issues

• Frequency domain
• Jitter window
• Other distance metrics
• Analytical expression of the distance with respect

to the parameters

• Events not totally synchronized with sampling
• Spatiotemporal regression
• Hidden process model, stochastic design matrix

Thank Y

• u!

Questions?