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 120 120 100 100 80 80 + 60 60 40 40 20 20 0 0 0 5 10 15 20 25 0 5 10 15 20 25 = 300 200 100 0 − 100 − 200 0 5 10 15 20 25

Simple linear regression review Y i = β 0 + β 1 x i + � i x i , Y i • Known/observed: • Unknown ( want to estimate ) : β 0 , β 1 � i ∼ N (0 , σ 2 )

Simple linear regression, least - squares solution Squared error: n � ( y i − ( β 0 + β 1 x i )) 2 E = i =1 Minimizing the squared error: ∂ E ∂ E = 0 = 0 ∂β 0 ∂β 1 Least - squares estimators: � n i =1 x i y i − n ¯ x n ¯ y n ˆ ˆ y n − ˆ β 1 = β 0 = ¯ β 1 ¯ x n � n i =1 x 2 x 2 i − n ¯ n

Simple linear regression, maximum - likelihood solution � � n L ∝ 1 − 1 � ( y i − ( β 0 + β 1 x i )) 2 σ exp 2 σ 2 i =1 n 1 � ( y i − ( β 0 + β 1 x i )) 2 � ∝ − n log σ − 2 σ 2 i =1 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 , σ 2 I )

More about the design matrix X • Row: time point in the time series • Column: parameter of the time series • Binary elements ( 0 and 1 ) Example:   1 0 0 0 0 0 0 1 0 1 0 0     0 0 1 0 1 0   0 0 0 0 0 1

Ordinary least - squares, least - squares solution Squared error: E = ( y − Xh ) T ( y − Xh ) = y T y − 2 y T Xh + h T X T Xh Minimizing the squared error: ∂ E ∂ h = 0 X T Xh = X T y Least - squares estimator: ˆ h = ( X T X ) − 1 X T y

Ordinary least - squares, maximum - likelihood solution ( y − Xh ) T ( y − Xh ) 1 � − 1 � L = (2 πσ 2 ) n/ 2 exp 2 σ 2 ( y − Xh ) T ( y − Xh ) 2 log σ 2 − 1 � = − n 2 log 2 π − n 2 σ 2 The maximum - likelihood estimators are the least - squares estimators.

Generalized least - squares y = Xh + u u ∼ N ( 0 , σ 2 Σ ) • • Otherwise similar to OLS

Generalized least - squares, continued Σ Because u is Gaussian, is positive de fi nite, there exists a nonsingular matrix L such that Σ − 1 = L T L T ransform u using L : u = Lu ˜ E ( ˜ u ) = 0 u ) = σ 2 I V ( ˜ 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: X T ˜ X ) − 1 ˜ h = ( ˜ ˆ X˜ y = ( X T Σ − 1 X ) − 1 X T Σ − 1 y Can also be veri fi ed that this is the maximum - likelihood estimator ( see [ Hamilton 1994 ])

Estimating the covariance matrix In fMRI time series, the covariance matrix is unknown. The AR ( 1 ) model closely matches data in fMRI time series [ Purdon, W eissko ff 1998 ] : � t ∼ N (0 , σ 2 ) | ρ | < 1 u t = ρ u t − 1 + � t ρ 2 ρ n − 1   1 ρ · · · ρ n − 2 1 ρ ρ · · ·     . . . . Σ = . . . .   . . . . · · ·     . . ρ n − 1 ρ n − 2 ρ n − 3 . 1

Estimating the covariance matrix, continued �   1 − ρ 2 0 0 0 0 · · · 1 0 0 0 − ρ   · · ·   0 1 0 0 − ρ L =   · · ·   . . . . . . . . . .   . . . . .   · · · 0 0 0 1 − ρ · · · Use L to transform the model with � n i =2 ˆ u i ˆ u i − 1 u = y − Xˆ ˆ h OLS ρ = ˆ 2 � i =1 ˆ u i

Summary of algorithm • Do an OLS regression on y = Xh + u to obtain residuals . u ˆ ˆ • Calculate from . u ˆ ρ • Use to create L . T ˆ ransform the original ρ regression model y = Xh + u using L . • Do an OLS regression on the transformed model, ˆ h giving .

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 ] � � t h ( t ) = At 8 . 60 exp − 0 . 547 Use A = 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 4 2 x 10 1.5 MSE 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 sampling 2500 2000 1500 MSE 1000 500 0 0 0.5 1 1.5 2 2.5 3 3.5 sampling

OLS, sampling jitter vs. non - jitter 4 2 x 10 1.5 MSE 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 sampling

OLS, ISI 4 3 x 10 2.5 2 MSE 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 ISI

OLS, number of stimuli 14000 12000 10000 MSE 8000 6000 4000 2000 0 20 40 60 80 100 120 no of events 2000 1500 MSE 1000 500 0 0 20 40 60 80 100 120 no of events

OLS, number of stimuli jitter vs. non - jitter 14000 12000 10000 8000 MSE 6000 4000 2000 0 0 20 40 60 80 100 120 no of events

GLS, sampling 4 2 x 10 1.5 MSE 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 sampling 3000 2500 2000 MSE 1500 1000 500 0 0 0.5 1 1.5 2 2.5 3 3.5 sampling

GLS, sampling jitter vs. non - jitter 4 2 x 10 1.5 MSE 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 sampling

GLS, ISI 4 3 x 10 2.5 2 MSE 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 ISI

GLS, number of stimuli 16000 14000 12000 10000 MSE 8000 6000 4000 2000 0 20 40 60 80 100 120 no of events 2000 1500 MSE 1000 500 0 0 20 40 60 80 100 120 no of events

GLS, number of stimuli jitter vs. non - jitter 16000 14000 12000 10000 MSE 8000 6000 4000 2000 0 0 20 40 60 80 100 120 no of events

Brainlex 0.5 0.5 0 0 − 0.5 − 0.5 5 10 15 5 10 15 0.5 0.5 0 0 − 0.5 − 0.5 5 10 15 5 10 15 0.5 0.5 0 0 − 0.5 − 0.5 5 10 15 5 10 15

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 ou! Questions?