A longitudinal Bayesian model for spectral analysis of neuroimaging time series data. Joint work with Ning Dai

Transcription

1 A longitudinal Bayesian model for spectral analysis of neuroimaging time series data Joint work with Ning Dai

2 Resting-state fmri is popular for functional connectivity studies.

3 Another analysis approach is to look at how much variation in the data is driven by the low frequencies of the signal.

4 There is evidence that the low frequencies fluctuations in the BOLD signal differ in those with Alzheimer s.

5 Under mild assumptions, we can represent our data with respect to time or frequencies. Figure prepared by Cristina Gorrostieta

6 Under mild assumptions, we can represent our data with respect to time or frequencies. Time domain data: X 8, t = 1,, T

7 Under mild assumptions, we can represent our data with respect to time or frequencies. Frequency domain data: M d? ω = T AB/D E X 8 exp ( i2πωt) 8NB

8 Under mild assumptions, the spectral density uniquely determines the properties of a time series. Heuristically, f? ω Var(d? ω )

9 Under mild assumptions, the spectral density uniquely determines the properties of a time series. A natural estimator is the periodogram: I M ω = d? ω D

10 Under mild assumptions, the spectral density uniquely determines the properties of a time series. The periodogram is a noisy estimator, so we smooth it: fv M ω = Z.[ W h M ω α I M ω dα AZ.[

11 Under mild assumptions, the spectral density uniquely determines the properties of a time series.

12 Under mild assumptions, the spectral density uniquely determines the properties of a time series.

13 Under mild assumptions, the spectral density uniquely determines the properties of a time series.

14 Under mild assumptions, the spectral density uniquely determines the properties of a time series.

15 Under mild assumptions, the spectral density uniquely determines the properties of a time series.

16 Estimating the spectral density is challenging, and there are still many open problems. Some form of smoothing is necessary to yield good estimators.

17 Some form of smoothing is necessary to yield good estimators. fv M ω = Z.[ W h M ω α I M ω dα AZ.[

18 Some form of smoothing is necessary to yield good estimators. Another approach is to use smoothing splines: log f ω a = α + E β _ cos (2πωk) _NB

19 Some form of smoothing is necessary to yield good estimators. Recall the Fourier coefficients: M d? ω = T AB/D E X 8 exp ( i2πωt) 8NB

20 Some form of smoothing is necessary to yield good estimators. The distribution of the Fourier coefficients can be approximated by the Whittle likelihood: MAB p d f E exp ( 0.5 log f ω h + I M ω /f ω h ) hnz

21 The Whittle likelihood gives us a way to construct a Bayesian model. log f ω a = α + E β _ cos (2πωk) _NB α~n(0, σ l D ) β B,, β _ ~N 0, τ D τ~unif(0, c o )

22 What about longitudinal data?

23 What about longitudinal data? log f p ω a = α p + E β p_ cos (2πωk) _NB α B,, α q ~N 0, σ l D (β B_,, β q_ )~N μ1, τ D R(ρ) μ~n(0, σ v D ) τ~unif 0, c o ρ~unif 0,1

24

25 Our analysis: ADNI resting-state fmri data set (17 AD, 13 CN; 65+ years old; up to five visits) Longitudinal fractional amplitude of low frequencies fluctuations (falff) for multiple regions of interest

26 What is falff?

27 What is falff? yz{ } ~ ƒ {{ ~ ƒ f(ω) f(ω)

28 How do you estimate falff? Estimate the spectrum (longitudinally) Compute falff within (0.01,0.08) Hertz

29 Spectral estimates Log periodogram, smoothed periodogram, Bayesian longitudinal

30 Spectral estimates Log periodogram, smoothed periodogram, Bayesian longitudinal

31 The degree of within-subject correlation varies across subjects and across regions.

32 We generally see lower falffs among those with AD and a decreasing trend over time. ROI Effect Estimate Precuneus Left Precuneus Right Putamen Left Putamen Right Age (Year) AD Age:AD Age (Year) AD Age:AD Age (Year) AD Age:AD Age (Year) AD Age:AD ( , ) ( , ) (0.0010, ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , )

33 Estimating the spectral density is challenging, and there are still many open problems. Optimal number of splines, incorporate spatial structure, generalizability to multivariate time series, nonstationary time series,

34 Estimating the spectral density is challenging, and there are still many open problems. Incorporation of covariates (e.g., MMSE scores, motion parameters) in the model, missing data,