A spatiotemporal second-order algorithm for fmri data analysis

Transcription

1 A spatiotemporal second-order algorithm for fmri data analysis F. J. Theis 1 P. Gruber 1 I. R. Keck 1 A.M. Tomé 2 E. W. Lang 1 1 Institute of Biophysics University of Regensburg 2 Dept. de Electrónica e Telecomunicações/IEETA Universidade de Aveiro CIMED 2005, Lisbon, Portugal

2 Outline BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi Motivation Models An Algorithm for spatiotemporal BSS Data Source One-Dimensional stsobi Comparisons

3 Outline BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi Motivation Models An Algorithm for spatiotemporal BSS Data Source One-Dimensional stsobi Comparisons

4 Linear BSS BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi (BSS) problem X = AS + N X observed m-dimensional random vector A (unknown) full-rank real matrix S (unknown) n-dimensional source signals n m N (unknown) noise (often assumed to be white Gaussian) Goal: recover unknown A and S given only X additional assumptions necessary without, problem ill-posed depending on assumptions FA, PCA, ICA, SCA, NMF remark: often simplification N = 0

5 Linear BSS BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi (BSS) problem X = AS + N X observed m-dimensional random vector A (unknown) full-rank real matrix S (unknown) n-dimensional source signals n m N (unknown) noise (often assumed to be white Gaussian) Goal: recover unknown A and S given only X additional assumptions necessary without, problem ill-posed depending on assumptions FA, PCA, ICA, SCA, NMF remark: often simplification N = 0

6 Joint Diagonalization BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi BSS algorithms often employ diagonalization techniques on source conditions to identify a mixing matrix Given a set of symmetric matrices R := {R 1,..., R K }, find an invertible matrix A such that A R i A is diagonal Matrix can be found by minimizing K (Â ) k=1 off R i Â, where off denotes the sum of the off-diagonal terms Exact diagonalization is only possible if all matrices commute therefore often approximate joint diagonalization Common algorithms gradient descend on off iteratively use Givens rotations ACDC

7 Joint Diagonalization BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi BSS algorithms often employ diagonalization techniques on source conditions to identify a mixing matrix Given a set of symmetric matrices R := {R 1,..., R K }, find an invertible matrix A such that A R i A is diagonal Matrix can be found by minimizing K (Â ) k=1 off R i Â, where off denotes the sum of the off-diagonal terms Exact diagonalization is only possible if all matrices commute therefore often approximate joint diagonalization Common algorithms gradient descend on off iteratively use Givens rotations ACDC

8 Joint Diagonalization BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi BSS algorithms often employ diagonalization techniques on source conditions to identify a mixing matrix Given a set of symmetric matrices R := {R 1,..., R K }, find an invertible matrix A such that A R i A is diagonal Matrix can be found by minimizing K (Â ) k=1 off R i Â, where off denotes the sum of the off-diagonal terms Exact diagonalization is only possible if all matrices commute therefore often approximate joint diagonalization Common algorithms gradient descend on off iteratively use Givens rotations ACDC

9 AMUSE and SOBI BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi Source assumption translates into diagonallity of certain matrices C i (s) Behavior under linear transformation: C i (As) = AC i (s)a Identification of A by joint diagonalization of C i (x) Example algorithms: AMUSE: C τ (s) := R s (τ) := 1 2 (R s(τ) + R s (τ) ) symmetrized autocovariance matrix at shift τ Using several shifts: SOBI [Belouchrani et al., 1997]

10 BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi Multidimensional autocovariances and mdsobi For spatial data sets use M-dimensional autocovariance R s (τ 1,..., τ M ) := E ( s(z 1 + τ 1,..., z M + τ M )s(z 1,..., z M ) ) Estimation from samples as usual (here centered s) Depends on M shifts τ i Very useful for (3d) fmri analysis Advantage: random processes s and x depend on multiple variables (z 1,..., z M ) get multidimensional SOBI [Theis et al., 2004] d cov 2d cov τ respectively (τ1, τ2) (rescaled to N)

11 Outline Motivation Models An Algorithm for spatiotemporal BSS BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi Motivation Models An Algorithm for spatiotemporal BSS Data Source One-Dimensional stsobi Comparisons

12 Spatial and temporal BSS Motivation Models An Algorithm for spatiotemporal BSS Real-world data sets (e.g. fmri) often possess additional structure use it! spatial or temporal BSS require separating condition either in space or in time spatiotemporal BSS tries to achieve both spatial and temporal separation Algorithm for spatiotemporal BSS first proposed [Stone et al., 2002] real data often has spatiotemporal structure (not i.i.d.) potential applications in biomedical data analysis Stone s algorithm is derived from infomax many parameters and difficult convergence Propose (batch) algebraic BSS algorithms for the spatiotemporal case

13 Spatial and temporal BSS Motivation Models An Algorithm for spatiotemporal BSS Real-world data sets (e.g. fmri) often possess additional structure use it! spatial or temporal BSS require separating condition either in space or in time spatiotemporal BSS tries to achieve both spatial and temporal separation Algorithm for spatiotemporal BSS first proposed [Stone et al., 2002] real data often has spatiotemporal structure (not i.i.d.) potential applications in biomedical data analysis Stone s algorithm is derived from infomax many parameters and difficult convergence Propose (batch) algebraic BSS algorithms for the spatiotemporal case

14 Spatial and temporal BSS Motivation Models An Algorithm for spatiotemporal BSS Real-world data sets (e.g. fmri) often possess additional structure use it! spatial or temporal BSS require separating condition either in space or in time spatiotemporal BSS tries to achieve both spatial and temporal separation Algorithm for spatiotemporal BSS first proposed [Stone et al., 2002] real data often has spatiotemporal structure (not i.i.d.) potential applications in biomedical data analysis Stone s algorithm is derived from infomax many parameters and difficult convergence Propose (batch) algebraic BSS algorithms for the spatiotemporal case

15 Spatial and temporal BSS Motivation Models An Algorithm for spatiotemporal BSS Assume data x(r, t) depends on two indices r R n (space) and t R (time) Number of spatial observations: s m Number of temporal observations: t m Contract vector r into 1d index r get data matrix X := (x(r, t)) of dimension s m t m Details on spatial and temporal BSS

16 Motivation Models An Algorithm for spatiotemporal BSS temporal BSS: X = t A t S spatial BSS: X = s A s S Theorem The factorization X = t AS s A into three terms has the trivial solution S = I if the source conditions contain PCA. from above X = t A t S = s S s A spatiotemporal BSS decompose X = s S t S spatial source matrix s S and temporal source matrix t S both have to fulfill some source conditions as much as possible indeterminacies contain at least scaling and permutation

17 Algebraic spatiotemporal BSS Motivation Models An Algorithm for spatiotemporal BSS Spatiotemporal source model temporal condition matrix Ci (X) := C i ( t x(t)) spatial condition matrix Ci (X ) := C i ( s x(r)) Spatiotemporal model C i (X) = C i ( s S t S) = s S C i ( t S) s S C i (X ) = C i ( t S s S) = t S C i ( s S) t S C i ( t S) = s S C i (X) s S C i ( s S) = t S C i (X ) t S find one source matrix by jointly diagonalizing either C i (X) (temporal BSS) or C i (X ) (spatial BSS) use both by double-sided approximate joint diagonalization

18 Motivation Models An Algorithm for spatiotemporal BSS Double-sided approximate joint diagonalization Assume now s m = t m = n, then s S = X t S 1 and C i ( t S) = t S X C i (X)X t S C i ( s S) 1 = t S C i (X ) 1 t S. Perform double-sided JD by JD of {X C i (X)X, C i (X ) 1 i = 1,...} Favor either spatial or temporal separation by weighting factor α [0, 1] and diagonalize {αx C i (X)X, (1 α)c i (X ) 1 i = 1,...} If A is such a diagonalizer, estimate sources by t Ŝ = A 1 and s Ŝ = A X

19 Algorithm Motivation Models An Algorithm for spatiotemporal BSS s m = t m = n yields far to small sample set dimension reduction to n min{ s m, t m} necessary Consider approximate singular value decomposition X = UDV of X pseudo-orthogonal matrices U R s m n and V R t m n diagonal matrix D R n n Theorem 1. Double-sided approximate JD in dimension n is achieved by approximate JD of {αc i (D 1/2 V ), (1 α)c i (D 1/2 U ) 1 i = 1,...} 2. If A is such a joint diagonalizer, the sources are estimated by t Ŝ = A D 1/2 V and s Ŝ = A 1 D 1/2 U

20 Outline Data Source One-Dimensional stsobi Comparisons BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi Motivation Models An Algorithm for spatiotemporal BSS Data Source One-Dimensional stsobi Comparisons

21 Data Source Data Source One-Dimensional stsobi Comparisons fmri data from 6 subjects performing a visual task 5 slices with 100 images were acquired (TR/TE = 3000/60 msec) Spatial resolution mm Oriented parallel to the calcarine fissure 10 repetitions of simulation and rest periods Photic stimulation using a 8 Hz alternating checkerboard with central fixation point The first scans were discarded for remaining saturation effects Here results only from the first subject are presented We used the (multidimensional) SOBI source condition

22 One-Dimensional stsobi Data Source One-Dimensional stsobi Comparisons cc: cc: cc: cc: 0.90 stsobi with α = 1/2 on first 4 principal components Component 4 is the desired stimulus component crosscorrelation with stimulus cc = 0.9, with a delay of roughly 6 seconds (BOLD effect)

23 Stone s Data Source One-Dimensional stsobi Comparisons cc: cc: cc: cc: 0.66 fmri analysis using Stone s algorithm Task component not recovered very well components 1 and 4 crosscorrelation of 0.66 Possibly due to convergence problems of the employed entropy/infomax rule

24 Fa Data Source One-Dimensional stsobi Comparisons cc: cc: cc: cc: 0.48 Fa, see [Hyvärinen and Oja, 1997] to identify spatially independent components Stimulus component not recovered Poor result due to the dimension reduction by PCA to only 4 components is obviously much more flexible

25 Data Source One-Dimensional stsobi Comparisons Component Maps for different weighting factors 1 cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: 0.90 α = 0: Spatial separation Stimulus cc = 0.8 component 4 see also the Fa experiment α = 0.5 Stimulus cc = 0.9 component 4 α = 1: Temporal separation Stimulus cc = 0.9 component 4

26 Data Source One-Dimensional stsobi Comparisons Component Maps for different weighting factors 1 cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: 0.90 α = 0: Spatial separation Stimulus cc = 0.8 component 4 see also the Fa experiment α = 0.5 Stimulus cc = 0.9 component 4 α = 1: Temporal separation Stimulus cc = 0.9 component 4

27 Data Source One-Dimensional stsobi Comparisons Component Maps for different weighting factors 1 cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: cc: 0.90 α = 0: Spatial separation Stimulus cc = 0.8 component 4 see also the Fa experiment α = 0.5 Stimulus cc = 0.9 component 4 α = 1: Temporal separation Stimulus cc = 0.9 component 4

28 Time-courses Data Source One-Dimensional stsobi Comparisons Compare the recovered time courses of the stimulus component for different α Similar when at least some temporal separation is performed Indicates that the algorithm is rather robust to the choice of the weight α Confirms the advantages of spatiotemporal separation

29 Outline BSS and ICA Algebraic BSS algorithms AMUSE, SOBI and mdsobi Motivation Models An Algorithm for spatiotemporal BSS Data Source One-Dimensional stsobi Comparisons

30 Novel spatiotemporal BSS algorithm Based on double-sided joint diagonalization Using multidimensional autocovariances Result for fmri data where it outperforms spatial only recovery Further development in the analysis of fmri data sets

31 Acknowledgements Acknowledgements Appendix References The authors gratefully acknowledge partial financial support by the DFG 1 and the BMBF 2. 1 graduate college: Nonlinearity and Nonequilibrium in Condensed Matter 2 project ModKog

32 Acknowledgements Appendix References Bibliography References A. Belouchrani, K. A. Meraim, J.-F. Cardoso, and E. Moulines. A blind source separation technique based on second order statistics. IEEE Transactions on Signal Processing, 45(2): , A. Hyvärinen and E. Oja. A fast fixed-point algorithm for independent component analysis. Neural Computation, 9: , J. Stone, J. Porrill, N. Porter, and I. Wilkinson. Spatiotemporal independent component analysis of event-related fmri data using skewed probability density functions. NeuroImage, 15(2): , F. Theis, A. Meyer-Bäse, and E. Lang. Second-order blind source separation based on multi-dimensional autocovariances. In Proc. ICA 2004, volume 3195 of Lecture Notes in Computer Science, pages , Granada, Spain, URL

33 Acknowledgements Appendix References Spatial and temporal BSS temporal BSS interpret data to contain a measured time series x r (t) := x(r, t) for each spatial location r apply BSS to s m-dimensional temporal observation vector t x(t) := ( x r1 (t),..., x r( s m) (t) ) find decomposition t x(t) = t A t s(t) with temporal mixing matrix t A and temporal sources t s(t) spatial BSS data is considered to be composed of t m spatial patterns x t (r) := x(r, t) apply BSS to t m-dimensional spatial observation vector t x(t) := ( x t1 (r),..., x t( t m) (r) ) find decomposition s x(r) = s A s s(r) with spatial mixing matrix s A and spatial sources s s(r)

34 Acknowledgements Appendix References Principal and independent component analysis PCA assumption S is uncorrelated without loss of generality white existence: eigenvalue decomposition Cov(X): D = V Cov(X)V with diagonal D and orthogonal V PCA-matrix W is constructed by W := D 1/2 V and A = VD 1/2 uniqueness and indeterminacies: A is unique up to right transformation in orthogonal group ICA assumption S independent i.e. I (S) = 0 indeterminacies: only permutation and scaling (!) if Cov(S) exists and S contains at most one Gaussian

35 Acknowledgements Appendix References Principal and independent component analysis PCA assumption S is uncorrelated without loss of generality white existence: eigenvalue decomposition Cov(X): D = V Cov(X)V with diagonal D and orthogonal V PCA-matrix W is constructed by W := D 1/2 V and A = VD 1/2 uniqueness and indeterminacies: A is unique up to right transformation in orthogonal group ICA assumption S independent i.e. I (S) = 0 indeterminacies: only permutation and scaling (!) if Cov(S) exists and S contains at most one Gaussian

36 Component Maps for different weighting factors cc: cc: cc: cc: 0.80

37 Component Maps for different weighting factors cc: cc: cc: cc: 0.89

38 Component Maps for different weighting factors cc: cc: cc: cc: 0.90