Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices Bin Nan Department of Biostatistics University of Michigan Joint work with Hai Shu January 31 – February 5, 2016 BIRS Workshop on Neuroimaging, Banff Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Notation ◮ Data matrix: X p × n = ( X ij ) p × n ◮ Columns of X p × n : X 1 , ..., X n , a series of brain images (resting state fMRI) acquired at n consecutive time points, where each image contains p voxels. ◮ Parameters of interest: ◮ Covariance matrix: Σ = cov ( X k ) = ( cov ( X ik , X jk )) p × p = ( σ ij ) p × p (or its normalization, the correlation matrix R ), which is the same for all k = 1 , . . . , n . ◮ Precision matrix: Ω = Σ − 1 . Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Temporal dependence ◮ There is a rich literature about estimating large covariance matrix Σ (with p > n ) when X 1 , ..., X n are i.i.d. ◮ Recently we have obtained the convergence results for large covariance/precision matrix estimates for temporally dependent data under the following assumption: k , l | ρ ij kl | ≤ C 0 | i − j | − α max kl = cov ( X ki , X lj ) / √ σ kk σ ll is the for all i � = j , where ρ ij cross-correlation, and C 0 and α are fixed positive constants. ◮ We name it polynomial-decay-dominated (PDD) temporal dependence. Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach An image data example rs-fMRI from the Human Connectome Project to assess brain functional connectivity: 20 ρ = m ax i | ˆ ρ i ( t ) | ρ = | ˆ ρ 1 ( t ) | ρ = 10 8 t − 3 ρ = 10 7 t − 3 15 ρ = t − 0 . 25 ρ = 0 . 26 t − 0 . 50 10 5 log( ρ ) 0 −5 −10 −15 0 1 2 3 4 5 6 7 log ( t ) , t > 0 Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Generalized thresholding for covariance matrix ◮ For a thresholding parameter τ ≥ 0, define a generalized thresholding function by s τ : R → R satisfying: (i) | s τ ( z ) | ≤ | z | ; (ii) s τ ( z ) = 0 for | z | ≤ τ ; (iii) | s τ ( z ) − z | ≤ τ . τ ( z ) = z 1 ( | z | > τ ) ◮ Here we only consider hard thresholding s H and soft thresholding s S τ ( z ) = sign ( z )( | z | − τ ) + ◮ Thresholding estimation: S τ ( ˆ σ ij )) p × p , where ˆ Σ ) = ( s τ ( ˆ Σ is the sample covariance matrix. Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach SPICE estimation for precision matrix ◮ SPICE is a modification of the graphic lasso method. ◮ Consider W − 1 ˆ W − 1 K λ ˆ Ω λ = ˆ ˆ with � � K λ = arg min K ≻ 0 , K = K T tr ( K ˆ R ) − log det ( K ) + λ | K | 1 , off ˆ , W = diag {√ ˆ σ 11 , . . . , � ˆ R is the sample correlation ˆ σ pp } , and ˆ matrix. Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Tuning parameter ◮ We write x n ≍ y n if there exists some constant C > 1 such that C − 1 ≤ lim inf y n / x n ≤ lim sup y n / x n ≤ C . ◮ For any fixed α , if p ≥ n c for some constant c , define n − α 1 2 ( log p ) 0 < α < 1 , 2 , τ ′ ≍ n − 1 1 2 [( log n )( log p )] α = 1 , 2 , n − 1 1 2 ( log p ) α > 1 . 2 , Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Covariance matrix estimator ◮ For sufficiently large constant M > 0, if τ = M τ ′ , then � c 0 ( p ) τ ′ 1 − q � � S τ ( ˆ Σ ) − Σ � 2 = O p , � c 0 ( p ) τ ′ 2 − q � 1 p � S τ ( ˆ Σ ) − Σ � 2 F = O p R , R ◮ The results above hold when ˆ Σ , Σ is replaced by ˆ R is the sample correlation matrix. respectively, where ˆ � � R ) = ρ ij ) 1 ( i � = j ) + 1 ( i = j ) Note: S τ ( ˆ s τ ( ˆ p × p . Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Precision matrix estimator ◮ For sufficiently large constant M > 0, if λ = M τ ′ and � τ ′ = o ( 1 / 1 + s p ) , then (without assuming irrepresentability) � τ ′ � � � ˆ Ω λ − Ω � 2 = O P 1 + s p , τ ′ � � � 1 √ p � ˆ Ω λ − Ω � F = O P 1 + s p / p . Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Gap -block cross-validation 1. Split the data X p × n into H 1 ≥ 4 (almost) equal-size nonoverlapping blocks X ∗ i , i = 1 , ..., H 1 such that X p × n = ( X ∗ 1 , X ∗ 2 , ..., X ∗ H 1 ) . For i = 1 , ..., H 1 , block X ∗ i is used as the validation data, and the remaining data excluding ( X ∗ i − 1 , X ∗ i + 1 ) , denoted by X ∗∗ i , are the training data. 2. Randomly subsample H 2 blocks X ∗ H 1 + 1 , ..., X ∗ H 1 + H 2 from X p × n , where X ∗ H 1 + j consists of ⌈ n / H 1 ⌉ consecutive columns of X p × n for each j = 1 , ..., H 2 . Note that these subsampled blocks can be overlapping. For i = H 1 + 1 , ..., H 1 + H 2 , block X ∗ i is used as the validation data, and the remaining data excluding the ⌈ n / H 1 ⌉ columns on either side of X ∗ i , denoted by X ∗∗ i , are the training data. Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Gap -block cross-validation (cont’d) 3. Set H = H 1 + H 2 . ◮ For covariance matrix estimation, select optimal τ from { τ j : j = 1 , . . . , J } by H 1 Σ ∗∗ Σ ∗ τ Σ � S τ j ( ˆ i ) − ˆ i � 2 ∑ s = arg min F . H 1 ≤ j ≤ J i = 1 ◮ For the estimation of precision matrix, we choose the optimal tuning parameter using the loss function Ω ∗∗ Ω ∗∗ Σ ∗ ) − log det ( ˆ tr ( ˆ λ ˆ λ ) . Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach A serious issue The cross-validation is infeasible for large p ! Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Choosing tuning parameter via submatrices ◮ Based on the forms of tuning parameter sizes, we propose a method based on estimations of submatrices ◮ Denote the candidate values of tuning parameter η ( η = τ for covariance matrix estimation and η = λ for precision matrix estimation) to be η 1 < ... < η m , and the submatrix dimension to be p s × p s . In practice, we could choose appropriate p s such that p / p s is (almost) an integer. Assume p / p s is an integer for simplicity. The submatrix approach is implemented as follows: Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
Introduction Covariance matrix Precision matrix Theoretical Results A feasible approach Step 1: Randomly partition the p random variables into p / p s groups. For the i -th group, choose the optimal η from { η j } m j = 1 using the gap-block CV for estimating the i -th sub-covariance-matrix or sub-precision-matrix, and denote it as η ( 1 ) . i Step 2: Randomly select p s random variables p / p s times without replacement from the total of p random variables. For the i -th sample, choose the optimal η from { η j } m j = 1 using the gap-block CV as in step 1, and denote it as η ( 2 ) . i η be the average of { η ( 1 ) , η ( 2 ) Step 3: Let ¯ : i = 1 , . . . , p / p s } . Then scale ¯ η i i by � log p η ¯ log p s and use it as the tuning parameter for the original p × p matrix estimation. Bin Nan Tuning parameter selection for voxel-wise brain connectivity estimation via low dimensional submatrices
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