Fast Methods for Spatially Correlated Multilevel Functional Data

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1 Fast Methods for Spatially Correlated Multilevel Functional Data Ana-Maria Staicu, Department of Statistics, North Carolina State University, 23 Stinson Drive Raleigh, NC , USA Ciprian M. Crainiceanu, Department of Biostatistics, Johns Hopkins University, 65 N. Wolfe St. Baltimore, MD 2205, USA. and Raymond J. Carroll, Department of Statistics, Texas A&M University, College Station, TX , USA. Appendix A: Computational Complements A. Matrix inversion Let Σ dr denote the variance of the M dr i= N dri vector Y dr. Similarly, let ǫ dr be the M dr i= N dri vector of regression errors. Define dr = ( dr,,..., dr,k ) T and define ζ dr = (ζ dr,,..., ζ dr,k2, ζ dr2,,..., ζ drmdr,k 2 ) T Define Σ = cov( dr ) = diag(λ (),..., λ () K ), Σ β = diag(λ (2),..., λ (2) K 2 ) and Σ ζ = cov(ζ dr ) = I Σ β. Let B T dr = (Φ()T dr,...,φ()t drm dr ) be the M dr i= N dri K matrix with elements {φ () (t),..., φ() K (t)}, where the arguments for t match those of the corresponding row of Y dr and let B dr = diag(φ (2) dr,...,φ(2) drm dr ) be the M dr i= N dri K 2 M dr matrix of φ (2) l (t) s. Let dri be the N dri vector of ones, and E dr = diag( dr,..., drmdr ). Finally let U dr be the M dr vector of {U( dri ) : i =,...M dr }

2 and Σ U,dr be the M dr M dr variance covariance matrix of U dr. Then our model is Y dr = µ dr +B dr dr + B dr ζ dr + E dr U dr +ǫ dr, where µ dr is the vector obtained by stacking µ d (t drip ) over p s and i s. Clearly, Σ dr = cov(y dr ) = σ 2 ǫ I + B drσ B T dr + B drσ ζ B T dr + E drσ U,dr E T dr. (S.) Here we show how using matrix and determinant equalities used in (Harville, 977) can simplify both the inversion of and the determinant calculation of Σ dr. For simplicity of notation only, since there will be no confusion, we drop the indices and write M dr = M and N dri = N, etc. Thus, we denote Φ () dr = Φ() and Φ (2) dri = Φ(2), E = E dr, B dr = B and B dr = B. To invert Σ dr, we repeatedly use the formula (A + CB C T ) = A AC(C T AC + B) C T A. (S.2) Define S = (σ 2 ǫ I + BΣ B T + BΣ ζ B T ). Then Σ dr = (S + EΣ U,dr E T ) = S SE(E T SE + Σ U,dr ) E T S. Note that E T SE+Σ U,dr is M M. To compute S without taking a large matrix inverse, denote by J = (σ 2 ǫ I+BΣ ζb T ). Then S = (J +BΣ B T ) = J JB(B T JB+Σ ) B T J. The matrix B T JB +Σ is K K, however, still J is the inverse of a MN MN matrix. Fortunately, note that σ 2 ǫ I+BΣ ζb T = diag(σ 2 ǫ I N +Φ (2) Σ β Φ (2)T,...,σ 2 ǫ I N +Φ (2) Σ β Φ (2)T ), where I N is the N N dimensional matrix. Hence, J = diag { (σ 2 ǫ I N + Φ (2) Σ β Φ (2)T ),...,(σ 2 ǫ I N + Φ (2) Σ β Φ (2)T ) }. Using (S.2) once again, we write (σ 2 ǫi N + Φ (2) Σ β Φ (2)T ) = σ 2 ǫ I N σ 2 ǫ Φ (2) (Φ (2)T Φ (2) + σ 2 ǫσ β ) Φ (2)T. Hence: () we can compute J with nothing more than K 2 K 2 inversion, (2) we can compute S with nothing more than K K and K 2 K 2 inversion and (3) we can compute Σ dr more than K K, K 2 K 2 and M M inversion. with nothing A.2 Matrix determinant Here is use the formula A + CBC T = A I + C T A CB. (S.3) 2

3 Of course, Σ dr = S + EΣ U,dr E T = S I + Σ /2 U,dr ET SEΣ /2 U,dr, where I + Σ/2 U,dr ET SEΣ /2 U,dr is M M. However, S = J I +Σ /2 B T JBΣ /2, where I +Σ /2 B T JBΣ /2 is K K. Finally, remember that J = σ 2 ǫ I + BΣ γb T = diag(σ 2 ǫ I N + Φ (2) Σ β Φ (2)T,...,σ 2 ǫ I N + Φ (2) Σ β Φ (2)T ). Hence J = det{σ 2 ǫi N + Φ (2) Σ β Φ (2)T } M, which involves the determinant of a N N matrix. It follows, that we can compute Σ dr without large matrix inversions or the need to take determinants of large matrices. References Harville, D.A. (977). Maximum likelihood approaches to variance component estimation and to related problems. Journal of the American Statistical Association 72, Appendix B: Additional Tables and Figures subjects subjects unit location (µm) unit location (µm) Figure S.: Simulation study: the spatial location of the units used in the uniform design (left panel) and in the colon carcinogenesis study design (right panel) for 2 subjects. 3

4 Figure S.2: Simulation study: the mean of the estimated covariance functions along with their pointwise 90% confidence interval in the case of the uniform design (top panel) and the CCS design (bottom panel) of the units location. The true covariance functions (blue line) are ν = σ 2 U ρ (left), ν 2 = σ 2 U ρ 2 (middle) and ν 3 = σ 2 U ρ 3 (right); the estimates are obtained using k-nearest neighbor with positive semi-definite adjustment (solid lines). 4

5 σ 2 U σ 2 U σ 2 U Figure S.3: Simulation study: boxplots of the estimated variance of the spatial process U dr for the three correlation functions ρ (left panel), ρ 2 (middle panel) and ρ 3 (right panel) in the case of uniform and CCS spatial design. 5

6 λ () σ ǫ 0 0 σ ǫ 0 0 σ ǫ λ () () (2) (3) correlation type () (2) (3) correlation type () (2) (3) correlation type Figure S.4: Simulation study: boxplots of the estimated eigenvalues at each hierarchical level. The top panel corresponds to the correlation function ρ ( ) under the uniform design (left) and CCS design (right) for two magnitudes of measurement error σ ǫ = 0 and σ ǫ =. The bottom panel corresponds to correlation functions ρ in (), ρ 2 in (2) and ρ 3 in (3), under CCS design and σ ǫ =. 6

7 σ 2 ǫ σ 2 ǫ σ 2 ǫ Figure S.5: Simulation study: boxplots of the estimated variance of the measurement error, σ 2 ǫ, for the correlation functions ρ (left), ρ 2 (middle) and ρ 3 (right), and the two designs for the crypt locations. mean log p27 per diet co+b co b fo+b fo b mean log p27 per diet co+b co b fo+b fo b mean log p27 per diet co+b co b fo+b fo b relative cell position relative cell position relative cell position Figure S.6: The estimated mean functions for the 4 diet groups by: (left panel) penalized spline smoothing under a working independence assumption, (middle panel) ordinary least squares quadratic estimation along with the independence assumption and (right panel) weighted least squares quadratic estimation, accounting for the dependence considered by the model. 7