likelihood approximation with parallel hierarchical matrices for large spatial datasets
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Center for UncertaintyQuantification
Center for UncertaintyQuantification
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LikelihoodApproximationWithParallelHierarchicalMatricesForLargeSpatialDatasets
A. Litvinenko, Y. Sun, M. Genton, D. Keyes, CEMSE, KAUST
HIERARCHICAL LIKELIHOOD APPROXIMATIONSuppose we observe a mean-zero, stationary and isotropic Gaussian process Z with a Matérn covari-
ance at n irregularly spaced locations. Let Z = (Z(s1), ..., Z(sn))T then Z ∼ N (0,C(θ)), θ ∈ Rq is anunknown parameter vector of interest, where
Cij(θ) = cov(Z(si), Z(sj)) = C(‖si − sj‖,θ), and
C(r) := Cθ(r) =2σ2
Γ(ν)
( r2`
)νKν
(r`
), θ = (σ2, ν, `)T
is the Matérn covariance function. The MLE ofθ is obtained by maximizing the Gaussian log-likelihood function:
L(θ) = −n2
log(2π)− 1
2log |C(θ)|− 1
2Z>C(θ)−1Z.
We approximate C ≈ C̃ in the H-matrix formatwith cost and storage O(kn log n), k � n.
Theorem 1 1. Let ρ(C̃−1C̃− I) < ε < 1. It holds| log |C| − log |C̃|| ≤ −n log(1− ε). Let ‖C−1‖ ≤ c1,then |L̃(θ; k)− L(θ)| ≤ c20 · c1 · ε+ n log(1− ε)
Operation Sequen. Compl. Parallel Compl. (shared mem.)
building(C̃) O(n logn) O(n logn)p
+O(|V (T )\L(T )|)storage(C̃) O(kn logn) O(kn logn)
C̃z O(kn logn) O(kn logn)p
+ n√p
H-Cholesky O(k2n log2 n) O(n logn)p
+O( k2n log2 n
n1/d )
Daily soil moisture, Mississippi basin.
H-matrix rank
3 7 9
cov.
leng
th
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
Box-plots for differentH-ranks k = {3, 7, 9}, ` = 0.0334.
ℓ, ν = 0.325, σ2 = 0.980 0.2 0.4 0.6 0.8 1
×104
-5
-4
-3
-2
-1
0
1
2
3
log(|C̃|)
zTC̃
−1z
−L̃
Moisture, n = 66049, rank k = 11.
PARALLEL HIERARCHICAL MATRICES (HACKBUSCH, KRIEMANN’05)Advantages to approximate C by C̃: H-approximation is cheap; storage and matrix-vector productcost O(kn log n); LU and inverse cost O(k2n log2 n); efficient parallel implementations exists.
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(1st) Matérn H-matrix approximations for moisture example, n = 8000, ε = 10−3, ` = 0.64, ν = 0.325,σ2 = 0.98, 29.3MB vs 488.3MB for dense, set up time 0.4 sec.; (2nd) Cholesky factor L, with accuracy ineach block ε = 10−8, 4.8 sec., storage 52.8 MB.; (3rd) Distribution across p processors; (4) Kronecker prod-uct ofH-matrices, n = 381K; (5) Discretization of Mississippi basin, [−84.8◦−72.9◦]×[32.446◦, 43.4044◦].
NUMERICAL EXAMPLES
H-matrix approximation, ν = 0.5, domain G = [0, 1]2, ‖C̃(0.25,0.75)‖2 = {212, 568}, n = 16049.
k KLD ‖C− C̃‖2 ‖C̃C̃−1 − I‖2` = 0.25 ` = 0.75 ` = 0.25 ` = 0.75 ` = 0.25 ` = 0.75
10 2.6e-3 0.2 7.7e-4 7.0e-4 6.0e-2 3.150 3.4e-13 5e-12 2.0e-13 2.4e-13 4e-11 2.7e-9
Computing time and number of iterations for maximization of log-likelihood L̃(θ; k), n = 66049.k size, GB C̃, set up time, s. compute L̃, s. maximizing, s. # iters10 1 7 115 1994 1320 1.7 11 370 5445 9
dense 38 42 657 ∞ -
Moisture data. We used adaptive rank arithmetics with ε = 10−4 for each block of C̃ and ε = 10−8 foreach block of C̃−1. Number of processing cores is 40.
n compute C̃ L̃L̃T inverseCompr. time size time size ‖I− (L̃L̃T)−1C̃‖2 time size ‖I− C̃−1C̃‖2rate % sec. MB sec. MB sec. MB
10000 86% 0.9 106 4.1 109 7.7e-6 44 230 7.8e-530000 92.5% 4.3 515 25 557 1.1e-3 316 1168 1.1e-1
n = 512K, accuracy inside each block 10−8, matrix setup 261 sec., compression rate 99.98% (0.4GB against 2006 GB).H-LU is done in 843 sec., required 5.8 GB RAM, inversion LU error 2 · 10−3.
(1st) −L vs. ν; (2nd) with nuggets {0.01, 0.005, 0.001} for Gaussian covariance, n = 2000, k = 14,σ2 = 1; (3rd) Zoom of 2nd figure; (4th) box-plots for ν vs number of locations n.
REFERENCES AND ACKNOWLEDGEMENTS
[1] B. N. KHOROMSKIJ, A. LITVINENKO, H. G. MATTHIES, Application of hierarchical matrices for computing theKarhunen-Loéve expan-sion, Computing, Vol. 84, Issue 1-2, pp 49-67, 2008.
[2] Y. SUN, M. STEIN, Statistically and computationally efficient estimating equations for large spatial datasets, JCGS, 2016,[3] A. LITVINENKO, M. GENTON, Y. SUN, D. KEYES, ??matrix techniques for approximating large covariance matrices
and estimating its parameters, PAMM 16 (1), 731-732, 2016[4] W. NOWAK, A. LITVINENKO, Kriging and spatial design accelerated by orders of magnitude: combining low-rank
covariance approximations with FFT-techniques, J. Mathematical Geosciences, Vol. 45, N4, pp 411-435, 2013.
Work supported by SRI-UQ and ECRC, KAUST. Thanks to Ronald Kriemann for HLIBPro .
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