Just some ideas how low-rank matrices/tensors can be useful in spatial and environmental statistics, where one usually has to deal with very large data

1. Possible applications of low-rank tensors in statistics
and UQ
Alexander Litvinenko,
Extreme Computing Research Center and Uncertainty
Quantification Center, KAUST
(joint work with H.G. Matthies, MIT and KAUST)
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http://sri-uq.kaust.edu.sa/

2. 4*
Problem 1. Predict temperature, velocity, salinity
Grid: 50Mi locations on 50 levels, 4*(X*Y*Z) = 4*500*500*50=
50Mi.
High-resolution time-dependent data about Red Sea: zonal velocity and
temperature
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3. 4*
Problem 1. Apply low-rank tensor for
1. Kriging estimate
ˆs := Csy C−1
yy y
2. Estimation of variance ˆσ, is the diagonal of conditional cov.
matrix
Css|y = diag Css − Csy C−1
yy Cys
,
3. Gestatistical optimal design
ϕA := n−1
trace{Css|y }
ϕC := cT
Css − Csy C−1
yy Cys c
,
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4. 4*
Problem 2. Stochastic Galerkin Operator
Problem 2. Stochastic Galerkin Operator
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5. 4*
Discretization of stoch. PDE − div(κ(p, x) u(p, x)) = f(x, p)
Pictures 1, 2 (poor and rich discretization of p):
(
i=1
∆i ⊗ Ki) · (x ⊗ e) = (f ⊗ e) (1)
Picture 3:
(
i=1
Ki ⊗ ∆i) · (x ⊗ e) = (f ⊗ e) (2)
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6. 4*
Problem 3. Predict moisture, estimate covariance parameters
Grid: 1830 × 1329 = 2, 432, 070 locations with 2,153,888
observations and 278,182 missing values.
−120 −110 −100 −90 −80 −70
253035404550
Soil moisture
longitude
latitude
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
High-resolution daily soil moisture data at the top layer of the Mississippi
basin, U.S.A., 01.01.2014 (Chaney et al., in review).
Important for agriculture, defense. Moisture is very heterogeneous.
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7. 4*
Problem 4: Identifying uncertain parameters
Given: a vector of measurements z = (z1, ..., zn)T with a
covariance matrix C(θ∗) = C(σ2, ν, ).
To identify: uncertain parameters (σ2, ν, ).
Plan: Maximize the log-likelihood function
L(θ) = −
1
2
Nlog2π + log det{C(θ)} + zT
C(θ)−1
z ,
On each iteration i we have a new matrix C(θi ).
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8. 4*
Solution: Estimation of uncertain parameters
H-matrix rank
3 7 9
cov.length
0.02
0.025
0.03
0.035
0.04
0.045
0.05
0.055
0.06
Box-plots for = 0.0334 (domain [0, 1]2) vs different H-matrix
ranks k = {3, 7, 9}.
Which H-matrix rank is sufficient for identification of parameters
of a particular type of cov. matrix?
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9. 0 10 20 30 40
−4000
−3000
−2000
−1000
0
1000
2000
parameter θ, truth θ*=12
Log−likelihood(θ)
Shape of Log−likelihood(θ)
log(det(C))
zT
C−1
z
Log−likelihood
Figure : Minimum of negative log-likelihood (black) is at
θ = (·, ·, ) ≈ 12 (σ2
and ν are fixed)
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10. 4*
Problem 5: Multivariate characteristic function
Multivariate characteristic function
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11. 4*
Problem 5: Multivariate characteristic function
The multivariate characteristic function ϕX(t) of a d-dimensional
random vector X = (X1, ..., Xd ) with X1,...,Xd independent, is
ϕX(t) =
Rd
pX(y)exp(i y, t )dy, t = (t1, ..., td ) ∈ Rd
, (1)
The probability density is
pX(y) =
1
(2π)d
Rd
exp(−i y, t )ϕX(t)dt, y ∈ Rd
(2)
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12. 4*
Elliptically contoured multivariate stable distribution
The characteristic function ϕX(t) of the elliptically contoured
multivariate stable distribution is defined as follow:
ϕX(t) = exp i(t1, t2) · (µ1, µ2)T
− (t1, t2)
σ2
1 0
0 σ2
2
(t1, t2)T
α/2
(3)
Now the question is to find a separation of
(t1, t2)
σ2
1 0
0 σ2
2
(t1, t2)T
α/2
≈
R
ν=1
φν,1(t1) · φν,2(t2), (4)
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13. 4*
Multivariate distribution
Let ϕX(t) of some multivariate d-dimensional distribution is
approximated as follow:
ϕX(t) ≈
R
=1
d
µ=1
ϕX ,µ
(tµ). (5)
pX(y) ≈
Rd
exp(−i y, t )ϕX(t)dt (6)
≈
Rd
exp(−i
d
j=1
yj tj )
R
=1
d
µ=1
ϕX ,µ
(tµ)dt1...dtd (7)
≈
R
=1
d
µ=1 R
exp(−iyµtµ)ϕX ,µ
(tµ)dtµ ≈
R
=1
d
µ=1
pX ,µ
(yµ)
(8)
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14. 4*
Literature
1. PCE of random coefficients and the solution of stochastic partial
differential equations in the Tensor Train format, S. Dolgov, B. N.
Khoromskij, A. Litvinenko, H. G. Matthies, 2015/3/11, arXiv:1503.03210
2. Efficient analysis of high dimensional data in tensor formats, M. Espig,
W. Hackbusch, A. Litvinenko, H.G. Matthies, E. Zander Sparse Grids and
Applications, 31-56, 40, 2013
3. Application of hierarchical matrices for computing the Karhunen-Loeve
expansion, B.N. Khoromskij, A. Litvinenko, H.G. Matthies, Computing
84 (1-2), 49-67, 31, 2009
4. Efficient low-rank approximation of the stochastic Galerkin matrix in
tensor formats, M. Espig, W. Hackbusch, A. Litvinenko, H.G. Matthies,
P. Waehnert, Comp. & Math. with Appl. 67 (4), 818-829, 2012
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