γλώσσες
Σελίδες
Νομικός
Multi-resolution models
for large data sets
Douglas Nychka,
National Center for Atmospheric Research
National Science Foundation IAM Retreat, NCAR, August 2013
Outline
D. Nychka LatticeKrig 2
• Surface observations of rainfall
• Compact basis functions (Φ),
Markov Random fields (H)
• The multi-resolution model
• Mean summer rainfall
• Trends in rainfall
Key idea: Introduce sparse basis and precisionmatrices without compromising the spatial model.
Estimating a curve or surface.
D. Nychka LatticeKrig 3
An additive statistical model:
Given n pairs of observations (xi, yi), i = 1, . . . , n
yi = g(xi) + εi
εi’s are random errors and g is an unknown, smooth realization of a
Gaussian process.
Observed mean summer precipitation
D. Nychka LatticeKrig 4
1720 stations reporting, ”mean” for 1950-2010
1000
2000
3000
4000
5000
6000
7000
Observed JJA Precipitation ( .1 mm)
A 1-d cartoon ...
D. Nychka LatticeKrig 5
8 basis functions 8 (random) weights
0 2 4 6 8 0 2 4 6 8
−1.
00.
01.
0
weighted basis Random curve
0 2 4 6 8 0 2 4 6 8
−1.
00.
01.
0
A Multiresolution
D. Nychka LatticeKrig 6
8 basis functions
0 2 4 6 8
16 basis functions
0 2 4 6 8
...
Adding them up
D. Nychka LatticeKrig 7
0 2 4 6 8 0 2 4 6 8
0 2 4 6 8 0 2 4 6 8
0 2 4 6 8
Distributions of coefficients
D. Nychka LatticeKrig 8
Uncorrelated (stationary)
0 10 20 30 40 50 60 0 10 20 30 40 50 60
Different variability
0 10 20 30 40 50 60 0 10 20 30 40 50 60
Different Correlation
0 10 20 30 40 50 60 0 10 20 30 40 50 60
D. Nychka LatticeKrig 9
Back to climate data
Some details for observed data:
D. Nychka LatticeKrig 10
• Used log transformation and weighted by number of observations
• Used stereographic projection for locations
• Elevation included as linear fixed effect.
• Covariance parameters found by maximum likelihood
Predicted surface
D. Nychka LatticeKrig 11
Predicted surface Pointwise standard errors
5 10 15 20 25 30cm
(a)
6 8 10 14 18Percent
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(b)
Breaking down into multi resolution
D. Nychka LatticeKrig 12
lon/lat Levels: 1 2 3
6.2 6.4 6.6 6.8 7.0log cm
(a)
−1.0 −0.5 0.0log cm
(b)
−1.0 −0.5 0.0log cm
(c)
−1.0 −0.5 0.0log cm
(d)
More on Uncertainty
D. Nychka LatticeKrig 13
EstimatedTrend in Summer Rainfall Average for Domain
−1
0
1
2
Per
cent
cha
nge
per
year
Change in summer rainfall 1950−2010
Percent change/year
Fre
quen
cy
0.06 0.10 0.14
05
1015
20
5% Lower Limit (.071) 95% Upper Limit (.121)
Summary
D. Nychka LatticeKrig 14
• Computational efficiency gained by compact basisfunctions and sparse precision matrix.
• Flexibility in model to account for nonstationary spa-tial dependence.
• Multi-resolution can approximate standard covariancefamilies (e.g. Matern)
See LatticeKrig package in R
Thank you!
D. Nychka LatticeKrig 15
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