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) + �i

�i’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

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●●

●

●

●●

●

●

●●

●

●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●●

●●

●

●

●

●

●●

● ●●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●●

●

●

●

●

●●

●●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●●

●●

●●

●●

●●

● ●●

●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

● ●●

●●●

●●

●

●

●

●

● ●●

● ●

● ●

●●

●

●

●

●

●

●●● ●

●●● ●

●●● ●

●●

●

●

●●

●●

●●

●

●

●

●

●

●

●●●

●●

●

●

●

●

●●

●●●

● ●

●●●

●

●

●

● ●●

●●●

●

●●

●

●

●●

●●

●

●

●●

●

●

●●

●●●

●●●

●●

●

● ●

●●

●

●

●● ●

●

●● ●

●

●

●● ●

●●

●●

●

●

●

●

●

●●●

●

●

●●

●●

●●

●●

●

●

●●

● ●●● ●

●

●

●

●

●

●●

●●● ●

●●

●

●

●

●

●

●

● ●●●

●

●

●

●

● ●

●●

●

●

●

●

●

●

● ●

●●

●

●

●

●●

●

●

●

● ●●●

●

●●

●●

●

●

●●

●●

●●

●

●●

●

●

●● ●

●

●

●

● ●

●

●●

●

●●

●

●

●●

●

●●●

●●

●

●

●

●●

●

●●

● ●

●

●

●

●

● ●●

●

●

●

●

●●

●

●

●●

● ●●●

●

● ●

●●

● ●● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●●

●

●

●

●

●

●●●

●

●●

●

●●

●

●●

●

●

● ● ●

●● ●

● ●

●

●●●

●

●

●

●●

●

●

●

●●

●

●● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●●

●●

●

●●

●

(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