Uncertainty in contour lines Peter Guttorp University of Washington Norwegian Computing Center.
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Transcript of Uncertainty in contour lines Peter Guttorp University of Washington Norwegian Computing Center.
Uncertainty in contour lines
Peter Guttorp
University of WashingtonNorwegian Computing Center
Outline
Contour lines and their uncertainty
Default lines
Using kriging
How many lines?
Lindgren-Rychlik
Bolin-Lindgren
Ozone data set
Built in data set in “maps” library in R
NW US ozone data
1974 June-August median daily maximum ground level ozone data from 41 stations in New Jersey, New York, Connecticut and Massachusetts
Contour plot using bilinear interpolation
Kriging with exponential covariance function and nugget
Data
Bilinear interpolation
Kriging
In order to take spatial dependence into account we need a spatial interpolation that reflects the dependence structure.
The kriging estimator is the conditional expectation of the random field, given the observations.
Danie Krige1919-2013
A Gaussian formula
If
then
Simple krigingLet X = (Z(s1),...,Z(sn))T, Y = Z(s0), so that
μX=μ1n, μY=μ,
ΣXX=[C(si-sj)], ΣYY=C(0), and
ΣYX=[C(si-s0)].
Then
This is the best unbiased linear predictor when μ and C are known (simple kriging).
The prediction variance is
Kriging the ozone
How many contours?
A Gaussian prediction falls between contour lines between a and b with probability
where q=(b-a)/s and r=
If q=.5 the probability is at most 0.2 that a statement about the level of Z(s) is correct (Polfeldt, 1999).
If q=2 it is at most 2/3
If q=4 it is at most 0.95.
Consequences
Points close to contour lines are always very uncertain as to whether they should be above or below the line.
If the contour lines are well separated there are high probabilities of correctness in the middle between them.
Revisit kriging contours
Confidence bands for contour lines
Lindgren & Rychlik (1995)
Isotropic Gaussian random field ξ(t)
Observe xk= ξ(tk) + e(tk), k=1,...,n
By kriging (ξ(t)|x) is a Gaussian process with mean mn(t) and covariance
Contour lines as level crossings
Let ηn(t) = ξ(t) – mn(t). Our best estimate of the level curve at u over a set A, given data x, solves mn(t)=u,
To make statements about level crossings of
take sections through the surface. A level curve for mn(t) is the union of the solutions to mn(t)=u over line segments in A.
Ozone data
50% CB
But we need simultaneous inference
The confidence band by Lindgren and Rychlik appears to be level 1-α at each point of the contour line. David Bolin’s excursion package allows us to compute simultaneous inference for the entire contour.
Bolin sets
The idea is to calculate excursion sets above and below a given level (a set such that the process is above at all points with probability 1-α). A contour set is the intersection of the (interiors of) complements of excursions above and below. The computations are done sequentially using fast integration methods for Gaussian integrals.
Ozone sets
Paraná rainfall
604 stations, average daily January rainfall (mm)
Kriging
Contour line confidence
References
Lindgren, G. and I. Rychlik (1995): How reliable are contour lines? Confidence sets for level contours. Bernoulli 1: 301-319.
Polfeldt, T. (1999): On the quality of contour lines. Environmetrics 10: 785-790.
Bolin, D. and F. Lindgren (2015): Excursion and contour uncertainty regions for latent Gaussian models. JRSS B 77: 85-106.