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.