1. Uncertainty in contour lines Peter Guttorp University of Washington Norwegian Computing Center

2. Outline Contour lines and their uncertainty Default lines Using kriging How many lines? Lindgren-Rychlik Bolin-Lindgren

3. 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

4. Data

5. Bilinear interpolation

6. 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 Krige 1919-2013

7. A Gaussian formula If then

8. Simple kriging Let X = (Z(s 1 ),...,Z(s n )) T, Y = Z(s 0 ), so that  X =  1 n,  Y = ,  XX =[C(s i -s j )],  YY =C(0), and  YX =[C(s i -s 0 )]. Then This is the best unbiased linear predictor when  and C are known (simple kriging). The prediction variance is

9. Kriging the ozone

10. How many contours? A Gaussian prediction falls between contour lines between a and b with probability where q=(b-a)/  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.

11. 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.

12. Revisit kriging contours

13. Confidence bands for contour lines Lindgren & Rychlik (1995) Isotropic Gaussian random field  (t) Observe x k =  (t k ) +  (t k ), k=1,...,n By kriging (  (t)  x) is a Gaussian process with mean m n (t) and covariance

14. Contour lines as level crossings Let  n (t) =  (t) – m n (t). Our best estimate of the level curve at u over a set A, given data x, solves m n (t)=u, To make statements about level crossings of take sections through the surface. A level curve for m n (t) is the union of the solutions to m n (t)=u over line segments in A.

15. Ozone data 50% CB

16. 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.

17. 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.

18. Ozone sets

19. Paraná rainfall 604 stations, average daily January rainfall (mm)

20. Kriging

21. Contour line confidence

22. 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.