Geostatistical Approaches to Change of Support ... - Semantic Scholar

Geostatistical Approaches to Change of Support Problems

– Theoretical Framework – C. L AJAUNIE , H. WACKERNAGEL

Report No 19 of Contract IST-1999-11313

December 2000

Technical Report N–30/01/G ENSMP - ARMINES, Centre de Géostatistique 35 rue Saint Honoré, F-77305 Fontainebleau, France http://cg.ensmp.fr

Contents Summary

4

1 Introduction

5

2 Geostatistical model-based approach

7

2.1

Classical presentation of change-of-support models . . . . . . . . . . . . . . . . . .

7

2.2

Basic principle of the geostatistical model . . . . . . . . . . . . . . . . . . . . . . .

8

2.3

Spatial models and support effect . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

3 Support effect models and data assimilation

12

3.1

Gaussian model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12

3.2

Lognormal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

4 Monte-Carlo approach

15

4.1

Change of support using conditional simulations . . . . . . . . . . . . . . . . . . . .

15

4.2

Conditioning upon the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.3

Short scales behavior and truncations . . . . . . . . . . . . . . . . . . . . . . . . . .

16

4.4

Monte-Carlo approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

5 Application to AIRPARIF air quality data

18

5.1

Presentation of 1999 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

5.2

Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19

6 Conclusion

24

A An example: the Hermitian model

25

B Monte-Carlo approach - simulation of Gamma model

29

B.1 Simulation by inverse Lévy measure . . . . . . . . . . . . . . . . . . . . . . . . . .

29

B.2 Infinitely divisible process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

B.3 Simulation algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

B.4 Conditioning to data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2

CONTENTS

Bibliography

3

35

Summary1 This report is a deliverable of the EC funded “IMPACT” IST project on “Estimation of Human Impact in Presence of Natural Fluctuations2 . Chapter 1 gives a general introduction. Chapter 2 presents the geostatistical change of support model. Chapter 3 discusses the importance of support effect models in data assimilation. Chapter 4 presents a Monte Carlo simulation approach as an alternative to conventional change of support models. Chapter 5 discusses first ideas in connection with the Paris area air pollution case study being set up within the IMPACT project. Chapter 6 summarizes the conclusions of the report. In the appendix Chapter A is giving details about the Hermitian change of support model and Chapter B is dwelling about recent work by Wolpert and Ickstadt in the framework of a Gamma model.

1 The report is available as a pdf file and is best viewed with the Acrobat-reader, which allows to take full advantage of the internal links. 2 See the Website: http://www.mai.liu.se/impact/.

4

Chapter 1

Introduction When dealing with spatial fields or time processes, it is necessary to assess the information carried by the data in a precise way. It is acknowledged that data have a limited precision and therefore measurement error models are frequently used to account for this. Another important aspect of data which is less often considered is their support. It turns out that very often data can be considered as spatial averages over fixed volumes, or time averages over intervals of fixed duration. This is obviously the case for soil samples, at least as long as each sample which has a well defined volume, is properly mixed up before analysis. But this is true also for air and water quality measurement, at least to a good approximation, because the analysis is based on accumulation of material through filters during fixed time intervals and under specific flux condition. This volume is termed in the geostatistics literature the support of the data. Why should we be concerned about the support of data, or in other words, on what problem does the support play a significant role? Spatial fields generally have a large spectrum of variability, including short scale components. This means that in spite of the fact that they have usually a long range smooth tendency, they vary in a very erratic way at microscopic scale. The range of variability of a spatial field or a time process can be analyzed for instance by Fourier spectrum, or equivalently by looking at the spatial covariance. But this analysis is limited by the support of the data available. Scales shorter than the data support are filtered out when only spatial averages at that volume are considered. This means that a part of the natural dispersion is not reflected by the data. To sum this up, while the overall average estimation is not biased by taking data at various support, the dispersion is underestimated in a way which depends on the sample volume considered. This is affecting the variance in an easily predictable way, but also the extremes of the statistical distributions and the percentages of exceeding over fixed thresholds for instance. The change of distribution with the data volume (or analyzed length in case of time data), is termed support effect in geostatistics. This effect has long been recognized in the literature on air quality data. In order to make corrections on the distributions according to the integration time of the measurement devices, which vary from case to case, the well known Larsen model had been proposed [8]. It has been used as a standard to normalize air quality measurements, relative to the integration time. However this model is known to rely on particular hypotheses, namely lognormality of the distributions and a de Wijs variogram 5

6

Introduction

model for the logarithmic variance, which are by no means general, and since more general models are available, the Larsen model can be considered as outdated (see [13], [18]). The support effect is not only important when comparing statistical distributions of sample data, but also when deriving estimates defined in terms of spatial averages. For instance, we might be interested in estimating the conditional probability that the average over a specified area exceeds a critical value. In agriculture, the probability that the average over a farming field is below a critical threshold, given the sample data, might be a criteria to consider in order to decide if nutrients need to be added to the soil. While the spatial averages can be estimated by classical kriging with no need to model the support effect, and the kriging variance can be used to assess the precision of this estimation, other quantities which depend on the conditional distribution, such as probabilities over thresholds, require a handling the support effect. Another situation, more of concern on account of the IMPACT project objectives, is encountered when doing data assimilation. In this situation, a mechanistic model is available, which is supposed to be able to handle the main physical process governing the dynamics of the natural system. As a rule, such models suffer from several shortcomings, which limit their prediction capability. Among these shortcomings, are the requirement on initial and on boundary conditions which are in practice imprecisely known. The lack of knowledge of the physical parameters involved, and also perhaps model inadequacy in some respect, are other ones. Finally sensitivity to initial conditions is known to be a more fundamental obstacle to such time forecast. On the other hand, as measurements become available with time, they can be used to improve the estimation of the state of the system provided by the model. This process of state estimation update is termed data assimilation in the literature. The derivation of this estimate is based on the specification of a model linking the unknown state to measurements. It can often reasonably be assumed that this link is adequately represented by a Gaussian error model. It will be argued in this document that a much more appropriate representation would be through a change of support model. One obvious reason for this is that mechanistic models are aimed at representing the physical process at the spatial resolution of the discretization grid. The shorter scales are simply not accounted for, and the physical parameters are supposed to be known at the scale resolution required by the model. On the other hand, the data are available at the measurement support, and this support is usually considerably smaller. Moreover, the data are rarely at the same location as the nodes of the discretization grid. This situation seems ideal for geostatistical change of support models.

Chapter 2

Geostatistical model-based approach In environmental studies an important problem is to characterize the amount of trespassing of a prescribed environmental threshold. This amount depends on the statistical distribution of the environmental variable at hand. The distribution will actually be significantly modified by a change of the support of the environmental variable. This phenomenon was at the heart of the development of geostatistics half a century ago. We begin developing change-of-support models in their initial historical setting, which was in mining.

2.1 Classical presentation of change-of-support models Geostatistical change of support models have originally been setup to allow recoverable reserve estimation in mining applications. This problem arises in selective exploitation, in which an attempt is made to separate the economically valuable parts of the orebody to the rest in an effective way, at a fine level. This methodology is essentially the result of work done by Georges Matheron (see [9], [10] and [11] and references mentioned there). The problem can be described as follows. The orebody is first divided into panels. Each panel is then cut into blocks of equal size, this size being limited roughly by the dimension and maneuverability of the trucks used to move the minerals. Now each block should be led to the plant only if it’s grade is worth the treatment cost, otherwise it is left to the waste. From this, result the definition of a cutoff grade zc , and the recoverable tonnage of a panel is proportional to the percentage of blocks which have a grade above this cutoff. Furthermore, the selected quantity of material (metal) is proportional to the grade of the selected blocks. Hence, the two following quantities have typically to be estimated to obtain the panel economic estimation:

1 X 1 N i2P Z (vi )>zc 1 X Q(zc ) = Z (vi ) 1Z (vi )>zc N i2P T (zc ) =

In these formulae, vi ; i

(2.1) (2.2)

2 fPg are the blocs within the panel P , and their number is N = Card(P ). 7

8

Geostatistical model-based approach

We have noted 1A the indicator function, which is equal to 1 if the condition A is satisfied and otherwise. Z (vi ) denotes the grade of the block vi , which is the spatial average:

Z (vi ) =

Z

1

0

1x2vi Z (dx)

jvij

We have written Z (dx) instead of Z (x)dx to emphasis the high spatial irregularity of grades, which imply that point values Z (x) do not have in general a precise meaning. Two points are worth mentioning concerning formulae (2.1) and (2.2): 1. The first one is that the quantities involved in the formulae are at block support, v. This support has usually an order of magnitude which is a few 10 m3 , to be compared to the few litters for the core data support. In other words a considerable change of support is involved to go from the data support to the support of concerns for the estimation. 2. The second one is that both quantities are non linear transforms of the block grades. This means that we have to be concerned with the conditional distribution of the blocks, and not merely by their conditional expectation.

2.2 Basic principle of the geostatistical model The building block of the geostatistical approach to the handling of support effect is bivariate model aimed at describing pairs in which a sample is randomly located within a larger unit (block). We shall denote by v and v respectively the sample and the block volumes, and we shall assume that these two are compatible in the sense that v can be partitioned into units equal to v up to a translation. From this covering by a finite number of disjoint samples, a random uniform sample within v can be considered. This can be done by labeling the samples and by picking a label at random with equal probability. The random concentration of this block will be denoted by Z (v). The model we are looking for is thus the probability distribution of the pair (Z (v); Z (v)). What can be assumed known about this distribution? Three guidelines can be used:

The trend (x), and the covariance C (h) can be estimated from standard methods. The first and second momentum can be calculated from these, since:

(v)

= E [Z (v)] =

Z

1

jvj

1x2v (x) dx

(2.3)

(v)2 = E [(Z (v) (v))2 ] 1 Z Z = 2 1x2v 1y2v C (x; y) dx dy = C (v; v) jvj

(2.4)

and the same holds for the mean and variance of Z (v). Concerning the covariance we have: CovfZ (v) ; Z (v)g

= E [C (v; v)] = 1 = N

X Z Z

i

vi

v

1 N

X

i

C (vi ; v)

C (x; y) dx dy = (v)2


9

Thus, the covariance between a randomly located sample and a block is equal to the block variance.

Now due to the additive nature of the variable, the concentration over v is the arithmetic average of the sample concentrations. The consequence of this on the bivariate pair Z (v); Z (v) is the following:

E [Z (v) jZ (v)] = Z (v)

This relation, which is known as Cartier’s relation, has profound consequences on the bivariate distribution model, despite its trivial appearance. In fact, it can be shown that more generally, for any pair of probability distributions FX and FY defined on R+ (of positive random variables), the following statements are equivalent: 1. One can find a pair of random variables X and Y , having probability distributions FX and FY , which satisfy Cartier’s relation E [X jY ] = Y . 2. For every convex function , we have:

E [(X )] =

1

Z

(x) FX (dx)

0

1

Z

0

(y) FY (dy) = E [(Y )]

3. For every cutoff s 0, we have:

E [(X

s)+ ] =

Z

1

0

(x s)+ FX (dx)

1

Z

0

(y

= E [(Y

where we have used the standard notation u+

s)+ FY (dy) s)+ ]

= u 1u>0 = max(u; 0).

4. If xc and yc give the same proportions for the distributions FX and FY respectively, that is if FX (xc ) = FY (yc ), then we have:

E [X 1X xc ] =

Z

1

xc

x FX (dx)

Z

1 yc

y FY (dy) = E [Y 1Y yc ]

Note that if any of this is true, in virtue of the first condition, we must have E [X ] = E [Y ]. Let be this common mean value. The second condition applied to the function (x) = (x )2 implies that the variance of X is not less than that of Y , and moreover that the same is true for any convex dispersion measure. The third means that sums in excess of cutoff is always decreasing with the support, and the last that if a fixed proportion is selected, the recovered quantity is always less for the highest support. In other words, selectivity is always decreasing with support.

It can be expected that for very large supports, distributions tend to normality due to the central limit theorem. While conditions that ensure theoretically this result do not necessarily hold in practice, this can be accepted as an heuristic and a guide in the design of model.

10


2.3 Spatial models and support effect In spatial applications, estimation is often required at large support, and thus some form of change of support is needed. When only the average Z (v) and estimation variance is required, classical linear kriging is called for, and only the variogram is necessary to this aim. But, as discussed in the introduction, when quantities which depends in a non linear way of Z (v) are to be estimated, such as (Z (v)), for a given function , change of support models are required. Two forms of estimators are classically used in geostatistics:

Conditional expectation E [(Z (v)) jZ (v1 ); :::; Z (vn )], which require the specification of the multivariate distribution L(Z (v); Z (v1 ); :::; Z (vn )). Disjunctive kriging, which have the following form:

fZ (v)gdk =

X

i

fi (Z (vi ))

In this expression, the weighting functions fi have to be optimized in order to obtain the least estimation variance. The advantage of disjunctive kriging over conditional expectation, is that it only requires the specification of the bivariate distributions: 8
> > < > > > :

=

= Var(in+1 ), since:

Cov(Z (tn+1 ; vi ) ; Z (tn+1 ; vi )

Var(Yn+1 (xi ))

= v2 + 2

2

= v2

= v2

v2 = S 2 [vjv]

This variance should be specified by the dispersion variance, which can be obtained from the variogram, according to: S 2 [v jv] = (v; v) (v; v): where we used the following standard notation:

(v; v) =

1

jvj2

Z Z

v v

(x y) dx dy

In practice, since the unconditional variances at the support v are calculated by the filter, following in the linear case the equation:

n+1 = F n F t + A At the model is non stationary, and the variances Var(Zni +1 ) = n+1 (i; i) depend on the node i. If we apply a proportional effect model, which amount to say that the short scale spectrum is in every cell proportional to a reference spectrum, the dispersion variance has to be calculated from a reference variogram, and then corrected by the variance ratio to obtain the error variance.

14

Support effect models and data assimilation

3.2 Lognormal model The pure Gaussian case is relatively uninteresting, since the distributions at any support are obtained by merely applying a variance correction. More interesting models are obtained when the distributions are assumed log-normal:

Z (vi ) = mi expfsiv Yvi Z (vi )

= mi expfsiv Yvi

1 i2 s g 2 v 1 i2 s g 2 v

The simpler log-normal model assumes that the pair (Yvi ; Yvi ) is Gaussian with correlation coefficient r, and standard margins. A more general model, termed hermitian model is outlined in the appendix. The variance relation Cov(Z (vi ) ; Z (vi )) = Var(Z (vi )) can be shown to impose the following equation:

siv r = siv

For data assimilation, the conditional distribution :

L(Z (v1); ::; Z (vn) jZ (v1) ::; Z (vk )) must be specified. According to the model, these variables are independent lognormal. The associated density is explicitely given by:

L(Z (v) jZ (v))

Y

i

fi (zi jZ (v)) =

1 2

n=2

Y

i

1 exp i zi

1 y (z ) 2 2 i i

The log-normal parameters and transformation involved in this expression are given by: 8 > > > > > < > > > > > :

i

p

= siv 1 r2

1 i2 2 = mi expfsiv rYvi sv r g 2 1 z yi (z ) = log + i i i 2 i

and the conditioning variables are:

Yvi =

1 Z (vi ) log i mi sv

+

1 i s 2 v

These expression make possible to obtain the posterior distribution, and thus to do Bayesian data assimilation.

Chapter 4

Monte-Carlo approach Monte-Carlo approaches for the change of support of Gaussian transformed random functions have been considered for a long time (see [16]). Some twenty years ago computer cost was still a serious limitation of the method, but the advent of fast and cheap computers makes this limitation less and less relevant. Simulation based methods are now widely used, partly in the context of Bayesian geostatistics (see [3]). Recently, Monte-Carlo change of support was considered in [2], for a Gaussian transformed random function. More generally, the approach could benefit from the advances in conditional simulations of random fields, and other models can be considered. This chapter takes a closer look at that perspective.

4.1 Change of support using conditional simulations When conditional simulation algorithms capable of handling the data at hand are available, an obvious alternative to change of support models is simulation at fine scale of the spatial field. The desired quantities are then obtained by numerical integration, and conditional expectation can be approximated by repeated independent simulations and standard Monte-Carlo approximations. Explicitely, if Zck (x); k

= 1; ::; m are independent conditional simulations of Z (x), we have to:

1. approximate the spatial averages by numerical integration:

Zck (v)

=

1

jvj

Z

v

Zck (x) dx

X

i

wi Zck (xi )

2. and approximate the desired conditional expectations by averaging over the realizations:

E [h(Z (v)) jdata]

m X m1 h(Zck (v)) k=1

The apparent simplicity of the method makes it very attractive, but a collection of conditions to its success have to be satisfied. Let us review these. 15

16

Monte-Carlo approach

4.2 Conditioning upon the data The first objective of the method is the generation of random field which respect the available data. Pure Gaussian random functions can be elegantly conditioned on data which are linear transforms, like spatial averages, or more generally, convolutions. The idea underlying the algorithm is the following split-up: P

Y (x) = Y k (x) + (Y (x) Y k (x))

where Y k (x) = i (x)Y (xi ) is the simple kriging estimate of Y (x) and Y (x) Y k (x) is the associated estimation error. We use the letter Y here to denote Gaussian random function instead of the notation Z used so far for reasons that will be apparent latter. The key property endowed by this Gaussian model is that the two terms in the above decomposition are independent random functions. Thus, if Y s (x) is another Gaussian function with the same distribution that Y (same mean and same P covariance), then the estimation error Y s (x) i (x)Y s (xi ) has the same distribution than the true error. This shows that a conditional simulation can be obtained using this simulated error, as in the following expression:

Ycs (x) = Y k (x) + (Y s (x)

X

i

i (x)Y s (xi ))

This procedure applies to any data which are linear transforms of Y , and not only to point data Y (xi ) as considered above. Simulation of Gaussian anamorphozed spatial field -this terminology is used to denote functions of the form Z (x) = (Y (x)) where Y is as above a Gaussian spatial function- is standard when data are at point support and when can be inverted. The values of Y (xi ) are then known and we are back to the previous situation. When data is still at point support, but cannot be inverted, Gibbs sampling can be used [4]. This situation is encountered for instance in the Gaussian representation of lithofacies, and then is an indicator function of a set Z (x) = 1Y (x)2A . It should be noted that this solution is at the price of an approximation, since Gibbs sampling is only an asymptotic method. The more complex case of data at non point support is obviously even more complicated, since data now have the form: Z

Z (v) =

1

jvj

v

(Yx ) dx

Simulated annealing is a possibility to handle such complex constraint. It has been used effectively to data which are quadratic means, a case which happens in dealing with seismic data [6] and [7]. But again this is only an asymptotic method, and convergence is more difficult to assess than in the case of Gibbs sampling considered previously.

4.3 Short scales behavior and truncations A second point to consider is truncation effects. Generally truncation effect happens at high frequencies, because simulation is calculated only on a finite grid, which by its discrete nature cannot account

Monte-Carlo approach

17

for short scales. In the case of turning band method, another asymptotic to consider, is the fact that isotropy and multinormality are only achieved for an infinite number of directions generated. Finally, in the case of the gamma random field considered in more details in the appendix, the simulation require an infinite number of components, and thus truncation occurs again. The third point point to discuss, which has obvious connections with the previous one, is the approximation of spatial integrals by finite sums, following a numerical integration procedure. Because of these two approximations, a part of the support effect is not accounted for by MonteCarlo simulations.

4.4 Monte-Carlo approximation And finally, we have to consider the convergence of the Monte-Carlo procedure itself. The rate of convergence is known to be very poor, the error variance being proportional to the inverse of the number of simulations performed. The order of magnitude of the error is thus

q

Var(hZ (v))

p1m

which is indeed very slow. A classical mean to improve Monte-Carlo methods is importance sampling. This involves the use of an alternative simulation model in which the variance would be less. But importance sampling would require the density function at block support to be known. Another classical idea is to generate anti-correlated rather than independent samples to speed up the convergence. Neither of these techniques seems directly applicable to the generation of random field. A more realistic alternative to speed up convergence could be through the addition of simulated conditioning points, for which the required densities are available. If these points are chosen in an appropriate way, for instance at the block centers, they can conditioned strongly enough the simulations to speed up convergence. To conclude this section, we stress that Monte-Carlo methods offer advantages which can pay-off their computational complexity:

They are exact asymptotically. By this we means that once the spatial model is specified at point support, no modeling approximation is required to obtain the distributions at various support. Only numerical approximations which can to some extend be controlled, are necessary. They offer full flexibility concerning the geometry of the supports. No compatibility constraint exists, and various supports can be simultaneously considered.

Chapter 5

Application to AIRPARIF air quality data Air quality data of Paris region have been made available to the participants of the IMPACT project by the AIRPARIF association. Contacts have been taken with the Air Pollution group at Laboratoire de Météorologie Dynamique (LMD) so that the data can be supplemented with runs from the model CHIMERE (see [14], [15] for a description and validation of the model and [12] for a use in sensitivity analysis).

5.1 Presentation of 1999 data This study considers only data from the year 1999, measured at 23 fixed stations for NO and NO2 , as well as at 19 stations for O3 . The location of the stations is shown on Figure 5.1. Data are hourly measurements. Since our aim is only to illustrate the support and models, we have to select one variable to study. Which one to use? It is known that due to the fast dynamic of the chemical reaction:

NO + O3

!

NO2 + O2

ozone is consumed by this reaction as long as there is NO available, and conversely. As a result, we expect these two to be highly anticorrelated, in the sense that they rarely coexist. The Figure 5.2 shows that this is indeed the case. On this figure we have separated the plots concerning the seasons, and also represented by different characters the days of week, in order to explore these effects meanwhile. This figure is limited to the data from the station labeled Paris 6, but the observed effects are typical. It is interesting to analyze the periodicities in the data. Yearly and dayly cycles are of course expected, and this will be referred to as seasonal and hourly effects. But in addition to this, we want to explore the day-of-week effect, since human activity is depending on it, and can be considered as entirely responsible of this effect. To this aim, we calculated the yearly averages under each specific Season Day Hour combination. The results, for the Paris 6 station are displayed on Figures 5.3 to 5.5 for ozone, NO and NO2 respectively. On these graphs, presented in matrix form, each line represents a season, and each column a day-of-week. The seasonal effect appears clearly on these 18

Application to AIRPARIF air quality data

19

Analysers locations (details)

2450000

2460000

Analysers locations

FremainV

Saints

Tremblay

Argenteuil SaintDenis GenneV

Y

2420000 2400000

Y

Prunay

2440000

Tremblay Argenteuil SaintDenis GenneV Bobigny AuberV Paris18 Neuilly Montgeron Paris7 Paris6 Garches Paris12 IssyParis13 IvryS Versailles VitryS

Evry

Neuilly

2430000

2440000

Montgé Mantes

Rambouillet Melun

Garches

Paris7 Paris6 Issy

2420000

2380000

NOx and O3 measured NOx only Fontainebleau

2360000

O3 only

560000

580000

600000

620000

640000

660000

X

AuberV Bobigny Paris18 Montgeron

Paris12 Paris13 IvryS

Versailles VitryS

580000

590000

600000

610000

620000

X

Figure 5.1: Locations of NOx and O3 stations are displayed in black, Stations measuring NOx only are represented in red, and O3 stations are in blue.

graphs, where higher O3 concentrations are observed in Summer. The maximum daily concentrations are observed around 12 am, an effect known to be due to thermal convection. As a consequence we observe a minimum of NO at the same time. The day-of-week effect is particularly evident in Winter and in Autumn, the increase in terms of NO, as well as NO2 , from Sunday to Saturday, is followed by a decrease, the minimum being observed on Sunday for this station. The accumulation effect of the air pollution is obvious if it is assumed that this results from the decrease of activity in the week-end. To reduce the effect of this chemical reaction we decided to study for the future the sum NOx = NO + NO2 , which can be expected more stable than either variable considered separately.

5.2 Future work The planned work on these data is the following 1. Estimate the three periodicities, that is the season the day of week and the hour effects. These will be considered further as mean terms. 2. Tailor a change of support model in the time dimension. Empirical regularization of data make it possible to validate a model. 3. Make the assumption that spatial variability can be compared to time variation, and thus obtain a change of support model in the space domain.

20


Paris6 50

100

150

200 x x* x x* x x x

400

0

200 100 0

400

NO

300

Autumn x x x** +x* x** + x ++ x** +x x x* x * x + xx x + + xxx**x*+x* x+ x xxx ++ x++x x+ x*x+xx*+ + + x **x+xxxxx x++ + x x++xx+ xx x+ x + *+ x x x + * * x + x x o + + * * x + + x+*x+*xx+ o + + x*x+ x*xx+o *xx+ x*+x+ x*++ x x+ x+o*o x +++ ++ **oxx *x+ o+ o+ o o+ + oo x+ x+ x*o o o *o + + xxxx x*o xx+ + x + + ***xx + o + + *o+ x++ o ++ ++ + + x+ x*++ xxx+ x+ o o + ++ *o x x*x*o+ o + + oo xx+ o xx+ x*x+ o + + x x xo x+ + ++ + + * x x x x++++xoo+xx+ o o + + * x x x o o xoo + x o + ++ + xx x x*o+ o+ o o + * x xo o o o o + + x o + + + + x oo x x x + + ++ + + * * ** x x oo o o o + +++ ++ + + + + * * x x x xx x x x oo o o o o o o + + + * * * * xx o x x x + ++ + ++ + + * x x xo o x**+ x o o + + + + + * * * * x x x++ x++ o o++ o o o o o o + ++ + + + + + * * xxx x x x x xxx x x xx xx x*x+ o o o o o o oo o + + + + + + ++ + + + + + + + + + * * * * x o x x o o x o xx o x o o x x xooo x x+ + + xx+ xx++++ x+ x+ o o oxo+++ x+ x+ x++ o **o *o **o *oo xx *o **ox+ *** x+ xxo*+ x+ xxx o +++ + *o *o *o *oo x+ x+ xx xx x*xoxo*xoxo oxx o +*o *x++ xxx oo o ox+ xo*x+ o xxoxoo ++ + + +++ + *o+ *xxx **xxx xxx x++ x+ o**+ oo xx x+++ xxox*oxx ++ + ++ + + + + +++ ++ ++ + ++ ++ + + + + ++ ++ + + + + + + +o+ ++ + + ++ ++ + + + ++ + ++ + + + + ++ + + *xx xx x+ xxx xxxx xxx x+ x**o xo x*+ xo x+++ xx x+ xx xx x++ xx xx x*o+ x++ x*o x++ xx xo xxxxxx x** x+*oo x+ x+ xx x++ xx x+ x+ x*+ xx x+ x*o++ x++ xx x*o xxx x++ x*x+ x*oo xxxx x*+ x*o+ x*o x+ x+*o x+ xx xxx xx xx xo+ xo+ x*o x*o+ x*o x+ x*oo* o oo x+++ x++ xxox* *ooo o+ o o o oo oo oo oo o+ o** o oo oo oo o ooo oo ooo o o o*oo o*o oo o+ o o oo o+ oo oo o*x+ oxxx o*xx o+*oo*o oo o o*++ o+ o o o o** oo o oo oo o o o o o o o oo oo ooo oo oo o*x+ oo oo oo *o+ *** *x+x+ **o x+ *o++ **x+ *o*o+ *xox* + **x+ *xxx *o *o ** *xo *** *xxx *xxx *oo *oo *x++ ** *o *x+ *o **o *oo *oxox** *oo+ *o+++ *o *o *xxx *** *o **o *xx *++ * * *** +**o+++ *x+++ *** **xo **++ ***xxx *o+ *o++ ** *oo ** ** *** *** *ox+ *o+x+ *++ *o ** *ooo *o *x+ ** *oxx ***** ** ** *oo *x+++ Summer

Spring

0

100

200

300

Winter * + ++ x+ + + + x +*++ x+ ++++ + x x*x*x + x*+ + x+x**x x + x**o + + x o o x o o+ o + x*xo x*o+o o x *+ o x+ o x*o+ x x+ + o ** x x *o + *xxxo+ o+ + *x x+ xx*o x x+ xx+ x++ xooo o **o o o+ x+xo + *o x x + o o x x+ + + xx+ *o ***o+ + x x++ + * + o oo x+ x x x + + o*x+ o ** **+ o + + * x xx+o + + * o o o x x+ xxx + * x+ xxx x xx+ o oo+ *o oo x+ o o+ xo + xxx o o oo + + ooo o+ oo ++ o o o o o x+o xxo x+ x+ xxx + + o x** x+ + +++ *o *o+ x*o x xx+ x++ x** * ***** xx*+ xx xx+ o + + ++ *xox+ oxo o x ++ ++ ++ + ++ **o o o+ oo ooo xo x xx ++ + +++ + + + + + ++ + +o x+ xx x+ xxx x** xxx x+ xx o o oo o ooo o** ooo o oo o+ o oo + + + *o x++ x*o xx xx x++ x+ xoo x+ o oo oo oo++++ oo ** *o ** ***o *** *x*o **xxx *x*o **oo *** *o *o *x+ *x** *x+ ** **o *oo *++ **oo *xx ** ** *o

o +x + + xx x x x+ x * x xx *+ xx*+ o * ** x +*xx+ *x+ + x ** x*x*xx+ o *x++ + + + * + + x o * x x x + * + + + *+ * o x**+ *o+ *xx+*+ x+ *o+ + ox+ *oxxo x*+ + o ox+*xx+ + *o o+x*+x+*oo x++x +x xxx xx+x+ + + x*o ++ x++ o + xxx+ + + *o **xxxo **x++ x+ o+ + *o xx+ xxxx o+ + + o xx++ x+ o ++ x+ x+ xx+ o + + + + xoo xx++ o + + + *o xxxx o + + xxx xxoo x+ x+ xx xx x o o o o++ + o + + xx x oo o ++ * * x+ x*x xx x xx+ o + + *o x+ xo x+ o o o oo ox*x+ ++ + + + * xx x+ xxx o + + + + + + xx x+ x xo*xoo x*o x x o o o o o + ++ + + + + + x xx xo x+ xx+ x oo x oo o o o o o o +++ * **xx+ x xx + + + * * x xx + + * * * *x*o xxx xx oo oo ooo o o + + + + * xx x+oo xo xx x o ++ + + + + + * x xxxx xo*o xx xx+ x xxx o o o o oo +++ ++ +++ + ++ ++ +++ ++ ++ +++ + + + ++ + ++ ++ + + *** * xx x xo x x x xx xx x+ xoo x x+ xx x xx x xxx xx x x x xxx xx xx oo oo o oo o oo o o o oo o oo oo o o oo o oo o o o o o o o o o o o oo o o o **o *o * * * ********* ** *** *oo **o *x+ *x++ **oo *o ** ** *oo *x+ **xxx ********* **xxx **** *x+ *xxx *xo+ **xo*x+ *x+ *x+ ** **x+ *xxx *** *ooo *** ** 0

50

100

Sun−Mon Tue−Wed Thu−Fri Sat

150

200

O3

Figure 5.2: Scatterplots of NO and O3 hourly concentrations at Paris 6, during the year 1999. As expected when one element is present, the other one is almost absent.


Sunday 15

Tuesday 5

Wednesday

15

Thursday 5

Friday

15

Saturday 5

15

80 100 60

Autumn Winter

20

40

Spring

Summer

60 40 20 80 100 60 40 20

O3 at Paris6 in 1999

80 100

20

40

60

80 100

5

Monday

21

5

15

5

15

5

15

Hour

Figure 5.3: Hourly averages of O3 at Paris 6 station.

22


Sunday 15

Tuesday 5

Wednesday

15

Thursday 5

Friday

15

Saturday 5

15

Autumn Summer

60 40 20

5

15

5

15

5

15

Hour

Figure 5.4: Hourly averages of NO at Paris 6 station.

Spring Winter

0

20

40

60

80

0 80 60 40 20 0

NO at Paris6 in 1999

80

0

20

40

60

80

5

Monday


Sunday 15

Tuesday 5

Wednesday

15

Thursday 5

Friday

15

Saturday 5

15

Autumn Summer

60 40

5

15

5

15

5

15

Hour

Figure 5.5: Hourly averages of NO2 at Paris 6 station.

Spring Winter

20

40

60

80

20 80 60 40 20

NO2 at Paris6 in 1999

80

20

40

60

80

5

Monday

23

Chapter 6

Conclusion In this report, only a general theoretical review has been given. Change of support can be viewed as statistical change of scale for additive variables, and there could be interesting connections with self-similar models. We did not attempt to explore these connections here. We preferred to review briefly the existing approaches, in order to make it possible to identify the needs concerning support effect models in the course of the IMPACT project. We emphasized the interest of change of support models when a simulation based on physical model is compared to data, as this is the case in data assimilation. But the situation is probably relevant to other problems in which such comparisons are attempted.

24

Appendix A

An example: the Hermitian model In this section we describe a fairly general model for continuous distributions. It should be mentioned that alternative models exist, some of them have continuous marginals, others are discrete, and mixed models exists as well. So this should really be considered as an illustrative example. The starting point of this model is a Gaussian representation of the concentrations at sample support. It is assumed that a monotonous transform is known, termed anamorphosis : R+ ! R, such that Z (v) = (Yv ) where Yv is a standard Gaussian variable. This transform is determined from the distribution of Z (v) by:

Fv (z ) = P [Z (v) < z ] = G[ 1 (z )] where G is the standard Gaussian distribution function. Thus we have = (Fv ) 1 G, which is always possible when Fv is continuous. The distribution at support v is then continuous as well, and we have the same representation Z (v) = v (Yv ) with a transform noted v . It should be understood that Yv is by no means the spatial average of Yv , since the average operates on the Z scale and not on the Gaussian transform. This is the reason why we used the notation Yv instead of Y (v). The model assumption concerns the bivariate pair (Yv ; Yv ) where Z (v) = (Yv ). It will be assumed that this pair follows a mixture of standard Gaussian distributions, termed Hermitian model. Specifically, if gr (u; v ) is the density of a standard Gaussian pair with correlation r, the hermitian distribution can be expressed as:

gh (u; v) =

Z

gr (u; v) !(dr)

where ! is a distribution on [ 1; +1]. In practice negative correlations are very unrealistic and the domain of ! can be assumed restricted to [0; 1]. The introduction of the mixing offer a higher level of flexibility to the model, compared to the pure Gaussian model (obtained by taking ! = r ). For instance, if ! = p 0 + (1 p)1 , we obtain the mosaic distribution, for which Yv and Yv are independent with probability p, and equal with probability 1 p. The change of support generated by this bivariate model is in fact the affine change of support, according to which block and sample density are the same up to an affine rescaling 1 . Models in which ! is a beta distribution have been considered 1 The terminology in use is unfortunately a bit confusing, for what is termed mosaic change of support model is in fact something different.

25

26

An example: the Hermitian model

by L .Hu [5]. Calculations are greatly simplified by considering hermitian polynomials:

n (y) =

g(n) (y) n! g(y)

p1

In this formula, g is the standard Gaussian density, and g (n) its n-derivative. It turns out that these expressions are indeed polynomials, which can be calculated from a three terms recurrence. Hermite polynomials are standardized and orthogonal relatively to the Gaussian distribution: R

R

n (y) g(y) dy = 0 n (y) m (y) g(y) dy = nm

Moreover, the orthogonality holds also for the bivariate Gaussian distribution: Z

0

0

0

n (y) m (y ) gr (y; y ) dy dy = nm rn

This result is a consequence of the following relation concerning the conditional expectation of a Hermite polynomial: if (Y; Y

0

)

gr

0

0

E [n (Y ) jY ] = rn n (Y )

then

In other words, the Hermite polynomials are the factors of the Gaussian model. A consequence of this is the following expression of the bivariate density:

0

X

0

gr (y; y ) = g(y) g(y )

k0

0

rk k (y) k (y )

This development holds for jrj < 1 only.

These expressions translate readily to the hermitian model. If now the pair (Yv ; Yv ) follows a R hermitian model, we have the following expressions, in which E [Rn ] = rn ! (dr):

E [n (Yv ) jYv ] = E [Rn ] n (Yv ) E [n (Yv ) jYv ] = E [Rn ] n (Yv )

(A.1) (A.2)

The interest of these expressions follows from the fact that Hermite polynomials allows representations of functions having finite order two moment relative to the Gauss density: Z

if

(y)k g(y) dy
0: 2 3 X k k 1 P 4 r k (Y ) > 5 ! 0 k2 1 In fact, since the variance is simply: 8 0) we have v2 variances:

= (1 p)2

P

2 k1 k

= (1 p)2 v2 , and the parameter p is therefore determined by the (1 p) = v v

The transformation at the larger support is given by:

X k k (Yv ) = m + v ((Yv ) m) Z (v) = m + v v k1 v and, the distributions are such that:

Z (v) m

v (Z (v) m) v

In other words, according to this model, the distribution at large support is obtained merely by sharpening around the mean. The algorithm resulting from the application of this model is termed affine correction, and it is known to have poor quality. In particular, convergence towards normality cannot obviously be satisfied by this model. We have just seen that asymptotic normality cannot be obtained by arbitrary choices of Hermitian models. A sufficient condition is the convergence of E [Rk ]=E [R] ! 0, uniform in k .

Appendix B

Monte-Carlo approach - simulation of Gamma model This section is based mainly on a collection of papers published by Robert Wolpert and Katja Ickstadt (see for instance [20], [19], and [1]). The applications considered by the authors concerns principally epidemiology, where data can be considered as Poisson distributed, with intensity determined by a latent spatial field to be estimated. The data augmentation algorithm proposed by the authors is particularly tailored to this situation. Conditional simulations where data are the spatial average at various support is a somewhat different problem.

B.1 Simulation by inverse Lévy measure Let us recall that a random variable follows an infinitely divisible distributions if it can broken into an arbitrary number of independent identically distributed random variable. So for each integer n > 0, the variable X can be decomposed into:

X = X1 + X2 + ::: + Xn where Xi are independent and identically distributed. Applied to spatial fields, if we were ignoring the spatial correlation, the distributions at every support should be infinitely divisible. In practice, correlation is more often the rule than the exception, and the argument does not have so firm foundations. Nevertheless, it remains that spatial fields with infinitely divisible distributions are very interesting from coherence point of view. It is known from Lévy’s representation theorem, that positive infinitely divisible distributions, if their mean is normalized in a suitable way, have positive Laplace transform, satisfying the formula:

log E [e

sX ]

=

1

Z

0

(1 e

su ) (du)

where is a positive measure, termed Lévy’s measure, satisfying: Z

0

1

min(1; u) (du) < 29

1:

30

Monte-Carlo approach - simulation of Gamma model

This condition ensures the convergence of the above integral, but in general, the measure of a neighborhood of the origin is not finite. Let c (u) = [u; 1[, which is finite for u > 0. Lévy’s theorem being granted, X has a representation in terms of a Poisson process, which we describe now. Let H be a Poisson process on R R+ , having intensity dt (du). By this we means that for each measurable disjoint sets Ai 2 R R+ , the random variables H (Ai ) are independent Poisson R variables with parameters i = Ai dt (du). Moreover, given the number H (A), the points of the process in A are distributed independently, following the probability

dt (du) (dt )(A)

1(t;u)2A

Then we shall see that the process defined for t > 0 by:

Xt =

1

Z tZ

0

0

X

u H (dt0 ; du) =

i

Ui 1Ti 2[0;t[

is such that X1 is infinitely divisible, with Lévy measure . In the above formula, we have denoted by (Ui ; Ti ) the points of the process H .

Proof:

Let > 0, and Xt be the truncated integral:

Xt

=

1

Z tZ

0

u H (dt; du):

Then, due to the fact that c () is finite, there is a finite number of points Nt of the process in the set [0; t[[; 1[. The Laplace transform:

E [e

sXt ]

= E [expf s

X

i

Ui 1Ti g]

can be calculated by conditioning on Nt , since each point follow independently each one another the distribution: 0

(T; U )

P (dt0; du) =

h

i

dt (du) 10 1 t c () t 2[0;t[ u2[;1[

We have thus:

E e

sXt

= E = e

"Z

1

c () t

= exp

e

X

k

(du) c ()

1

Nt #

( c () t)k k!

c () t + t

Z

= exp t

su

(e

su

Z

1

Z

1

e

e

(du) c ()

su

su (du)

1) (du)

k


31

Hence, we obtain the log characteristic transform:

log(Ee

sX1 )

=

1

Z

(1 e

su ) (du)

and in the limit, we have the desired result.

Simulation: A simple simulation method can be derived by observing that the projections on the u axis of the process points in the band t0 2 [0; t[, follow a Poisson process with intensity (du) = t: (du): Ui such that 0 Ti < t have distribution Ui Po(t: ) The associated Ti are uniformly distributed in the interval t 2 [0; t[. The first step requires a method to simulate Poisson process with intensity t . The key to this is the following change of variable: (

y = t [u; 1[ = t c (u) dy = t (du)

which transforms a process Po[t ] into a standard Poisson process (independence of counts over disjoint sets is ensured by monotonicity of the transform, and if NY (dy ) Po[dy ], then NU (du) is also Poisson distributed with parameter t: (du)). Hence the algorithm: 8 > > < > > :

Yi Po(1) Ui = c ( Yti ) Ti U [0; t[

where c is the inverse of c .

B.2 Infinitely divisible process We consider only positive spatial process, and we denote them by Z . For test functions having the P P form = i i 1Ai where Ai are disjoint measurable sets, we write Z () = i i Z (Ai ). This notation is extended to test functions which are limits of this form when possible. The Laplace functional, which is defined by: L() = log E [e Z ()] can be used to characterize positive spatial fields, in a similar way Laplace transform characterize the distribution of positive random variables. It turns out that infinitely divisible and positive spatial process, with independence over disjoint sets have Laplace transform which can be written as: Z

L() = (1

e

u(x) ) (dx; du)

where satisfies some constraints, analogous to the one specified in the previous section. Gamma process are obtained when:

(dx; du) = (dx)

e

u

u

du

32


The Laplace transform of gamma process can be calculated by first considering test functions (x) = P i 1x2Ai , for which we have: Z

since:

(1 e Z

1

0 we obtain, after taking the limit:

u

P

i 1Ai ) (dx)

ui ) e

(1 e

Z

L() =

u

u

=

X

i

(1 e

ui ) (A ) i

du = log 1 +

i

(x) (dx) log 1 +

Remarks:

The contributions from disjoint sets Ai enter additively into the Laplace transform, which ensures the independence of the random variables Z (Ai ).

For = s 1A , we obtain:

E [e which shows that the average of and scale 1= :

s Z (A) ]

= 1+

s

(A)

Z over A is gamma distributed, with shape parameter (A) Z (A)

Ga((A); 1 )

Wolpert and Ickstadt propose the following extension of the gamma random field:

(dx; du) = (dx)

e

(x)u

u

du

in which the scale parameter vary through space. However, in this model, the spatial averages Z (A) are no longer gamma-distributed. Only if is slowly varying through space, and if the dimension of A is small do we have approximate gamma distributions. The simulation algorithm have to be modified to deal with this generalization.

B.3 Simulation algorithm. The positive Lévy spatial process can be generated from a Poisson process intensity (dx; du) using the algorithm described in the previous paragraph: Z

Z (A) = and more generally:

Z () =

x2A Z

u H (dx; du) =

(x) u H (dx; du) =

X

i

1Xi 2A Ui

X

i

(Xi ) Ui

H on R+ Rn with


33

The proof is the same. The generation of H is almost unmodified if is the product of two measures, such as (dx; du) = (dx) (du). We describe now the algorithm proposed by the two authors, in a slightly simplified version, for simulation of points in R+ A, in the extended gamma case (space-varying ):

Let Yi ; i = 1; :: be a standard Poisson process on R+ .

Determine Ui from the following equation:

Let Xi follow the distribution:

Xi

) ((dx 1 A) x2A

1 e (Xi )u (A) du = Yi u Ui R which is equivalent, if Ei (x) = x1 e u =u du is the exponential integral function, to: Yi 1 1 E Ui = (Xi ) i (A) Z

In practice, only a finite number of points can be generated, and it is necessary to stop at some point (note that Ui ! 0)

B.4 Conditioning to data. Spatial fields generated by the algorithm considered so far in this section do not have spatial correlation, and only the parameters and can break the spatial homogeneity. It is very easy to obtain spatially correlated fields by moving averages. For example, Y being a gamma spatial process generated according to the model described in the previous paragraph, Y Ga((dx); (x) 1 ), we can consider: Z

Z (y) =

(y; x) Y (dx) =

X

i

(y; Xi ) Ui

Note that simulation at support v are obtained simply by integration the kernel relatively to y . If:

(v; x) = then:

Z (v) =

1

jvj X

i

Z

v

(y; x) dy

(v ; Xi ) Ui

Monte-Carlo simulations at various support seems easy to implement. The difficult problem remaining is the conditioning to data. The authors have considered the case of conditionally Poisson distributed data, that is, conditioned on Z , the data are:

Ni

Poisson(Z (vi ))

with independence over disjoint supports vi . The ingredient to simulate Y conditioned on N1 ; ::Nk is data augmentation.

Bibliography [1] B EST, N., I CKSTADT, K., AND W OLPERT, R. Spatial poisson regression for health and exposure data measured at disparate spatial scales. Journal of the American Statistical Association 95 (2000), 1076–1088. 29 [2] C HRISTENSEN , O. F., J, D. P., AND R IBEIRO , P. J. Analysing positive-valued spatial data: the transformed gaussian model. In geoENV – Geostatistics for Environmental Applications (Amsterdam, 2001), P. Monestiez, D. Allard, and R. Froidevaux, Eds., Kluwer, pp. 287–298. 15 [3] D IGGLE , P., TAWN , J., (1998), 299–350. 15

AND

M OYEED , R. Model-based geostatistics. Applied Statistics 47, 3

[4] F REULON , X. Conditionnement du Modèle Gaussien par des inégalités ou des randomisées. Doctoral thesis, Ecole des Mines de Paris, Fontainebleau, 1992. 16 [5] H U , L. Y. Mise en oeuvre du modèle gamma pour l’estimation de distributions spatiales. doctoral thesis, Ecole des Mines de Paris, Fontainebleau, 1988. 26 [6] L AJAUNIE , C. Simulation of velocity fields under constraints on the stack values. Tech. Rep. N8/98/G - confidential, Centre de Géostatistique, Ecole des Mines de Paris, Fontainebleau, 1998. 16 [7] L AJAUNIE , C. Simulation of velocity constrained by stack analysis model, methods and a case study. Tech. Rep. N-20/99/G - confidential, Centre de Géostatistique, Ecole des Mines de Paris, Fontainebleau, 1999. 16 [8] L ARSEN , R. I. A new mathematical model of air pollutant concentration averaging time and frequency. Journal of the Air Pollution Control Association 19 (1969), 449–467. 5 [9] M ATHERON , G. Isofactorial models and change of support. In Verly et al. [17], pp. 449–467. 7 [10] M ATHERON , G. The selectivity of the distributions and the second principle of geostatistics. In Verly et al. [17], pp. 421–434. 7 [11] M ATHERON , G. Change of support for diffusion type random functions. Mathematical Geology 17 (1985), 137–165. 7 34

Bibliography

35

[12] M ENUT, L., VAUTARD , R., B EEKMANN , M., AND H ONORÉ , C. Sensitivity of photochemical pollution using the adjoint of a simplified chemistry-transport model. Journal of Geophysical Research 105, D 12 (2000). 18 [13] O RFEUIL , J. P. Interprétation géostatistique du modèle de Larsen. Tech. Rep. N-413, Centre de Géostatistique, Ecole des mines de Paris, Fontainebleau, 1975. 6 [14] VAUTARD , R., B EEKMANN , M., D ELEUZE , I., AND H ONORÉ , C. La pollution photochimique en région parisienne simulée par le modèle Chimère et l’influence du transport régional d’ozone. Tech. rep., Laboratoire de Météorologie Dynamique, Ecole Polytechnique, 1997. 18 [15] VAUTARD , R., B EEKMANN , M., ROUX , J., AND G OMBERT, D. Validation of a deterministic forecasting system for the ozone concentrations over the Paris area. Atmospheric Environment 35 (2001), 2449–2461. 18 [16] V ERLY, G. The block distribution given a point multivariate normal distribution. In Verly et al. [17], pp. 495–516. 15 [17] V ERLY, G., DAVID , M., AND J OURNEL , A. G., Eds. Geostatistics for Natural Resources Characterization, vol. C-122 of NATO ASI Series C-122. Reidel, Dordrecht, 1984. 34, 35 [18] WACKERNAGEL , H., T HIÉRY, L., AND G RZEBYK , M. The Larsen model from a de Wijsian perspective. In geoENV II – Geostatistics for Environmental Applications (Amsterdam, 1999), J. Gomez-Hernandez, A. Soares, and R. Froidevaux, Eds., Kluwer, pp. 125–135. 6 [19] W OLPERT, R., AND I CKSTADT, K. Simulation of Lévy random fields. In Practical Nonparametric and semiparametric Bayesian Statistics (Berlin), Müller, Dey, and Sinha, Eds., SpringerVerlag, pp. 227–242. 29 [20] W OLPERT, R., AND I CKSTADT, K. Poisson/gamma random field models for spatial statistics. Biometrika 85 (1998), 251–267. 29