Ubiquity of, and geostatistics for, nonstationary increment random fields

Authors

  • Daniel O'Malley,

    Corresponding author
    1. Department of Earth, Atmospheric and Planetary Sciences, Purdue University, West Lafayette, Indiana, USA
    • Corresponding author: D. O'Malley, Department of Earth, Atmospheric and Planetary Sciences, Purdue University, West Lafayette, IN 47907, USA. (omalled@gmail.com)

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  • John H. Cushman

    1. Department of Earth, Atmospheric and Planetary Sciences, Purdue University, West Lafayette, Indiana, USA
    2. Department of Mathematics, Purdue University, West Lafayette, Indiana, USA
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Abstract

[1] Nonstationary random fields such as fractional Brownian motion and fractional Lévy motion have been studied extensively in the hydrology literature. On the other hand, random fields that have nonstationary increments have seen little study. A mathematical argument is presented that demonstrates processes with stationary increments are the exception and processes with nonstationary increments are far more abundant. The abundance of nonstationary increment processes has important implications, e.g., in kriging where a translation-invariant variogram implicitly assumes stationarity of the increments. An approach to kriging for processes with nonstationary increments is presented and accompanied by some numerical results.

1. Introduction

[2] Let math formula be a random field. When nonstationarity is mentioned in the context of random fields, it usually refers to nonstationarity of the process values, i.e., the distribution of math formula and math formula may differ. This could arise, for example, if math formula is a log-fractional Brownian conductivity field [Mandelbrot and Ness, 1968]. Another type of nonstationarity that can arise is the nonstationarity of the increments, i.e., the distribution of math formula may depend on x. Such nonstationarity can arise, for example, if math formula is a fractional Gaussian noise with nonlinear space [O'Malley et al., 2012].

[3] Stochastic processes with nonstationary field values have been studied extensively in hydrology using models such as fractional Brownian motion (fBm, which has a power-law variogram) [Molz et al., 1997] and fractional Lévy motion [Lu et al., 2003; Painter, 1996]. Stochastic processes with nonstationary increments have been scrutinized much less intensely. The primary purpose of this note is to demonstrate the ubiquity of processes with nonstationary increments via a mathematical argument showing that processes with stationary increments are the exception, rather than the rule. The implications of this are then considered in the context of kriging, and a method for kriging in the presence of nonstationary increments is presented along with some numerical results.

2. Nonstationary Increment Processes

[4] If a constant mean process has nonstationary increments, then the variance of the increments is generally given by a function of two variables,

display math(1)

[5] When the increments are stationary, math formula becomes a function of a single spatial variable, math formula. It is not hard to imagine that just as there are many more functions of two variables than functions of one variable, there are many more processes with nonstationary increments than processes with stationary increments. This speaks to the sparsity of models with stationary increments relative to the abundance of models with nonstationary increments. This idea can be made more precise mathematically by observing that the set of stochastic processes with stationary increments is nowhere dense, and the set of processes with nonstationary increments is dense everywhere. To clarify what this means, some definitions may be helpful. A limit point, x, of a set, C, is a point such that math formula where each math formula. The closure of C (denoted math formula) is the set containing C plus all the limit points of C. The interior of a set is the complement of the closure of the complement of the set. A set is called nowhere dense if it has empty interior. A set, math formula, is called dense everywhere if the closure of C is X.

[6] Let S denote the set of stochastic processes defined on the domain math formula that have suitably smooth probability density functions for every finite dimensional distribution. If math formula, let it be said that

display math(2)

or in other symbols,

display math(3)

if

display math(4)

for any m, for all math formula where d is the dimension of the math formula, and for any measurable set math formula. In other words, math formula if the distribution functions for the finite dimensional distributions of math formula converge pointwise to the distribution function for the same finite dimensional distribution of math formula.

[7] Let Ss denote the set of elements of S with stationary increments, and Sns denote the set of elements of S with nonstationary increments. The objective is to show that Ss is a nowhere dense set in S using the definition of limit points given above. The first step in the proof is to show that the closure of Ss is Ss, i.e., math formula. The second step is to show that for any math formula, there is a sequence of elements math formula such that math formula for math formula. This shows that math formula is not in the interior of math formula, where math formula denotes the set difference operator. Since math formula was an arbitrary element in math formula, the interior of math formula must be the empty set. Therefore, Ss is nowhere dense in S.

[8] Other mathematical definitions for convergence can be given, and other sets S can be chosen. However, the basic outline given above would provide a reasonable approach to proving that the set of stochastic processes with stationary increments is nowhere dense with a different set of assumptions. Depending on these assumptions, different choices of math formula may become necessary. The one that will be employed here is to let math formula where the math formula have nonstationary increments and become smaller as n becomes larger. This amounts to adding a small perturbation to the original process.

[9] To begin the proof, it must be shown that math formula. Let math formula. This implies that there is a sequence math formula such that math formula, and

display math(5)

so that math formula has stationary increments as well. Therefore, math formula, and math formula.

[10] Let math formula, so that math formula is a trivial stochastic process with no uncertainty, but with nonstationary increments where math formula. From the fact that

display math(6)

it can be seen that the increments of math formula have a spatially dependent mean and are thus nonstationary. Hence, math formula. Let C be an open set in math formula, and observe that

display math

where math formula is the math formula standard basis element in math formula. The last equality follows from the assumption of smoothness on the joint probability density function for math formula, math formula. This string of equalities implies that math formula. Since math formula, math formula is not in the interior of math formula and because math formula was an arbitrary element of math formula, the interior of math formula must be empty. Hence, the interior of the closure of Ss is empty, so Ss is nowhere dense in S. This argument also demonstrates that the closure of Sns is S, so Sns is everywhere dense in S.

[11] The proof presented here concerns an abstraction of spatial statistical processes, and it is natural to wonder whether this abstraction bears any resemblance to stochastic processes found in nature. Recent studies of diverse phenomena such as soil properties [Haskard and Lark, 2009], air pollution [Fuentes, 2002], and ocean temperatures [Higdon, 1998] indicate that nonstationarity does frequently occur in nature. The argument given here can be seen to provide some theoretical motivation for why this is the case.

2.1. Nonstationary Increments Via Nonlinear Spatial Transformations

[12] The mathematical simplicity of processes with stationary increments has made them a rich area of study that has produced models based on Brownian motion, fBm, Lévy motion, and fractional Lévy motion. Assuming zero mean, the variogram for a stationary increment model takes the form

display math(7)

[13] Many varieties of processes with nonstationary increments can be constructed, and the argument in the previous section demonstrates the need for more effort in this direction.

[14] Models with stationary increments can be used as building blocks for models with nonstationary increments. Suppose that math formula is a random field with stationary increments (such as those just mentioned), and math formula is a deterministic function that transforms the spatial coordinate, x. That is, math formula is a function that maps math formula into math formula. These two components can be combined to define a new random field,

display math(8)

that has nonstationary increments. A more detailed discussion of these processes can be found in O'Malley et al. [2012]. This approach has been employed in the context of diffusion and dispersion to introduce nonstationarity [O'Malley and Cushman, 2010].

[15] If math formula is normally distributed, math formula will be normally distributed as well. If the process math formula has mean zero and covariance function math formula, then math formula will also have mean zero, but the covariance function is given by math formula. This follows from the fact that

display math(9)

and

display math(10)

[16] It can be shown similarly that the variogram for math formula, math formula, can be written in terms of the variogram for math formula,

display math(11)

[17] An important point to observe is that if math formula is nonlinear then math formula cannot be expected to take the form in equation (7) even if (or, especially if) math formula does take that form.

[18] Random fields such as math formula with math formula being a process with stationary increments and math formula being nonlinear provide a powerfully descriptive approach to studying processes with nonstationary increments for two reasons. First, the choice of the base process, math formula, allows for many different types of statistics (e.g., heavy tailed or not and skewed or not). Second, the choice of the nonlinear function allows for many different types of nonstationarity (e.g., power laws, exponentials, and polar coordinates). For these reasons, these types of processes will be used as geostatistical models for the kriging process in the absence of stationary increments.

3. Kriging With Nonstationary Increments

[19] If either nonstationarity of the field values or the increments arise, the variogram in equation (7) does not provide sufficient information to enable proper kriging. In the case of nonstationary field values, information about the point variance is needed. In the case of nonstationary increments, it does not even make sense to write equation (7). Therefore, in the presence of either of these types of nonstationarity, something more informative than the variogram is needed.

[20] Guttorp and Sampson [1994] review a number of methods for estimating heterogeneous spatial covariance functions in the context of spatiotemporal stochastic processes. Their approach [Sampson and Guttorp, 1992] assumes temporal stationarity and relies on temporal averages. The approach described here is more appropriate when temporal variability is not present in the data. Stein [2005] provides a review of some analytical approaches to constructing covariance functions for processes with nonstationary variations that builds on the work of Pintore and Holmes [2004], Paciorek [2003], and Higdon [1998] and includes some practical advice for applying these covariance functions. Haskard and Lark [2009] build upon the work of Pintore and Holmes [2004] in dealing with nonstationary covariance by tempering an empirical spectrum. A review of numerous methods for estimating covariance functions for nonstationary spatial processes is provided by Sampson [2010].

[21] Suppose that there is a field, math formula, of random variables that has a constant, known mean and the values of this field are known at locations math formula. The increments and the variation about the mean of the field are not assumed to be stationary. The goal is to determine an estimate, math formula, for math formula at some point math formula. Subtracting this mean from each value in the field results in a field with zero mean. By kriging on this modified field and adding the mean on afterward, we may assume without loss of generality that the mean is zero. A set of candidate covariance functions, math formula, is devised where math formula is a vector of length ni containing the parameters for the math formula covariance function and math formula. The choice of M and the covariances is specified at run time by the user.

[22] For each of the candidate models, a maximum likelihood set of parameters, math formula, is determined. In order to determine these parameters a likelihood function is needed, and for this purpose the field is assumed to obey a multivariate normal distribution with zero mean and covariance math formula. Although the assumption of normality is made here to compute the likelihood functions, the same procedure can be used for other distributions provided that the joint probability density function can be computed. A constrained optimization procedure is used to determine the maximum likelihood parameters. This procedure requires an initial set of parameters and a range of parameters to be specified at run time.

[23] Upon determining the maximum likelihood parameters for each candidate covariance function, the Akaike information criteria (AIC) and the “corrected” AIC (AICc) [Burnham and Anderson, 2002] are used to determine weightings for models associated with each of the covariance functions,

display math(12)

where Ai is either the AIC or AICc for the ith model, and

display math(13)

[24] The AIC and AICc were chosen because they are simpler than some other information criteria such as the Bayesian and Kashyap information criteria, and they have proven to be effective for our purposes. These weightings are based on the probability that model i minimizes the information loss. See Ye et al. [2008] for a review of model selection criteria in a hydrological context. Two different approaches can be used to determine math formula as well as metaparameters. One option is to krige using the covariance function math formula that corresponds to the minimum Ai. The other is to use the weights to average the kriging estimates based on each of the covariance functions. Simple kriging is used because it employs the covariance function rather than the variogram [Kitanidis, 1997]. A presentation of kriging techniques can be found in Chilès and Delfiner [2012] or Kitanidis [1997].

3.1. Results

[25] Two types of random fields were chosen to measure the effectiveness of the methodology. One has stationary increments, the other does not, and both are functions of two spatial variables. These choices were made to answer two basic questions: Can this approach effectively distinguish between models that are stationary and models that are nonstationary based on a relatively small sample from a single realization of the random field? Does weighting the kriging results from two models (one with stationary increments, the other with nonstationary increments) via the model selection criteria result in better estimates than using only one of the models? As will be seen, the answer to the first question is unequivocally in the affirmative. The answer to the second question is clearly in the affirmative for estimating the mean, but somewhat ambiguous for estimating the kriging error.

[26] The field with stationary increments is a fBm. As an example, the Hurst exponent, H, was chosen to be 0.4, and the scaling coefficient, σ2, was chosen to be 1. When searching for the maximum likelihood set of parameters, the initial point was chosen so that the Hurst exponent was 0.5 and the scaling coefficient was 1. The lower and upper bounds for the Hurst exponent were chosen to be 0.1 and 0.9, respectively. The lower and upper bounds for the scaling coefficient were chosen to be 0.1 and 10, respectively.

[27] The field with nonstationary increments is a fractional Brownian field with a nonlinear spatial transformation that takes the form of equation (8) with math formula being a fBm and

display math(14)

and R is a rotation matrix that rotates the plane through the angle θ. This makes the coordinate system obey power laws in two perpendicular directions that need not be aligned with the coordinate axes. For this reason, we refer to such a field as fBm with power-law space (fBm-pls). As an example, the parameters were chosen to be 0.6 for the Hurst exponent (H), 1 for the scaling coefficient math formula, 1.5 for math formula, 0.5 for math formula, 10 for x0, 25 for y0, and math formula for θ. The parameters H, math formula, math formula, math formula, x0, y0, and θ were assumed to be contained in the intervals [0.1, 0.9], [0.1, 10], [0.25, 2], [0.25, 2], [0, 100], [0, 100], and math formula, respectively.

[28] For both the fields with stationary and nonstationary increments, 10,000 realizations were tested using a combination of the Cholesky decomposition and random midpoint displacement methods [O'Malley et al., 2012]. In each realization, the field values were sampled at the 25 points given by math formula. Kriging was used to estimate the mean and variance at the point (2.5,2.5). See Figure 1 for a pictorial representation. The model selection criteria were used to determine the most appropriate model based on each individual realization. The known field values were limited to 25 points for two reasons. One is that it is important that this approach is able to determine the correct model based only on a modest data set from a single field. The other is more practical. Computing the likelihood function requires math formula operations where N is the number of points at which the field value is known. Many likelihood function evaluations are required to determine the maximum likelihood set of parameters, and therefore, this procedure can have significant computational requirements if the field values are known at many points. In practice, this is not a significant limitation, because the maximum likelihood set of parameters for a given field need only be determined once. However, for the present purposes, the maximum likelihood parameters must be determined for a large number of random fields in order to compute accurate statistics. For each random field, two covariance models were considered as candidate models: the covariance for fBm and the covariance for fBm-pls. In practice, it would be prudent to use a greater number of points to improve the accuracy of the parameter estimation and to deal with a greater number of covariance functions.

Figure 1.

A realization of a fractional Brownian field with a power-law transformation with crosses denoting the locations where the field value is known and circles denoting the position at which kriging was tested.

[29] When the field was an fBm, both model selection criteria performed well in selecting this as the model preferred over fBm-pls. The AIC selected fBm over fBm-pls 99.44% of the time, and the AICc selected fBm 100% of the time over 10,000 realizations. This reliability is very good considering that only 25 points were used in each case. In estimating the mean and variance via kriging, the fBm covariance outperformed the fBm-pls covariance model, and the weightings determined by both model selection criteria tracked the performance of the fBm model closely. Table 1 summarizes the results.

Table 1. Performance for fBm Modela
 AICAICcMeanVariance
  1. a

    The first two columns give the number of times out of 10,000 that each covariance model was chosen when the underlying field is a 2-D fBm based on AIC and AICc. The latter two columns give the mean-square error for the mean and absolute value of the estimated kriging error both relative to the kriging error based on the true covariance model.

fBm994410,0000.02270.3168
fBm-pls5600.04430.3359
AIC-weightedN/AN/A0.02290.3170
AICc-weightedN/AN/A0.02270.3166

[30] When the field was an fBm-pls, both model selection criteria performed well in selecting this as the model preferred over fBm. The AIC selected fBm-pls over fBm 100% of the time, and the AICc selected fBm-pls 100% of the time over 10,000 realizations. Once again, this reliability is very good considering that only 25 points were used in each case. In estimating the mean, the fBm-pls covariance outperformed the fBm covariance model but was outperformed by the fBm model in estimating the variance. The weightings determined by both model selection criteria tracked the performance of the fBm-pls model closely. Table 2 summarizes the results.

Table 2. Performance for fBm-pls Modela
 AICAICcMeanVariance
  1. a

    The first two columns give the number of times out of 10,000 that each covariance model was chosen when the underlying field is a 2-D fBm with a power-law spatial transformation based on AIC and AICc. The latter two columns give the mean-square error for the mean and absolute value of the estimated kriging error both relative to the kriging error based on the true covariance model.

fBm000.04360.5681
fBm-pls10,00010,0000.01460.6249
AIC-weightedN/AN/A0.01460.6249
AICc-weightedN/AN/A0.01460.6249

4. Conclusion

[31] The fact that processes with stationary increments are nowhere dense in the set of stochastic processes provides a rigorous mathematical backing to the notion that the occurrence of such a process in nature is an exceptional event. On the other hand, the fact that processes with nonstationary increments are dense everywhere implies that these processes should be expected to occur with great frequency.

[32] There are innumerable ways to devise a process with nonstationary increments, but one particularly simple method is to apply a nonlinear spatial transformation to a process that does have stationary increments. This approach allows for a wide variety of field statistics and nonstationary behaviors.

[33] Kriging on processes with nonstationary increments requires a more careful approach than kriging on processes with stationary increments, because a translation-invariant variogram does not exist in the former case. The approach to kriging these processes presented here relies on maximum likelihood methods and model selection criteria to determine the covariance structure and simple kriging to estimate field values. This approach proved effective at differentiating between stationary and nonstationary increments in the fields tested and estimating the mean field values.

Acknowledgments

[34] The authors wish to thank the National Science Foundation for supporting this work under contracts CMG-0934806 and EAR-0838224.