Modelling spatio-temporal data: A new variogram and covariance structure proposal

ARTICLE IN PRESS

0167-7152/$ - se

doi:10.1016/j.sp

$Work part�CorrespondE-mail addr

Statistics & Probability Letters 77 (2007) 83–89

www.elsevier.com/locate/stapro

Modelling spatio-temporal data: A new variogram andcovariance structure proposal$

E. Porcua, J. Mateua,�, A. Zinib, R. Pinib

aDepartment of Mathematics, Universitat Jaume I, Campus Riu Sec, E-12071 Castellon, SpainbDipartimento di Metodi Quantitativi per le Scienze Economiche ed Aziendali, Universita di Milano Bicocca, Piazza dell’Ateneo Nuovo,

1, I-20126 Milano, Italy

Received 2 August 2005; accepted 23 May 2006

Available online 7 July 2006

Abstract

We adapt the Dagum survival function to become a function of space and time and study its theoretical properties as a

covariance in the isotropic case. The resulting Dagum class is proved to have certain interesting mathematical properties

and shows smooth behaviour at the origin, which has considerable applicability.

A simple extension to the spatio-temporal case is provided and interesting points of comparison arise with other models

appearing in literature.

r 2006 Elsevier B.V. All rights reserved.

MSC: 60G10; 60E05

Keywords: Criteria of Polya type; Dagum survival function; Nonseparability; Positive-definite radial functions

1. Introduction

The Dagum distribution (Dagum, 1977, 1980, 1983) has been successfully used in economic studies. Aparametric model of the Dagum type has been applied to contribute to the study of income policy effects onthe distribution of personal income. The model is particularly interesting because it shows the associationbetween the probability density function of a population of individuals in the labour force, with certainparameters representing the level of unemployment and other economic conditions. In fact, unemploymentmay operate over the distribution of income through channels that are more complex and indirect thanpreviously thought. The distribution of income may indeed deteriorate or improve as a result of a number offactors. For example, a deterioration is said to exist when the relative distance between the first and the lastdecile of individuals in the labour force widens, irrespective of movements in the unemployment rate.Similarly, when the distribution moves as a whole to the left, the mean income of each decile decreases (as does

e front matter r 2006 Elsevier B.V. All rights reserved.

l.2006.05.013

ially funded by Grant MTM2004-06231 from the Spanish Ministry of Science and Culture.

ing author. Tel.: +34964 728391; fax: +34 964 728429.

ess: [email protected] (J. Mateu).

ARTICLE IN PRESSE. Porcu et al. / Statistics & Probability Letters 77 (2007) 83–8984

the total distribution mean) and this type of movement is also interpreted as a deterioration, that might not benecessarily associated with changes in the level of unemployment.

However, the Dagum model can be seen and explored as a valid class for spatio-temporal modellingproviding a rich family of variogram functions for data evolving in space and time. And this is the key pointand main concern in this paper. We explore the characteristics of the Dagum survival function as a radialfunction and obtain sufficient conditions so that the positive definiteness condition is satisfied. The techniquesused to show this result come from Gneiting’s (2001) criteria of Polya type. As far as positive definite radialfunctions are concerned, extensive studies can be found in Polya (1949) and Sasvari (1994). Other interestingresults can be found in Askey (1973), Mittal (1976) and Zastavnyi (2000).

Directly related to our work and key references are Gneiting and Schlather (2004) for the so-called Cauchyclass, having an interesting liaison with our model, Gneiting et al. (2001) for the relationship betweenvariogram and covariance functions and Ehm et al. (2004) for radial positive definite functions with compactsupport.

As far as the spatio-temporal literature related with the work reported here, we can name De Iaco et al.(2001), Ma (2003b), Gneiting (2002) or Stein (1999, 2005). Particularly, analogies and differences between thefirst two authors’ models and the models here proposed will be highlighted.

In this paper we build new spatial-isotropic covariance and variogram functions, analysing theirmathematical properties. We next consider two very simple extensions to the spatio-temporal case, obtaininga separable and a nonseparable structure presenting interesting features. We conclude the paper with someconclusions and discussions.

2. A new spatio-temporal covariance and variogram

2.1. Background and set-up

Before introducing the new class of spatial and spatio-temporal covariances and variograms, let us beginwith some introductory geostatistical concepts. For the sake of conciseness, we reduce our discussion to thecase of spatial weakly stationary-isotropic standard Gaussian random fields, ZðxÞ, x 2 Rd , such thatE½ZðxÞ� ¼ 0, E½Z2ðxÞ� ¼ 1 and having autocovariance function (Stein, 1999, p. 17) E½ZðxÞZðxþ hÞ� ¼ CðkhkÞ

which does not depend on x but only on the Euclidean distance k:k between two points of the domain, where his the so-called separation vector. A random field having these characteristics is often called weakly isotropic,as the strong isotropy refers to invariance over all rigid motions of the finite-dimensional joint distribution ofthe random field. Recall that covariance functions fulfil the statements of Bochner’s theorem, i.e. they must bepositive definite. See complete details of positive definite functions in Sasvari (1994). For positive definiteradial functions, a formal representation can be obtained through the Hankel transforms, i.e. integralrepresentations of ordinary Bessel functions (Abramowitz and Stegun, 1965). For an extensive treatment ofpositive definiteness, see the leading work of Sasvari (1994). For a simpler introduction, we refer to Cressie(1993) or Chiles and Delfiner (1999).

Another important tool which shall be used in this paper is the variogram, gð:Þ, which represents one half thevariance of the increments of an intrinsically stationary random field. The term intrinsic was coined byMatheron (1971, 1973) in its tour de force regarding random fields whose associated spectral density is notintegrable in a neighbourhood of the origin. In his work, Matheron also showed that the variogram is aconditionally negative definite function. Weak stationarity implies the intrinsic one, so that in the former casethe covariance and the variogram of the random field exist with an explicit relation between them.Particularly, if a standard Gaussian random field is weakly stationary-isotropic and the variogram is bounded,then it is easy to show that gðkhkÞ ¼ 1� CðkhkÞ. For a simple introduction about intrinsically stationaryrandom fields, see Stein (1999, pp. 36–39).

The extension of the aforementioned concepts to standard Gaussian weakly stationary continuous spatial-isotropic temporal-isotropic random fields is straightforward as far as covariance and variogram areconcerned. From now on, the isotropic spatio-temporal covariance will be indicated with the notationCðkhk; jujÞ, with u the temporal lag and h as indicated above, where j:j denotes the Euclidean distance in R.

ARTICLE IN PRESSE. Porcu et al. / Statistics & Probability Letters 77 (2007) 83–89 85

By analogical extension, the isotropic spatio-temporal variogram will be indicated with the compact notationgðkhk; jujÞ. All the previously stated properties and relations are preserved in the spatio-temporal domain.

2.2. The Dagum spatio-temporal covariance and variogram

In this section we consider the Dagum survival function (Zenga and Zini, 2001), denoted here by cð:Þ, andwith equation

cðtÞ ¼

1 if t ¼ 0;

1�1

ð1þ lt�yÞe

� �if t40;

8><>: (1)

where the parameters l; y; e are strictly positive. Here, e and y act as smoothing parameters and l as a scalingparameter.

Our aim is to study the function in (1) as a radial function to analyse if there is a feasible parameters rangeallowing cð:Þ to be positive definite on Rd , with d ¼ 1; 2; . . . : For this purpose, the following result is crucial.

Proposition 1. Consider the function h 7!cðkhkÞ where cð:Þ is defined as in (1). If yoð7� eÞ=ð1þ 5eÞ and eo7,then cðkhkÞ is positive definite (indeed a permissible covariance function) in R3.

Proof. The proof of this sufficient condition can be obtained applying Polya (1949) theorem for radialfunctions and the corresponding extension to Rd , which can be found in Gneiting (2001). Define

Z1ðtÞ ¼ �d

du

� �k

cðu1=2Þju¼t2 , (2)

for t40 and k a nonnegative integer. Due to Gneiting’s theorem, if there exist aX 12so that

Z2ðtÞ ¼ �d

dt

� �kþl�1

½�Z01ðtaÞ� (3)

is convex for t40 and l a nonnegative integer, then c is positive definite on Rd , d ¼ 1; . . . ; 2l þ 1.Imposing k ¼ 0; l ¼ 1; a ¼ 1, we find that if

�Z0

1ðtÞ ¼ ½ð1þ lt�yÞ�e�1elt�y�1y� (4)

is convex, then (1) is valid on R3.To show this, we take the second derivative of (4), which has the following expression:

lyet�3

ð1þ lt�yÞ�e�3l2t�3yy2e2 þ 3ð1þ lt�yÞ�e�3l2t�3yy2e

þ2ð1þ lt�yÞ�e�3l2t�3yy2 � 3ð1þ lt�yÞ�e�2lt�2yy2e

�3ð1þ lt�yÞ�e�2lt�2yy2 � 3ð1þ lt�yÞ�e�2lt�2yye

�3ð1þ lt�yÞ�e�2lt�2yyþ ð1þ lt�yÞ�e�1t�yy2

þ3ð1þ lt�yÞ�e�1t�yyþ 2ð1þ lt�yÞ�e�1t�y

0BBBBBB@

1CCCCCCA. (5)

Collecting the factor y�ð1þ t�yÞ���3t�3y from the terms in parenthesis, we easily get that (5) can be writtenas follows (we choose l ¼ 1 for the sake of simplicity, since it does not affect the computations):

y2e2ð1þ t�yÞ���3t�3y�3ðAðy; �Þt2y þ Bðy; �Þty þ Cðy; �ÞÞ,

where

Aðy; �Þ ¼ y2 þ 3yþ 2,

Bðy; �Þ ¼ � 3y2�� 3y�� y2 þ 3yþ 4,

Cðy; �Þ ¼ y2�2 � 3y�þ 2.

ARTICLE IN PRESSE. Porcu et al. / Statistics & Probability Letters 77 (2007) 83–8986

In particular, the expression

Aðy; �Þt2y þ Bðy; �Þty þ Cðy; �Þ

is a second degree polynomial with respect to ty; a sufficient condition provided that the sign is positive is that

B2ðy; �Þ � 4Aðy; �ÞCðy; �Þo0.

Easy computations show that this happens if

7� y� e� 5ye40,

so that

yo7� e1þ 5e

. (6)

Taking now into account that y must be positive, we trivially get the corresponding condition on e, and thusthe proof is completed. &

It is interesting to observe that from Lemma (1.10.16) of Sasvari (1994) it can be deduced that the Dagumclass is not valid if y42=e. However, it can be trivially verified that this condition is compatible with thecondition here obtained. Another interesting point arises from the fact that the Dagum class coincides with theso-called Cauchy class (Gneiting and Schlather, 2004) if we set e ¼ 1. Any other parameter setting impliesdifferent characteristics and behaviours of the two classes. Finally, observe that these results do not depend onthe spatial dimension d.

Note that in Proposition 1, we obtain a sufficient condition for the range of parameters by which cðkhkÞ is avalid covariance on R3. The validity on three-dimensional spaces implies validity on less-dimensional ones.Thus, from Proposition 1 we can obtain a spatial covariance CðhÞ ¼ cðkhkÞ, for h 2 Rd , d ¼ 1; 2; 3, and atemporal covariance CðuÞ ¼ cðjujÞ, u 2 R. Besides, as we are dealing with standard Gaussian weaklystationary-isotropic random fields, it is possible to obtain the corresponding spatial and temporal variograms,with equations gðhÞ ¼ 1� cðkhkÞ and gðuÞ ¼ 1� cðjujÞ, u 2 R, respectively.

The result in Proposition 1 does not exclude that there exists a bigger range of parameters implying validityof the covariance. Direct calculation of the Hankel transform is ‘‘often impossible’’ (Gneiting, 2001), so wemust recur to a sufficient condition. As the variogram class is wider than the covariance one, the existenceof a larger parameters range by which the validity condition is satisfied cannot be excluded. However,as this fact cannot be proved, from now on we proceed by considering only the parameters range obtained inProposition 1.

Proposition 2. The so-obtained covariance function CðhÞ ¼ cðkhkÞ, with cð:Þ as defined in (1), is L1- and L2-integrable, respectively, under the conditions y4d and y4d=2 for d ¼ 1; 2; 3, where d is the dimension of the

spatial domain.

Proof. The proof is trivial. It is sufficient to notice that cð:Þ is continuous in every ball centred at the originwith radius R, say BRð0Þ. Thus, the integral over BRð0Þ is finite. Integrating in polar coordinates and usingwell-known integrability criteria, one gets easily the result. &

The limited range of parameter for the permissibility of the Dagum covariance function reflects in some wayin the level of smoothing of the covariance. For this reason, it would be desirable to obtain a smoothercovariance starting from the Dagum one. Next result shows how we can do this.

Proposition 3. Under the same constraints in the range of parameters y and e as in Proposition 1, the function

c�ðtÞ ¼

1 if t ¼ 0;

1

1þ ð1þ lt�yÞ�e

� �if t40

8><>: (7)

is a permissible covariance function in Rd , d ¼ 1; 2; 3.

ARTICLE IN PRESSE. Porcu et al. / Statistics & Probability Letters 77 (2007) 83–89 87

Proof. Consider the completely monotone function jðtÞ ¼ ð1þ tÞ�1. Observe that (7) can be seen as acomposition of the type j � g, where gðtÞ ¼ 1� cðtÞ, the Dagum variogram obtained as complementary to oneof the Dagum covariance function, as in Eq. (1)—note that the Dagum covariance has unitary sill. Thus, itsuffices to show that the composition j � g is a covariance function, as it is indeed. To see this, recall that dueto Bernstein theorem (see Gneiting, 2002, and references therein), a function j is completely monotone if andonly if it admits the representation

jðtÞ ¼Z 10

e�rt dF ðrÞ, (8)

with F a nondecreasing and bounded measure. So, the composition j � g is of the form

jðgðtÞÞ ¼Z 10

e�rgðtÞ dF ðrÞ. (9)

Now, remind that for Schoemberg (1938) theorem, expð�gÞ is a covariance function, and (9) is a positivemixture of covariance functions, indeed a covariance function. And so the proof is completed. &

The covariance function in (7) decays to zero less rapidly than the Dagum covariance in (1), ensuring ahigher level of smoothing away from the origin. In order to obtain spatio-temporal covariance and variogramstructures, we consider the following two alternatives:

�

C1ðh; uÞ ¼ cðkhkÞcðjujÞ ¼ 1� gðhÞ � gðuÞ þ gðhÞgðuÞ. (10)

�

C2ðh; uÞ ¼ WcðkhkÞ þ ð1� WÞcðjujÞ, (11)

where W 2 ½0; 1�.

Both (10) and (11) are valid spatio-temporal covariance functions (under the restriction on the parameters asin Proposition 1), for the well-known properties of convexity and stability (Chiles and Delfiner, 1999). Due tothe assumption of weak stationarity, the respective associated variograms

g1ðh; uÞ ¼ gðhÞ þ gðuÞ � gðhÞgðuÞ, (12)

g2ðh; uÞ ¼ WðgðhÞ � gðuÞÞ þ gðuÞ (13)

are still valid structures on space and time, under the restriction on the parameter values, for the permissibilityof the covariance.

The so-obtained permissibility condition imposes some further analysis and comments. Observe that theparameters y and e can vary in the same range, i.e. the open interval ð0; 7Þ, but they are inversely correlated. Ingeneral, for eo1, we get higher values of y. The opposite happens for e41.

It is worth remarking that the integrability on L1 is not compulsory for the permissibility of the covariancefunction. If we wanted to match integrability and validity conditions, the parameters range would bedramatically restricted, above all if working on three-dimensional spaces. In fact, it is easy to see that theranges for y would be ð3; 7Þ, ð2; 7Þ and ð1; 7Þ, and for parameter e, the condition in Proposition 1 would becomeeo 1

4, eo 5

11and eo1, respectively, for R3, R2 and R1.

It is interesting to note that the spatial variogram gðhÞ ¼ 1� cðkhkÞ never reaches the sill, but its practicalrange (i.e. the quantile of order p, 0ppp1) can be easily calculated from the expression

xp ¼p�1=e � 1

l

� ��1=y. (14)

Some comments about the spatio-temporal structures proposed in Eqs. (10) and (11) follow. The separablestructure seems to be very flexible, but does not take into account the interaction between the spatial and

ARTICLE IN PRESSE. Porcu et al. / Statistics & Probability Letters 77 (2007) 83–8988

temporal components, while the nonseparable one seems to be smoother and allows to take into account theinteraction between the spatial and temporal components through the parameter W. Observe that this structureis somehow similar to the so-called product– sum model of De Cesare et al. (2001), even if they do not imposeany restriction on the parameters of the linear combination (they only need to be positive). On the other hand,Ma (2003a) proposed a linear combination of type (11), imposing a bigger range for W, which can be negativeunder some conditions. In this case Ma (2003a) theorem cannot be applied, as it is easy to show that (1) is notcompletely monotone. The nondifferentiability of our spatio-temporal structure is not surprising, as we areworking with linear combinations. But, neither are the structures proposed by De Iaco et al. (2001) and Ma(2003a).

It is trivial to show that the covariance function in (10) is L1- and L2-integrable under the same conditionsconsidered in remark (2). Observe that the integrability condition is not satisfied for (11). In fact a functiondefined on Rd is not integrable on Rd

�R. Thus, the same conclusions can be achieved for the product– sum

model of De Cesare et al. (2001).

3. Conclusions and discussion

Modern literature emphasizes the need for new contributions for spatial and spatio-temporal covarianceand variogram models. In this work, we have proposed an original approach, considering a survivalprobability function and studying its properties as a covariance.

Some remarks are necessary. It is very difficult, often impossible, to show directly the validity of an isotropicvariogram. The use of Hankel transforms requires integration with respect to Bessel functions; this kind ofintegrals can sometimes be solved through computer intensive calculation, but very often it is not possible toobtain a result (and this was our case). So, it maybe preferable to avoid the problem of direct calculus of aHankel transform and use some known properties which can guarantee the validity of the covariance or thevariogram.

Nevertheless, the efforts put in this paper allow to have a new valid structure for space and space–time.Besides, these kind of approaches do not respect the separate literature’s development for space and timemodelling. So, we tried to avoid the unpleasant features induced by using exponential covariance functions.

The class we propose does not represent a subset of Gneiting class, as it is obtained with a different criteriaand with a function which is not completely monotone. As previously said, the only comparison can be carriedout with respect to the product– sum model and to the Ma model, where the differences have been alreadyemphasized.

References

Abramowitz, M., Stegun, I.A., 1965. Handbook of Mathematical Functions. Dover, New York.

Askey, R., 1973. Radial Characteristic Functions. Mathematics Research Center, University of Wisconsin-Madison, p. 1262.

Chiles, J.P., Delfiner, P., 1999. Geostatistics: Modelling Spatial Uncertainty. Wiley, New York.

Cressie, N., 1993. Statistics for Spatial Data, revised version. Wiley, New York.

Dagum, C., 1977. A new model of personal income distribution: specification and estimation. Economie Appl. XXX, 413–437.

Dagum, C., 1980. The generation and distribution of income, the Lorenz curve and Gini ratio. Economie Appl. XXXIII, 327–367.

Dagum, C., 1983. Income distribution models. In: Johnson, N.L., Kotz, S. (Eds.), Encyclopedia of Statistical Sciences, vol. 4, pp. 27–34.

De Cesare, L., Myers, D., Posa, D., 2001. Product–sum covariance for space–time modeling: an environmental application.

Environmetrics 12, 11–23.

De Iaco, S., Myers, D.E., Posa, D., 2001. Space–time analysis using a general product–sum model. Statist. Probab. Lett. 52, 21–28.

Ehm, W., Gneiting, T., Richards, D., 2004. Convolution roots of radial positive definite functions with compact support. Trans. Amer.

Math. Soc. 356, 4655–4685.

Gneiting, T., 2001. Criteria of Polya type for radial positive definite functions. Proc. Amer. Math. Soc. 129, 2309–2318.

Gneiting, T., 2002. Stationary covariance functions for space–time data. J. Amer. Statist. Assoc. 97, 590–600.

Gneiting, T., Schlather, M., 2004. Stochastic models that separate fractal dimension and the Hurst effect. SIAM Rev. 46, 269–282.

Gneiting, T., Sasvari, Z., Schlather, M., 2001. Analogies and correspondences between variogram and covariance functions. Adv. Appl.

Probab. 33, 617–630.

Ma, C., 2003a. Linear combinations of spatio-temporal covariance functions and variograms. Paper presented at METMA Conference in

Granada, Spain.

ARTICLE IN PRESSE. Porcu et al. / Statistics & Probability Letters 77 (2007) 83–89 89

Ma, C., 2003b. Families of spatio-temporal stationary covariance models. J. Statist. Plann. Inference 116, 489–501.

Matheron, G., 1971. The theory of regionalized variables and its applications. Cahiers Centre Morphologie Math. 5.

Matheron, G., 1973. The intrinsic random functions and their applications. Adv. Appl. Probab. 5, 439–468.

Mittal, Y., 1976. A class of isotropic covariance functions. Pacific J. Math. 64, 517–538.

Polya, G., 1949. Remarks on characteristic functions. In: Neyman, J. (Ed.), Proceedings of the Berkeley Symposium on Mathematical

Statistics and Probability. University of California Press, Berkeley, CA, pp. 115–123.

Sasvari, Z., 1994. Positive Definite and Definitizable Functions. Akademie Verlag, Berlin.

Schoemberg, I.J., 1938. Metric spaces and completely monotone functions. Ann. of Math. 39, 811–841.

Stein, M., 1999. Interpolation of Spatial Data. Some Theory of Kriging. Springer, New York.

Stein, M., 2005. Space–time covariance functions. J. Amer. Statist. Assoc. 100, 310–319.

Zastavnyi, V.P., 2000. On positive definiteness of some functions. J. Multivariate Anal. 73, 55–81.

Zenga, M., Zini, A., 2001. A modification of the right tail for heavy-tailed income distributions. Metron 59, 17–25.