Approach and methodology

As mentioned above, spatially distributed data values at different locations are frequently not completely random, but usually exhibit some form of spatial correlation structure, where close points in space tend to be highly correlated. A phenomenon spreads in space and ex-hibits a certain spatial correlation structure is called as term ”regionalized”. Iff(x)is the value at the pointx of a characteristicf of this phenomenon then f(x) is called ”region-alized variable”. Geostatistics can be defined as ”the application of probabilistic methods to regionalized variables” which designates a function displayed in real space in order to exhibit a structure of spatial correlation (Matheron,1971).

In the context of geostatistics, the regionalized variables are assumed to be a single realiza-tion of a random variable, which is characterized by its distriburealiza-tion funcrealiza-tion. The distribu-tion funcdistribu-tion that completely describes a spatially distributed variable can thus be defined by assigning ann-dimensional probability distributionF_nto the set of points in the domain of interest. Using only one realization for counting frequencies to find a multivariate prob-ability distributionF that fits to the data is not possible without an additional assumption about the spatial process, that is the spatial strong stationarity (Guthke,2013).

Under the assumption of the spatial strong stationarity, the hypothesis that the random function is dependent on the location is relaxed towards a dependence of the spatial config-uration of the regionalized variables. In other words, the random function remains the same if a certain configuration of points is only shifted by a length of separating distanceh(this also called translation invariant). The bivariate distribution function can now be inferred by counting frequencies of values in the same configuration (Guthke,2013) and the spatial correlation model finally can be estimated.

The spatial correlations are calculated under the isotropic assumption. The isotropic as-sumption means that the spatial correlation structure is the same for all directions. The isotropic spatial correlation depends only on the length of the separating distance between two points and not on their relative directions. The isotropic assumption is commonly made in any practical spatial application. This is because the interpretation of the correlation be-havior becomes simple and the estimation process of the correlation coefficient is also easy (Maity and Sherman,2012).

The spatial correlations derived in the context of spatial geostatistics require regionalization in order to construct an empirical bivariate distribution. The spatial Spearman’s rank corre-lations are calculated based on underlying bivariate normal distribution on the unit domain U(0,1). This censored bivariate Gaussian copula is fitted to the empirical bivariate copulas.

The empirical bivariate copulas for the spatial precipitation observationszat any given loca-tionsofngaugesz(s1), ..., z(sn)can be constructed using the procedures given byB´ardossy (2006). The procedure is conducted for each selected time interval of precipitation values

separately due to the assumption of a single realization of a spatial random variable as fol-lows.

1. Precipitation data from all spatial observation points at any given time step are trans-formed into the uniform distributionu(0,1)using the empirical marginal distribution functionF_z(z(s_i))or the normalized ranks from the observation points as follows.

u= n−R(z(si))−¹₂

n (5.5)

where z(s_i) is a univariate random variable at location s_i, n is the number of data points,R(z(Si))is the rank ofz(si)in the setz(s1), ..., z(sn),zs1 is the maximum value, andz_sn is the minimum value.

2. For any given length of the separating distanceh, all gauge stations are grouped with similar lagsh and the set of pairsS(h) of gauge stations can be created. The set of pairs S(h) consists of the empirical distribution function values of the precipitation variables of any pair at locations separated by the length of distanceh. For the isotropic assumption, the length of distance h does not depend on any direction, no matter whether the two gauges at separating length of distanceh are located in the north-south direction, the west-east direction, or any other directions. The set of pairsS(h) are formulated as follows.

S(h) =c

Fz z(s)

, Fz z(s+h)

(5.6) Again here,S(h)is a set of points in the unit square. Note thatS(h)is by definition symmetrical regarding the major axisu₁ =u₂of the unit square, namely, if(u₁, u₂) ∈ S(h), then(u₂, u₁)∈S(h).

3. From the set of pairsS(h), the spatial copula functionC_Sis implemented as a function of the length of distancehextended from Equation4.4. For any two selected quantiles u1andu2, Equation4.4becomes as follows.

C_s(u₁, u₂;h) =P rob

F_z Z(s)

≤u₁, F_z Z(s+h)

≤u₁

=C

u₁ =F_z Z(s)

, u₂ =F_z Z(s+h) (5.7) Here,F_zis the marginal distribution of the variableZand is supposed to be the same for each locations. Any two locations separated by the length of distancehare related to each other by the bivariate distribution function whose dependence is described by the bivariate copula. As with variograms, it is assumed that this copula does not depend on the locationxbut only on the separating distanceh.

4. The unit square is divided into k x k grids. The empirical copula density can then be constructed by calculating the empirical frequencies on each grid resulting in two dimensional histograms of the marginalsu1andu2.

50 5.3. GEOSTATISTICAL CONTEXT Since the empirical bivariate copula density was calculated, the Spearman’s rank correla-tion coefficient can be estimated via fitting of the censored bivariate Gaussian copulas to the empirical bivariate copulas by maximizing numerically the likelihood function as given by Equation5.4. The fitted parameter of the censored Gaussian copulas is Spearman’s rank cor-relation. The detailed procedure of estimating Spearman’s rank correlation can be described as follows.

1. The uniformly distributed valuesu1 andu2 are transformed into the normal domain using the inverse of the standard normal distribution.

2. The likelihood function of the censored bivariate normal distribution, as shown in Equation5.4, is maximized numerically to get the parameterθ.

5.3.1.2 Anisotropic assumption

The assumption of isotropy is frequently applied to describe the spatial correlation struc-ture because the interpretation of the correlation behavior is simple and the process of esti-mation is straightforward as mentioned. The assumption of isotropy, however, can give a bias estimation when not appropriate to the data. If the spatial correlation structure is not isotropic, then the anisotropic assumption is suggested. The anisotropic assumption means that the spatial correlation between any two observations depends not only on the distances between those sites but also on their relative directions (Maity and Sherman, 2012), thus different spatial correlation structures are calculated for the different directions.

In general, some procedures of estimating of the spatial correlation function under the anisotropic assumption in the case of geostatistical approach is similar to the isotropic as-sumption above. For example, for each selected time interval, the set of pairsS(h)which consists of the empirical distribution function values of the interested variables at differ-ent locations separated by the length of distance vectorhshould be determined. However, unlike the isotropic assumption where all directions are considered to be the same for esti-mating the spatial correlation function, the anisotropic approach in this study proposes four different orientations that can be taken into account for estimating the spatial correlation function, for example, 0^o, 45^o, 90^o and 135^o from the North direction with a tolerance of 22.5^o. By definition, the spatial correlations at those directions are the same as the directions 180^o, 225^o,270^o, and315^o from the direction of north, respectively, due to the symmetrical condition. For each selected time interval, a different correlation function is estimated for each direction. In principle, for each direction, this anisotropic approach employs the same procedure for estimating the spatial correlation function as the isotropic approach above.

The available number of pairs of gauge stations at any given distance vectorh is a crucial issue for the geostatistical approach since the inferential processes for this approach use the data sets collected from those pairs. Just for example in the region of Singapore, all possible combinations of gauge pairs for the directions of0^o,45^o,90^o and135^o or directions of180^o, 225^o, 270^o, and 315^o from the directions of North are shown in Figure 5.2a, and the total number of all possible pairs for each direction at any given distance vectorhis presented in Figure5.2b.

The configuration and number of pairs in each direction at any given distance vectorhcan be different from the isotropic approach. For example, in the direction of0^ofrom the North, the total number of pairs within the distance groups of 6 km, 10 km, 15 km, and 20 km is 70, 117, 143, and 90, respectively. Those numbers are sufficient to be used for estimating the spatial correlation functions. For the directions of45^oand90^o, the total number of pairs within the distance classes of 6 km, 10 km, 15 km, and 20 km is 13, 27, 28, 16 for the direction of 45^o, respectively, and 16, 26, 44, and 35 for the direction of90^o, respectively. Then, the total number of possible pairs for the direction of135^owithin inter-gauge distances of 6 km, 10 km, 15 km, and 20 km is 15, 28, 29, 21, respectively. The total number of all possible gauge pairs at certain directions in Singapore might not be sufficient for investigating the correlation behavior accurately. Nevertheless, this is an alternative to the isotropic approach for the comparison of correlation behaviors.

(a) All possible combinations of pairs (b) Number of all possible combinations of pairs

Figure 5.2: All possible combinations of gauge pairs with different orientations, namely,0^o, 45^o,90^oand135^ofrom the direction of North in the Singapore region.