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Characterization of Multi-scale Invariant Random Fields

Applying certain flexible geometric sampling of a multi-scale invariant (MSI) field we provide a multi-dimensional multi-selfsimilar field which has a one to one correspondence with such sampled MSI field. This sampling enables us to characterize harmonic-like representation and spectral density function of the sampled MSI field. Imposing Markov property for the MSI field, we find that the covariance function and spectral density matrix of such sampled Markov MSI field are characterized by the covariance functions of samples of the first scale rectangle. We present an example of MSI field as two-dimensional simple fractional Brownian motion. We consider a real data example of the precipitation in some area of Brisbane in Australia for some special period. We show that precipitation on this area has MSI property and estimate time dependent scale and Hurst parameters of this MSI field in three dimension as latitude, longitude and time. Our method enables one to predict precipitation in time and place.


Introduction
Gaussian self-similar fields have been extensively studied and applied in various areas such as hydrology, biology, economics, finance and image processing [28]. Multivariate random fields are mainly interested in spatial statistics as well as in many natural applications, say environmental, agricultural, and ecological sciences. Modeling of multivariate data that have been measured in space, such as temperature and pressure, are challenging tasks in environmental sciences. Genton et al. [6] studied multi-selfsimilar (MSS) fields that have self-similar property and characterized their structure via their stationary counterpart by using some quasi Lamperti transformation. They defined a random field {X(t); t ∈ R d } to be MSS with index H = (H 1 , . . . , H d ) T ∈ R d + (H-MSS) if for any Λ = (λ 1 , . . . , λ d ) T ∈ R d + , where Λt = (λ 1 t 1 , . . . , λ d t d ) T and d = denotes equality of finite-dimensional distributions. If the above equality holds for some fixed Λ then the field is called multi-scale invariant (MSI) with index H and scale vector Λ and is denoted by (H, Λ)-MSI. The family of distribution of such random fields are invariant up to certain dilation factors. Applying some flexible geometric sampling of MSI field, adopted from Modarresi and Rezakhah [17,18], we provide a corresponding multi-dimensional MSS field. Then applying some quasi Lamperti transform, the result of Gladyshev [10] enables us to characterize the spectral density and spectral representation of such sampled MSI field. Imposing Markov property for the MSI field, we show that the covariance function and spectral density of such sampled Markov MSI field are characterized by the covariance function of the first scale rectangles. For the estimation of Hurst parameters of MSI fields we extend the method of Rezakhah et al. [23], [24] for estimation of the Hurst parameter of discrete scale invariant (DSI) processes. This paper is motivated by applications in environmental and climate phenomena. As an example of MSI field, real data of the precipitation in some part of Brisbane area of Australia for some special period of time are considered and the MSI behavior of these precipitation in three dimension as latitude, longitude and time are verified [2]. Also the corresponding time dependent scale and Hurst parameters of this MSI field are estimated. By estimating scale and Hurst parameters, it is possible to predict precipitation in time and place. Section 2 is devoted to some preliminary definitions and also definitions of MSS and MSI fields. Our sampling scheme and a quasi-Lamperti transform are defined in this section. The definition of a two-dimensional sfBm as an example of MSI field is given in section 2. Section 3 is devoted to a harmonic-like representation and spectral density function of n-dimensional MSS fields. The characterization of covariance and spectral density functions of two dimensional scale invariant wide-sense Markov fields are presented in section 4. In section 5 we present some heuristic method for the estimation of Hurst parameters of MSI fields. Implying the rainfall data of Brisbane area which has been presented by Australian bureau of meteorology, their MSI property is verified and also the scale and Hurst parameters of this fields are estimated.

Theoretical Structure
A random field is simply a stochastic process, taking values in a Euclidean space, and defined over a parameter space of dimensionality at least one. In this section, we introduce multiselfsimilar (MSS) and multi-scale invariant (MSI) fields. Then we present quasi-Lamperti transform which provides a one to one correspondence between MSS and stationary fields and also between MSI and periodic fields respectively. First we present the definition of periodic filed and we use them as the Lamperty counterpart of self-similar field to obtain the harmonic representation and spectral density of MSI fileds. According to Hurd et al. [9], we have the following definition. Definition 1. Given τ ∈ R d , the shift operator S τ operates on random field {X(t); t ∈ R d } according to The random field is called to be periodic field with period τ 0 if the above equality holds for some τ = τ 0 .

Definition 2. Given some
We present the following definition for MSS field from Genton et al. [6].
Extending definition of Modarresi et al. [17] for DSI process with some parameter space, we present the following definition.
This field is called wide-sense (H, a )-MSI of index H and scaling factor a = (a 1 , . . . , a d ) T where a i > 0, if the above conditions hold for some a = a .
In the rest of the paper we consider MSS and MSI in the wide-sense fields, so for simplicity we omit the term "in the wide sense" henceforth. Following Modarresi and Rezakhah [17], we present the following the quasi-Lamperti transformation. where It is easy to verify that L H,α L −1 H,α X(t) = X(t) and L −1 Note that in the above definition, if α = (e, . . . , e) T , we have the usual Lamperti transformation L H . Proposition 1. The quasi-Lamperti transform assurances an equivalence between the shift operator S Log α k and the renormalized dilation operator D H,k in the sense that, for any k > 0 Proof. By a similar method as in [17], the validation of (2.5) follows.
So we have the following corollaries.
As an example of MSI field we introduce two-dimensional sfBm by the followings.
where B(., .), I(.) are the two-dimensional fractional Brownian motion indexed by H = (H 1 , H 2 ) T and indicator function respectively and Hence the covariance function of the field is is the two-dimensional Brownian motion.
(ii) The spectral representation of the covariance function is where R j (τ ) is defined by (3.14).
. Hence as the spectral distribution function of Q n (τ ) is D j (dλ), so using (3.11) and (3.12) and the relation of Q H m (τ ) and Q m (τ ) as above, we find by (3.15) that the corresponding spectral density is

Data modeling
In this section we study the precipitation of rainfall data on a region of Brisbane area in Australia for two days (25 and 26 January 2013). This study follows by the evaluation of such precipitation on squares with side length 2km of a grids over a 512 km × 512 km in this region. Thus the precipitation values are considered as a 256 × 256 matrix. The precipitation of rainfall in the area is depicted in Figure 1. The region was affected by extreme rainfall and subsequent flood. This rainfall data is provided by the Australian bureau of meteorology [2]. We consider MSI structure for precipitation in this field and estimate corresponding scales and Hurst parameters along three axes (horizontal, vertical and time) for certain specified area in this region. This circumscription is specified by yellow rectangle in Figure 1 and re-plotted in Figure 2. The selected part is a 120 km × 100 km area which corresponce to 60 × 50 squares with side length 2km. Let X ij be the value of precipitation on ij-th square with vertices in (i, j), (i, j −1), (i−1, j), (i− 1, j − 1) in this area in 25th and 26th January 2013, where i = 1, . . . , 60 and j = 1, . . . , 50. We consider sum of accumulated precipitation on these two days on horizontal and vertical strips with side length 2km in Figure 2. Let X i. = 50 k=1 X ik be sum of precipitation data on i-th vertical strip and X .j = 60 l=1 X lj is sum of precipitation data on j-th horizontal strip, where i = 1, . . . , 60 and j = 1, . . . , 50. Table 1 shows precipitation on vertical strips by their orders as cosecutive data on succesive rows of this table. Also the precipitation on horizontal strips are recorded on succesive rows of Table 2 as well. These precipitation on vertical and horizontal strips are plotted in Figures 3 and 4 respectivaly by blue. These accumulated precipitation on vertical and horizontal strips which are plotted by Figures 5 and 6 provide DSI processes. Following [1], [18] and [24] we fit some appropriate curve to these plots to detect the corresponding scale intervals, which are pointed out by vertical red lines. Thus we have three scale intervals for the accumulated percipitation on vertical strips in Figure 3 with end points a 1 = 0, a 2 = 14, a 3 = 31, a 4 = 52, and three scale intervals for precipitation on horizontal strips in Figure 4 with end points b 1 = 0, b 2 = 10, b 3 = 21, b 4 = 39. So we evaluate the corresponding time varying scale parameters by λ 1,n−1 = a n+1 −an an−a n−1 and λ 2,n−1 = b n+1 −bn  11169 11448 11812 12174 12454 12620 12673 12673 12663 12636  12590 12545 12516 12499 12511 12567 12692 12855 13013 13201  13406 13561 13646 13694 13750 13802 13813 13754 13613 13460  13382 13409 13532 13696 13896 14138 14377 14560 14672 14730  14765 14788 14820 14876 14956 15051 15152 15235 15299 15365  15433 15496 15572 15687 15850 16044 16276 16536 16787 17009    dashed lines in Figures 5, 6. We consider 7 equally spaced samples in each subintervals in Figure 5; and 5 equally spaced samples in subintervals of Figure 6. These eqully spaced samples provide some partitions for each subinterval. The accumulated precipitation on vertical and horizontal strips corresponding to these strips are evaluated as y (n,m)k which denote the sum of precipitation on the k-th partition of the m-th subinterval in the n-th scale interval. Following [19,24] we estimate the Hurst parameters of these DSI processes by calculation of quadratic variations of the m-th subinterval in the n-th scale interval as