Characterization of Multi-scale Invariant Fields H. Ghasemia , S. Rezakhaha,
arXiv:1710.01470v1 [stat.AP] 4 Oct 2017
a
b
∗
and N. Modarresib
Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran. Faculty of Mathematics and Computer Science, Allameh Tabataba’i University, Tehran, Iran.
Abstract 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. Mathematics Subject Classification MSC 2010: 60G18; 60G22; 62M15; 62H05. Keywords: Multi-scale invariant; Quasi-Lamperti Transformation; Self-Similarity; Spectral Representation; Wide-Sense Markov Fields.
1
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 ∗ Address correspondence to S. Rezakhah, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran ; E-mails:
[email protected],
[email protected],
[email protected].
1
Lamperti transformation. They defined a random field {X(t); t ∈ Rd } to be MSS with index H = (H1 , . . . , Hd )T ∈ Rd+ (H-MSS) if for any Λ = (λ1 , . . . , λd )T ∈ Rd+ , {
d Y
d λi−Hi X(Λt); t ∈ Rd } = {X(t); t ∈ Rd }
i=1 d
where Λt = (λ1 t1 , . . . , λd td )T and = 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.
2
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 ∈ Rd , the shift operator Sτ operates on random field {X(t); t ∈ Rd } 2
according to Sτ X(t) := X(t + τ ). A random field {X(t); t ∈ Rd } is said to be stationary field, if for any t, τ ∈ Rd d
{Sτ X(t)} = {X(t)} d
where = is the equality of finite dimensional distributions. 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 H = (H1 , . . . , Hd )T and Λ = (λ1 , . . . , λd )T where Hi > 0, λi > 0 for i = 1, . . . , d the renormalized dilation operator DH,Λ operates on random field {X(t); t ∈ Rd+ } as d Y i DH,Λ X(t) := λ−H X(Λt), i i=1
where Λt = (λ1 t1 , . . . , λd td )T . We present the following definition for MSS field from Genton et al. [6]. Definition 3. A random field {X(t); t ∈ Rd+ } is said to be MSS of index H = (H1 , . . . , Hd )T (H-MSS), if for any Λ = (λ1 , . . . , λd )T , λi > 0, i = 1, . . . , d d
{DH,Λ X(t)} = {X(t)},
(2.1)
The random field is said to be MSI of index H and scaling factor Λ0 = (λ01 , . . . , λ0d )T or (H, Λ0 )-MSI, if (2.1) holds for some Λ = Λ0 . Extending definition of Modarresi et al. [17] for DSI process with some parameter space, we present the following definition. ˇ is called H-MSS with parameter space T, ˇ Definition 4. A random field {X(k); k ∈ T} d T ˇ where T is any subset of distinct points of R+ , if for any k1 = (k11 , k12 , . . . , k1d ) , k2 = ˇ (k21 , k22 , . . . , k2d )T ∈ T d Y k2i Hi d {X(k2 )} = {X(k1 )}. (2.2) k1i i=1
ˇ and vector scale L = A random field X(.) is called (H, L)-MSI with parameter space T T ˇ (l1 , . . . , ld ) where li > 0 for i = 1, . . . , d, if for any k1 , k2 ∈ T where k2i = li k1i , (2.2) holds. Remark 1. If the random field {X(t); t ∈ Rd+ } is (H, L)-MSI with vector scale L = (l1 , . . . , ld )T where li = αiui for fixed αi > 1, ui ∈ N, i = 1, . . . , d, then by sampling of the field at points αk , k ∈ Wd where αk = (α1k1 , . . . , αdkd )T , W = {0, 1, . . .}, we have X(.) as a (H, L)-MSI with ˇ = {αk ; k ∈ Wd } and scale L. If we consider sampling of X(.) at points parameter space T n1 u1 +k1 nu+k α = (α1 , . . . , αdnd ud +kd )T , n ∈ Wd , for i = 1, . . . , d and fixed ki = 0, 1, . . . , ui − 1, ˇ = {αnu+k ; n ∈ Wd }. then X(.) is H-MSS field with parameter space T
3
Following the definition of the wide-sense self-similar process presented by Nuzman et al. [21], we present the following definition. Definition 5. A random field {X(t); t ∈ Rd+ } is said to be wide-sense H-MSS, if the following properties are satisfied for each a = (a1 , . . . , ad )T where ai > 0 (i)E[X 2 (t)] < ∞ Q d Hi (ii)E[X(at)] = E[X(t)] i=1 ai Q d 2Hi E[X(t1 )X(t2 )] (iii)E[X(at1 )X(at2 )] = i=1 ai 0 This field is called wide-sense (H, a )-MSI of index H and scaling factor a0 = (a01 , . . . , a0d )T where a0i > 0, if the above conditions hold for some a = a0 . 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. Definition 6. The quasi-Lamperti transform with positive index H and α, denoted by LH,α operates on a rendom field {Y (t); t ∈ Rd+ } as LH,α Y (t) =
d Y
i tH Y (Logα t) i
(2.3)
i=1
where Logα t = (logα1 t1 , . . . , logαd td )T and α > 1 i.e. αi > 1 for i = 1, . . . , d. The corred sponding inverse quasi-Lamperti transform L−1 H,α on field {X(t); t ∈ R+ } acts as d Y X(t) = (2.4) L−1 αi−ti Hi X(αt ) H,α i=1
where αt = (α1t1 , . . . , αdtd )T . −1 It is easy to verify that LH,α L−1 H,α X(t) = X(t) and LH,α LH,α Y (t) = Y (t). Note that in the above definition, if α = (e, . . . , e)T , we have the usual Lamperti transformation LH .
Proposition 1. The quasi-Lamperti transform assurances an equivalence between the shift operator SLogα k and the renormalized dilation operator DH,k in the sense that, for any k > 0 L−1 H,α DH,k LH,α = SLogα k .
(2.5)
Proof. By a similar method as in [17], the validation of (2.5) follows. So we have the following corollaries. Corollary 1. If {Y (t); t ∈ Rd } is a stationary field, its quasi-Lamperti transformation {LH,α Y (t); t ∈ Rd+ } is H-MSS. Conversely if {X(t); t ∈ Rd+ } is a H-MSS, its inverse quasid Lamperti transformation {L−1 H,α X(t); t ∈ R } is stationary. Corollary 2. If {X(t); t ∈ Rd+ } is a (H, αU )-MSI then L−1 H,α X(t) = Y (t) is periodic field with d period U > 0. Conversely if {Y (t); t ∈ R } is periodic field with period U then LH,α Y (t) = X(t) is (H, αU )-MSI.
4
ˇ = {Ln ; n ∈ Wd } and Hurst vector Remark 2. If X(.) is a H-MSS with parameter space T U H = (H1 , . . . , Hd )T and L = (α1U1 , . . . , αd d )T , then it is easy to show that its stationary ˜ = {nU; n ∈ Wd } where nU = (n1 U1 , . . . , nd Ud )T . counterpart Y (.) has parameter space T As an example of MSI field we introduce two-dimensional sfBm by the followings. Example 1. A two dimensional sfBm {X(t); t ∈ R2+ } with Hurst and scale vectors H = (H1 , H2 )T and λ = (λ1 , λ2 )T is defined by X(t1 , t2 ) =
∞ X ∞ X
n (H1 −H10 ) n2 (H2 −H20 ) λ2 I[λn1 −1 ,λn1 ) (t1 )I[λn2 −1 ,λn2 ) (t2 )B(t1 , t2 ),
λ1 1
1
n1 =1 n2 =1
1
2
2
where B(., .), I(.) are the two-dimensional fractional Brownian motion indexed by H0 = (H10 , H20 )T and indicator function respectively and Hi > 0, Hi0 > 0, λi > 1 for i = 1, 2. Let Ai1 = [1, λi ), Ai2 = [λi , λ2i ) and Ain = [λn−1 , λni ) for i = 1, 2 are disjoint sets. The i field X(.) is MSI. Let’s first we find the covariance function that is used to prove. For (t1 , t2 ) ∈ A1n1 × A2n2 and (s1 , s2 ) ∈ A1n01 × A2n02 , B(., .) is the two-dimensional fractional Brownian motion with covariance 2 1Y 0 2H 0 2H 0 Cov B(t1 , t2 ), B(s1 , s2 ) = ti i + si i − | ti − si |2Hi . 4 i=1
Hence the covariance function of the field is 2 1Y 0 (n +n0 )(H −H 0 ) 2H 0 2H 0 Cov X(t1 , t2 ), X(s1 , s2 ) = λi i i i i ti i + si i − | ti − si |2Hi . 4 i=1
If (t1 , t2 ) ∈ A1n × A2m , then (λ1 t1 , λ2 t2 ) ∈ A1(n+1) × A2(m+1) . Thus, for (λ1 t1 , λ2 t2 ) ∈ A1(n1 +1) × A2(n2 +1) and (λ1 s1 , λ2 s2 ) ∈ A1(n01 +1) × A2(n02 +1) we have Cov X(λ1 t1 , λ2 t2 ), X(λ1 s1 , λ2 s2 ) =
2 1 Y (ni +n0i +2)(Hi −Hi0 ) Cov B(λ1 t1 , λ2 t2 ), B(λ1 s1 , λ2 s2 ) λi 4 i=1
=
=
=
2 1Y
4 1 4
i=1 2 Y
(ni +n0i +2)(Hi −Hi0 ) 2Hi0 λi
λi
(ni +n0i )(Hi −Hi0 )
i λ2H λi i
2Hi0
ti
2Hi0
ti
2Hi0
+ si
2Hi0
+ si
0
− | ti − si |2Hi 0
− | ti − si |2Hi
i=1
2 Y
i λ2H Cov X(t1 , t2 ), X(s1 , s2 ) . i
i=1
So X(t1 , t2 ) is H, λ -MSI field. In the case H0 = ( 21 , 12 )T , B(., .) is the two-dimensional Brownian motion.
5
3
Spectral Representation
In this section we characterize the harmonic-like representation and spectral density matrix of the MSI field. Following Hurd et al. [9] and Haghbin et al. [8], let {Y (m); m = (m1 , m2 ) ∈ Z2 } be a periodic field with period U = (U1 , U2 ) and define DU = {j := (j1 , j2 ), j1 = 0, 1, . . . , U1 − 1, j2 = 0, 1, . . . , U2 − 1}, then there exists a collection {Z j1 j2 (m), j1 = 0, 1, . . . , U1 − 1, j2 = 0, 1, . . . , U2 − 1, m ∈ Z2 } of jointly stationary fields for which X
Y (m) =
Z j1 j2 (m)e
−2πi(
m1 j1 m j + U2 2 ) U1 2
(3.6)
j=(j1 ,j2 )∈DU
where Z j1 j2 (m) has the spectral representation Z j1 j2 Z (m) = e−i(m1 j1 +m2 j2 ) ψω(j) (dλ)
(3.7)
2π 2π [0, U ]×[0, U ] 1
2
where ψω(j) is a random spectral measure with orthogonal increments and the function ω : DU → {1, . . . , U1 U2 } is defined by ω(k1 , k2 ) = k2 U1 + k1 + 1. Also Y (m) can be represented by the integral Z Y (m) = e−i(m1 λ1 +m2 λ2 ) φ(dλ) (3.8) [0,2π]2 n 1 2π(n1 +1) ), for λ ∈ [ 2πn where the spectral random measure φ is given by φ(dλ) = ψω(n) (dλ−2π U ]× U1 , U1 2 2π(n2 +1) [ 2πn ]. U2 , U2
Furthermore, the spectral distribution F (., .), specified by F (dλ, dλ0 ) = E[φ(dλ)φ(dλ0 )], is a measure on [0, 2π]2 × [0, 2π]2 which is supported by Sn = {(λ, λ0 ) ∈ [0, 2π]2 × [0, 2π]2 , λ − λ0 = n 2π U } for n1 = −U1 + 1, . . . , U1 − 1 and n2 = −U2 + 1, . . . , U2 − 1. Considering Fn as the restriction of F on the set Sn , then E[ψω(m) (dz)ψω(n) (dz)] = Fm−n (dz + 2π
m ); U
z ∈ [0,
2π 2π ] × [0, ] U1 U2
(3.9)
where the corresponding square matrix F(dz) = [Fω(m),ω(n) (dz)]m,n∈DU is defined by Fω(m),ω(n) (dz) = Fm−n (dz + 2π
m ). U
(3.10)
Let Qn (τ ) = Cov(Y (n), Y (n+τ )) be the covariance function which is periodic with respect to n and has the representation −2πi(
X
Qn (τ ) =
e
n1 j1 n j + U2 2 ) U1 2
Rj (τ )
(3.11)
j=(j1 ,j2 )∈DU
where Rj (τ ) is represented as Z Rj (τ ) =
e−i(τ1 λ1 +τ2 λ2 ) Dj (dλ)
[0,2π]2
6
(3.12)
and each Dj (λ) is defined by 2πj2 1 Fj−U (λ), λ ∈ A1 = [0, 2πj U1 ] × [0, U2 ] F 2πj1 2πj2 λ ∈ A2 = [0, U1 ] × [ U2 , 2π] j−U1 (λ), Dj (λ) = 2πj2 1 F (λ), λ ∈ A3 = [ 2πj j−U2 U2 , 2π] × [0, U2 ] 2πj2 1 Fj (λ), λ ∈ A4 = [ 2πj U2 , 2π] × [ U2 , 2π] where U = (U1 , U2 ), U1 = (U1 , 1), U2 = (1, U2 ). Hence for τ = (τ1 , τ2 ), by (3.11) we have the Fourier series representation n j n j X 1 2πi( U1 1 + U2 2 ) 1 2 Qn (τ ) Rj (τ ) = e U1 U2
(3.13)
(3.14)
(n1 ,n2 )∈DU
and by (3.12) for A ⊂ Ak , k = 1, 2, 3, 4 Z 1 Dj (A) = ( )2 2π A
X
ei(τ1 δ1 +τ2 δ2 ) Rj (τ )dδ
(3.15)
(τ1 ,τ2 )∈DU
and the corresponding spectral density is denoted by dj (λ) which is for A = (λ, λ + dλ] = (λ1 , λ1 + dλ1 ] × (λ2 , λ2 + dλ2 ], X 1 Z Dj (dλ) 1 2 =( ) ei(τ1 δ1 +τ2 δ2 ) dδ Rj (τ ) dj (λ) : = dλ 2π dλ (λ,λ+dλ] (τ1 ,τ2 )∈DU
= (
= (
1 2 ) 2π 1 2 ) 2π
1 = ( )2 2π
X (τ1 ,τ2 )∈DU
X (τ1 ,τ2 )∈DU
X
2 Y 1 ei(λn +dλn )τn − eiλn τn ( ) lim Rj (τ ) iτn dλn →0 dλn n=1
2 Y 1 iτn λn Rj (τ ) ( )(iτn )e iτn n=1
ei(τ1 λ1 +τ2 λ2 ) Rj (τ )
(τ1 ,τ2 )∈DU
Proposition 2. If {X(α1n1 , α2n2 ); (n1 , n2 ) ∈ W2 } is ((H1 , H2 ), (αU1 , αU2 ))-MSI field, then (i) The harmonic-like representation of the field is Z n1 n2 n1 H1 n2 H2 e−i(n1 λ1 +n2 λ2 ) φ(dλ) (3.16) X(α1 , α2 ) = α1 α2 [0,2π]2
where φ is the corresponding spectral measure. (ii) The spectral representation of the covariance function is m j m j X −2πi( U1 1 + U2 2 ) (2m1 +τ1 )H1 (2m2 +τ2 )H2 1 2 α e Rj (τ ) QH (τ ) = α m 1 2
(3.17)
(j1 ,j2 )∈DU
where Rj (τ ) is defined by (3.14). (iii) The spectral density of X(α1j1 , α2j2 ) for λ ∈ Ak , k = 1, 2, 3, 4, is X 1 (2τ +λ )H (2τ +λ )H dH α1 1 1 1 α2 2 2 2 ei(λ1 τ1 +λ2 τ2 ) Rj (τ ) j (λ) = (2π)2 (τ1 ,τ2 )∈DU
7
(3.18)
Proof. (i) According to corollary 2, the corresponding quasi-Lamperti transformation of X(α1n1 , α2n2 ) is periodic field Y (n1 , n2 ) with period U = (U1 , U2 ). By (3.8), the spectral representation of field has the form X(α1n1 , α2n2 ) = LH,α Y (α1n1 , α2n2 ) = α1n1 H1 α2n2 H2 Y (n1 , n2 ) Z n1 H1 n2 H2 e−i(n1 λ1 +n2 λ2 ) φ(dλ) α2 = α1 [0,2π]2
(ii) The covariance function is m1 +τ1 m2 m1 , α2m2 +τ2 ))] QH m (τ ) = Cov[X(α1 , α2 ), X(α1
= Cov[LH,α Y (α1m1 , α2m2 ), LH,α Y (α1m1 +τ1 , α2m2 +τ2 )] (2m1 +τ1 )H1
= α1 =
(2m2 +τ2 )H2
α2
Cov[Y (m), Y (m + τ )]
(2m +τ )H (2m +τ )H α1 1 1 1 α2 2 2 2 Qm (τ )
So by (3.11), the proof is completed. (2m1 +τ1 )H1 (2m2 +τ2 )H2 α2 Rj (τ ). Hence as the spectral distribution (iii) In (3.17), let RH j (τ ) = α1 function of Qn (τ ) is Dj (dλ), so using (3.11) and (3.12) and the relation of QH m (τ ) and Qm (τ ) as above, we find by (3.15) that the corresponding spectral density is
dH j (λ) = =
4
1 (2π)2 1 (2π)2
X
ei(λ1 τ1 +λ2 τ2 ) RH j (τ )
(τ1 ,τ2 )∈DU
X
(2τ1 +λ1 )H1
α1
(2τ2 +λ2 )H2 i(λ1 τ1 +λ2 τ2 )
α2
e
Rj (τ ).
(τ1 ,τ2 )∈DU
Two dimensional Scale Invariant Markov Fields
The covariance function of a Markov random field {X(s, t), (s, t) ∈ R2 } has separable property if it satistifies R(s1 , s2 ; t1 , t2 ) = R1 (s1 , t1 )R2 (s2 , t2 ) where R(s1 , s2 ; t1 , t2 ) = Cov(X(s1 , s2 ), X(t1 , t2 )) and R1 (s1 , t1 ), R2 (s2 , t2 ) both have nonnegative definite property and can be considered as the covariance function of some Markov process, see [7] and [25, p. 320]. Also there exist some formal methods to test the separability of the covariance function of random fields, see [5]. By considering some rectangle grid over full area of consideration, if the covariance function R(s1 , s2 ; t1 , t2 ) has MSI property, then the covariance functions R1 (s1 , t1 ), R2 (s2 , t2 ) can be considered as the covariance functions of DSI Markov proceses X1 (.) and X2 (.) corresponding to the vertical and horizontal strips respectively. So R1 (s1 , t1 ) = Cov(X1 (s1 ), X1 (t1 )) and R2 (s2 , t2 ) = Cov(X2 (s2 ), X2 (t2 )). In this section we show that the covariance function of wide sense Markov MSI (WM-MSI) field with separable property is characterized by the covariance functions of the field at points of the first scale rectangle. 8
Now we consider two dimensional scale invariant wide-sense Markov field {X(α1n1 , α2n2 ), (n1 , n2 ) ∈ Z2 } which has separable covariance function with Hurst H = (H1 , H2 ) and scale αT = (α1T1 , α2T2 ). Let RnH (τ ) := R(n, n + τ ) = Cov[X(α1n1 +τ1 , α2n2 +τ2 ), X(α1n1 , α2n2 )] for τ ∈ Z2 . Also assume that {Xi (αik ), k ∈ Z} be a DSI wide-sense Markov process with parameters (Hi , αiTi ) Hi and covariance function Ri,n (τi ) = Cov[Xi (αini +τi ), Xi (αini )] for i = 1, 2. So we have i H2 H1 (τ2 ). (τ1 )R2,n RnH (τ ) = R1,n 2 1
Thus by Theorem 3.2 in [17], we have that ˜ T−1 )]k h(α ˜ ν+n−1 )[h(α ˜ n−1 )]−1 RH (0), RnH (kT + ν) = [h(α n
(4.19)
and H RnH (−kT + ν) = α−2kTH Rn+ν (kT − ν)
where ˜ r) = h ˜ 1 (αr1 )h ˜ 2 (αr2 ), h(α 1 2
α−2kTH = α1−2k1 T1 H1 α2−2k2 T2 H2
˜ i (αri ) = Qri h(αj ) = Qri and for i = 1, 2; ki ∈ {0, 1, 2, . . .}, νi = 0, 1, . . . , Ti − 1 and h j=0 j=0 i i
H
Rj i (1) H
Rj i (0)
,
˜ i (α−1 ) = 1. h i Proposition 3. Let {X(α1n1 , α2n2 ); (n1 , n2 ) ∈ Z2 } be a TS-MSI field with wide-sense Markov property which has Hurst parameter H = (H1 , H2 ) and scale vector αT = (α1T1 , α2T2 ). Then the covariance function can be characterized by the covariance functions in (4.19). According to Remark 1 and corresponding to the TS-MSI field with wide-sense Markov property, {X(α1n1 , α2n2 ); (n1 , n2 ) ∈ Z2 } with the scale L = (α1T1 , α2T2 ), there exists a T1 T2 -dimensional self-similar wide-sense Markov field Y (t1 , t2 ) = Y0,0 (t1 , t2 ), Y0,1 (t1 , t2 ), . . . , Y(T1 −1),(T2 −1) (t1 , t2 ) ˇ = {Ln ; n ∈ W2 , L = (αT1 , αT2 )}, where with parameter space T 1 2 Yk1 ,k2 (Ln ) = Yk1 ,k2 (α1n1 T1 , α2n2 T2 ) = X(α1n1 T1 +k1 , α2n2 T2 +k2 )
(4.20)
and for i = 1, 2; ki = 0, 1, . . . , Ti − 1. Hence, the cross covariance function of Yj1 ,j2 and Yk1 ,k2 at points Ln and Ln+τ can be written as n+τ QH , Ln ) = Cov[Yj1 ,j2 (Ln+τ ), Yk1 ,k2 (Ln )] j1 ,j2 ;k1 ,k2 (L
= α2nTH Cov[X(α1τ1 T1 +j1 , α2τ2 T2 +j2 ), X(α1k1 , α2k2 )] = α2nTH RkH (τ T + j − k) ˜ T−1 )]τ h(α ˜ j−1 )[h(α ˜ k−1 )]−1 RH (0) = α2nTH [h(α k
where
α2nTH
=
α1 2n1 T1 H1 α2 2n2 T2 H2 .
Proposition 4. Let {X(α1n1 , α2n2 ); (n1 , n2 ) ∈ Z2 } be a TS-MSI field with wide-sense Markov property with the covariance function RnH (.) and {Y (Ln ); n ∈ W2 }, defined in (4.20), be its T1 T2 -dimensional self-similar wide-sense Markov field with the cross covariance function QH (., .). Then the cross covariance function of Yj1 ,j2 and Yk1 ,k2 at points Ln and Ln+τ is n+τ ˜ T−1 )]τ h(α ˜ j−1 )[h(α ˜ k−1 )]−1 RH (0) QH , Ln ) = α2nTH [h(α j1 ,j2 ;k1 ,k2 (L k
9
(4.21)
Figure 1: Two dimensional image of precipitation data (data unit is mm) over a 512 km × 512 km region in the Brisbane area for two days (25th and 26th January 2013) and the selected area that Figure 2: Two dimensional image of precipitation data for the selected area. specified by yellow rectangle.
5
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 Xij 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 P these two days on horizontal and vertical strips with side length 2km inP Figure 2. Let Xi. = 50 k=1 Xik be sum of precipitation data on 60 i-th vertical strip and X.j = l=1 Xlj 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 a1 = 0, a2 = 14, a3 = 31, a4 = 52, and three scale intervals for precipitation on horizontal strips in Figure 4 with end points b1 = 0, b2 = 10, b3 = 21, b4 = 39. So we evaluate −an bn+1 −bn the corresponding time varying scale parameters by λ1,n−1 = aann+1 −an−1 and λ2,n−1 = bn −bn−1 for n = 2, 3. This leads to find λ1,1 = 1.214, λ1,2 = 1.235 or Λ1 := (λ1,1 , λ1,2 ) = (1.214, 1.235), as the values of scale parameter for DSI process of accumulated precipitation on the vertical strips and λ2,1 = 1.1, λ2,2 = 1.636 or Λ2 := (λ2,1 , λ2,2 ) = (1.1, 1.636), as the values of scale parameter for DSI process of accumulated precipitation on the horizontal strips. For each scale interval we consider two equal subintervals which are indicated with green 10
11169 12590 13406 13382 14765 15433
11448 12545 13561 13409 14788 15496
11812 12516 13646 13532 14820 15572
12174 12499 13694 13696 14876 15687
12454 12511 13750 13896 14956 15850
12620 12567 13802 14138 15051 16044
12673 12692 13813 14377 15152 16276
12673 12855 13754 14560 15235 16536
12663 13013 13613 14672 15299 16787
12636 13201 13460 14730 15365 17009
Table 1: Sum of precipitation data on vertical strips as millimeters
10593 14171 17161 18728 19246
10964 14131 17257 19068 19017
11379 14184 17546 19215 18882
11862 14395 17927 19214 18772
12443 14835 18188 19152 18575
13051 15431 18246 19194 18340
13578 16064 18303 19419 18184
13927 16638 18371 19634 18045
14105 17039 18387 19668 17809
14187 17179 18451 19491 17553
Table 2: Sum of precipitation data on horizontal strips as millimeters
Figure 3:
Fitted black curves for precipitation data on vertical Figure 4: Fitted black curves for precipitation data on horizontal strips and revealing the corresponding scale intervals by red lines. strips and revealing the corresponding scale intervals by red lines.
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 SSn,m
lm 1 X 2 = y(n,m)k lm
(5.22)
k=1
where lm is the number of partitions in the m-th subinterval for n = 1, 2, 3 and m = 1, 2. Then the corresponding Hurst parameters are estimated by H(n,m)1 =
log(SSn+1,m /SSn,m ) 2 log λ1,m
H(n,m)2 =
log(SSn+1,m /SSn,m ) 2 log λ2,m
and
11
Figure 5:
Two subintervals with equally spaced red points are indi- Figure 6: Two subintervals with equally spaced red points are incated with green dashed lines in each scale interval for precipitation dicated with green dashed lines in each scale interval for precipitation data for vertical strips. data for horizontal strips.
for precipitation data in subintervals along vertical and horizontal strips respectively. These method provide different estimation of Hurst parameters for the first subinterval (m = 1) and second subintervals (m = 2) Hence, H1 := (H(1,1)1 , H(1,2)1 , H(2,1)1 , H(2,2)1 ) = (1.40, 1.42, 1.42, 1.49) and H2 := (H(1,1)2 , H(1,2)2 , H(2,1)2 , H(2,2)2 ) = (3.42, 3.02, 1.46, 1.29) are the Hurst parameters for precipitation in subintervals along vertical and horizontal strips respectively. So for vertical strips we can evaluate the Hurst parameters of second subinterval with respect to first subinterval and for the third subinterval with respect to the second as H1,1 =
H(1,1)1 + H(1,2)1 = 1.41, 2
H1,2 =
H(2,1)1 + H(2,2)1 = 1.46, 2
Also for horizontal strips these Hurst parameters for the second subinterval with respect to the first and third subinterval with respect to the second are estimated as H2,1 =
H(1,1)2 + H(1,2)2 = 3.22, 2
H2,2 =
H(2,1)2 + H(2,2)2 = 1.375 2
Now we are to consider the variation of precipitation with respect to time in the selected area shown in Figure 2 which shows another DSI behavior. For this we have considered the accumulated perticipation in this area for every 30 minutes on 25th and 26th January 2013 and depicted in Table 3 by their orders. In Figure 7, Precipitation per 30 minutes at the two days are indicated by blue. The fitted black curves and red lines reveal the corresponding scale intervals which illuminates the DSI behavior of precipitation. The end points of these scale intervals are c1 = 38, c2 = 44, c3 = 60, −cn c4 = 79. Thus, the scale parametere can be evaluated by values λ3,n−1 = ccnn+1 −cn−1 for n = 2, 3. So, we have λ3,1 = 2.667, λ3,2 = 1.187 or Λ3 := (λ3,1 , λ3,2 ) = (2.667, 1.187). According to prior method and dividing each scale interval to two equal subintervals which are indicated with green dashed lines in Figure 8 and the equally spaced red points in each subinterval, the corresponding Hurst parameters are obtained by H(n,m)3 =
log(SSn+1,m /SSn,m ) 2 log λ3,m 12
2311 1453 3511 3064 1084 4130 15266 14326 21492 13795
2568 1823 3954 2455 2163 5911 15964 14867 21567 11567
2702 2415 2339 1515 4603 8518 15842 16331 16331 9122
2802 2596 1525 1516 6409 8876 16711 16070 22416 7488
2351 2885 1527 1527 6157 9693 17760 15880 23215 6425
2248 2947 2071 848 5246 11377 17274 18072 23309 4745
2496 2936 2907 1837 4998 10841 15813 21312 21192
2552 3113 2545 2110 5610 12442 14470 22709 17992
2061 2877 1965 1064 3880 14649 13623 24026 17550
1780 2974 2310 975 2292 16939 13521 22112 15295
Table 3: The precipitation data on the selected region in the Brisbane area for the two days (25 and 26 January 2013) per 30 minutes as millimeters.
Figure 7:
The fitted black curves shows scale intervals correspond- Figure 8: Two equally length subintervals for each scale interval are ing to DSI behavior of precipitation data on 25th and 26th January indicated with green dashed lines for precipitation in the area per 30 2013 per 30 minutes. Scale intervals are shown by red lines. minutes in 25th and 26th January 2013.
Thus, H3 := (H(1,1)3 , H(1,2)3 , H(2,1)3 , H(2,2)3 ) = (2.56, 1.94, 7.93, 3.48) and H3,1 =
H(1,1)3 + H(1,2)3 = 2.25, 2
H3,2 =
H(2,1)3 + H(2,2)3 = 5.7 2
Acknowledgement We would like to thank Professor Alan Seed for providing the data used in this paper and the Australian Bureau of Meteorology as the source of the data.
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