# A Class of Negatively Fractal Dimensional Gaussian Random Functions

Authors: Ming Li
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

Letx(t) be a locally self-similar Gaussian random function. Denote by rxx(τ) the autocorrelation function (ACF) of x(t). For x(t) that is sufficiently smooth on (0,∞), there is an asymptotic expression given by rxx(0)-rxx(τ)~c|τ|α for |τ|→0, where c is a constant and α is the fractal index of x(t). If the above is true, the fractal dimension of x(t), denoted by D, is given by D=D(α)=2−α/2. Conventionally, α is strictly restricted to 0<α≤2 so as to make sure that D∈[1,2). The generalized Cauchy (GC) process is an instance of this type of random functions. Another instance is fractional Brownian motion (fBm) and its increment process, that is, fractional Gaussian noise (fGn), which strictly follow the case of D∈[1,2) or 0<α≤2. In this paper, I claim that the fractal index α of x(t) may be relaxed to the range α>0 as long as its ACF keeps valid for α>0. With this claim, I extend the GC process to allow α>0 and call this extension, for simplicity, the extended GC (EGC for short) process. I will address that there are dimensions 0≤D(α)<1 for 2<α≤4 and further D(α)<0 for 4<α for the EGC processes. I will explain that x(t) with 1≤D<2 is locally rougher than that with 0≤D<1. Moreover, x(t) with D<0 is locally smoother than that with 0≤D<1. The local smoothest x(t) occurs in the limit D→−∞. The focus of this paper is on the fractal dimensions of random functions. The EGC processes presented in this paper can be either long-range dependent (LRD) or short-range dependent (SRD). Though applications of such class of random functions for D<1 remain unknown, I will demonstrate the realizations of the EGC processes for D<1. The above result regarding negatively fractal dimension on random functions can be further extended to describe a class of random fields with negative dimensions, which are also briefed in this paper.


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## Body

## 1. Introduction

Conventionally, for a time series, or a random functionx(t), such as fully developed ocean wave series, we need not discuss its fractal dimension, as can be judged from the power spectra discussed by Massel [1], the Specialist Committee on Waves of the 23rd ITTC [2], Li [3]. However, in some cases, we have to consider the fractal dimension of a random function, such as oceanic monthly temperature; see Alvarez-Ramirez et al. [4]. As a matter of fact, time series with fractal dimensions are observed in many fields of sciences and technologies; see, for example, Beran [5], Mandelbrot [6], Korvin [7], West and Deering [8], Schreiber [9], Abry et al. [10], Werner [11], Levy-Vehel [12], Cattani [13, 14], and references therein.Denote the fractal dimension ofx(t) by D, which measures the local roughness or local irregularity or local self-similarity of x(t); see Mandelbrot [15], Li [16]. Then, in the standard fractal time series, one has the positive dimension given by
(1.1)D∈[1,2).The literature with respect to the time series withD∈[1,2) is rich; see, for example, [4–68], simply a drop in the bucket in the field. However, how to represent D∈(0,1) and D<0 in particular for time series remains an open problem. This paper gives a solution to this problem in the case that x(t) is a locally self-similar Gaussian random function the ACF of which follows the form of the GC process.There are various definitions of dimensions (Mandelbrot [15]), such as the Minkowski dimension, the Rényi dimension, the Hausdorff dimension, the packing dimension, the box-counting dimension, the correlation dimension. Those dimensions may not be equal for a specific object but this does not matter. What an important thing is whether there are objects the dimensions of which are negative and how to represent negative dimensions of objects (Mandelbrot [69]). In this regard, the research conducted by Mandelbrot and his colleagues reveals a new outlook in negative dimensions in geometry, see Mandelbrot [69–75], applications of which were found to turbulence, see Molenaar et al. [76], Chhabra and Sreenivasan [77, 78].Note that the negative dimensions described based on the Duplantier’s function are from a view of geometry. Differing from the work by Mandelbrot [69–75], this paper addresses time series or random functions with negative dimensions. It is well known that commonly used models of fractal time series are fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) that is the increment process of fBm (Mandelbrot [17]). For fBm as well as fGn, dimensions are restricted to be positive. As a matter of fact, denote by DfGn and HfGn the fractal dimension of fGn and its Hurst parameter, respectively. Denote fGn by xfGn(t). Then, the autocorrelation function (ACF) of fGn follows
(1.2)E[xfGn(t)xfGn(t+τ)]≈HfGn(2HfGn-1)τ(2HfGn-2).
The right side of the above expression requires 0<HfGn<1; see Mandelbrot [17]. Thus, 1<DfGn<2 because DfGn=2-HfGn. In the LRD case, 0.5<HfGn<1 and 1.5<DfGn<2. As a result, I have the following note.Note 1.
The dimension of fGn is never negative. That is,1≤DfGn<2.There are various types of fractal time series, such as fGn, alpha-stable processes, Levy processes. However, not all time series have negative dimensions. For example, fGn does not have negative dimensions. However, there may exist time series that have negative dimensions. This research of mine restricts my study to a specific class of Gaussian random functions. It is the extension of the GC process that were reported by Gneiting and Schlather [79], Lim and Li [80], Li and Lim [81], and Li et al. [82]. The extended GC (EGC) processes may have negative dimensions.LetDGC and HGC be the fractal dimension and the Hurst parameter of the GC process, respectively. Then, DGC and HGC are independent of each other. In the previous research regarding the GC process, only the case of 1<DGC<2 was discussed, see [79–81], Li [52], Li and Lim [83, 84]. In this paper, I extend the GC process such that DEGC∈(0,1) and DEGC<0, where DEGC is the fractal dimension of the EGC process in addition to the traditional case of DEGC∈[1,2). The term EGC process means that it is a class of processes that is based on the GC process but extended to the case of DEGC<1, simply for the purpose of distinguishing it from the standard GC process.The rest of paper is organized as follows. The class of negatively dimensional random functions, that is, EGC processes, is addressed in Section2. Discussions are given in Section 3. Extending the presented class of negative dimensional random functions to the corresponding random fields is briefed in Section 4. Finally, Section 5 concludes the paper.


## 2. Extended Generalized Cauchy (EGC) Processes

Denote by(Ω,T,P) the probability space. Then, x(t,ζ) is said to be a stochastic process when the random variable x represents the value of the outcome of an experiment T for every time t, where Ω represents the sample space, T is the event space or sigma algebra, and P the probability measure.As usual,x(t,ζ) is simplified to be written as x(t), that is, the event space is usually omitted. Denote by P(x) the probability function of x. Then, one can define the general nth order, time varying, joint distribution function P(x1,…,xn;t1,…,tn) for the random variables x(t1),…,x(tn). The joint distribution density function is written by
(2.1)p(x1,…,xn;t1,…,tn)=∂nP(x1,…,xn;t1,…,tn)∂x1⋯∂xn.In this paper, only the first- and second-order properties of processes are considered instead of the complete joint distribution function. Moreover, this research only considers Gaussian processes. Gaussian processes can be completely determined by the second-order properties, more precisely, mean and ACF, see Papoulis [85]. Without generality losing, this paper only considers processes with mean zero.Note that(1+|τ|α)-β/α is a valid ACF for α>0 and β>0. Denote (1+|τ|α)-β/α by rEGC(τ) that is the ACF of a Gaussian random function denoted by xEGC(t). That is, (2.2)rEGC(τ)=E[xEGC(t+τ)xEGC(t)]=(1+|τ|α)-β/α,α>0,β>0.
Then, we call a random function x(t) as an EGC process if it is a stationary Gaussian centred process with the ACF given by (2.2).Usually, the norm of a random functionx(t) is expressed by E[x(t)x(t)]=∥x(t)∥2, see, for example, Cramer [86, 87], Liu [88], Gelfand and Vilenkin [89], and Adler et al. [90]. However, for the EGC process, E[xEGC(t)xEGC(t)]=rEGC(0)=1 regardless of the values of α and β. Therefore, it is inconvenient to use ∥xEGC(t)∥. For this reason, I utilize ∥rEGC(τ)∥ rather than ∥xEGC(t)∥ in this paper. The norm ∥rEGC(τ)∥ is suitable for our research purpose. In fact, since xEGC(t) is Gaussian, it is uniquely determined by rEGC(τ) and vice versa. Note that
(2.3)∫-∞∞|rEGC(τ)|2dτ=∞for0<β<0.5.
Thus, I need to express ∥rEGC(τ)∥ in the domain of generalized functions.Definition 2.1 (see Griffel [91]).
A function of rapid decay is a smooth functionϕ:ℝ→ℂ such that tnϕ(r)(t)→0 as t→±∞ for all n,r≥0, where ℂ  is the space of complex numbers. The set of all functions of rapid decay is denoted by S.Lemma 2.2 (see Griffel [91]).
Every function belonging toS is absolutely integrable.Now, define the norm and inner product ofr∈HEGC by
(2.4)‖rEGC‖2=(rEGC,rEGC)=∫-∞+∞[rEGC(u)]2g(u)du,
where g∈S. Combining any r∈HEGC with its limit yields that HEGC is a Hilbert space. Denote it by
(2.5)HEGC={rEGC;‖rEGC‖2<∞;α>0,β>0}.For0<α≤2, I express the subspace of HEGC by
(2.6)HEGC1={rEGC;‖rEGC‖2<∞;0<α≤2,0<β}.Denote byHGC the space of the standard GC process. Then, one immediately sees that HGC is a subspace of HEGC. More precisely,
(2.7)HGC=HEGC1.In the case of2<α≤4, I denote another subspace of HEGC by
(2.8)HEGC2={rEGC;‖rEGC‖2<∞;2<α≤4,0<β}.
Further, I denote the subspace of HEGC for α>4 by
(2.9)HEGC3={rEGC;‖rEGC‖2<∞;4<α,0<β}.
Then, I have the remark below.Remark 2.3.
One hasHEGC=HEGC1∪HEGC2∪HEGC3.Recall that eachrEGC∈HEGC corresponds to a Gaussian process, see Gelfand and Vilenkin [89, Chapter 4], Kanwal [92]. Its ACF is given by (2.2). However, the dimensions of processes in HEGC2 and HEGC3 are undefined, more precisely, unknown. The following theorems will describe the dimensions of processes in HEGC2 and HEGC3.Theorem 2.4.
Processes inHEGC2 have the dimensions less than one and greater than or equal to zero.Proof.
Denote a process inHEGC2 by xEGC2(t). Let an ACF in HEGC2 be rEGC2. Then, taking into account the definition of the local self-similarity provided by Kent and Wood [93], Hall and Roy [94], Chan et al. [95], Adler [96], one says that a Gaussian stationary process is locally self-similar of order α if its ACF satisfies for τ→0(2.10)rEGC2(τ)=E[xEGC2(t+τ)xEGC2(t)]=1-βα|τ|α{1+O(|τ|α)},2<α≤4,0<β.
Therefore, according to [93–96], the fractal dimension denoted by DEGC2 is given by
(2.11)0≤DEGC2=(1-α2)<1.
This yields Theorem 2.4.LetxEGC3(t) be a process in HEGC3. Denote the ACF of xEGC3(t) by rEGC3. Then, the following theorem gives the negative dimensions of xEGC3(t).Theorem 2.5.
Processes inHEGC3 have negative dimensions.Proof.
Similar to (2.10), one has
(2.12)rEGC3(τ)=E[xEGC3(t+τ)xEGC3(t)]=1-βα|τ|α{1+O(|τ|α)},4<α,0<β.
Then, following (2.12), we have
(2.13)DEGC3=(1-α2)<0.
This completes the proof of Theorem 2.5.Remark 2.6.
For an EGC processxEGC(t) with 0<α and 0<β, the fractal dimension of xEGC(t) in general satisfies
(2.14)DEGC<2.Theorem 2.7.
The power spectrum density (PSD) function of the EGC process is given(2.15)SEGC(ω)=∑k=0∞(-1)kΓ(β/α+k)πΓ(β/α)Γ(1+k)I1(ω)*Sa(ω)+∑k=0∞(-1)kΓ(β/α+k)πΓ(β/α)Γ(1+k)[πI2(ω)-I2(ω)*Sa(ω)],
where ω is angular frequency, * implies the convolution, Sa(ω)=sin(ω)/ω and
(2.16)I1(ω)=-2sin(αkπ2)Γ(αk+1)|ω|-αk-1,I2(ω)=2sin[(β+αk)π2]Γ[1-(β+αk)]|ω|(β+αk)-1.Proof.
Note that(1+x)ν can be expanded as a binomial series given by
(2.17)(1+x)ν=∑k=0∞(νk)xk=∑k=0∞Γ(ν+k)Γ(ν)Γ(1+k)xkfor|x|<1,
where ν∈ℝ. and (νk) is the binomial coefficient.
Now, I expand the ACF of the EGC process by(2.18)C(τ)={∑k=0∞(-1)kΓ[(β/α)+k]Γ(β/α)Γ(1+k)|τ|αk}[u(τ+1)-u(τ-1)]+{∑k=0∞(-1)kΓ[(β/α)+k]Γ(β/α)Γ(1+k)|τ|-(β+αk)}[u(τ-1)+u(-τ-1)],
where u(τ) is the Heaviside unit step function (Li and Lim [84]).
Because the Fourier transform (FT) of|t|λ is expressed by(2.19)F(|t|λ)=-2sin(λπ2)Γ(λ+1)|ω|-λ-1,
where λ≠-1,-3,…, I have
(2.20)F[|τ|αk]=-2sin(αkπ2)Γ(αk+1)|ω|-αk-1=I1(ω).
Similarly,(2.21)F[|τ|-(β+αk)]=2sin[(β+αk)π2]Γ[1-(β+αk)]|ω|(β+αk)-1=I2(ω).
Note thatF[u(τ+1)-u(τ-1)]=2Sa(ω). Doing the FT of the first term on the right side of (2.18) term-by-term yields the following:(2.22)F{[∑k=0∞(-1)kΓ[(β/α)+k]Γ(β/α)Γ(1+k)|τ|αk][u(τ+1)-u(τ-1)]}=12π{∑k=0∞(-1)kΓ[(β/α)+k]Γ(β/α)Γ(1+k)F(|τ|αk)}*F[u(τ+1)-u(τ-1)]={∑k=0∞(-1)kΓ[(β/α)+k]πΓ(β/α)Γ(1+k)I1(ω)}*Sa(ω)=∑k=0∞(-1)kΓ[(β/α)+k]πΓ(β/α)Γ(1+k)I1(ω)*Sa(ω).
In addition, computing the FT of the second term on the right side of (2.18) term-by-term yields(2.23)F{{∑k=0∞(-1)kΓ[(β/α)+k]Γ(β/α)Γ(1+k)|τ|-(β+αk)}[u(τ-1)+u(-τ-1)]}=12π{∑k=0∞(-1)kΓ[(β/α)+k]Γ(β/α)Γ(1+k)F[|τ|-(β+αk)]}*F[u(τ-1)+u(-τ-1)]={∑k=0∞(-1)kΓ[(β/α)+k]2πΓ(β/α)Γ(1+k)I2(ω)}*[2πδ(ω)-2Sa(ω)]=∑k=0∞(-1)kΓ[(β/α)+k]πΓ(β/α)Γ(1+k)[πI2(ω)-I2(ω)*Sa(ω)].
Adding the right sides of (2.22) and (2.23) yields the result of this theorem.Remark 2.8.
The EGC processes are non-Markovian sincerEGC(t1,t2) does not satisfy the triangular relation given by
(2.24)rEGC(t1,t3)=rEGC(t1,t2)rEGC(t2,t3)rEGC(t2,t2),t1<t2<t3,
which is a necessary condition for a Gaussian process to be Markovian; see Todorovic [97]. In fact, up to a multiplicative constant, the Ornstein-Uhlenbeck process is the only stationary Gaussian Markov process; see Lim and Muniandy [98], Wolperta and Taqqu [99].Note 1.
SincerEGC(τ)~|τ|-β for τ→∞, one has the Hurst parameter of the EGC processes given by
(2.25)HEGC=1-β2.
An EGC process is LRD if 0<β<1. It is short-range dependent (SRD) if β>1.Note 2.
Parameterα is independent of β and vice versa in the ACF of an EGC process.Note 3.
Dimensions of an EGC process relies on the value ofα, irrelevant with β. That is, dimensions of an EGC process is irrelevant with its statistic dependenc.I will discuss the meaning ofDEGC2 and DEGC3 in the next section.


## 3. Discussions

The emphasized point I will explain is that the ACF, (2.2), of the EGC process differs significantly from that of the GC process because I relax the restriction of α to be α>0 instead of 0<α≤2 as that in the GC process, though two ACFs appear the similar, referring [80, 81] for the details of the GC process. Relaxing the range of α from 0<α≤2 in the GC model to α>0 in this paper makes a considerable step further in the aspect of dimensions of random functions. To exhibit this step, I should explain the meaning of dimensions less than one and negative for a random function.The fractal indexα of a random function x(t) is considered for τ→0 in (2.10) or in the following expression if the ACF of x(t) is sufficiently smooth on (0,∞):(3.1)rxx(0)-rxx(τ)~c|τ|αfor|τ|⟶0,
where α relates to D by D=1-α/2. Obviously, rEGC is sufficiently smooth on (0,∞). As implied by (3.1), one sees that the larger the value of α, the smoother the sample path of a random function. The following notes become apparent, accordingly.Note 1.
The present fractal dimensions, sayDEGC2 and DEGC3, imply that conventionally random functions are not the locally smoothest because there are random functions with dimensions less than one, for example, DEGC2, or even negative, for example, DEGC3.Note 2.
The zero dimension occurs whenα=4. That is,
(3.2)DEGC=DEGC2=0forα=4.Note 3.
Random functions withDEGC=0 are not the locally smoothest. They are locally rougher than those with DEGC3<0.In the extreme case ofα→∞, I have
(3.3)lim⁡α→∞DEGC=-∞.
Because
(3.4)lim⁡α→∞|τ|α=0for|τ|⟶0,
we say that the locally smoothest random function is that with DEGC→-∞.Note 4.
ForDEGC→-∞, xEGC(t) is locally uncorrelated at any point of t. However, it may not be a white noise because it may globally be LRD if 0<β<1. As a matter of fact, there are the Hurst effects on xEGC(t) regardless of the value of DEGC.Note 5.
The standard GC process is a special case of the EGC process for0<α≤2 and β>0, which has applications to relaxation description in physics (Lim and Li [80]), the Internet traffic (Li and Lim [81], Li and Zhao [100]), chromatin morphologies in breast cancer cells (Muniandy and Stanslas [101]).Note 6.
The usual Cauchy process is a special case of the EGC process. In fact, whenα=β=2, one gets the ACF of the usual Cauchy process. Denote by rC(τ) the ACF of the usual Cauchy process. Then,
(3.5)rC(τ)=(1+|τ|2)-1.It is easily seen that the fractal dimension of the usual Cauchy process is one. It is SRD sincerC(τ)~|τ|-2 for τ→∞.In what follows, I denoterEGC(τ) by rEGC(τ;α,β) for facilitating the explanation. When α=2 and β=1, one has
(3.6)rEGC(τ;2,1)=(1+|τ|2)-1/2.
The above rEGC(τ;2,1) is the ACF proposed by Spector and Grant [102] for interpreting areomagnetic data, which is a special case of the EGC process. When α=2 and β=3, rEGC(τ) reduces to
(3.7)rEGC(τ;2,3)=(1+|τ|2)-3/2,
which has applications to magnetic fields; see Chilés and Delfiner [103]. The ACF of the Cauchy type stated by Chilés and Delfiner [103, page 86] is a reduced case of rEGC(τ) in the cases of α=2 and β>2b for b>0. That is,
(3.8)rEGC(τ;2,2b)=(1+|τ|2)-bforb>0.At moment, I am unaware what practical data may have the fractal dimensions less than one or negative but we are able to synthesize such data following the simulation method by Li [104]. Denote by w(t) the standard white noise. Let F-1 be the operator of the inverse Fourier transform. Denote by y(t) the synthesized random function that follows the ACF of the EGC process. Then,
(3.9)y(t)=w(t)*F-1{[F(1+|t|α)-β/α]0.5}.Replacingα and β by DEGC and HEGC, respectively, yields(3.10)y(t)=w(t)*F-1{[F(1+|t|4-2DEGC)-(1-HEGC)/(2-DEGC)]0.5}.
In the discrete case, we have
(3.11)y(i)=w(i)*IFFT{[FFT(1+|i|α)-β/α]0.5}=w(i)*IFFT{[FFT(1+|i|4-2DEGC)-(1-HEGC)/(2-DEGC)]0.5},
where FFT represents the fast Fourier transform and IFFT stands for its inverse. Figure 1 indicates the realizations of the EGC process with various values of α for β=0.8 (the LRD case).Realizations of the EGC process forβ=0.8, that is, HEGC=0.6. (a) Realization for α=0.1, that is, DEGC=1.95. (b) Realization for α=1, that is, DEGC=1.5. (c) Realization for α=2, that is, DEGC=1. (d) Realization for α=4, that is, DEGC=0. (e) Realization for α=8, that is, DEGC=-2. (f) Realization for α=16, that is, DEGC=-6.
(a)(b)(c)(d)(e)(f)


## 4. Extension to Corresponding Random Field with Negative Dimension

Denote byℝn the n-dimensional Euclidean space. The bold letters t and τ represent vectors belonging to ℝn, respectively denoting t=(t1,…,tn) and τ=(τ1,…,τn). The symbols ∥t∥ and ∥τ∥ represent their Euclidean norms.In the work by Lim and Teo [105], a random field X(t) is called a Gaussian field with the GC’s covariance function if its covariance is given by
(4.1)R(τ)=E[X(t)X(t+τ)]=(1+‖τ‖α)-β/α,
where β>0 and α is restricted by 0<α≤2. Lim and Teo termed X(t) as the GFGCC (Gaussian field with the generalized Cauchy covariance) in short. We denote X(t) by XGFGCC(t) for simplicity. Over the hyperrectangle C=∏i=1n[ai,bi], they obtained the positive fractal dimension of XGFGCC(t) by the following expression:
(4.2)DGFGCC=n+1-α2,0<α≤2.
Clearly,
(4.3)n≤DGFGCC<n+1.As previously discussed in Sections2 and 3, the restriction of α can be relaxed to α>0. Therefore, GFGCC can be extended to the case of α>0. We call such an extension by EGFGCC (extended Gaussian field with the generalized Cauchy covariance) and denote it by XEGFGCC(t). The fractal dimension of XEGFGCC(t) is expressed by(4.4)DEGFGCC=n+1-α2,α>0.The difference betweenXEGFGCC(t) and XGFGCC(t) is considerable because DEGFGCC may be negative while DGFGCC is always positive. As a matter of fact, we have
(4.5)DEGFGCC<0ifα>2(n+1).The meaning of negative dimension forXEGFGCC(t) is similar to that explained in Section 3. That is, locally, XEGFGCC(t) is more regular for smaller DEGFGCC.


## 5. Conclusions

I have explained that the GC process can be extended to the EGC process with dimensions less than 1 and negative. The EGC process is rich such that it takes the standard GC process as its special case. I have explained that the EGC process with smaller fractal dimensions is smoother than that with larger ones. The present results are theoretic but data with negative dimensions have been synthesized in this paper. One interesting thing, as a consequence of this paper, is to explore such a class of data in various fields, for example, either in time series as those in [13, 14, 106–127], or random fields [103].



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