A Class of Negatively Fractal Dimensional Gaussian Random Functions

Abstract

Let 𝑥(𝑡) be a locally self-similar Gaussian random function. Denote by 𝑟_𝑥𝑥(𝜏) the autocorrelation function (ACF) of 𝑥(𝑡). For 𝑥(𝑡) that is sufficiently smooth on (0,∞), there is an asymptotic expression given by 𝑟_𝑥𝑥(0)−𝑟_𝑥𝑥(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.

1. Introduction

Conventionally, for a time series, or a random function 𝑥(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 of 𝑥(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 𝐷∈[1,2). (1.1)

The literature with respect to the time series with 𝐷∈[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 𝐷_fGn and 𝐻_fGn the fractal dimension of fGn and its Hurst parameter, respectively. Denote fGn by 𝑥_fGn(𝑡). Then, the autocorrelation function (ACF) of fGn follows 𝐸𝑥_fGn(𝑡)𝑥_fGn(𝑡+𝜏)≈𝐻_fGn2𝐻_fGn−1𝜏^{(2𝐻fGn−2)}. (1.2) The right side of the above expression requires 0<𝐻_fGn<1; see Mandelbrot [17]. Thus, 1<𝐷_fGn<2 because 𝐷_fGn=2−𝐻_fGn. In the LRD case, 0.5<𝐻_fGn<1 and 1.5<𝐷_fGn<2. As a result, I have the following note.

Note 1. The dimension of fGn is never negative. That is, 1≤𝐷_fGn<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.

Let 𝐷_GC and 𝐻_GC be the fractal dimension and the Hurst parameter of the GC process, respectively. Then, 𝐷_GC and 𝐻_GC are independent of each other. In the previous research regarding the GC process, only the case of 1<𝐷_GC<2 was discussed, see [79–81], Li [52], Li and Lim [83, 84]. In this paper, I extend the GC process such that 𝐷_EGC∈(0,1) and 𝐷_EGC<0, where 𝐷_EGC is the fractal dimension of the EGC process in addition to the traditional case of 𝐷_EGC∈[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 𝐷_EGC<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 Section 2. 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,…,𝑥_𝑛;𝑡₁,…,𝑡_𝑛) for the random variables 𝑥(𝑡₁),…,𝑥(𝑡_𝑛). The joint distribution density function is written by 𝑝𝑥₁,…,𝑥_𝑛;𝑡₁,…,𝑡_𝑛=𝜕^𝑛𝑃𝑥₁,…,𝑥_𝑛;𝑡₁,…,𝑡_𝑛𝜕𝑥₁⋯𝜕𝑥_𝑛. (2.1)

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 𝑟_EGC(𝜏) that is the ACF of a Gaussian random function denoted by 𝑥_EGC(𝑡). That is, 𝑟_EGC(𝜏)=𝐸𝑥_EGC(𝑡+𝜏)𝑥_EGC(𝑡)=1+|𝜏|^{𝛼−𝛽/𝛼},𝛼>0,𝛽>0. (2.2) 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 function 𝑥(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[𝑥_EGC(𝑡)𝑥_EGC(𝑡)]=𝑟_EGC(0)=1 regardless of the values of 𝛼 and 𝛽. Therefore, it is inconvenient to use ‖𝑥_EGC(𝑡)‖. For this reason, I utilize ‖𝑟_EGC(𝜏)‖ rather than ‖𝑥_EGC(𝑡)‖ in this paper. The norm ‖𝑟_EGC(𝜏)‖ is suitable for our research purpose. In fact, since 𝑥_EGC(𝑡) is Gaussian, it is uniquely determined by 𝑟_EGC(𝜏) and vice versa. Note that ^∞_−∞𝑟_EGC(𝜏)²𝑑𝜏=∞for0<𝛽<0.5. (2.3) Thus, I need to express ‖𝑟_EGC(𝜏)‖ in the domain of generalized functions.

Definition 2.1 (see Griffel [91]). A function of rapid decay is a smooth function 𝜙∶ℝ→ℂ such that 𝑡^𝑛𝜙^(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 to 𝐒 is absolutely integrable.

Now, define the norm and inner product of 𝑟∈𝐻_EGC by 𝑟_EGC²=𝑟_EGC,𝑟_EGC=^+∞_−∞𝑟_EGC(𝑢)²𝑔(𝑢)𝑑𝑢, (2.4) where 𝑔∈𝐒. Combining any 𝑟∈𝐻_EGC with its limit yields that 𝐻_EGC is a Hilbert space. Denote it by 𝐻_EGC=𝑟_EGC;𝑟_EGC²<∞;𝛼>0,𝛽>0. (2.5)

For 0<𝛼≤2, I express the subspace of 𝐻_EGC by 𝐻_EGC1=𝑟_EGC;𝑟_EGC²<∞;0<𝛼≤2,0<𝛽. (2.6)

Denote by 𝐻_GC the space of the standard GC process. Then, one immediately sees that 𝐻_GC is a subspace of 𝐻_EGC. More precisely, 𝐻_GC=𝐻_EGC1. (2.7)

In the case of 2<𝛼≤4, I denote another subspace of 𝐻_EGC by 𝐻_EGC2=𝑟_EGC;𝑟_EGC²<∞;2<𝛼≤4,0<𝛽. (2.8) Further, I denote the subspace of 𝐻_EGC for 𝛼>4 by 𝐻_EGC3=𝑟_EGC;𝑟_EGC²<∞;4<𝛼,0<𝛽. (2.9) Then, I have the remark below.

Remark 2.3. One has 𝐻_EGC=𝐻_EGC1∪𝐻_EGC2∪𝐻_EGC3.

Recall that each 𝑟_EGC∈𝐻_EGC 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 𝐻_EGC2 and 𝐻_EGC3 are undefined, more precisely, unknown. The following theorems will describe the dimensions of processes in 𝐻_EGC2 and 𝐻_EGC3.

Theorem 2.4. Processes in 𝐻_EGC2 have the dimensions less than one and greater than or equal to zero.

Proof. Denote a process in 𝐻_EGC2 by 𝑥_EGC2(𝑡). Let an ACF in 𝐻_EGC2 be 𝑟_EGC2. 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→0𝑟_EGC2(𝜏)=𝐸𝑥_EGC2(𝑡+𝜏)𝑥_EGC2(𝑡𝛽)=1−𝛼|𝜏|^𝛼1+𝑂|𝜏|^𝛼,2<𝛼≤4,0<𝛽. (2.10) Therefore, according to [93–96], the fractal dimension denoted by 𝐷_EGC2 is given by 0≤𝐷_EGC2=𝛼1−2<1. (2.11) This yields Theorem 2.4.

Let 𝑥_EGC3(𝑡) be a process in 𝐻_EGC3. Denote the ACF of 𝑥_EGC3(𝑡) by 𝑟_EGC3. Then, the following theorem gives the negative dimensions of 𝑥_EGC3(𝑡).

Theorem 2.5. Processes in 𝐻_EGC3 have negative dimensions.

Proof. Similar to (2.10), one has 𝑟_EGC3(𝜏)=𝐸𝑥_EGC3(𝑡+𝜏)𝑥_EGC3(𝑡𝛽)=1−𝛼|𝜏|^𝛼1+𝑂|𝜏|^𝛼,4<𝛼,0<𝛽. (2.12) Then, following (2.12), we have 𝐷_EGC3=𝛼1−2<0. (2.13) This completes the proof of Theorem 2.5.

Remark 2.6. For an EGC process 𝑥_EGC(𝑡) with 0<𝛼 and 0<𝛽, the fractal dimension of 𝑥_EGC(𝑡) in general satisfies 𝐷_EGC<2. (2.14)

Theorem 2.7. The power spectrum density (PSD) function of the EGC process is given 𝑆_EGC(𝜔)=^∞_𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘Γ()𝐼𝜋Γ𝛽/𝛼1+𝑘¹(𝜔)(𝜔)+∗𝑆𝑎_∞𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘𝜋Γ𝛽/𝛼Γ(1+𝑘)𝜋𝐼₂(𝜔)−𝐼₂(𝜔)∗𝑆𝑎(𝜔), (2.15) where 𝜔 is angular frequency, * implies the convolution, 𝑆𝑎(𝜔)=sin(𝜔)/𝜔 and 𝐼₁(𝜔)=−2sin𝛼𝑘𝜋2Γ(𝛼𝑘+1)|𝜔|^{−𝛼𝑘−1},𝐼₂(𝜔)𝜋=2sin𝛽+𝛼𝑘2Γ1−𝛽+𝛼𝑘|𝜔|^{(𝛽+𝛼𝑘)−1}. (2.16)

Proof. Note that (1+𝑥)^𝜈 can be expanded as a binomial series given by (1+𝑥)^𝜈=^∞_𝑘=0𝜈𝑘𝑥^𝑘=^∞_𝑘=0Γ(𝜈+𝑘)Γ(𝜈)Γ()𝑥1+𝑘^𝑘for|𝑥|<1, (2.17) where 𝜈∈ℝ. and (^𝜈_𝑘) is the binomial coefficient.
Now, I expand the ACF of the EGC process by𝐶(𝜏)=^∞_𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘ΓΓ()|𝜏|𝛽/𝛼1+𝑘^𝛼𝑘𝑢()(+[𝜏+1−𝑢𝜏−1)]_∞𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘ΓΓ()|𝜏|𝛽/𝛼1+𝑘^{−(𝛽+𝛼𝑘)}[𝑢(𝜏−1)+𝑢(−𝜏−1)], (2.18) where 𝑢(𝜏) is the Heaviside unit step function (Li and Lim [84]).
Because the Fourier transform (FT) of |𝑡|^𝜆 is expressed by𝐹|𝑡|^𝜆=−2sin𝜆𝜋2Γ(𝜆+1)|𝜔|^−𝜆−1, (2.19) where 𝜆≠−1,−3,…, I have 𝐹|𝜏|^𝛼𝑘=−2sin𝛼𝑘𝜋2Γ(𝛼𝑘+1)|𝜔|^{−𝛼𝑘−1}=𝐼₁(𝜔). (2.20)
Similarly,𝐹|𝜏|^{−(𝛽+𝛼𝑘)}𝜋=2sin𝛽+𝛼𝑘2Γ1−𝛽+𝛼𝑘|𝜔|^{(𝛽+𝛼𝑘)−1}=𝐼₂(𝜔). (2.21)
Note that 𝐹[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:𝐹^∞_𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘ΓΓ()|𝜏|𝛽/𝛼1+𝑘^𝛼𝑘𝑢()(=1[𝜏+1−𝑢𝜏−1)]2𝜋^∞_𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘ΓΓ()𝐹|𝜏|𝛽/𝛼1+𝑘^𝛼𝑘[𝑢()(=∗𝐹𝜏+1−𝑢𝜏−1)]_∞𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘Γ()𝐼𝜋Γ𝛽/𝛼1+𝑘¹(𝜔(𝜔)=)∗𝑆𝑎_∞𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘Γ()𝐼𝜋Γ𝛽/𝛼1+𝑘¹(𝜔)∗𝑆𝑎(𝜔). (2.22)
In addition, computing the FT of the second term on the right side of (2.18) term-by-term yields𝐹^∞_𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘ΓΓ()|𝜏|𝛽/𝛼1+𝑘^{−(𝛽+𝛼𝑘)}𝑢()(=1[𝜏−1+𝑢−𝜏−1)]2𝜋^∞_𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘ΓΓ()𝐹𝜏|𝛽/𝛼1+𝑘|^{−(𝛽+𝛼𝑘)}[𝑢()(=∗𝐹𝜏−1+𝑢−𝜏−1)]_∞𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘Γ()𝐼2𝜋Γ𝛽/𝛼1+𝑘²(𝜔∗[(𝜔)(𝜔=)2𝜋𝛿−2Sa)]_∞𝑘=0(−1)^𝑘Γ𝛽/𝛼+𝑘𝜋Γ𝛽/𝛼Γ(1+𝑘)𝜋𝐼₂(𝜔)−𝐼₂(𝜔)∗𝑆𝑎(𝜔). (2.23) 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 since 𝑟_EGC(𝑡₁,𝑡₂) does not satisfy the triangular relation given by 𝑟_EGC𝑡₁,𝑡₃=𝑟_EGC𝑡₁,𝑡₂𝑟_EGC𝑡₂,𝑡₃𝑟_EGC𝑡₂,𝑡₂,𝑡₁<𝑡₂<𝑡₃, (2.24) 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. Since 𝑟_EGC(𝜏)∼|𝜏|^−𝛽 for 𝜏→∞, one has the Hurst parameter of the EGC processes given by 𝐻_EGC𝛽=1−2. (2.25) An EGC process is LRD if 0<𝛽<1. It is short-range dependent (SRD) if 𝛽>1.

Note 2. Parameter 𝛼 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 of 𝛼, irrelevant with 𝛽. That is, dimensions of an EGC process is irrelevant with its statistic dependenc.

I will discuss the meaning of 𝐷_EGC2 and 𝐷_EGC3 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 index 𝛼 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,∞):𝑟_𝑥𝑥(0)−𝑟_𝑥𝑥(𝜏)∼𝑐|𝜏|^𝛼for|𝜏|⟶0, (3.1) where 𝛼 relates to 𝐷 by 𝐷=1−𝛼/2. Obviously, 𝑟_EGC 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, say 𝐷_EGC2 and 𝐷_EGC3, imply that conventionally random functions are not the locally smoothest because there are random functions with dimensions less than one, for example, 𝐷_EGC2, or even negative, for example, 𝐷_EGC3.

Note 2. The zero dimension occurs when 𝛼=4. That is, 𝐷_EGC=𝐷_EGC2=0for𝛼=4. (3.2)

Note 3. Random functions with 𝐷_EGC=0 are not the locally smoothest. They are locally rougher than those with 𝐷_EGC3<0.

In the extreme case of 𝛼→∞, I have lim_𝛼→∞𝐷_EGC=−∞. (3.3) Because lim_𝛼→∞|𝜏|^𝛼=0for|𝜏|⟶0, (3.4) we say that the locally smoothest random function is that with 𝐷_EGC→−∞.

Note 4. For 𝐷_EGC→−∞, 𝑥_EGC(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 𝑥_EGC(𝑡) regardless of the value of 𝐷_EGC.

Note 5. The standard GC process is a special case of the EGC process for 0<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, when 𝛼=𝛽=2, one gets the ACF of the usual Cauchy process. Denote by 𝑟_C(𝜏) the ACF of the usual Cauchy process. Then, 𝑟_C(𝜏)=1+|𝜏|²⁻¹. (3.5)

It is easily seen that the fractal dimension of the usual Cauchy process is one. It is SRD since 𝑟_C(𝜏)∼|𝜏|⁻² for 𝜏→∞.

In what follows, I denote 𝑟_EGC(𝜏) by 𝑟_EGC(𝜏;𝛼,𝛽) for facilitating the explanation. When 𝛼=2 and 𝛽=1, one has 𝑟_EGC(𝜏;2,1)=1+|𝜏|^2−1/2. (3.6) The above 𝑟_EGC(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, 𝑟_EGC(𝜏) reduces to 𝑟_EGC(𝜏;2,3)=1+|𝜏|^2−3/2, (3.7) 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 𝑟_EGC(𝜏) in the cases of 𝛼=2 and 𝛽>2𝑏 for 𝑏>0. That is, 𝑟_EGC(𝜏;2,2𝑏)=1+|𝜏|^2−𝑏for𝑏>0. (3.8)

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, 𝑦(𝑡)=𝑤(𝑡)∗𝐹⁻¹𝐹1+|𝑡|^{𝛼−𝛽/𝛼0.5}. (3.9)

Replacing 𝛼 and 𝛽 by 𝐷_EGC and 𝐻_EGC, respectively, yields𝑦(𝑡)=𝑤(𝑡)∗𝐹⁻¹𝐹1+|𝑡|^{4−2𝐷EGC−(1−𝐻EGC)/(2−𝐷EGC)0.5}. (3.10) In the discrete case, we have 𝑦(𝑖)=𝑤(𝑖)∗IFFTFFT1+|𝑖|^{𝛼−𝛽/𝛼0.5}=𝑤(𝑖)∗IFFTFFT1+|𝑖|^{4−2𝐷EGC−(1−𝐻EGC)/(2−𝐷EGC)0.5}, (3.11) 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).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 1 

Realizations of the EGC process for 𝛽=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.

4. Extension to Corresponding Random Field with Negative Dimension

Denote by ℝ^𝑛 the 𝑛-dimensional Euclidean space. The bold letters 𝐭 and 𝝉 represent vectors belonging to ℝ^𝑛, respectively denoting 𝐭=(𝑡₁,…,𝑡_𝑛) and 𝝉=(𝜏₁,…,𝜏_𝑛). 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 𝐑(𝝉)=𝐸[𝐗(𝐭)𝐗(𝐭+𝝉)]=1+‖𝝉‖^{𝛼−𝛽/𝛼}, (4.1) 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 𝐗_GFGCC(𝐭) for simplicity. Over the hyperrectangle 𝐶=∏^𝑛_𝑖=1[𝑎_𝑖,𝑏_𝑖], they obtained the positive fractal dimension of 𝐗_GFGCC(𝐭) by the following expression: 𝐷_GFGCC𝛼=𝑛+1−2,0<𝛼≤2. (4.2) Clearly, 𝑛≤𝐷_GFGCC<𝑛+1. (4.3)

As previously discussed in Sections 2 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 𝐗_EGFGCC(𝐭). The fractal dimension of 𝐗_EGFGCC(𝐭) is expressed by𝐷_EGFGCC𝛼=𝑛+1−2,𝛼>0. (4.4)

The difference between 𝐗_EGFGCC(𝐭) and 𝐗_GFGCC(𝐭) is considerable because 𝐷_EGFGCC may be negative while 𝐷_GFGCC is always positive. As a matter of fact, we have 𝐷_EGFGCC<0if𝛼>2(𝑛+1). (4.5)

The meaning of negative dimension for 𝐗_EGFGCC(𝐭) is similar to that explained in Section 3. That is, locally, 𝐗_EGFGCC(𝐭) is more regular for smaller 𝐷_EGFGCC.

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].

Acknowledgment

This paper was supported in part by the National Natural Science Foundation of China under the project Grant nos. 60573125, 60873264, 61070214, and the 973 plan under the project no. 2011CB302800/2011CB302802.