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Fractal time series substantially differs from conventional one in its statistic properties. For instance, it may have a heavy-tailed probability distribution function (PDF), a slowly decayed autocorrelation function (ACF), and a power spectrum function (PSD) of

Denote by

Denote an element belonging to

Note that the nature is rich and colorful (Mandelbrot [

We now turn to time series. Intuitively, we say that

The theory of conventional series is relatively mature; see, for example, Fuller [

The remaining article is organized as follows. In Section

A time series can be taken as a solution to a differential equation. In terms of engineering, it is often called signal while a differential equation is usually termed system, or filter. Therefore, without confusions, equation, system, or filter is taken as synonyms in what follows.

A stationary time series can be regarded as the output

A stochastic filter can be written by

Denote the Laplace transform of

Taking into account

In the discrete case, the system function is expressed by the

A realization

Let

Let

Without loss of the generality to explain the concept of fractal time series, we reduce (

A realization

For a stochastically fractional differential equation, Note

We shall further explain Note

Replacing

FBm is a special case as a realization of a fractional filter driven with

Other articles discussing fBm from the point of view of systems or filters of fractional order can be seen in Ortigueira [

Fractal time series has its particular properties in comparison with the conventional one. Its power law in general is closely related to the concept of memory. A particular point, which has to be paid attention to, is that there may usually not exist mean and/or variance in such a series. This may be a main reason why measures of fractal dimension and the Hurst parameter play a role in the field of fractal time series.

Denote the ACF of

Denote the PSD of

The PSD of an LRD fractal series is divergent for

Denote the PDF of

Denote

One thing remarkable in LRD fractal time series is that the tail of

Note that

In fractal time series, one, respectively, uses the fractal dimension and the Hurst parameter of

On the other side, expressing

In passing, we mention that the estimation of

In the end of this section, we note that self-similarity of a stationary process is a concept closely relating to fractal time series. Fractional Gaussian noise (fGn) is an only stationary increment process with self-similarity (Samorodnitsky and Taqqu [

Fractal time series can be classified into two classes from a view of statistical dependence. One is LRD and the other is SRD. It can be also classified into Gaussian series or nonGaussian ones. I shall discuss the models of fractal time series of Gaussian type in Sections

FBm is commonly used in modeling nonstationary fractal time series. It is Gaussian (Sinai [

Note that the increment process of the fBm of the Riemann-Liouville type is nonstationary (Lim and Muniandy [

The Weyl integral of order

Either the fBm of the Riemann-Liouville type or the one of the Weyl type is nonstationary as can be seen from (

Both the fBm of the Riemann-Liouville type and the one of the Weyl type are self-similar because they have the property expressed by

The PSD of fBm is divergent at

The process fBm reduces to the standard Brownian motion when

A consequence of Note

The fractal dimension of fBm is given by

Recall that the fractal dimension of a sample path represents its self-similarity. For fBm, however,

By using

Assume that

Based on the local growth of the increment process, one may write a sequence expressed by

The continuous fGn is the derivative of the smoothed fBm that is in the domain of generalized functions. Its ACF denoted by

FGn includes three classes of time series. When

The PSD of fGn is given by (Li and Lim [

Denote the discrete fGn by dfGn. Then, the ACF of dfGn is given by

Note that the expression

The fGn as the increment process of the fBm of the Weyl type is stationary. It is exactly self-similar with the global self-similarity described by (

The PSD of the fGn is divergent at

Again, we remark that the fGn may be too strict for modeling a real series in practice. Hence, generalized versions of fGn are expected. One of the generalization of fGn is to replace

As discussed in Section

A series

The function

When considering the multiscale property of a series, one may utilize the time varying

In practice, the asymptotic expressions of

The GC process is LRD if

The GC process has the local self-similarity measured by

The GC process is nonMarkovian since

The above discussions exhibit that the GC model can be used to decouple the local behavior and the global one of fractal time series, flexibly better agreement with the real data for both short-term and long-term lags. Li and Lim gave an analysis of the modeling performance of the GC model in Hilbert space [

As previously mentioned, two-parameter models are useful as they can separately characterize the local irregularity and global persistence. The CG process is one of such models and it is Gaussian. In some applications, for example, network traffic at small scales, a series is nonGaussian; see, for example, Scherrer et al. [

Stable distributions imply a family of distributions. They are defined by their characteristic functions given by [

The parameters in

The parameter

The parameter

The parameter

The family of

The property of heavy tail is described as follows.

When

Alpha-stable processes are in general nonGaussian. They include two. One is linear fractional stable noise (LFSN) and the other log-fractional stable noise (Log-FSN).

The model of linear fractional stable motion (LFSM) is defined by the following stochastic integral [

Denote by

LSFN is the increments process of LSFM while Log-FSN is the increment process of Log-FSM. Denote the LSFN and Log-FSN respectively by

LSFN is nonGaussian except

In the above subsections, the series may be LRD. We now turn to a type of SRD fractal time series called OU processes.

Following the idea addressed by Uhlenbeck and Ornstein [

Denote the Fourier transforms of

The ordinary OU process is obviously SRD. It is one-dimensional. What interests people in the field of fractal time series is the generalized OU processes described hereinafter.

Consider the following fractional Langevin equation with a single parameter

Note that the PSD of

Let

Keep in mind that the Langevin equation is in the sense of generalized functions since we take

The generalized OU process governed by (

According to (

We now further extend the Langevin equation by indexing it with two fractions

Note that

The local irregularity of series relies on the fractal dimension instead of the statistical dependence. The local irregularity of an SRD series may be strong if its fractal dimension is large.

The concepts, such as power law in PDF, ACF, and PSD in fractal time series, have been discussed. Both LRD and SRD series have been explained. Several models, fBm, fGn, the GC process, alpha-stable processes, and generalized OU processes have been interpreted. Note that several models revisited above are a few in the family of fractal time series. There are others; see, for example, [

This work was partly supported by the National Natural Science Foundation of China (NSFC) under the project Grant nos. 60573125 and 60873264.