Next, we recall how FD algorithms works for fractal dimension calculation purposes. The next remark becomes especially appropriate to tackle with computational applications involving fractal structures.

Let *d*_{n} denote the mean of {diam (*A*)}_{A∈Δn}, that is a sample of *M*(2^{-n}, *ω*) (c.f. Remark 2). Thus, *d*_{n} approaches the mean of *M*(2^{-n}, *ω*). Then by Theorem 7 (7), ${r}_{n}=\frac{{d}_{n+1}}{{d}_{n}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{s}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}r=\frac{1}{{2}^{H}}$, a constant. Thus, *H* = -log_{2} *r*, and hence, Theorem 7 gives $\mathrm{d}\mathrm{i}\mathrm{m}(\alpha )\simeq -\frac{1}{{\mathrm{log}}_{2}r}$. The previous arguments lead to a first approach we shall refer to as FD1 algorithm.

#### Algorithm (FD1)

*[3, Algorithm 1]*

*Let d*_{n} be the mean of {diam(*A*)}_{A∈Δn} *for* 1 ≤ *n* ≤ log_{2} *l*.

*Define* ${r}_{n}=\frac{{d}_{n+1}}{{d}_{n}}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}for\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}1\le n\le {\mathrm{log}}_{2}l-1$.

*Calculate r as the mean of* {*r*_{n}}_{1 ≤n≤log2l-1}.

*Return* $\mathrm{d}\mathrm{i}\mathrm{m}(\alpha )=-\frac{1}{{\mathrm{log}}_{2}r},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}H=-{\mathrm{log}}_{2}r$.

It is worth noting that FD1 is valid to calculate the parameter of random processes satisfying E[

*X*_{n}] = 2

^{H} · E[

*X*_{n+1}]. In particular, it works if

*X*_{n} ∼ 2

^{H} ·

*X*_{n+1}, which is the case of the processes under the hypotheses of

Theorem 7.

It is worth pointing out that GM2 algorithm (first provided in [21], and revisited afterwards in [15] from the viewpoint of fractal structures) becomes also valid to calculate the parameter of random processes lying under the condition E[*X*_{1}] = 2^{(n-l)H} · E[*X*_{n}]. Moreover, since that equality is equivalent to E[*X*_{n}] = 2^{H} · E[*X*_{n+1}], then the validity of GM2 approach is equivalent to the validity of FD1. For a description of GM2 algorithm in terms of fractal structures, we refer the reader to [15, Algorithm 3.6].

In Figure 4, we illustrate how the parameter of a time series may be calculated via FD1 algorithm. In that case, they have been plotted the values of the coefficients *r*_{n} and *r* (their mean value) for a 2^{11} point time series from a BM. This makes a total amount of 11 levels in the induced fractal structure *Δ*.

Figure 4 The markers in the plot above correspond to the coefficients *r*_{n}, and the straight line depicts their mean. Such a graphical representation has been carried out for a 2048 point BM

On the other hand, recall that the (absolute) *s*-moment of a random variable *X* (under the assumption that such a value exists) is defined as *m*_{s}(*X*) = E[*X*^{s}]. In addition, let {*x*_{k} : *k* = 1,...,*l*} be a sample of length *l* from a random variable *X*. Its sample *s*-moment will be calculated by the following expression, as usual:
$${m}_{s}(X)=\frac{1}{l}\cdot \sum _{i=1}^{l}{x}_{i}^{s}.$$

#### Theorem 9

*(c*.*f*. *[3*, *Theorem 3]) Let α* : [0, 1] → ℝ *be a sample function of 𝐗*, *Γ be the natural fractal structure on* [0, 1], *Δ be the fractal structure induced by Γ on α*([0, 1]), *and X*_{n} = *M*(2^{-n}, *ω*). *If there exists s* ≥ 0 *satisfying the two following properties:*

*The s-moment of X*_{n}, *m*_{s}(*X*_{n}), *is finite*, *and*

${m}_{s}({X}_{n})=2\cdot {m}_{s}({X}_{n+1})$,

*then s* = dim(

*α*).

The following result contains sufficient conditions to verify the hypothesis (9) in Theorem 9.

#### Theorem 10

*Let 𝐗 be a random process with parameter H*, *α* : [0, 1] → ℝ *be a sample function of 𝐗*, *X*_{n} = *M*(2^{-n}, *ω*), *and assume that*$${X}_{n}\sim {\mathcal{T}}_{n}^{H}\cdot {X}_{0},$$(6)*for* 𝒯_{n} = 2^{-n}. *Then the hypothesis (9) in Theorem 9 stands for* $H=\frac{1}{s}$, *i*.*e*.,
$${m}_{\frac{1}{H}}({X}_{n})=2\cdot {m}_{\frac{1}{H}}({X}_{n+1}).$$

*In particular*, $H=\frac{1}{\mathrm{d}\mathrm{i}\mathrm{m}(\alpha )}$.

Accordingly, the next result follows immediately from Theorem 10.

#### Corollary 11

*Let 𝐗 be a random process with self-affine increments with parameter H and stationary*. *In addition*, *let α* : [0, 1] → ℝ *be a sample function of 𝐗*. *Then the hypothesis (9) in Theorem 9 stands for* $H=\frac{1}{s}$. *In particular*, $H=\frac{1}{\mathrm{d}\mathrm{i}\mathrm{m}(\alpha )}$.

Notice that FBMs and FLSMs lie under the hypothesis (9) of Theorem 9.

#### Corollary 12

*(c*.*f*. *[3*, *Corollaries 1 and 2]) Let α* : [0, 1] → ℝ *be a sample function of either a FBM or a FLSM with parameter H*. *Then the hypothesis (9) in Theorem 9 stands for* $H=\frac{1}{s}$. *In particular*, $H=\frac{1}{\mathrm{d}\mathrm{i}\mathrm{m}(\alpha )}$.

#### Algorithm (FD2)

(*c*.*f*. *[3*, *Algorithm 2]) For s* > 0,

*Calculate y*_{s} = {*y*_{k,s}}_{k = 1,..., log2l−1}, *where*$${y}_{k,s}=\frac{{m}_{s}({X}_{k})}{{m}_{s}({X}_{k+1})}.$$

*Let* $\frac{}{{y}_{s}}$ *be the mean of each list y*_{s}.

*Calculate the point s*_{0} *such that* $\frac{}{{y}_{{s}_{0}}}$ = 2. *Observe that* {(*s*, $\frac{}{{y}_{s}}$) : *s* > 0} *is s-increasing*.

*Thus*, dim(*α*) = *s*_{0} (*due to Theorem 9*), *and* $H=\frac{1}{{s}_{0}}$ (*by Theorem 7*).

It turns out that FD2 approach is valid to properly calculate the parameter of processes lying under the condition
$$\mathrm{E}[{X}_{n}^{\frac{1}{H}}]=2\cdot \mathrm{E}[{X}_{n+1}^{\frac{1}{H}}],$$
i.e., the hypothesis (9) in Theorem 9. In particular, it holds for random functions such that
$${X}_{n}^{\frac{1}{H}}\sim 2\cdot {X}_{n+1}^{\frac{1}{H}}.$$
Going beyond, if ${X}_{n}\sim {2}^{H}\cdot {X}_{n+1},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{X}_{n}^{\frac{1}{H}}\sim 2\cdot {X}_{n+1}^{\frac{1}{H}}$, and hence, both Theorems 7 and 9 guarantee that FD2 is valid to estimate the parameter *H* of processes with self-affine and stationary increments. Figure 5 graphically shows how to computationally deal with the calculations involved in FD2 algorithm. It depicts the graph of $\frac{}{{y}_{s}}$ in terms of *s*. The *s*-increasing nature of the function $\frac{}{{y}_{s}}$ makes easy to find out the value *s*_{0} such that $\frac{}{{y}_{{s}_{0}}}$ = 2, namely, the fractal dimension of such a sample function. In this case, a 2048 point BM was considered for illustration purposes.

Figure 5 Example of a graph representation of *s* vs. $\frac{}{{y}_{s}}$ for a 2048 point BM (c.f. Algorithm FD2)

Next step is to describe the so-called FD3 algorithm, an alternative to FD2 approach, which is also based on Theorem 9. First, let us sketch some theoretical notes. In fact, it is clear that the condition *m*_{s}(*X*_{n}) = 2 · *m*_{s}(*X*_{n+1}) is equivalent to
$${m}_{s}({X}_{n})=\frac{1}{{2}^{n-1}}\cdot {m}_{s}({X}_{1}).$$(7)
Taking 2-base logarithms in Eq. (7), we have that
$${\mathrm{log}}_{2}{m}_{s}({X}_{n})=-n+\gamma ,$$(8)
where *γ* = 1 + log_{2} *m*_{s}(*X*_{1}) remains constant. Thus, Eq. (8) provides a linear relationship between *n* and log_{2} *m*_{s}(*X*_{n}). The algorithm based on the ideas described above is stated as follows.

#### Algorithm (FD3)

*(c*.*f*. *[3*, *Algorithm 3]) For s* > 0,

*Calculate* {(*k*,*β*_{k,s})}_{k = 1,..., log2l}, *where β*_{k,s} = log_{2} *m*_{s}(*X*_{k}). *Let β*_{s} be the slope of the regression line of {(*k*, *β*_{k,s})}_{k = 1,..., log2l}.

*Calculate s*_{1} *such that β*_{sl} = −1.

*Hence*, dim(*α*) = *s*_{1} *(due to both Theorem 9 and Eq*. *(8))*, *and* $H=\frac{1}{{S}_{1}}$ *(by Theorem 7)*.

It is worth noting that FD3 approach is valid to calculate the parameter of random processes satisfying the equality
$$\mathrm{E}[{X}_{n}^{\frac{1}{H}}]=\frac{1}{{2}^{n-1}}\cdot \mathrm{E}[{X}_{1}^{\frac{1}{H}}].$$

Since that expression is equivalent to
$$\mathrm{E}[{X}_{n}^{\frac{1}{H}}]=2\cdot \mathrm{E}[{X}_{n+1}^{\frac{1}{H}}],$$
then FD3 is valid to calculate the parameter of 𝐗, if and only if, FD2 is valid for that purpose, where the increments of 𝐗 are self-affine and stationary.

Additionally, it holds that FD3 procedure also allows to verify the condition *m*_{s}(*X*_{n}) = 2 · *m*_{s}(*X*_{n+1}) (c.f. hypothesis (9) in Theorem 9). In fact, since Eq. (8) is equivalent to such a condition, then we can check that some empirical data lie under such a condition if Eq. (8) stands, i.e., if the regression coefficient in Eq. (8) is close to 1.

#### Theorem 13

*(c*.*f*. *[4*, *Theorem 3.1]) Assume that the increments of 𝐗 are stationary*. *If the next relationship stands for some H* > 0*:*$$M(\mathcal{T},\omega )\sim {\mathcal{T}}^{H}\cdot \phantom{\rule{thinmathspace}{0ex}}M(1,\omega ),$$(9)*then FD algorithms as well as GM2 approach are valid to calculate H*.

Theorem 13 leads to the following corollaries.

#### Corollary 14

*FD algorithms and GM2 procedure are valid to calculate the parameter of random processes with self-affine and stationary increments*.

#### Corollary 15

*Let 𝐗 be a FBM or a FLSM with parameter H*. *Then FD algorithms and GM2 approach are valid to calculate H*.

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