*1.2. Examination of Multifractality*

Copious methods have been proposed to examine the multifractality in a certain process {*X*(*t*), 0 ≤ *t* ≤ *<sup>T</sup>*}, including the *multifractal detrended fluctuation analysis method* (MF-DFA) by Kantelhardt et al. (2002) and the *wavelet transform modulus maxima method* (WTMM). In this paper, we employ the *standard partition method* introduced by Mandelbrot et al. (1997). This method attempts to detect multifractality based on concavity in the scaling function. From Definition 1, one can easily derive

$$\log \mathbb{E}|X(t)|^q = \log c(q) + \tau(q)\log t,\text{ for all } q \in [0,Q] \text{ and } t \in [0,T]. \tag{3}$$

where the *q*-th moment of the process *<sup>X</sup>*(*t*), <sup>E</sup>|*X*(*t*)|*<sup>q</sup>*, can be estimated by the sample statistic

$$S\_q(T, \Delta t) = \frac{1}{N} \sum\_{i=1}^{N} |X((i-1)\Delta t) - X(i\Delta t)|^q,\tag{4}$$

with *N* = *T*/Δ*t*.

The scaling function can be derived by fitting a linear regression between log *Sq*(*<sup>T</sup>*, Δ*t*) and log *t* with various values of *q* ∈ [0, Q]. For a fixed value of *q*, we obtain an estimate *<sup>τ</sup>*<sup>ˆ</sup>(*q*), the slope of our linear regression. A plot of *τ*<sup>ˆ</sup>(*q*) against *q* then provides visual display of the scaling function. The function of ln *Sq*(*<sup>T</sup>*, Δ*t*) for various values of ln Δ*t* is called *the partition function*. If the scaling function shows concavity, we have evidence that multifractality is present in the process of interest.
