Definition of the fluctuation function in the detrended fluctuation analysis and its variants

Abstract

The detrended fluctuation analysis (DFA) and its variants are popular methods to analyze the self-similarity of a signal. Two steps characterize them: firstly, the trend of the centered integrated signal is estimated and removed. Secondly, the properties of the so-called fluctuation function which is an approximation of the standard deviation of the resulting process is analyzed. However, it appears that the statistical mean was assumed to be equal to zero to obtain it. As there is no guarantee that this assumption is true a priori, this hypothesis is debatable. The purpose of this paper is to propose two alternative definitions of the fluctuation function. Then, we compare all of them based on a matrix formulation and the filter-based interpretation we recently proposed. This analysis will be useful to show that the approach proposed in the original paper remains a good compromise in terms of accuracy and computational cost.

Graphic abstract

Appendices

Appendix A: About the pointwise Hölder exponent

The pointwise Hölder exponent can be defined as follows. Let y be a function of ${\mathbb {R}}$ in ${\mathbb {R}}$, $s>0$, $s \in {\mathbb {R}}\backslash {\mathbb {N}}$ and $t_0 \in {\mathbb {R}}$. Then, $y \in C^s(t_0)$ if and only if there is a real $\eta >0$, P a polynomial of degree smaller than s and c a constant such that

$$\begin{aligned} \forall t \in [t_0-\eta ,t_0+\eta ], \ \ |y(t)-P(t-t_0)|\le c |t-t_0|^s. \end{aligned}$$

By definition, the pointwise exponent of y in $t_0$, noted $\alpha _p(t_0)$ is the supremum of the s such as $y \in C^s(t_0)$.

An equivalent definition can be given to the without directly displaying the $C^s$ space.

$$\begin{aligned} \alpha _p(t_0)=\liminf _{h \rightarrow 0} \frac{\log |y(t_{0}+h)-y(t_{0})|}{\log |h|} \end{aligned}$$

(50)

This definition is valid if y is not derivable in $t_0$, otherwise one has to remove its regular part [65].

Geometrically, the equation 50 means that the graph of the y function around $t_0$ is included in a envelope which will be called the Hölderian envelope (see figure 16). For every $\epsilon >0$, there is a neighborhood of $t_0$ such that the y graph in this neighborhood is all included in the space defined by the two curves that associate t respectively $y(t_0)+c|t-t_0|^{\alpha _p(t_0)- \epsilon }$ and $y(t_0)-c|t-t_0|^{\alpha _p(t_0)-\epsilon }$ and such that this property is no longer true for the space defined by curves that associate to t respectively $y(t_0)+c|t-t_0|^{\alpha _p(t_0)+ \epsilon }$ and $y(t_0)-c|t-t_0|^{\alpha _p(t_0)+\epsilon }$. We see that the higher the $\alpha _p(t_0)$, the more the signal is smooth and inversely that the more $\alpha _p(t_0)$ is small the more the signal is irregular in $t_0$.

Fig. 16

Hölderian envelope of a signal y at the point $t_0$

Appendix B: Some additional information about the DFA of order d, the CDFA and the RDFA

The DFA has several variants, such as the higher-order DFA (DFA$_d$), the so-called continuous DFA (CDFA) where the consecutive local trends are continuous [54] and the regularized DFA (RDFA) [21] the purpose of which is to reduce the discontinuities that appear in the global trend by using the Tikhonov regularization of parameter¹⁰$\lambda$ follow the same scheme. However, in these cases, the matrices $B_{DFA,d}$, $B_{CDFA}$ and $B_{RDFA}(\lambda )$ defined similarly as $B_{DFA}$ are given by:

$$\begin{aligned} B_{DFA,d}= & {} \big (I_{LN}-A_{DFA,d}(A_{DFA,d}^TA_{DFA,d})^{-1} A_{DFA,d}^T\big )\nonumber \\&\times C_{LN,0}\text{ H}_{M}J_M \end{aligned}$$

(51)

where the $(LN \times (d+1)L)$ matrix $A_{DFA,d}$ is block diagonal defined from the set of matrices $\{A_{l,d}\}_{l=1,...,L}$, with $A_{l,d}$ a $N\times (d+1)$ matrix whose $c^{th}$ column is defined by the set of values $\{[(l-1)N+n]^{c-1}\}_{n=1,...,N}$, with $c=1,...,d+1$.

$$\begin{aligned} B_{CDFA}= & {} \big (I_{LN}-A_{CDFA}(A_{CDFA}^TA_{CDFA})^{-1}A_{CDFA}^T\big )\nonumber \\&C_{LN,0}\text{ H}_{M}J_M \end{aligned}$$

(52)

where the $LN \times (L+1)$ matrix $A_{CDFA}$ is defined as follows:

$$\begin{aligned} A_{CDFA}(1:N,1:L+1)=\begin{bmatrix} 1 &{} 0 &{}\ldots &{} 0 &{} 1\\ 2 &{} \vdots &{} &{}\vdots &{} 1\\ \vdots &{} \vdots &{} &{} \vdots &{} \vdots \\ N &{}0&{}\ldots &{}0 &{} 1 \\ \end{bmatrix} \end{aligned}$$

(53)

and $\forall l \in [2;L]$:

$$\begin{aligned}&A_{CDFA}((l-1) \times N+1:lN,1:L+1)\nonumber \\&\quad =\begin{bmatrix} \beta (1) &{} N \,\cdots \,N &{}(l-1)N+1-\beta (l-1) &{} 0 \,\cdots \,0 &{}1\\ \beta (1) &{} N \,\cdots \,N &{} (l-1)N+2-\beta (l-1) &{} \vdots &{}\vdots \\ \vdots &{} \vdots &{}\vdots &{}\vdots &{} \vdots \\ \beta (1) &{} \underbrace{N \,\cdots \,N}_{l-2} &{}lN-\beta (l-1) &{} \underbrace{0 \,\cdots 0\,}_{L-l} &{}1\\ \end{bmatrix} \end{aligned}$$

(54)

with $\beta (l)=lN+1$.

$$\begin{aligned}&B_{RDFA}(\lambda )\nonumber \\&\quad =\left( I_{LN}-A_{DFA} \left( A_{DFA}^TA_{DFA} +\lambda ^2A_{DFA}^T\varDelta _2^T\varDelta _2A_{DFA}\right) ^{-1}\right. \nonumber \\&\qquad \qquad \left. A_{DFA}^T\right) C_{LN,0}\text{ H}_{M}J_M \end{aligned}$$

(55)

with the $2^{nd}$-derivative operator $\varDelta _2$ of size $(LN-2)\times LN$ defined by:

$$\begin{aligned} \varDelta _2=\begin{bmatrix} 1 &{} -2 &{} 1 &{}\cdots &{}0 \\ \vdots &{} \ddots &{} \ddots &{} \ddots &{} \vdots \\ 0 &{} \cdots &{} 1 &{} -2 &{} 1 \end{bmatrix} \end{aligned}$$

(56)

Consequently, for these methods, the trend estimation is based on the minimization of the following criteria (Table 3)

Table 3 Expression of $B_{\bullet }$ for each approach

For more details about the variants, the reader may refer to the papers where they were proposed for the first time.