key: cord-0448418-h38oi4de
authors: Gil-Maqueda, Luis; Itz'a-Ortiz, Benjam'in A.
title: Detrended Fluctuation Analysis for Continuous Real Variable Functions
date: 2022-03-29
journal: nan
DOI: nan
sha: 896b049b2b3b8d9c17a1035b9ccf237922fbb396
doc_id: 448418
cord_uid: h38oi4de

Based on the well-known Detrended Fluctuation Analysis (DFA) for time series, in this work we describe a DFA for continuous real variable functions. Under certain conditions, DFA accurately predicts the long-term auto-correlation of the time series, depending on the value of certain scaling parameter. We show that for continuous functions, the proposed continuous DFA also exhibits fractal properties and approximates a power law with scaling exponent one.

Time series analysis is a useful study area as it provides means to understand the dynamics of data collected from diverse research areas such as business, economics, medicine, volcanology , among many others [10, 3, 12] . One of the important aspects in a time series is to determine whether the data has autocorrelation. This property is related to the concept of long-memory [8] , i.e it helps to determine if there exists a relation between the data in the past and the data in the future. One way to quantify this relation is through the computation of an exponent called the Hurst exponent [9, 18] . In this sense, the Detrended Fluctuation Analysis (DFA) may be regarded as a powerful method to detect self-similar patterns in non-stationary time series. It works by transforming a time series into a new time series which approximates a power law [16] ; depending on the scaling exponent, one may conclude that the original time series was autocorrelated or not. This method is widely used today, for example, in [1, 11, 5, 3, 14, 15, 13] , just to mention a few. Roughly speaking, DFA is a modified root mean square analysis so it seemed plausible to adapt its discrete context to the continuous setting. The ensuing natural question, which we answer in the affirmative in this paper, was whether a continuous version of the DFA would also manifest a power law. It is worth mentioning that the classical DFA model presents some problems in mitigating non-stationaries [2] .

In this work we propose a DFA that is applied to continuous real functions. As in the classical DFA for time series, it will consist of two steps. In the first step we define the function called the integral process associated to a integrable function, and establish that this is a sum of two self similar fractals functions, where we regard a fractal function in the sense of Hutchinson [6, 7] . The second step consists in removing the trend and define a detrended function F , and prove that this function F is approximately a power law.

We divide this work in two sections. In the Section 1, we review the concept of self similar fractal functions and provide some examples of such functions. In Section 2 we present the main results of the paper, namely, we give the DFA version for continuous functions and prove that it approximates a power law with scaling exponent one.

The first author gratefully acknowledges support from CONACyT grant 1002291.

In this section we will introduce the definition of self similar fractal function, and give some examples. Since in the classical DFA the integrated process, a time series, is a self similar process, the definition of self similar fractal real variable function is important, as we would expect the integrated process for continuous functions to have self similar or fractal properties. The definition given here was introduced by Hutchinson [6, 7] . There are, in the literature, other definitions for self similar functions; however, the one presented here seems to give a more natural generalization of the concept of self similar sets, its formulation makes it easy to give examples and, as it will be shown here, preserves some properties that have the geometrical fractals, for example translation invariance.

We denote the Lipschitz constants by Lip(S i ). Definition 1.2. Let I = I 1 ⊔ · · · ⊔ I N be a partition of the interval I into disjoint subintervals. Given maps g i : 

We say that f satisfies the scaling law S, or that f is a self similar fractal function if

Note that in Definition 1.3 the use of Lipschitz maps may be seen as the analogous of verifying scaling properties in f , and by the disjoint union we can interpret that we are joining the pieces after the scaling, so this emulates the behavior of known fractals. Consider the following important example.

Proof. Consider the following disjoint partition of the interval [a, b] divided by = a, a+b 2 , a+b 2 , b and let

It is easy to see that φ 1 and φ 2 are Lipschitz maps because are differentiable, and derivative is positive. Consider the scaling law (S 1 ,

, then a straightforward computation shows that

and hence (1) follows. Then f is a self similar fractal function.

The following lemma is relevant to establish the translations invariance of self similar fractal function. Lemma 1.5. Let I be a bounded and closed interval in R and let S :

Then,S is a Lipschitz map, as wanted.

Theorem 1.6. Let I be a closed an bounded interval and let c ∈ R. If f : I → R is a self similar fractal function, then f + c is also a self similar fractal function.

Proof. By hypothesis f admits the following representation

for some scaling law (S 1 , . . . , S N ) and some family of Lipschitz maps φ i 's.

For every i ∈ {1, 2, . . . , N } consider the family of sets Y i = {f • φ −1 i (x) + c}, and let us define a function S i in these sets, byS

which is a Lipschitz by Lemma 1.5 for every i ∈ {1, 2 . . . , N }. Then:

Hence the function f + c is a self similar fractal function, with scaling law (S 1 ,S 2 , . . . ,S N ).

The classical DFA may be regarded as a procedure to transform a time series into a new time series which approximates a power law. It consists on two steps. This section contains the main contribution of the paper, namely, we present the corresponding two steps which will define a DFA for continuous functions. There is a correspondence between time series a(i) of size M ∈ N, and simple functions x : [0, M ] → R given by

Hence the definition of integrated function generalizes the notion of the integrated series.

One of the features in the DFA method is that the integrated time series is a self similar process. We wonder if the integrated function is a self similar fractal function as in Definition 1.3. We were unable to prove it. However, as we show in the next theorem, it is the sum of two such functions. 

with inverse maps: If we show that S 1 and S 2 are Lipschitz maps on the images of z(φ −1 1 (t)) and z(φ −1 2 (t)) respectively, then by Kirzbraun Theorem [4] , we can extend these maps to Lipschitz maps on R.

By continuity of x(s) in the compact set [0, M ], there exist K and k given by:

For t, τ ∈ [0, M 2 ] with t < τ we have the following:

note that by hypothesis x(s) ≥ δ > 0 so we have that K > 0 y k > 0. Then we may choose R > 0 such that: K ≤ 2Rk. Hence, using (3) and (4) we have that:

And using a variable change s = s 2 , to the integral on the right in (5) x(s) ds.

Since by hypothesis x(s) ≥ δ > 0, the integrals in the last inequality are non negatives, so we obtain

x(s) ds .

Then, if we add a zero to the integral on the left hand side of (6) we obtain With this (6) becomes:

This prove that S 1 is a Lipschitz map on the image of z • φ −1 1 . The proof that S 2 is a Lipschitz map on the image of z • φ −1 2 is analogous, so we omit it. Then we have that 

is a sum of a two self similar fractals functions.

The following corollary establish a sufficient condition for a function to be self similar fractal. Proof. Since by the hypothesis dx ds ≥ δ > 0, it is possible apply Theorem 2.2, then:

is a self similar fractal function, and then x(t) is a self similar fractal function by Theorem 1.6. 

For the continuous case, in this second step, we will require the use continuous functions on compact intervals. One reason for this hypothesis is that in order to prove lemma ??, which leads to the establishment of a power law, the fundamental theorem of calculus in needed. It is important to remark that the continuity of a function on a compact space, implies that the function is bounded, and so it is integrable. As consequence, we are able to define the integrated functions described in subsection 2.1, and apply Theorem 2.2.

We now propose one way of removing the trend from the integrated function y(t) of a given continuous function x : [0, M ] → R, where M is a positive number, as in Definition 2.1. One may try removing the trend by means of a least squares approximation, that is, we restrict the integrated function y(t) on subintervals of size n, with 1 < n < M ; on each window of size n, we approximate the graph by means of a linear approximation and proceed to remove the trend from y(t) by subtracting the value of the y-coordinate of the corresponding line. However, unlike the discrete case, this approach will not lead us to obtain a power law, even when an approximation by polynomials replaces the linear approximation. For the this reason, we consider a different way to remove the trend, by means of a methodology analogous to the technique of differentiation in the context of time series. Let 0 < m < M . Restrict the integrated function y(t) given in Definition 2.1, to the interval [m, M ] and let 0 < r < m. Consider the difference

The expression in (7) may be regarded as the process of removing the trend of y(t), taking into account the immediate past, instead of the tendency established by windows. Then as in the methodology introduced by Peng and his collaborators, [14, 13] , consider the square root of the average of the squared values obtained in (7) 

The following lemma will be useful in the sequel. Then, for all ε > 0, there exists 0 < δ < m such that if 0 < r < δ, then • in case x = 0, we have |F (r) − r|x|| < ε,

• in case x = 0 we have |F (r) − r| < ε.

Hence F (r) approximates a power law.

Proof. Let ǫ > 0. Since the square root function is uniformly continuous in [0, M ], there exists δ 1 > 0 such that if |s − t| < δ 1 then | √ s − √ t| < ǫ. On the other hand, by the continuity of x(s), there is a function X such that X ′ = x. Since X is uniformly continuous on the compact set [m, M ], there exists δ 2 > 0 such that if |s − t| < δ 2 then |X(s) − X(t)| < δ1 2 . We prove first the case whenx = 0. Let

Note that by (7) we have that 

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As in the traditional DFA model, we obtain a power law, but in this case we do not obtain a complete spectrum of values for the exponent of the power law [14] . In the classical DFA, the scaling exponent close to one indicated the existence of long-range correlations, while the scaling exponent equal one corresponds to the so-called 1/f noise [17] . In our case, the fact that we obtain the scaling exponent equal one may be due to the fact that a function may be regarded as a deterministic object. Finally, we point out that, by Example 1.4 and Theorem 2.4, the detrended function F is approximately a self similar fractal function.

Hence, for 0 < r < δ we haveand thusWe now prove the second case whenx = 0. LetNote that by (7) we have thatand thus