Extended Methodology for DFA and DCCA: Application of Automatic Search Procedure and Correlation Map to the Weierstrass-Mandelbrot Functions

Detrended fluctuation analysis and detrended cross-correlation analysis are used in this study to identify and characterize correlated data. The objective of these two techniques is to separate different fluctuations from the contributions due to external trends by evaluating the autocorrelation and cross-correlation exponents, in order to determine if scale properties persist with the size of the series. Two new methodologies were extended from cross-correlation coefficients for local analysis, which we call the \textit{automatic search procedure

Key words
Detrended fluctuation analysis; detrended cross correlation analysis; local analysis; Weierstrass-Mandelbrot functions

Authorship

EULER B.S. MARINHO

Universidade Federal da Bahia, CPGG/IGEO, Rua Barão de Jeremoabo, s/n, Ondina, 40170-115 Salvador, BA, BrazilUniversidade Federal da BahiaBrazilSalvador, BA, BrazilUniversidade Federal da Bahia, CPGG/IGEO, Rua Barão de Jeremoabo, s/n, Ondina, 40170-115 Salvador, BA, Brazil

https://orcid.org/0000-0002-0945-8742

AMIN BASSREI

Instituto Nacional de Ciência e Tecnologia de Geofísica do Petróleo, Rua Barão de Jeremoabo, s/n, Ondina, 40170-115 Salvador, BA, BrazilInstituto Nacional de Ciência e Tecnologia de Geofísica do PetróleoBrazilSalvador, BA, BrazilInstituto Nacional de Ciência e Tecnologia de Geofísica do Petróleo, Rua Barão de Jeremoabo, s/n, Ondina, 40170-115 Salvador, BA, Brazil

https://orcid.org/0000-0002-4653-2016

ROBERTO F.S. ANDRADE

Universidade Federal da Bahia, Instituto de Física, Rua Barão de Jeremoabo, s/n, Ondina, 40210-340 Salvador, BA, BrazilUniversidade Federal da BahiaBrazilSalvador, BA, BrazilUniversidade Federal da Bahia, Instituto de Física, Rua Barão de Jeremoabo, s/n, Ondina, 40210-340 Salvador, BA, Brazil

https://orcid.org/0000-0002-9323-1400

Correspondence to: Amin Bassrei E-mail: bassrei@ufba.br

AUTHOR CONTRIBUTIONS

Euler Marinho developed this study, processed the data, generated the figures, interpreted the results and wrote the first version of the manuscript. Amin Bassrei supervised the research, contributed to the results discussion, reviewed and translated the manuscript. Roberto Andrade co-supervised the research and contributed to the results discussion.

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables

Figures (14)
Tables (1)

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Figure 1
Example for autocorrelation analysis procedure: (a) fluctuation around the average and (b) fluctuations around the trend.

y_{k}

is the original input series (equivalent to the integrated series) and red lines represent, for each window

m

of size

ν

, the local averages in (a) and the local trends in (b).

Thumbnail

Figure 2
Example of transforming the series of successive differences into the integrated series: (a) the input series

y_{k}

; (b) the series of successive differences of

y_{k}

, that is,

x_{i} = y_{k + 1} - y_{k}

. The integrated series

y_{k} = \sum_{i = 1}^{k} x_{i}

, where

i, k \in {1, 2, . . ., N}

, is obviously equal to (a).

Thumbnail

Figure 3
(a) the series of successive difference magnitudes,

| x_{i} |

; (b) the series of successive difference signals,

s i g n (x_{i})

, where

s i g n (x_{i}) = 1

, if

x_{i} > 0

;

s i g n (x_{i}) = 0

, if

x_{i} = 0

and

s i g n (x_{i}) = - 1

, if

x_{i} < 0

; (c) the integrated series of magnitudes,

y_{k}^{M} = \sum_{i = 1}^{k} | x_{i} |

; (d) the integrated series of signals,

y_{k}^{S} = \sum_{i = 1}^{k} s i g n (x_{i})

Thumbnail

Figure 4
Records of 300 samples of the Weierstrass-Mandelbrot function: original series in (a) and (b) with various

α

exponent, signal series in (c) and (d) and magnitude series in (e) and (f). The numbers from (1) to (10) indicate the order in which each signal and magnitude series was obtained from the original series. For example, series (3) was obtained from the original series with

α = 0.3

Thumbnail

Figure 5

D F A_{0}

and

D F A_{1}

curves for the autocorrelation of the function W2. The original series is showed in (a), the signal series in (b) and the magnitude series in (c).

Thumbnail

Figure 6

D C C A_{0}

and

D C C A_{1}

curves for cross-correlation between the functions W3 and W7. The original series is showed in (a), the signal series in (b) and the magnitude series in (c).

Thumbnail

Figure 7
Procedure for the local analysis procedure: (a) Two series,

y_{k}

and

y_{k}^{'}

, with a moving box

S

with size

N^{'}

. (b) Detail of the same moving box

S

, now divided into

M_{ν} = 5

windows, that is,

m

varying from 1 to 5, with

ν = 20

in each sample.

Thumbnail

Figure 8
Automatic search procedure for the average coefficient

\overline{σ}

between the functions W2 and W3, as a function of the series size

N

and the relative displacement

D_{R}

. (a) With the condition

\overline{σ} 0.75

. (b) With the condition

\overline{σ} 0.80

in (b). The magnitude series is displayed in (1), the signal series in (2) and original series in (3).

Thumbnail

Figure 9
Automatic search procedure for the average coefficient

\overline{σ}

between the functions W1 and W3 as a function of the series size

N

and the relative displacement

D_{R}

, only for the magnitude series. (a) With the condition

\overline{σ} 0.75

. With condition

\overline{σ} 0.80

. Notice a much less populated diagram in this figure when compared to Figure 8. This is due to the fact that now the two functions are less close.

Thumbnail

Figure 10
Analysis between the functions W2 and W3 for the relative displacement

D_{R} = 0

point. In (a) the vertical axis represents the series size

N

and the horizontal axis represents the relative displacement

D_{R}

between the series for the condition

\overline{σ} > 0.80

. In (b) the vertical axis represents the window variation

ν

and the horizontal axis is the size of the series

N

. The colors represent the values of

σ

. The solid black arrow indicates that the map of correlations was generated from the automatic search procedure for

D_{R} = 0

Thumbnail

Figure 11
Analysis between the functions W2 and W3 for the relative displacement

D_{R} = 190

points. In (a) the vertical axis represents the series size

N

and the horizontal axis represents the relative displacement

D_{R}

between the series for the condition

\overline{σ} > 0.80

. In (b) the vertical axis represents the window variation

ν

and the horizontal axis represents the size of the series

N

. The colors represent the values of

σ

. The solid black arrow indicates that the map of correlations was generated from the automatic search procedure for

D_{R} = 190

. Each black dashed arrow shows the relationship between the coefficients

σ_{ν} > 0.80

in (b) with the average

\overline{σ}

of these coefficients in (a).

Thumbnail

Figure 12
Analysis between the functions W2 and W3 for the relative displacement

D_{R} = 690

points. In (a) the vertical axis represents the series size

N

and the horizontal axis represents the relative displacement

D_{R}

between the series for the condition

\overline{σ} > 0.80

. In (b) the vertical axis represents the window variation

ν

and the horizontal axis represents the size of the series,

N

. The colors represent the values of

σ

. The solid black arrow indicates that the map of correlations was generated from the automatic search procedure for

D_{R} = 690

. The black dashed arrow shows the relationship between the coefficients

σ_{ν} > 0.80

in (b) with the average

\overline{σ}

of these coefficients in (a).

Thumbnail

Figure 13
Power spectrum of the function W3 for the signal series, with the non-smoothed spectrum at the top and the smoothed one at the bottom. The values obtained for the spectral exponent

b

are indicated at lower left of each panel.

Thumbnail

Figure 14
Power spectrum of the function W3 for the original series, with the non-smoothed spectrum at the top and the smoothed one at the bottom. The values obtained for the spectral exponent

b

are indicated at lower left of each panel.

Thumbnail

Table I
Autocorrelation and cross-correlation exponents for W2, W3 and W7.

α_{O}

and

α_{S}

are the autocorrelation exponents for the original and signal series, respectively, while

λ_{O}

and

λ_{S}

are the cross-correlation exponents for the original and signal series, respectively.

Figure 1 Example for autocorrelation analysis procedure: (a) fluctuation around the average and (b) fluctuations around the trend. $y_{k}$ is the original input series (equivalent to the integrated series) and red lines represent, for each window $m$ of size $ν$ , the local averages in (a) and the local trends in (b).

Figure 2 Example of transforming the series of successive differences into the integrated series: (a) the input series $y_{k}$ ; (b) the series of successive differences of $y_{k}$ , that is, $x_{i} = y_{k + 1} - y_{k}$ . The integrated series $y_{k} = \sum_{i = 1}^{k} x_{i}$ , where $i, k \in {1, 2, . . ., N}$ , is obviously equal to (a).

Figure 3 (a) the series of successive difference magnitudes, $| x_{i} |$ ; (b) the series of successive difference signals, $s i g n (x_{i})$ , where $s i g n (x_{i}) = 1$ , if $x_{i} > 0$ ; $s i g n (x_{i}) = 0$ , if $x_{i} = 0$ and $s i g n (x_{i}) = - 1$ , if $x_{i} < 0$ ; (c) the integrated series of magnitudes, $y_{k}^{M} = \sum_{i = 1}^{k} | x_{i} |$ ; (d) the integrated series of signals, $y_{k}^{S} = \sum_{i = 1}^{k} s i g n (x_{i})$ .

Figure 4 Records of 300 samples of the Weierstrass-Mandelbrot function: original series in (a) and (b) with various $α$ exponent, signal series in (c) and (d) and magnitude series in (e) and (f). The numbers from (1) to (10) indicate the order in which each signal and magnitude series was obtained from the original series. For example, series (3) was obtained from the original series with $α = 0.3$ .

Figure 5 $D F A_{0}$ and $D F A_{1}$ curves for the autocorrelation of the function W2. The original series is showed in (a), the signal series in (b) and the magnitude series in (c).

Figure 6 $D C C A_{0}$ and $D C C A_{1}$ curves for cross-correlation between the functions W3 and W7. The original series is showed in (a), the signal series in (b) and the magnitude series in (c).

Figure 7 Procedure for the local analysis procedure: (a) Two series, $y_{k}$ and $y_{k}^{'}$ , with a moving box $S$ with size $N^{'}$ . (b) Detail of the same moving box $S$ , now divided into $M_{ν} = 5$ windows, that is, $m$ varying from 1 to 5, with $ν = 20$ in each sample.

Figure 8 Automatic search procedure for the average coefficient $\overline{σ}$ between the functions W2 and W3, as a function of the series size $N$ and the relative displacement $D_{R}$ . (a) With the condition $\overline{σ} 0.75$ . (b) With the condition $\overline{σ} 0.80$ in (b). The magnitude series is displayed in (1), the signal series in (2) and original series in (3).

Figure 9 Automatic search procedure for the average coefficient $\overline{σ}$ between the functions W1 and W3 as a function of the series size $N$ and the relative displacement $D_{R}$ , only for the magnitude series. (a) With the condition $\overline{σ} 0.75$ . With condition $\overline{σ} 0.80$ . Notice a much less populated diagram in this figure when compared to Figure 8. This is due to the fact that now the two functions are less close.

Figure 10 Analysis between the functions W2 and W3 for the relative displacement $D_{R} = 0$ point. In (a) the vertical axis represents the series size $N$ and the horizontal axis represents the relative displacement $D_{R}$ between the series for the condition $\overline{σ} > 0.80$ . In (b) the vertical axis represents the window variation $ν$ and the horizontal axis is the size of the series $N$ . The colors represent the values of $σ$ . The solid black arrow indicates that the map of correlations was generated from the automatic search procedure for $D_{R} = 0$ .

Figure 11 Analysis between the functions W2 and W3 for the relative displacement $D_{R} = 190$ points. In (a) the vertical axis represents the series size $N$ and the horizontal axis represents the relative displacement $D_{R}$ between the series for the condition $\overline{σ} > 0.80$ . In (b) the vertical axis represents the window variation $ν$ and the horizontal axis represents the size of the series $N$ . The colors represent the values of $σ$ . The solid black arrow indicates that the map of correlations was generated from the automatic search procedure for $D_{R} = 190$ . Each black dashed arrow shows the relationship between the coefficients $σ_{ν} > 0.80$ in (b) with the average $\overline{σ}$ of these coefficients in (a).

Figure 12 Analysis between the functions W2 and W3 for the relative displacement $D_{R} = 690$ points. In (a) the vertical axis represents the series size $N$ and the horizontal axis represents the relative displacement $D_{R}$ between the series for the condition $\overline{σ} > 0.80$ . In (b) the vertical axis represents the window variation $ν$ and the horizontal axis represents the size of the series, $N$ . The colors represent the values of $σ$ . The solid black arrow indicates that the map of correlations was generated from the automatic search procedure for $D_{R} = 690$ . The black dashed arrow shows the relationship between the coefficients $σ_{ν} > 0.80$ in (b) with the average $\overline{σ}$ of these coefficients in (a).

Figure 13 Power spectrum of the function W3 for the signal series, with the non-smoothed spectrum at the top and the smoothed one at the bottom. The values obtained for the spectral exponent $b$ are indicated at lower left of each panel.

Figure 14 Power spectrum of the function W3 for the original series, with the non-smoothed spectrum at the top and the smoothed one at the bottom. The values obtained for the spectral exponent $b$ are indicated at lower left of each panel.

Table I Autocorrelation and cross-correlation exponents for W2, W3 and W7. $α_{O}$ and $α_{S}$ are the autocorrelation exponents for the original and signal series, respectively, while $λ_{O}$ and $λ_{S}$ are the cross-correlation exponents for the original and signal series, respectively.

Exponents	W2	W2	W3	W3	W7	W7	W3 & W7	W3 & W7
Exponents	$α_{O}$	$α_{S}$	$α_{O}$	$α_{S}$	$α_{O}$	$α_{S}$	$λ_{O}$	$λ_{S}$
$D F A_{0}$ or $D C C A_{0}$	0.199	0.460	0.304	0.435	0.725	0.720	0.517	0.590
$D F A_{1}$ or $D C C A_{1}$	0.194	0.392	0.289	0.384	0.664	0.644	0.478	0.504

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Métricas

Extended Methodology for DFA and DCCA: Application of Automatic Search Procedure and Correlation Map to the Weierstrass-Mandelbrot Functions

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[1] Correspondence to: Amin Bassrei E-mail: bassrei@ufba.br