Detrended fluctuation analysis of autoregressive processes

Călin Vamoş
(original), ISI/JCR, Numerical Modeling, paper, Time Series

Abstract

Autoregressive processes (AR) have typical short-range memory. Detrended Fluctuation Analysis (DFA) was basically designed to reveal long-range correlations in non stationary processes. However DFA can also be regarded as a suitable method to investigate both long-range and short-range correlations in non stationary and stationary systems. Applying DFA to AR processes can help understanding the non-uniform correlation structure of such processes. We systematically investigated a first order autoregressive model AR(1) by DFA and established the relationship between the interaction constant of AR(1) and the DFA correlation exponent. The higher the interaction constant the higher is the short-range correlation exponent. They are exponentially related. The investigation was extended to AR(2) processes. The presence of an interaction between distant terms with characteristic time constant in the series, in addition to a near by interaction will increase the correlation exponent and the range of correlation while the effect of a distant negative interaction will significantly decrease the range of interaction, only. This analysis demonstrate the possibility to identify an AR(1) model in an unknown DFA plot or to distinguish between AR(1) and AR(2) models.

Authors

V.V. Morariu
Department of Molecular and Biomolecular Physics, National Institute of R&D for Isotopic and Molecular Technology

L. Buimaga-Iarinca
Department of Molecular and Biomolecular Physics, National Institute of R&D for Isotopic and Molecular Technology

C. Vamoș
-Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

S. Soltuz
-Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Keywords

Short-range correlation; autoregressive processes; detrended fluctuation analysis.

Cite this paper as:

V.V. Morariu, L. Buimaga-Iarinca, C. Vamoş, Ş.M. Şoltuz, Detrended fluctuation analysis of autoregressive processes, Fluctuations and noise letters, Vol. 7, No. 3, 2007. pp. L249-L255, doi: 10.1142/S0219477507003908

PDF

2007 Vamos Detrended fluctuation analysis of autoregressive processes

About this paper

Journal

Fluctuations and noise letters

Publisher Name

World Scientific Publishing Company

DOI

10.1142/S0219477507003908

Print ISSN

0219-4775

Online ISSN

1793-6780

MR

?

ZBL

?

[1] W. Li, A Bibliography on 1/f Noise (1996-present), http://www.nslij-genetics.org/wli/1fnoise/

[2] Wikipedia, Autoregressive moving average model, http://en.wikipedia.org/wiki/

[3] P. J. Brockwell and R. A. Davies, Time Series: Theory and Methods, 2nd edn. (Springer, New York, 1991).

[4] F. N. Hooge, T. G. M. Kleinpenning, L. K. J. Vandamme, Experimental studies on 1/f noise, Rep. Progr. Phys. 44 (1981) 479–532.

[5] M.B. Weissman, 1/f noise and other slow, nonexponential kinetics in condensed matter, Rev. Mod. Phys. 60 (1988) 537–532.

[6] C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley and A. L. Goldberger, Mosaic organization of DNA nucleotides, Phys. Rev. E 49 (1994) 1685–1689.

[7] K. Hu, P. Ch.Ivanov, Z. Chen, P. Carpena and H. Eugene Stanley, Effect of trends on detrended fluctuation analysis, Phys. Rev. E 64 (2001) 0111114–1.

[8] Z. Chen, P. Ch. Ivanov, K. Hu, H. Eugene Stanley, Effect of nonstationarities on detrended fluctuation analysis, Phys. Rev. E 65 (2002) 041107–1.

[9] A. Coza and V. V. Morariu, Generating 1/f β noise with a low dimensional attractor characteristic: its significance for atomic vibrations in proteins and cognitive data, Physica A 320 (2001) 449–460.

[10] M. König and J. Timmer, Analyzing X-ray variability by linear state space model, Astron. Astrophys. Suppl. Ser. 124 (1997) 589–596.

[11] Th. L. Thornton and D. L. Gilden, Provenance of correlations in psychological data, Psychonomic Bul. & Rev. 12 (2005) 409–441

Fluctuation and Noise Letters Vol. 7, No. 3 (2007) L249–L255 c World Scientific Publishing Company DETRENDED FLUCTUATION ANALYSIS OF AUTOREGRESSIVE PROCESSES VASILE V. MORARIU, and LUIZA BUIMAGA-IARINCA Department of Molecular and Biomolecular Physics, National Institute of R&D for Isotopic and Molecular Technology, R-400293 Cluj-Napoca, P.O.Box 700 Romania vvm@L40.itim-cj.ro CĂLIN VAMOŞ and ŞTEFAN M. ŞOLTUZ "T. Popoviciu" Institute of Numerical Analysis, Romanian Academy, P.O. Box 68, 400110 Cluj-Napoca, Romania Received 3 April 2007 Revised 22 August 2007 Accepted 25 August 2007 Communicated by Laszlo Kish Autoregressive processes (AR) have typical short-range memory. Detrended Fluctuation Analysis (DFA) was basically designed to reveal long-range correlations in non stationary processes. How- ever DFA can also be regarded as a suitable method to investigate both long-range and short-range correlations in non stationary and stationary systems. Applying DFA to AR processes can help un- derstanding the non-uniform correlation structure of such processes. We systematically investigated a first order autoregressive model AR(1) by DFA and established the relationship between the inter- action constant of AR(1) and the DFA correlation exponent. The higher the interaction constant the higher is the short-range correlation exponent. They are exponentially related. The investigation was extended to AR(2) processes. The presence of an interaction between distant terms with characteris- tic time constant in the series, in addition to a near by interaction will increase the correlation expo- nent and the range of correlation while the effect of a distant negative interaction will significantly decrease the range of interaction, only. This analysis demonstrate the possibility to identify an AR(1) model in an unknown DFA plot or to distinguish between AR(1) and AR(2) models. Keywords: Short-range correlation; autoregressive processes; detrended fluctuation analysis. 1. Introduction Many natural processes can be described by stochastic models. The time or scale depend- ent correlation characteristics of such processes are characterized by the autocorrelation function C(n) where the time coordinate n is often shown in a logarithmic scale. They may involve either long-range correlation or short-range correlation characteristics. The decay of C(n)for a long-range correlated process often lacks a characteristic time because of scaling with a power or logarithmic function. For some processes the de-correlation time is infinite. A well known class of long-range correlation processes is the 1/f 2 L249 Fluct. Noise Lett. 2007.07:L249-L255. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 12/25/14. For personal use only.
L250 V. V. Morariu et al. phenomena or Brownian noise [1]. On the other hand the decay of short-range correlation processes is usually described by an exponential function and therefore exhibits a charac- teristic time scale. Typical examples of characteristic time scale (short-range memory) are the autoregressive processes (AR) [2–3], or a large class of two-state fluctuation proc- esses in solids [4–5]. Detrended fluctuation analysis (DFA) was a method basically designed to investigate long-range correlations in non stationary series [6–8]. DFA produces an autocorrelation function F(n) as a function of log n. The plot of log F(n) vs. log n is a straight line if cor- relations with a power-law decay are present. The slope is the so called α scaling expo- nent which has values 0.5 and 1.5 for random (uncorrelated) series and Brownian noise respectively. However, very often, in practice the DFA plot is not a straight line because the process does not have a single long-range exponent but instead two or more correla- tion ranges with exponential cut-off. In such a case DFA can be used to investigate the short-range correlation behavior of a process. An appropriate way to investigate the behavior of DFA, in the case of short range memory, is to analyze AR processes. The purpose of this work is to perform DFA on the first order and second order autoregressive models known as AR(1) and AR(2) respec- tively. The main interest is to look for the relationship between the correlation exponent and the characteristic parameters of the AR(1) and AR(2) models. This may help to de- compose processes to the sum of short-range components. 2. Autoregressive Models An AR(1) model is given by the equation: t t t X X ε ϕ + = -1 (1) where ε t is a white noise process with zero mean and variance σ 2 , while φ is a parameter. The parameter values φ have to be restricted for the process to be stationary which means that | φ | < 1. If φ = 1 then X t can also be considered as a random walk. AR(1) model may apply to temporal or spatial processes and the significance of φ can be better understood when applied to such particular processes. For example in the case of a temporal process the parameter φ can be understood in terms of a relaxation time τ determined by [7]: () ϕ τ log 1 - = (2) The parameter φ can alternatively be regarded as the strength of interaction among the terms X i [9]. Obviously the more distant the terms of the series the lower the correlation. Regardless of a temporal or spatial process the parameter φ can be understood as the scale of short range memory of the system. Higher order models AR(p) are characterized by i parameters of φ i which indicate the strength of interaction between the first and the second term, between the first and the third term and so on. The model is given by the equation: ∑ = - + = p i t i t i t X X 1 ε ϕ (3) with φ i parameters where i = 1, …, p. Fluct. Noise Lett. 2007.07:L249-L255. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 12/25/14. For personal use only.
Detrended Fluctuation Analysis of Autogressive Processes L251 AR models have successfully been applied to astrophysical and psychological data [10–11]. More recently we found that various biophysical phenomena can be well described by AR(1) models or by higher order AR(p) models [work to be published]. They include the structure of proteins, flickering of the red blood cells and random number generation by human subjects. It was however felt that a systematic DFA of a short range memory model is needed to better understand how the correlation and the scale of memory are related. 3. Detrended Fluctuation Analysis of AR(1) Model Autoregressive series of 1000 terms were generated with a program written in MATLAB. The DFA method involves an integration of the series which is further divided into boxes of equal size n. In each box the integrated series is fitted by using a polynomial function which is called the local trend. The integrated series is detrended by subtracting the local trend in each box. For a given box size n a detrended fluctuation function is calculated and then the root mean square fluctuation F(n) is obtained. Finally a DFA plot log F(n) vs. log(n) is obtained. The slope of the plot represents the correlation exponent α. In our analysis we used the standard DFA-1 method which means that the local trend was fitted with a first degree polynomial. An example of DFA plot is shown in Fig. 1(a). It can be seen that within the range 0.6 < log n < 1.4 the system is characterized by a correlation exponent α 1 which describes a short range memory. This result was obtained from a single series of data. If different representations of AR(1) series are generated by starting from different random series then a bunch of curves which diverge at higher n values resulted (Fig. 1(b)). 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 -0.3 0.0 0.3 0.6 0.9 1.2 log F(n) log(n) 1a 0.6 0.9 1.2 1.5 1.8 2.1 2.4 -0.3 0.0 0.3 0.6 0.9 1.2 log F(n) log(n) 1b Fig. 1. (a) Detrended fluctuation analysis of AR(1) model for φ = 0.7. A correlation exponent α = 0.97 is calculated from the slope of straight line approximation of the first part of the plot. This correlation exponent is characteristic for the short range interval 0.6 < log(n) < 1.4. (b) The same plot generated with ten different random series. Each case of AR(1) model for a given φ was averaged over ten generated series. Averaged plots are represented in Fig. 2 for different values of φ. The correlation is limited to ranges shorter than about one order of magnitude of n. The correlation decreases gradually for distances longer than about one order of magnitude. Stronger correlation of the DFA plot is visible as a higher slope (higher α 1 value) for higher values of φ. The upper part of the DFA plots can also be described by a correlation exponent α 2 which is smaller than α 1 . The standard deviation for the end of each curve varied between 0.04 and 0.1. Fluct. Noise Lett. 2007.07:L249-L255. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 12/25/14. For personal use only.
L252 V. V. Morariu et al. 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 log F(n) log(n) φ=0.1 φ=0.2 φ=0.3 φ=0.4 φ=0.5 φ=0.6 φ=0.7 φ=0.8 φ=0.9 Fig. 2. Averaged detrended fluctuation analysis plots of AR(1) models for different values of parameter φ. The plot illustrate the short range correlation of the AR(1) model as the slope of the plot (correlation exponent) gradually decreases at higher n. 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 1.2 3a α φ 0.0 0.2 0.4 0.6 0.8 1.0 0.5 0.6 0.7 0.8 0.9 3b Delta log (n) φ Fig. 3 (a) The influence of the AR(1) parameter φ on the short range correlation DFA exponent α 1 . The data are fitted with an exponential. (b) The range of short range correlation Δlog(n) in the DFA plot corresponding to exponent α 1 for different values of the AR(1) parameter φ. The data are fitted by a straight line. Further we illustrate how the correlation exponent α 1 depends on the value of φ (Fig. 3(a)). We notice that the short-range correlation exponent α 1 is closely correlated to the AR(1) parameter φ. It should be stressed that the α 1 exponent is a correlation property while φ is a scale parameter describing either a characteristic time or the strength of interaction among the terms of the series. The range of short-range correlation Δlog(n), i.e. the linear domain on which exponent α 1 is defined as function of φ, is illustrated in Fig. 3(b). The higher the value of φ the longer the range of the short-range correlation. The upper part of the DFA plot (Fig. 1) can also be characterized by a slope which corresponds to an α 2 exponent. Its dependence on the value of parameter φ is illustrated in Fig. 4. It shows that the upper part of the DFA plot of an AR(1) model is practically uncorrelated (α 2 = 0.5). Correlation still persists at φ ≥ 0.8 where α 2 ≥ 0.6. On the other hand this range of correlation decreases with increase of φ, see Fig. 4(b), as opposed to the behavior over the range of correlation for α 1 (Fig. 3(b)). Fluct. Noise Lett. 2007.07:L249-L255. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 12/25/14. For personal use only.
Detrended Fluctuation Analysis of Autogressive Processes L253                                                                                                                                                                                                                                       Fluct. Noise Lett. 2007.07:L249-L255. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 12/25/14. For personal use only.
L254 V. V. Morariu et al. tion, it is likely that the short-range correlation property of the series can be described by an AR(1) model. Obviously an AR(2) model will not fulfill this condition as the range of correlation is different. 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 -0.3 0.0 0.3 0.6 0.9 1.2 1.5 log F(n) log(n) AR(1) φ = 0.6 AR(2) φ 1 = 0.6 φ 2 = 0.3 AR(2) φ 1 = 0.6 φ 2 = -0.3 Fig. 5. Example of DFA plots for an AR(2) model compared to AR(1) one. 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 -0.3 0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 log F(n) log(n) φ 1 =0.6 φ 2 =0.1 φ 1 =0.6 φ 2 =0.2 φ 1 =0.6 φ 2 =0.3 φ 1 =0.6 φ 2 =0.4 6a 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 -0.2 0.0 0.2 0.4 0.6 0.8 log F(n) log(n) φ 1 =0.6 φ 2 = -0.1 φ 1 =0.6 φ 2 = -0.2 φ 1 =0.6 φ 2 = -0.4 φ 1 =0.6 φ 2 = -0.8 6b Fig. 6. (a) The influence of a positive value of φ 2 on the DFA plot of an AR(2) model with φ 1 = 0.6. (b) The influence of a negative value of φ 2 on the DFA plot of an AR(2) model with φ 1 = 0.6. Other characteristics of AR(2), such as the effect of a negative value of φ 2 can be easily recognized by a significant inflexion of the DFA plot or by a less important inflexion at positive values of φ 2 . Further it shows that the local correlation varies continuously in a more complicated way for AR(2) with negative φ 2 values of about 0.7– 0.8. Analysis in terms of short range correlation and range of correlation gradually changes to local correlation properties as φ 2 becomes more negative (Fig. 6(b)). 5. Conclusions The DFA correlation exponent is nonlinearly related to the AR(1) interaction parameter φ and the range of correlation is linearly related to the same parameter. DFA of first order autoregressive processes shows that the correlation exponent and the range of correlation represent a pair of characteristic data for a given AR(1) process. Consequently an un- Fluct. Noise Lett. 2007.07:L249-L255. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 12/25/14. For personal use only.
Detrended Fluctuation Analysis of Autogressive Processes L255 known AR process can be identified as an AR(1) process by confronting its characteris- tics with theoretical values. A second order AR(2) model revealed that a positive distant interaction φ 2 increased both the correlation exponent and the range of correlation com- pared to an AR(1) with the same nearby interaction constant φ 1 . A negative distant inter- action – φ 2 significantly decreased the range of correlation. A careful qualitative analysis of the DFA plots may identify an AR(2) process or distinguish between AR(1) and AR(2) processes. Acknowledgements Funding of the work was provided by the Romanian Authority for Scientific Research. References [1] W. Li, A Bibliography on 1/f Noise (1996-present), http://www.nslij-genetics.org/wli/1fnoise/ [2] Wikipedia, Autoregressive moving average model, http://en.wikipedia.org/wiki/ [3] P. J. Brockwell and R. A. Davies, Time Series: Theory and Methods, 2nd edn. (Springer, New York, 1991). [4] F. N. Hooge, T. G. M. Kleinpenning, L. K. J. Vandamme, Experimental studies on 1/f noise, Rep. Progr. Phys. 44 (1981) 479–532. [5] M.B. Weissman, 1/f noise and other slow, nonexponential kinetics in condensed matter, Rev. Mod. Phys. 60 (1988) 537–532. [6] C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley and A. L. Goldberger, Mosaic organization of DNA nucleotides, Phys. Rev. E 49 (1994) 1685–1689. [7] K. Hu, P. Ch.Ivanov, Z. Chen, P. Carpena and H. Eugene Stanley, Effect of trends on detrended fluctuation analysis, Phys. Rev. E 64 (2001) 0111114–1. [8] Z. Chen, P. Ch. Ivanov, K. Hu, H. Eugene Stanley, Effect of nonstationarities on detrended fluctuation analysis, Phys. Rev. E 65 (2002) 041107–1. [9] A. Coza and V. V. Morariu, Generating 1/f β noise with a low dimensional attractor characteristic: its significance for atomic vibrations in proteins and cognitive data, Physica A 320 (2001) 449–460. [10] M. König and J. Timmer, Analyzing X-ray variability by linear state space model, Astron. Astrophys. Suppl. Ser. 124 (1997) 589–596. [11] Th. L. Thornton and D. L. Gilden, Provenance of correlations in psychological data, Psychonomic Bul. & Rev. 12 (2005) 409–441. Fluct. Noise Lett. 2007.07:L249-L255. Downloaded from www.worldscientific.com by MCMASTER UNIVERSITY on 12/25/14. For personal use only.
2007

Related Posts