We develop a method for the multifractal characterization of nonstationary time series, which is based on a generalization of the detrended fluctuation analysis (DFA). We relate our multifractal DFA method to the standard partition function-based multifractal formalism, and prove that both approaches are equivalent for stationary signals with compact support. By analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series. By comparing the multifractal DFA results for original series with those for shuffled series we can distinguish multifractality due to long-range correlations from multifractality due to a broad probability density function. We also compare our results with the wavelet transform modulus maxima method, and show that the results are equivalent.

Multifractal Detrended Fluctuation Analysis of Nonstationary Time Series

scaling behavior. We find that crossovers result from the competition between the scaling of the noise and the ‘‘apparent’’ scaling of the trend. We study how the characteristics of these crossovers depend on ~i! the slope of the linear trend; ~ii! the amplitude and period of the periodic trend; ~iii! the amplitude and power of the power-law trend, and ~iv! the length as well as the correlation properties of the noise. Surprisingly, we find that the crossovers in the scaling of noisy signals with trends also follow scaling laws—i.e., long-range power-law dependence of the position of the crossover on the parameters of the trends. We show that the DFA result of noise with a trend can be exactly determined by the superposition of the separate results of the DFA on the noise and on the trend, assuming that the noise and the trend are not correlated. If this superposition rule is not followed, this is an indication that the noise and the superposed trend are not independent, so that removing the trend could lead to changes in the correlation properties of the noise. In addition, we show how to use DFA appropriately to minimize the effects of trends, how to recognize if a crossover indicates indeed a transition from one type to a different type of underlying correlation, or if the crossover is due to a trend without any transition in the dynamical properties of the noise.

https://arxiv.org/pdf/physics/0103018.pdf

Effect of trends on detrended fluctuation analysis.

Detrended fluctuation analysis ~DFA! is a scaling analysis method used to quantify long-range power-law correlations in signals. Many physical and biological signals are ‘‘noisy,’’ heterogeneous, and exhibit different types of nonstationarities, which can affect the correlation properties of these signals. We systematically study the effects of three types of nonstationarities often encountered in real data. Specifically, we consider nonstationary sequences formed in three ways: ~i! stitching together segments of data obtained from discontinuous experimental recordings, or removing some noisy and unreliable parts from continuous recordings and stitching together the remaining parts—a ‘‘cutting’’ procedure commonly used in preparing data prior to signal analysis; ~ii! adding to a signal with known correlations a tunable concentration of random outliers or spikes with different amplitudes; and ~iii! generating a signal comprised of segments with different properties—e.g., different standard deviations or different correlation exponents. We compare the difference between the scaling results obtained for stationary correlated signals and correlated signals with these three types of nonstationarities. We find that introducing nonstationarities to stationary correlated signals leads to the appearance of crossovers in the scaling behavior and we study how the characteristics of these crossovers depend on ~a! the fraction and size of the parts cut out from the signal, ~b! the concentration of spikes and their amplitudes ~c! the proportion between segments with different standard deviations or different correlations and ~d! the correlation properties of the stationary signal. We show how to develop strategies for preprocessing ‘‘raw’’ data prior to analysis, which will minimize the effects of nonstationarities on the scaling properties of the data, and how to interpret the results of DFA for complex signals with different local characteristics.

Effect of nonstationarities on detrended fluctuation analysis.

Effect of Nonstationarities on Detrended Fluctuation Analysis

Physiological and behavioural phenomena are often complex, characterized by variations in time series. Variations in time series reflect how these phenomena organize into coherent structures by interactions that span multiple scales in both time and space. The present tutorial is an introduction to multifractal analyses that can identify these scale invariant interactions within time series by its multifractal spectrum. The multifractal spectrum can be estimated directly from scale-dependent measurements or from its q-order statistics. The tutorial emphasizes the most common scale-dependent measurements defined by the wavelet transforms and the detrended fluctuation analyses. The tutorial also emphasizes common features of all multifractal analyses, like the choice of linear regression method, scaling range and elimination of spurious singularities, which are important for a robust estimation of the multifractal spectrum. The tutorial ends with two brief examples where multifractal analyses are employed to time series from multifractal models and the complex phenomena of cognitive performance. References to available software for multifractal analyses are included at the end of the tutorial. The main aim of the tutorial is to give the reader an introduction to multifractal analyses without the extensive technicalities typically provided in mathematical journals.

https://www.ntnu.edu/documents/170234/1315232/Introduction_to_MFDFA.pdf

Introduction to Multifractal Detrended Fluctuation Analysis in Matlab

We examine the detrended fluctuation analysis (DFA), which is a well-established method for the detection of long-range correlations in time series. We show that deviations from scaling which appear at small time scales become stronger in higher orders of DFA, and suggest a modified DFA method to remove them. The improvement is necessary especially for short records that are affected by non-stationarities. Furthermore, we describe how crossovers in the correlation behavior can be detected reliably and determined quantitatively and show how several types of trends in the data affect the different orders of DFA.

Detecting long-range correlations with detrended fluctuation analysis

The paddlefish perceives its world through an array of electroreceptor organs spread over its rostrum. Using Detrended-Fluctuation Analysis (DFA) we find long-range anti- correlations in sequences of time intervals between neural action potentials recorded from spon- taneously firing electroreceptor afferents. In contrast, spontaneous discharges from mechano- receptor afferents in the crayfish tailfan lack long-range correlations altogether. We interpret this finding as a consequence of the self-sustained oscillatory nature of the electroreceptor organ compared with the threshold-type dynamics of the mechanoreceptor.

Long-range temporal anti-correlations in paddlefish electroreceptors

Using an agent-based model we examine the dynamics of stock price fluctuations and their rates of return in an artificial financial market composed of fundamentalist and chartist agents with and without confidence. We find that chartist agents who are confident generate higher price and rate of return volatilities than those who are not. We also find that kurtosis and skewness are lower in our simulation study of agents who are not confident. We show that the stock price and confidence index-both generated by our model-are cointegrated and that stock price affects confidence index but confidence index does not affect stock price. We next compare the results of our model with the S&P 500 index and its respective stock market confidence index using cointegration and Granger tests. As in our model, we find that stock prices drive their respective confidence indices, but that the opposite relationship, i.e., the assumption that confidence indices drive stock prices, is not significant.

/pdf/confidence-and-self-attribution-bias-in-an-artificial-stock-mijbp32j9v.pdf

Confidence and self-attribution bias in an artificial stock market.

Various social, financial, biological and technological systems can be modeled by interdependent networks. It has been assumed that in order to remain functional, nodes in one network must receive the support from nodes belonging to different networks. So far these models have been limited to the case in which the failure propagates across networks only if the nodes lose all their supply nodes. In this paper we develop a more realistic model for two interdependent networks in which each node has its own supply threshold, i.e., they need the support of a minimum number of supply nodes to remain functional. In addition, we analyze different conditions of internal node failure due to disconnection from nodes within its own network. We show that several local internal failure conditions lead to similar nontrivial results. When there are no internal failures the model is equivalent to a bipartite system, which can be useful to model a financial market. We explore the rich behaviors of these models that include discontinuous and continuous phase transitions. Using the generating functions formalism, we analytically solve all the models in the limit of infinitely large networks and find an excellent agreement with the stochastic simulations.

/pdf/cascading-failures-in-interdependent-networks-with-multiple-3ejddotzjk.pdf

Cascading Failures in Interdependent Networks with Multiple Supply-Demand Links and Functionality Thresholds

We present a Monte Carlo study of a linear chain (d=1) with long-range bonds whose occupancy probabilities are given by p ij =p/r ij α (0⩽p⩽1; α⩾0) where rij=1,2,… is the distance between sites. The α→∞ (α=0) corresponds to the first-neighbor (“mean field”) particular case. We exhibit that the order parameter P∞ equals unity ∀p>0 for 0⩽α⩽1, presents a familiar behavior (i.e., 0 for p⩽pc(α) and finite otherwise) for 1 2. Our results confirm recent conjecture, namely that the nonextensive region (0⩽α⩽1) can be meaningfully unfolded, as well as unified with the extensive region (α>1), by exhibiting P∞ as a function of p ∗ where (1−p ∗ )=(1−p) N ∗ (N ∗ ≡(N 1−α/d −1)/(1−α/d), N being the number of sites of the chain). A corollary of this conjecture, now numerically verified, is that pc∝(α−1) in the α→1+0 limit.

Henio H. A. Rego

Papers

Detecting long-range correlations with detrended fluctuation analysis

Long-range temporal anti-correlations in paddlefish electroreceptors

Confidence and self-attribution bias in an artificial stock market.

Cascading Failures in Interdependent Networks with Multiple Supply-Demand Links and Functionality Thresholds

Crossover from extensive to nonextensive behavior driven by long-range d=1 bond percolation