Detrending Moving Average Research Articles

Analysing the intrinsic multifractal nature of seismic sequences distributed along the Pacific Ring of Fire

A multifractal methodology was utilized to analyze a set of seismic sequences distributed along the Pacific Ring of Fire, sourced from the National Earthquake Information Center (NEIC) catalog. The analysis employed the Multifractal Detrended Moving Average (MFDMA) analysis method to characterize the presence of intrinsic multifractality using different multifractal indicators. The findings of this study can be summarized as follows: our statistical tests confidently assert that only the Kuriles, Peru, and Solomon Islands zones demonstrate an absence of intrinsic multifractality; tests were also applied using the exponent , but they were inconclusive. We believe these results contribute to an enhanced understanding of the multifractal characteristics of seismic activity and their implications for earthquake dynamics.

Multidimensional fractal scaling analysis using higher order moving average polynomials and its fast algorithm

The detrending moving average (DMA) analysis demonstrates excellent performance for the characterization of long-range correlations and fractal scaling and is performed in various research fields. The conventional DMA with a simple moving average can remove linear trends embedded in the observed time series. To improve the detrending ability of the DMA, higher-order DMA including a higher order polynomial detrending was also introduced using the Savitzky-Golay filter and its fast implementation algorithm was developed. However, the higher-order DMA applicable to higher dimensional data is yet to be well established. As the data dimension increases, an increase in the computational cost becomes a problem that needs to be resolved. Further, the implementation of the higher order DMA is a time-consuming procedure. To resolve this problem, we here proposed a fast algorithm for multidimensional DMA with higher order polynomial detrending. In the proposed algorithm, to reduce the computational complexity, parallel translation and recurrence techniques are introduced. Monte Carlo experiments for two-dimensional data show that the computational time of the proposed algorithm is approximately proportional to the cubic of the data length, whereas the computational time of the conventional implementation is approximately proportional to the quartic of the data length. Moreover, we evaluate the estimation accuracy of the Hurst exponent of the proposed method. Finally, we demonstrate the possible application of the proposed method by estimating the Hurst exponent of images.

Multifractal characterization as a function of timescale in the light curves with planetary signal observed by the Kepler mission

We investigate the evolution of five multifractal indicators of the Kepler Mission light curves of the moderately young Sun-like Kepler-30 star accompanied by a three-planet system. We focus not only on the fact that the rotational modulation has multifractality properties but also on some studies on these properties based on scale. Then, we systematically investigate the dynamic behaviors of the small and large fluctuations in two types of light curves (PDC and SAP). Those fluctuations indicate that the properties of stellar noise and rotation modulation are highlighted when we apply the Multifractal Detrended Moving Average (MFDMA) algorithm. Our results also demonstrated that the multifractality of the light curve is due to both long-range correlation and broad probability density function, but the main source of multifractality is the long-range correlation. This new approach can be used to develop theoretical and computational models for various stellar magnetic activity-related phenomena and their interactions with the planets, and it can greatly simplify spot modeling from current TESS and future PLATO data.

Multi-scale regression based on detrending moving average and its application to seismic data

We investigate the statistical properties of multi-scale regression model based on detrending moving average (DMA). The performance of the multi-scale regression estimator based on DMA is evaluated by varying the length, distribution and structure for different position parameters. Using different position parameters for the detrending windows in simulation, we find that the variance of the estimated regression coefficients for position parameter [Formula: see text] is the smallest. By changing series length, distribution and structure, the estimated regression coefficients are stably near the theoretical values. The method is applied to analyze the dependence of inter-earthquakes time (IET) on inter-earthquakes distances (IED) between consecutive earthquakes in the California region. Results suggest that the cross-correlation between the IET and IED series is statistically significant. Scale-dependent statistic of estimated DMA multi-scale regression coefficient demonstrates significant dependence between IET and IED series.

Detrending Moving Average, Power Spectral Density, and Coherence: Three EEG-Based Methods to Assess Emotion Irradiation during Facial Perception

Understanding brain reactions to facial expressions can help in explaining emotion-processing and memory mechanisms. The purpose of this research is to examine the dynamics of electrical brain activity caused by visual emotional stimuli. The focus is on detecting changes in cognitive mechanisms produced by negative, positive, and neutral expressions on human faces. Three methods were used to study brain reactions: power spectral density, detrending moving average (DMA), and coherence analysis. Using electroencephalogram (EEG) recordings from 48 subjects while presenting facial image stimuli from the International Affective Picture System, the topographic representation of the evoked responses was acquired and evaluated to disclose the specific EEG-based activity patterns in the cortex. The theta and beta systems are two key cognitive systems of the brain that are activated differently on the basis of gender. The obtained results also demonstrate that the DMA method can provide information about the cortical networks’ functioning stability, so it can be coupled with more prevalent methods of EEG analysis.

Detrending-moving-average-based multivariate regression model for nonstationary series.

Dependency between a response variable and the explanatory variables is a relationship of universal concern in various real-world problems. Multivariate linear regression (MLR) is a well-known method to focus on this issue. However, it is limited to dealing with stationary variables. In this work, we develop a MLR framework based on detrending moving average (DMA) analysis to reveal the actual dependency among variables with nonstationary measures. The DMA-based MLR can generate multiscale regression coefficients, which characterize different dependent behavior at different timescales. Artificial tests show that the DMA-MLR model can successfully resist the impact of trends on the studied series and produce more accurate regression coefficients with multiscale. Furthermore, some scale-dependent statistics are developed to deduce some important relationships in three typical DMA-based MLR models, which help us to deeply understand the DMA-MLR models in theory. The application of the proposed DMA-MLR framework to Beijing's air quality index system demonstrates that fine particulate matter with diameter ≤2.5μm (PM_{2.5}) is the dominant pollutant affecting the air quality of Beijing in recent years.

A novel Bayesian approach to estimate long memory parameter

The property of long memory and its parameter estimate are important in financial econometrics. This paper proposes a novel Bayesian approach to estimate the fractional differencing order and compares its performance with rescaled range analysis (R/S), detrended fluctuation analysis (DFA) and detrended moving average (DMA), based on the fractal Brownian motion sequence simulated by the ARFIMA model. The results indicate that the estimation accuracy of the Bayesian approach is significantly improved and is less volatile than R/S and DFA methods. We adopt mean absolute error and standard deviation as the criteria for measuring the finite sample performance of the estimators. To check robust results, four different methods (i.e. the Gelman–Rubin diagnostic, the Geweke diagnostic, the Heidelberger–Welch diagnostic, the Raftery–Lewis diagnostic) are simultaneously implemented to monitor the convergence of the posterior distribution in the Bayesian methodology. Therefore, the long memory estimator from the Bayesian model is preferred according to the results in this paper.

How Statistically Significant is the DMCA Coefficient?

In this paper, we proposed a statistical test for the Detrending Moving-Average Cross-Correlation Coefficient ([Formula: see text]). With this methodology, it is possible to evaluate the statistical significance of [Formula: see text] for different confidence levels. The test was applied to financial market and climatological data. Findings on this research show that rejection or non-rejection of the null hypothesis of [Formula: see text] depends on the size [Formula: see text] of the series and the moving average window length [Formula: see text] evaluated. Our findings also show a behavioral pattern in the critical values of [Formula: see text]. Fixing the size of the series [Formula: see text], as the size of the moving average window length [Formula: see text] increases, the critical values tend to increase.

Calculating Hurst Exponent with the Use of the Siroky Method in Developed and Emerging Markets

The purpose of the article This paper analysis Hurst exponents calculated with the use of the Siroky method in two time intervals of 625 (H625) and 1250 (H1260) sessions for the following assets: (the number of assets for a given group in brackets): Stock indices (74), currency pairs divided into segments: USD exchange rate in relation to 42 other currencies (USDXXX), EURO exchange rate in relation to 41 other currencies (EURXXX), JPY exchange rate in relation to 40 other currencies (JPYXXX) and other currency pairs (12). In total, 209 financial instruments were analyzed. Methodology: Hurst coefficient calculation with the use of the following methods; Siroky, Detrended Moving Average (DMA) and Detrended Fluctuation Analysis (DFA). Results of the research: The Hurst coefficient values calculated with the use of Siroky method are similar to the results obtained using DFA and DMA methods. The second main conclusion that was drawn from the research may be formulated as follows: exchange rates calculated for the developed-developed country currencies are more effective than in the case of the developed-emerging countries group.

Alternative ways to compare the detrended fluctuation analysis and its variants. Application to visual tunneling detection

The detrended fluctuation analysis (DFA) and its variants such as the detrended moving average (DMA) are widely used to estimate the Hurst exponent. These methods are very popular as they do not require advanced skills in the field of signal processing and statistics while providing accurate results. As a consequence, a great deal of interest has been paid to compare them and to better understand their behaviors from a mathematical point of view. In this paper, our contribution is threefold. Firstly, we propose another variant avoiding the discontinuities between consecutive local trends of the DFA by a priori constraining them to be continuous. Secondly, we show that, in all these approaches, the square of the fluctuation function can be presented in a similar matrix form. When the process is wide-sense stationary (w.s.s.), the latter can be seen as the power of the output of a linear filtering whose frequency response depends on the given method. In the general case, an interpretation of the square of the fluctuation function is also given by expressing it as the convolution between the 2D-Fourier transform of two matrices, one whose elements correspond to the instantaneous correlation function of the signal and the other which depends on the detrending method. To end up, an illustration is provided in the field of avionics for the detection of the visual tunneling, a deleterious cognitive state.

Identification of EMCI in MR Brainstem Structure Using Fractal Measures and Random Forest Approach.

In this work, an attempt has been made to assess the texture variations of brainstem occurring in the Early Mild Cognitive Impairment (EMCI) condition. Fractal dimension is calculated for the segmented brainstem volume using Higuchi's and Detrending moving average (DMA) method. These measures are validated using Random forest classifier. DMA shows better performance when compared to Higuchi's method in categorising EMCI stage. Thus, the proposed approach using brainstem image's DMA fractal measures with random forest can be used in diagnosis of EMCI.

Evaluation of monofractal and multifractal properties of inter‐beat (R‐R) intervals in cardiac signals for differentiation between the normal and pathology classes

In this study, the monofractal and multifractal properties of inter-beat (R-R) intervals in cardiac signals for normal and pathology classes were studied and applied to a dataset in PhysioNet which consists of 24 h R-R intervals from 54 healthy subjects (hs) and 92 patients with various diagnoses [44 with congestive heart failure (chf), 25 with atrial fibrillation (af) and 23 diagnosed with sudden death syndrome (sd)]. The results in this study indicate that the most suitable method for estimating the monofractal properties of R-R intervals is detrending moving average (DMA). The Hurst exponents ( H ) of the healthy and pathological groups, calculated using the DMA method, are shown to be statistically different by the Kruskal–Wallis test [p-value < 0.01; healthy H hs = 0.23(0.18–0.27); pathological group H chf = 0.06(0.03–0.10), H af = 0.05(0.03–0.07), H sd = 0.05(0.03–0.08)]. To study the multifractal properties of R-R intervals, the multifractal detrended fluctuation analysis was used. Formal statistical tests indicate statistically significant differences (p-value < 0.05, ANOVA and pairwise testing) between the width of the spectrum of the chf group W chf = 1.11 ± 0.12 and the width of the spectra of the af group W af = 0.58 ± 0.15, the sd group W sd = 0.60 ± 0.20, and the hs group W hs = 0.61 ± 0.08. Using these results, a logistic regression model was developed for differentiating chf from other pathologies (af and sd).

Theoretical foundation of detrending methods for fluctuation analysis such as detrended fluctuation analysis and detrending moving average.

We present a bottom-up derivation of fluctuation analysis with detrending for the detection of long-range correlations in the presence of additive trends or intrinsic nonstationarities. While the well-known detrended fluctuation analysis (DFA) and detrending moving average (DMA) were introduced ad hoc, we claim basic principles for such methods where DFA and DMA are then shown to be specific realizations. The mean-squared displacement of the summed time series contains the same information about long-range correlations as the autocorrelation function but has much better statistical properties for large time lags. However, the scaling exponent of its estimator on a single time series is affected not only by trends on the data but also by intrinsic nonstationarities. We therefore define the fluctuation function as mean-squared displacement with weighting kernel. We require that its estimator be unbiased and exhibit the correct scaling behavior for the random component of a signal, which is only achieved if the weighting kernel implies detrending. We show how DFA and DMA satisfy these requirements and we extract their kernel weights.

Fractality and singularity in CME linear speed signal: Cycle 23

In the recent past, coronal mass ejection (CME) has received much research attention for its geo-effectiveness. In this paper, an investigation has been made to identify the scaling pattern of the CME linear speed time series data (February 1999 to December 2007 of solar cycle 23) collected from the Solar and Heliospheric Observatory (SOHO) using Multi-Fractal Detrended Fluctuation Analysis (MFDFA) and Multi-Fractal Detrended Moving Average (MFDMA) method. The scaling exponent, generalized Hurst exponent, singularity strength and also the singularity spectrum have been computed to quantify the multifractality and to identify the singularities of the time series data. An effort has also been made to find out the possible sources which are responsible for the multifractality in the signal by studying the scaling patterns of the shuffled and surrogate version of the original data. It has been revealed in this paper that CME linear speed signal exhibit multifractal behaviour with long-term persistence. Both the long-range temporal correlation and the broad probability density function (pdf) are found to be the primary source of multifractality in the signal. The singularities or abruptness present in the signal are found to vary with time, and this fluctuation follows an AR (2) model.

Multifractality signatures in quasars time series – I. 3C 273

The presence of multifractality in a time series shows different correlations for different time scales as well as intermittent behaviour that cannot be captured by a single scaling exponent. The identification of a multifractal nature allows for a characterization of the dynamics and of the intermittency of the fluctuations in non-linear and complex systems. In this study, we search for a possible multifractal structure (multifractality signature) of the flux variability in the quasar 3C 273 time series for all electromagnetic wavebands at different observation points, and the origins for the observed multifractality. This study is intended to highlight how the scaling behaves across the different bands of the selected candidate which can be used as an additional new technique to group quasars based on the fractal signature observed in their time series and determine whether quasars are non-linear physical systems or not. The Multifractal Detrended Moving Average algorithm (MFDMA) has been used to study the scaling in non-linear, complex and dynamic systems. To achieve this goal, we applied the backward ({\theta} = 0) MFDMA method for one-dimensional signals. We observe weak multifractal (close to monofractal) behaviour in some of the time series of our candidate except in the mm, UV and X-ray bands. The non-linear temporal correlation is the main source of the observed multifractality in the time series whereas the heaviness of the distribution contributes less.

Comparative analysis of grey detrended fluctuation analysis methods based on empirical research on China’s interest rate market

This study empirically compares the development of China’s interest rate market in the past decade. The yield of Shibor market from 2006 to 2016 showed anti-persistence, and the later five-year trend was stronger than that in the previous half period, indicating a significantly growing trend of China’s interest rate market. We perform a comparative analysis of detrended fluctuation analysis (DFA), detrending moving average (DMA) algorithm, and the maximum overlap wavelet transform (DFA-MODWT) using grey and original sequences. The autoregressive integrated moving average model is adopted to generate set-point H sequences. Results illustrated that 1) for original sequences, both the DFA-MODWT and DMA (θ=0.5) shows outstanding performances when the set Hurst exponents are 0.6 and 0.7 and 0.8 and 0.9, respectively. 2) For grey sequences, G-DFA performs best among the three methods, followed by G-DMA and G-DFA-MODWT. 3) The long-range correlation of the original sequences is slightly lower than the set values. On the contrary, the long-range correlation of grey sequences is much higher than the set values. Thus, the grey sequence exerts a strong aggregation effect as it can accumulate the trend of the sequence. Our results show that the grey sequence can determine the regular pattern from the messy sequence. However, grey sequences tend to be overly analyzed and misleading when large volumes of data are involved. Empirical studies confirm that China’s interest rate market remains an inefficient market.

New Suns in the Cosmos. IV. The Multifractal Nature of Stellar Magnetic Activity in Kepler Cool Stars

Abstract In the present study, we investigate the multifractal nature of a long-cadence time series observed by the Kepler mission for a sample of 34 M dwarf stars and the Sun in its active phase. Using the Multifractal Detrending Moving Average algorithm, which enables the detection of multifractality in nonstationary time series, we define a set of multifractal indices based on the multifractal spectrum profile as a measure of the level of stellar magnetic activity. This set of indices is given by the (A, , C, H)-quartet, where A, , and C are related to geometric features from the multifractal spectrum and the global Hurst exponent H describes the global structure and memorability of time series dynamics. As a test, we measure these indices and compare them with a magnetic index defined as S ph and verify the degree of correlation among them. First, we apply the Poincaré plot method and find a strong correlation between the index and one of the descriptors that emerges from this method. As a result, we find that this index is strongly correlated with long-term features of the signal. From the multifractal perspective, the index is also strongly linked to the geometric properties of the multifractal spectrum except for the H index. Furthermore, our results emphasize that the rotation period of stars is scaled by the H index, which is consistent with Skumanich’s relationship. Finally, our approach suggests that the H index may be related to the evolution of stellar angular momentum and a star’s magnetic properties.

Long-range correlation and market segmentation in bond market

This paper investigates the long-range auto-correlations and cross-correlations in bond market. Based on Detrended Moving Average (DMA) method, empirical results present a clear evidence of long-range persistence that exists in one year scale. The degree of long-range correlation related to maturities has an upward tendency with a peak in short term. These findings confirm the expectations of fractal market hypothesis (FMH). Furthermore, we have developed a method based on a complex network to study the long-range cross-correlation structure and applied it to our data, and found a clear pattern of market segmentation in the long run. We also detected the nature of long-range correlation in the sub-period 2007–2012 and 2011–2016. The result from our research shows that long-range auto-correlations are decreasing in the recent years while long-range cross-correlations are strengthening.

Time and frequency domain characteristics of detrending-operation-based scaling analysis: Exact DFA and DMA frequency responses

We develop a general framework to study the time and frequency domain characteristics of detrending-operation-based scaling analysis methods, such as detrended fluctuation analysis (DFA) and detrending moving average (DMA) analysis. In this framework, using either the time or frequency domain approach, the frequency responses of detrending operations are calculated analytically. Although the frequency domain approach based on conventional linear analysis techniques is only applicable to linear detrending operations, the time domain approach presented here is applicable to both linear and nonlinear detrending operations. Furthermore, using the relationship between the time and frequency domain representations of the frequency responses, the frequency domain characteristics of nonlinear detrending operations can be obtained. Based on the calculated frequency responses, it is possible to establish a direct connection between the root-mean-square deviation of the detrending-operation-based scaling analysis and the power spectrum for linear stochastic processes. Here, by applying our methods to DFA and DMA, including higher-order cases, exact frequency responses are calculated. In addition, we analytically investigate the cutoff frequencies of DFA and DMA detrending operations and show that these frequencies are not optimally adjusted to coincide with the corresponding time scale.

Detrending moving average algorithm: Frequency response and scaling performances.

The Detrending Moving Average (DMA) algorithm has been widely used in its several variants for characterizing long-range correlations of random signals and sets (one-dimensional sequences or high-dimensional arrays) over either time or space. In this paper, mainly based on analytical arguments, the scaling performances of the centered DMA, including higher-order ones, are investigated by means of a continuous time approximation and a frequency response approach. Our results are also confirmed by numerical tests. The study is carried out for higher-order DMA operating with moving average polynomials of different degree. In particular, detrending power degree, frequency response, asymptotic scaling, upper limit of the detectable scaling exponent, and finite scale range behavior will be discussed.