Self-similarity of higher-order moving averages

Probabilistic properties of detrended fluctuation analysis for Gaussian processes

Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter.

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.

An efficient algorithm for extracting the magnitude of the measurement error for fractional dynamics

Modern live-imaging fluorescent microscopy techniques following the stochastic motion of labeled tracer particles, i.e. single particle tracking (SPT) experiments, have uncovered significant deviations from the laws of Brownian motion in a variety of biological systems. Accurately characterizing the anomalous diffusion for SPT experiments has become a central issue in biophysics. However, measurement errors raise difficulty in the analysis of single trajectories. In this paper, we introduce a novel surface calibration method based on a fractionally integrated moving average (FIMA) process as an effective tool for extracting both the magnitude of the measurement error and the anomalous exponent for autocorrelated processes of various origins. This method is developed using a toy model - fractional Brownian motion disturbed by independent Gaussian white noise - and is illustrated on both simulated and experimental biological data. We also compare this new method with the mean-squared displacement (MSD) technique, extended to capture the measurement noise in the toy model, which shows inferior results. The introduced procedure is expected to allow for more accurate analysis of fractional anomalous diffusion trajectories with measurement errors across different experimental fields and without the need for any calibration measurements.

Direct determination approach for the multifractal detrending moving average analysis

In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent τ(q) is related to the partition function and the multifractal spectrum f(α) can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional p-model, the two-dimensional p-model, and the fractional Brownian motions. We find that both approaches have comparable performances to unveil the fractal and multifractal nature. In other words, without loss of accuracy, the multifractal spectrum f(α) can be directly determined using the new approach with less computation cost. We also apply the new MF-DMA approach to the volatility time series of stock prices and confirm the presence of multifractality.

Response of vegetation phenology to urbanization in the conterminous United States

The influence of urbanization on vegetation phenology is gaining considerable attention due to its implications for human health, cycling of carbon and other nutrients in Earth system. In this study, we examined the relationship between change in vegetation phenology and urban size, an indicator of urbanization, for the conterminous United States. We studied more than 4500 urban clusters of varying size to determine the impact of urbanization on plant phenology, with the aids of remotely sensed observations since 2003-2012. We found that phenology cycle (changes in vegetation greenness) in urban areas starts earlier (start of season, SOS) and ends later (end of season, EOS), resulting in a longer growing season length (GSL), when compared to the respective surrounding urban areas. The average difference of GSL between urban and rural areas over all vegetation types, considered in this study, is about 9 days. Also, the extended GSL in urban area is consistent among different climate zones in the United States, whereas their magnitudes are varying across regions. We found that a tenfold increase in urban size could result in an earlier SOS of about 1.3 days and a later EOS of around 2.4 days. As a result, the GSL could be extended by approximately 3.6 days with a range of 1.6-6.5 days for 25th ~ 75th quantiles, with a median value of about 2.1 days. For different vegetation types, the phenology response to urbanization, as defined by GSL, ranges from 1 to 4 days. The quantitative relationship between phenology and urbanization is of great use for developing improved models of vegetation phenology dynamics under future urbanization, and for developing change indicators to assess the impacts of urbanization on vegetation phenology.

Detrended fluctuation analysis and the difference between external drifts and intrinsic diffusionlike nonstationarity

Detrended fluctuation analysis (DFA) has been shown to be an effective method to study long-range correlation of nonstationary series. In principle, DFA considers F_{DFA}^{2}(s), the mean of variance around the local polynomial fit in segments with length s, and then estimates the scaling exponent α_{DFA} in F_{DFA}(s)∼s^{α_{DFA}} with varying s. Usually, the methodological studies of DFA focus on its effect on removing the drift due to the external trends. Only few paid attention to nonstationary series without drift, such as fractional Brownian motion (FBM) with nonstationarity due to its intrinsic dynamics. Both of these types of nonstationarity can shift the local mean by drift or diffusion and can be treated as the additive nonstationarity eliminable by the additive decomposition. In this study, we limit our discussion to such additive nonstationarity and furthermore specifically distinguish these two types of nonstationarity, namely the drift and the intrinsic diffusionlike nonstationarity. To understand how DFA works for the intrinsic diffusionlike nonstationarity, we take FBM as the example and seek for the answers to two fundamental questions: (1) what DFA removes from FBM; and (2) why DFA can handle such intrinsic diffusionlike nonstationarity, in contrast to methods only applicable to stationary series such as the fluctuation analysis. A crucial condition, i.e., statistical equivalence among all segments, is proposed and checked in the fluctuation analysis and DFA. As shown, the crucial condition is a natural requirement for the connection between DFA and autocorrelation function. With the help of the crucial condition, our study analytically and numerically demonstrates for the intrinsic diffusionlike nonstationary series that (1) rather than the nonstationarity as thought, DFA actually removes the difference among all segments; (2) the detrended segments fulfill the crucial condition so that the average over segments becomes equivalent to the ensemble average over realizations. These answers are also true for series with a drift. Thus, we provide a unified perspective to refresh the understanding of how DFA works on nonstationarity and underpin the mathematical ground of DFA.

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.

Fast algorithm for scaling analysis with higher-order detrending moving average method

Among scaling analysis methods based on the root-mean-square deviation from the estimated trend, it has been demonstrated that centered detrending moving average (DMA) analysis with a simple moving average has good performance when characterizing long-range correlation or fractal scaling behavior. Furthermore, higher-order DMA has also been proposed; it is shown to have better detrending capabilities, removing higher-order polynomial trends than original DMA. However, a straightforward implementation of higher-order DMA requires a very high computational cost, which would prevent practical use of this method. To solve this issue, in this study, we introduce a fast algorithm for higher-order DMA, which consists of two techniques: (1) parallel translation of moving averaging windows by a fixed interval; (2) recurrence formulas for the calculation of summations. Our algorithm can significantly reduce computational cost. Monte Carlo experiments show that the computational time of our algorithm is approximately proportional to the data length, although that of the conventional algorithm is proportional to the square of the data length. The efficiency of our algorithm is also shown by a systematic study of the performance of higher-order DMA, such as the range of detectable scaling exponents and detrending capability for removing polynomial trends. In addition, through the analysis of heart-rate variability time series, we discuss possible applications of higher-order DMA.

Establishing a direct connection between detrended fluctuation analysis and Fourier analysis

To understand methodological features of the detrended fluctuation analysis (DFA) using a higher-order polynomial fitting, we establish the direct connection between DFA and Fourier analysis. Based on an exact calculation of the single-frequency response of the DFA, the following facts are shown analytically: (1) in the analysis of stochastic processes exhibiting a power-law scaling of the power spectral density (PSD), S(f)∼f(-β), a higher-order detrending in the DFA has no adverse effect in the estimation of the DFA scaling exponent α, which satisfies the scaling relation α=(β+1)/2; (2) the upper limit of the scaling exponents detectable by the DFA depends on the order of polynomial fit used in the DFA, and is bounded by m+1, where m is the order of the polynomial fit; (3) the relation between the time scale in the DFA and the corresponding frequency in the PSD are distorted depending on both the order of the DFA and the frequency dependence of the PSD. We can improve the scale distortion by introducing the corrected time scale in the DFA corresponding to the inverse of the frequency scale in the PSD. In addition, our analytical approach makes it possible to characterize variants of the DFA using different types of detrending. As an application, properties of the detrending moving average algorithm are discussed.

Estimating the anomalous diffusion exponent for single particle tracking data with measurement errors - An alternative approach

Accurately characterizing the anomalous diffusion of a tracer particle has become a central issue in biophysics. However, measurement errors raise difficulty in the characterization of single trajectories, which is usually performed through the time-averaged mean square displacement (TAMSD). In this paper, we study a fractionally integrated moving average (FIMA) process as an appropriate model for anomalous diffusion data with measurement errors. We compare FIMA and traditional TAMSD estimators for the anomalous diffusion exponent. The ability of the FIMA framework to characterize dynamics in a wide range of anomalous exponents and noise levels through the simulation of a toy model (fractional Brownian motion disturbed by Gaussian white noise) is discussed. Comparison to the TAMSD technique, shows that FIMA estimation is superior in many scenarios. This is expected to enable new measurement regimes for single particle tracking (SPT) experiments even in the presence of high measurement errors.

Detection of a long-range correlation with an adaptive detrending method

We propose a methodology of estimating the scaling exponent for a long-range correlation in a nonstationary time series from the perspective of the regression analysis. By an adaptive degree determination of a regression polynomial, the proposed methodology is designed to properly remove various types of trends embedded in the nonstationary signal so that the scaling exponent can be estimated without artificial crossovers. To show the validity of the proposed methodology, we applied it to the detrended fluctuation analysis and tested it out against correlated data superimposed by various types of trends. It turned out that, unlike the conventional technique, our approach was capable of eliminating artificial crossovers. We also discuss the statistical characteristics of the proposed method with regard to the estimation of the scaling exponent.