Multifractal Detrended Fluctuation Analysis Algorithm Research Articles

Multi-Signal Multifractal Detrended Fluctuation Analysis for Uncertain Systems —Application to the Energy Consumption of Software Programs in Microcontrollers

Uncertain systems are those wherein some variability is observed, meaning that different observations of the system will produce different measurements. Studying such systems demands the use of statistical methods over multiple measurements, which allows overcoming the uncertainty, based on the premise that a single measurement is not representative of the system’s behavior. In such cases, the current multifractal detrended fluctuation analysis (MFDFA) method cannot offer confident conclusions. This work presents multi-signal MFDFA (MS-MFDFA), a novel methodology for accurately characterizing uncertain systems using the MFDFA algorithm, which enables overcoming the uncertainty of the system by simultaneously considering a large set of signals. As a case study, we consider the problem of characterizing software (Sw) consumption. The difficulty of the problem mainly comes from the complexity of the interactions between Sw and hardware (Hw), as well as from the high uncertainty level of the consumption measurements, which are affected by concurrent Sw services, the Hw, and external factors such as ambient temperature. We apply MS-MFDFA to generate a signature of the Sw consumption profile, regardless of the execution time, the consumption levels, and uncertainty. Multiple consumption signals (or time series) are built from different Sw runs, obtaining a high frequency sampling of the instant input current for each of them while running the Sw. A benchmark of eight Sw programs for analysis is also proposed. Moreover, a fully functional application to automatically perform MS-MFDFA analysis has been made freely available. The results showed that the proposed methodology is a suitable approximation for the multifractal analysis of a large number of time series obtained from uncertain systems. Moreover, analysis of the multifractal properties showed that this approach was able to differentiate between the eight Sw programs studied, showing differences in the temporal scaling range where multifractal behavior is found.

Multifractal structure of the monthly rainfall regime in Catalonia (NE Spain): Evaluation of the non-linear structural complexity.

The complex non-linear regime of the monthly rainfall in Catalonia (NE Spain) is analyzed by means of the reconstruction fractal theorem and the multifractal detrended fluctuation analysis algorithm. Areas with a notable degree of complex physical mechanisms are detected by using the concepts of persistence (Hurst exponent), complexity (embedding dimension), predictive uncertainty (Lyapunov exponents), loss of memory of the mechanism (Kolmogorov exponent), and the set of multifractal parameters (Hölder exponents, spectral asymmetry, spectral width, and complexity index). Besides these analyses permitting a detailed description of monthly rainfall pattern characteristics, the obtained results should also be relevant for new research studies concerning monthly amounts forecasting at a monthly scale. On one hand, the number of necessary monthly data for autoregressive processes could change with the complexity of the multifractal structure of the monthly rainfall regime. On the other hand, the discrepancies between real monthly amounts and those generated by some autoregressive algorithms could be related to some parameters of the reconstruction fractal theorem, such as the Lyapunov and Kolmogorov exponents.

Extraction of Frictional Vibration Features with Multifractal Detrended Fluctuation Analysis and Friction State Recognition

For the purpose of extracting the frictional vibration characteristics of the friction pair during friction and wear in different friction states, the friction and wear tests of friction pair in different friction states were conducted on a testing machine. Higher-dimensional fractal and multifractal characteristics hidden in time series can be examined by multifractal detrended fluctuation analysis (MFDFA) method. The frictional vibration time-domain signals, the friction coefficient signals and the frictional vibration frequency-domain signals were analyzed and multifractal spectra were acquired by using the MFDFA algorithm. According to the spectra, the multifractal spectrum parameters of these signals were calculated to realize the quantitative characterization of frictional vibration characteristics in different friction states. The analysis shows that it is symmetric in the variation trends of the multifractal spectrum parameters of the frictional vibration signals and the friction coefficient data. Based on the multifractal spectrum parameters of frictional vibration, the principal component analysis (PCA) algorithm was applied to establish the friction state recognition method. The results show that the multifractal spectra and their parameters can characterize the frictional vibrations, and the friction state recognition can be realized based on the multifractal spectrum parameters of frictional vibrations.

A Novel MFDFA Algorithm and Its Application to Analysis of Harmonic Multifractal Features

A power grid harmonic signal is characterized as having both nonlinear and nonstationary features. A novel multifractal detrended fluctuation analysis (MFDFA) algorithm combined with the empirical mode decomposition (EMD) theory and template movement is proposed to overcome some shortcomings in the traditional MFDFA algorithm. The novel algorithm is used to study the multifractal feature of harmonic signals at different frequencies. Firstly, the signal is decomposed and the characteristics of wavelet transform multiresolution analysis are employed to obtain the components at different frequency bands. After this, the local fractal characteristic of the components is studied by utilizing the novel MFDFA algorithm. The experimental results show that the harmonic signals exhibit obvious multifractal characteristics and that the multifractal intensity is related to the signal frequency. Compared with the traditional MFDFA algorithm, the proposed method is more stable in curve fitting and can extract the multifractal features more accurately.

Random deposition with a power-law noise model: Multiaffine analysis.

We study the random deposition model with power-law distributed noise and rare-event dominated fluctuation. In this model instead of particles with unit sizes, rods with variable lengths are deposited onto the substrate. The length of each rod is chosen from a power-law distribution P(l)∼l^{-(μ+1)}, and the site at which each rod is deposited is chosen randomly. The results show that for μ<μ_{c}=3 the log-log diagram of roughness, W(t), versus deposition time, t, increases as a step function, where the roughness in each interval acts as W_{loc}(t)≈t^{β_{loc}}. The local growth exponent, β_{loc}, is zero for μ=1. By increasing the μ exponent, the value of β_{loc} is increased. It tends to the growth exponent of the random distribution model with Gaussian noise, β=1/2, at μ_{c}=3. The fractal analysis of the height fluctuations for this model was performed by multifractal detrended fluctuation analysis algorithm. The results show multiaffinity behavior for the height fluctuations at μ<μ_{c} and the multiaffinity strength is greater for smaller values of the μ exponent.

Multifractal Detrended Fluctuation Analysis of Frictional Vibration Signals in the Running-in Wear Process

Multifractal detrended fluctuation analysis (MFDFA) provides a valuable tool for extracting nonlinear characteristics of signals, which makes it very powerful for the status recognition of friction pair by analyzing frictional vibration signals. This paper presents an algorithm for denoising the frictional vibration signals by using the ensemble empirical mode decomposition. The denoised signals of frictional vibration were analyzed by utilizing the MFDFA algorithm to derive the q-order Hurst exponent as well as multifractal spectrum. The paper illustrates these issues by analyzing signals taken from the friction and wear experiments on CFT-I testing machine. The results show that the q-order Hurst exponent, as well as multifractal spectrum, presents a certain trend with the running-in wear process of friction pair. The MFDFA algorithm can extract effectively the fractal characteristics of the frictional vibration signals. The frictional vibration signals could be characterized by the q-order Hurst exponent and multifractal spectrum.

Multi-fractal detrended fluctuation analysis algorithm based identification method of scale-less range for multi-fractal charateristics of traffic flow

Scale-less range is an interval of measurement of time series in which fractional self-similarity exists statistically. In order to solve the problem of the lack of necessary steps to calculate fractal range in multi-fractal detrended fluctuation analysis algorithm (MF-DFA) in traffic flow, a new scale-less identification method based on MF-DFA is proposed through analyzing the characteristics of the mutation point in logistic curve of traffic flow wave function in steps of MF-DFA and the principles of the traditional fractal scale-less range identification method. Beijing's road network is taken for example to investigate the fractal scale-less range. Analysis results show that the identification method based on MF-DFA algorithm is valid, automatic and steady in identifying the fractal scale-less range in Beijing's traffic flow. Further, the reason why the scale-less range in traffic is limited is that small traffic flow waves account for a bigger percentage in scale-less range while big wave is bigger so that it is out of the scale-less range.

An Improved MapReduce Model for Computation-Intensive Task

MapReduce is a widely adopted parallel programming model. The standard MapReduce model is designed for data-intensive processing. However, some machine learning algorithms are computation-intensive and time-consuming tasks which process the same data set repeatedly. In this paper, we proposed an improved MapReduce model for computation-intensive algorithms. The model is constructed from a service combination perspective. In the model, the whole task is divided into lots of subtasks taking account into the algorithms parameters, and the datagram with acknowledgement mechanism is used as the communication channel among cluster workers. We took the multifractal detrended fluctuation analysis algorithm as an example to demonstrate the model.

Multifractal information fusion based condition diagnosis for process complex systems

Abnormal conditions are hazardous in process complex systems, and the aim of condition diagnosis is to detect abnormal conditions and thus avoid serious accidents. Comparing with conventional techniques of condition diagnosis without concerning the nonlinearity of complex system, multifractal analysis elaborately reveals scale-invariance or self-similarity properties of time series data, which is one of the intrinsic characteristics of complex systems. Moreover, the monitoring data within multiple feature variables should be investigated by combining multifractal analysis and information fusion techniques, so that significant patterns of the whole system would be discovered. In this article, a condition diagnosis framework is proposed for industrial complex systems, by which nonlinear features are extracted from univariate time series through multifractal analysis using multifractal detrended fluctuation analysis algorithm, and multiple feature variables are investigated through Mahalanobis–Taguchi system as an information fusion method to determine the condition of the whole system. The effectiveness of the approach is illustrated using data from both simulated model and real production system in an industrial enterprise.

Multifractality of laser beam spatial intensity in a turbulent medium

We present the results of a laser beam passing through a turbulent medium. First we measure the geometric parameters and the spatial coherence of the beam as a function of wind velocities. A multifractal detrended fluctuation analysis algorithm is applied to determine the multifractal scaling behavior of the intensity patterns. The measurements are fitted with models used in the analysis of river runoff records. We show the surprising result that the multifractality decreases when the spatial coherence of the laser is decreased. Universal scaling properties could be applied to the spatial characterization of a laser propagating in a turbulent medium or random medium.