Notwithstanding the significant efforts to develop estimators of long-range correlations (LRC) and to compare their performance, no clear consensus exists on what is the best method and under which conditions. In addition, synthetic tests suggest that the performance of LRC estimators varies when using different generators of LRC time series. Here, we compare the performances of four estimators [Fluctuation Analysis (FA), Detrended Fluctuation Analysis (DFA), Backward Detrending Moving Average (BDMA), and Centred Detrending Moving Average (CDMA)]. We use three different generators [Fractional Gaussian Noises, and two ways of generating Fractional Brownian Motions]. We find that CDMA has the best performance and DFA is only slightly worse in some situations, while FA performs the worst. In addition, CDMA and DFA are less sensitive to the scaling range than FA. Hence, CDMA and DFA remain “The Methods of Choice” in determining the Hurst index of time series.

The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. As many long-term correlated time series in real systems contain various trends, we investigate the effects of polynomial trends on the scaling behaviors and the performances of three widely used DMA methods including backward algorithm (BDMA), centered algorithm (CDMA) and forward algorithm (FDMA). We derive a general framework for polynomial trends and obtain analytical results for constant shifts and linear trends. We find that the behavior of the CDMA method is not influenced by constant shifts. In contrast, linear trends cause a crossover in the CDMA fluctuation functions. We also find that constant shifts and linear trends cause crossovers in the fluctuation functions obtained from the BDMA and FDMA methods. When a crossover exists, the scaling behavior at small scales comes from the intrinsic time series while that at large scales is dominated by § Corresponding author. 1550034-1Fractals Downloaded from www.worldscientific.com by EMORY UNIVERSITY on 07/28/15. For personal use only.Y. -H. Shao et al. the constant shifts or linear trends. We also derive analytically the expressions of crossover scales and show that the crossover scale depends on the strength of the polynomial trends, the Hurst index, and in some cases (linear trends for BDMA and FDMA) the length of the time series. In all cases, the BDMA and the FDMA behave almost the same under the influence of constant shifts or linear trends. Extensive numerical experiments confirm excellently the analytical derivations. We conclude that the CDMA method outperforms the BDMA and FDMA methods in the presence of polynomial trends.

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Background: The characteristics of plaque that ultimately lead to different subcortical infarctions remain unclear. We explored the differences in plaque characteristics between patients with small subcortical infarction (SSI) and large subcortical infarction (LSI) of the middle cerebral artery (MCA) using highresolution magnetic resonance vessel wall imaging (HR-MRVWI).Methods: The study group comprised 71 patients (mean age, 47.49±11.5 years; 55 male) with MCA territory ischemic stroke. Whole-brain HR-MRVWI was performed using a three-dimensional T1-weighted variable-flip-angle turbo spin echo (SPACE) sequence. Patients were divided into SSI and LSI groups based on routine MRI images. Plaque distribution was classified as the superior, inferior, ventral, or dorsal wall of the MCA. The number of quadrants with plaque formation, location of plaque, plaque burden (PB), arterial remodeling pattern (positive or negative), and degree of stenosis were analyzed and compared between groups.Results: Of the 71 patients, 43 (60.6%) and 28 (39.4%) were identified as the SSI and LSI groups, respectively. The proportion of plaques involving only one quadrant was significantly higher in the SSI group, and these plaques were located in the superior or dorsal MCA vessel wall. There was no significant difference between groups in the proportion of plaques involving two or more quadrants, plaque distribution, or PB. Most plaques in both groups showed positive remodeling, and the percentage of remodeling pattern was similar. A significantly higher incidence of low-grade stenosis (<50%) was observed in the SSI group.Conclusions: Both SSI and LSI may be associated with major intracranial artery atherosclerosis, but patients with SSI showed relatively fewer quadrants with plaque formation and a lesser degree of stenosis.