In the real world, every nonlinear system is inevitably affected by noise. As an example, a logistic map driven by white noise is studied. Unlike previous studies which focused on the behavior under local parameters to find analytical results, we investigate the whole driven logistic map. For a white noise driven logistic map, its nondivergent interval decreases with increasing white noise. The white noise does not change the equilibrium point and two-cycle intervals in statistics, if the driven logistic map is kept non-divergent. In particular, chaos can be excited by white noise only after the four-cycle bifurcation begins. The latest result is a necessary condition which has not been given in the literature [Int. J. Bifur. Chaos 18 ( 2008) 509], and it can be deduced from Sharkovsky's theorem. Numerical simulations prove these analytical results.
Objective: Serum magnesium (Mg2+) levels are associated with insulin resistance, hypertension, lipid abnormalities, and inflammation. However, limited studies have indicated the relationship between Mg2+ and multiple system indexes. The purpose of this study was to investigate the association between Mg2+ and allostatic load (AL) in hemodialysis patients.Methods: A cross-sectional survey was conducted on hemodialysis patients from different centers in Anhui Province, China, between January and December 2020. A total of 3,025 hemodialysis patients were recruited. Their clinical data were measured before hemodialysis. Information was collected by an online self-reported questionnaire and medical record. Serum Mg2+ was divided into three groups by tertiles. A score of AL greater than or equal to 3 was defined as high AL. A binary logistic regression model was applied to examine the relationship between serum Mg2+ and AL.Results: A total of 1,222 patients undergoing hemodialysis were included, 60% of whom were males (733/1,222). The mean (standard deviation) age of patients was 55.90 (12.75). The median level of serum Mg2+ was 1.22 mmol/L. The rate of high AL levels was 23.4%. Serum Mg2+ was negatively correlated with body mass index, fasting blood glucose (Glu), and C-reactive protein and positively correlated with high-density lipoprotein, low-density lipoprotein, total cholesterol, diastolic blood pressure (DBP), and serum phosphorus. After adjusting for gender, anxiety, diabetes, family residence, lipid-lowering agents, antihypertensive medications, albumin, and Glu, the binary logistic regression model showed that patients with lower levels of serum Mg2+ were more likely have high AL (OR for the T1 group of serum Mg2+:1.945, 95% CI: 1.365–2.773, and OR for the T2 group of serum Mg2+:1.556, 95% CI: 1.099–2.201).Conclusion: Our data support the hypothesis that higher serum Mg2+ concentrations may contribute to lower health risk in hemodialysis populations. Further randomized controlled trials and cohort studies are warranted to verify whether Mg2+ supplementation could be part of routine examinations in hemodialysis populations.
: In order to find the convergence rate of finite sample discrete entropies of a white Gaussian noise(WGN), Brown entropy algorithm is numerically tested.With the increase of sample size, the curves of these finite sample discrete entropies are asymptotically close to their theoretical values.The confidence intervals of the sample Brown entropy are narrower than those of the sample discrete entropy calculated from its differential entropy, which is valid only in the case of a small sample size of WGN. The differences between sample Brown entropies and their theoretical values are fitted by two rational functions exactly, and the revised Brown entropies are more efficient. The application to the prediction of wind speed indicates that the variances of resampled time series increase almost exponentially with the increase of resampling period. Entropy in information theory is a powerful conception to describing the indeterminacy of a non-stationary time series [1,2] , and it has been used in a wide range of fields [3][4][5][6][7][8][9][10][11][12] . Its estimators from time series have also been studied extensively [13] . The differential entropy of a "true" white Gaussian noise(WGN), which follows a normal distribution and has an infinite sample size, is determined exactly by its variance of normal distribution [2] . But in practice, WGNs have finite sample sizes, often contain outliers and deviate from the normal distributions [14] . The exact and robust estimators of entropy and its variance from observed data are needed all the time [15] . In the prediction of non-stationary time series containing a WGN, the WGN forms an inaccessible limit of prediction accuracy. The identification of correct distribution function of the WGN from observed data is still an open problem [15] . In general, if the level of confidence keeps the same, the confidence intervals of a statistic get broader with the diminishing sample size. The smaller the sample size, the greater the differences between the estimated values of entropy or variance of a WGN and their true values [16] .Many studies show that the minimum error entropy (MEE)criterion can outperform the traditional mean square error criterion in supervised machine learning, especially in nonlinear and non-Gaussian situations [3] . Here the mean square error has the same calculating formula as the variance. Although Chen and Principe [3] explained this phenomenon by the smooth characteristic of entropy under the pollution due to an undesired random variable, the reason why the entropy is more reliable than the mean square error or the variance is still unknown. The content of this paper is organized as follows: for a pseudo-WGN with outliers and deviation from the normal distribution, the divergent intervals of its sample discrete entropies and its sample variances are calculated numerically; the convergence rates of sample discrete entropies which are asymptotical to their theoretical discrete entropies are fitted by two rational functions; the lost rate of signal in a wi...
An analytical model has been developed in the present paper based on a square root transformation of white Gaussian noise. The mathematical expectation and variance of the new asymmetric distribution generated by white Gaussian noise after a square root transformation are analytically deduced from the preceding four terms of the Taylor expansion. The model was first evaluated against numerical experiments and a good agreement was obtained. The model was then used to predict time series of wind speeds and highway traffic flows. The simulation results from the new model indicate that the prediction accuracy could be improved by 0.1–1% by removing the mean errors. Further improvement could be obtained for non‐stationary time series, which had large trends. Copyright © 2016 John Wiley & Sons, Ltd.
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