Statistics and contrasts of order patterns in univariate time series.

Order patterns apply well to many fields, because of minimal stationarity assumptions. Here, we fix the methodology of patterns of length 3 by introducing an orthogonal system of four pattern contrasts, that is, weighted differences of pattern frequencies. These contrasts are statistically independent and turn up as eigenvectors of a covariance matrix both in the independence model and the random walk model. The most important contrast is the turning rate. It can be used to evaluate sleep depth directly from EEG (electroencephalographic brain data). The paper discusses fluctuations of permutation entropy, statistical tests, and the need of new models for noises like EEG.

Two new parameters for the ordinal analysis of images.

Local patterns play an important role in statistical physics as well as in image processing. Two-dimensional ordinal patterns were studied by Ribeiro et al. who determined permutation entropy and complexity in order to classify paintings and images of liquid crystals. Here, we find that the 2 × 2 patterns of neighboring pixels come in three types. The statistics of these types, expressed by two parameters, contains the relevant information to describe and distinguish textures. The parameters are most stable and informative for isotropic structures.

Entropy Ratio and Entropy Concentration Coefficient, with Application to the COVID-19 Pandemic.

In order to study the spread of an epidemic over a region as a function of time, we introduce an entropy ratio U describing the uniformity of infections over various states and their districts, and an entropy concentration coefficient C=1-U. The latter is a multiplicative version of the Kullback-Leibler distance, with values between 0 and 1. For product measures and self-similar phenomena, it does not depend on the measurement level. Hence, C is an alternative to Gini's concentration coefficient for measures with variation on different levels. Simple examples concern population density and gross domestic product. Application to time series patterns is indicated with a Markov chain. For the Covid-19 pandemic, entropy ratios indicate a homogeneous distribution of infections and the potential of local action when compared to measures for a whole region.

Small Order Patterns in Big Time Series: A Practical Guide.

The study of order patterns of three equally-spaced values x t , x t + d , x t + 2 d in a time series is a powerful tool. The lag d is changed in a wide range so that the differences of the frequencies of order patterns become autocorrelation functions. Similar to a spectrogram in speech analysis, four ordinal autocorrelation functions are used to visualize big data series, as for instance heart and brain activity over many hours. The method applies to real data without preprocessing, and outliers and missing data do not matter. On the theoretical side, we study the properties of order correlation functions and show that the four autocorrelation functions are orthogonal in a certain sense. An analysis of variance of a modified permutation entropy can be performed with four variance components associated with the functions.

Elementary fractal geometry. New relatives of the Sierpinski gasket.

By slight modification of the data of the Sierpinski gasket, keeping the open set condition fulfilled, we obtain self-similar sets with very dense parts, similar to fractals in nature and in random models. This is caused by a complicated structure of the open set and is revealed only under magnification. Thus, the family of self-similar sets with separation condition is much richer and has higher modelling potential than usually expected. An interactive computer search for such examples and new properties for their classification are discussed.

Quality Improvement Initiative for Severe Sepsis and Septic Shock Reduces 90-Day Mortality: A 7.5-Year Observational Study.

OBJECTIVE: To investigate the impact of a quality improvement initiative for severe sepsis and septic shock focused on the resuscitation bundle on 90-day mortality. Furthermore, effects on compliance rates for antiinfective therapy within the recommended 1-hour interval are evaluated. DESIGN: Prospective observational before-after cohort study. SETTING: Tertiary university hospital in Germany. PATIENTS: All adult medical and surgical ICU patients with severe sepsis and septic shock. INTERVENTION: Implementation of a quality improvement program over 7.5 years. MEASUREMENTS: The primary endpoint was 90-day mortality. Secondary endpoints included ICU and hospital mortality rates and length of stay, time to broad-spectrum antiinfective therapy, and compliance with resuscitation bundle elements. MAIN RESULTS: A total of 14,115 patients were screened. The incidence of severe sepsis and septic shock was 9.7%. Ninety-day mortality decreased from 64.2% to 45.0% (p < 0.001). Hospital length of stay decreased from 44 to 36 days (p < 0.05). Compliance with resuscitation bundle elements was significantly improved. Antibiotic therapy within the first hour after sepsis onset increased from 48.5% to 74.3% (p < 0.001). Multivariate analysis revealed blood cultures before antibiotic therapy (hazard ratio, 0.60-0.84; p < 0.001), adequate calculated antibiotic therapy (hazard ratio, 0.53-0.75; p < 0.001), 1-2 L crystalloids within the first 6 hours (hazard ratio 0.67-0.97; p = 0.025), and greater than or equal to 6 L during the first 24 hours (hazard ratio, 0.64-0.95; p = 0.012) as predictors for improved survival. CONCLUSIONS: The continuous quality improvement initiative focused on the resuscitation bundle was associated with increased compliance and a persistent reduction in 90-day mortality over a 7.5-year period. Based on the observational study design, a causal relationship cannot be proven, and respective limitations need to be considered.

A simple classification tool for single-trial analysis of ERP components.

Event-related potentials (ERPs) were recorded by measuring a dense sensor EEG from eight healthy volunteers in a visual oddball experiment. Single trials were analyzed with an extremely simple high-dimensional version of discriminant analysis. The question was how many of the target trials contribute to the average P3, and to test whether other components in the ERP are sensitive to discriminate between target and non-target trials. One common classification rule for all participants expressing the P3 component correctly classified 88% of the ERPs of all subjects in response to a target or non-target trial. For four of the eight participants, there were strong differences in an early ERP component over the occipital recording sites. Their individual classification rules, obtained from the training data in the time interval up to 200 ms, correctly classified 85% of the trials of the test data.

Women are periodontally healthier than men, but why don't they have more teeth than men?

OBJECTIVE: Periodontal diseases are more prevalent in men than in women. However, in a population-based epidemiological study, we found that, on average, women have fewer teeth than men. The aim of this study was to test different hypotheses that could explain this obvious paradox. DESIGN: In 4,290 randomly selected participants from the normal population (Study of Health in Pomerania), we determined diagnostic periodontal parameters, attachment loss, and number of teeth. Behavioral and environmental risk factors were assessed by interviews and questionnaires. Use of estrogens was assessed, and urinary excretion of collagen cross-links was determined. RESULTS: Multiple regression analyses adjusted for caries and periodontitis revealed that in the women of this population, there is an inverse association between the number of children born and the number of teeth (P < 0.01). This relationship depends on socioeconomic status, bone metabolism, and the use of estrogens. In the group of the youngest (20-40 years), the bone turnover rate is positively related to the number of children born (P < 0.01). In postmenopausal women treated with estrogens, the number of teeth was higher than in men of the same age group. Only in women without hormone treatment were there fewer teeth. CONCLUSION: The apparent paradox of having fewer teeth despite better periodontal health in women compared with men is related to an increased bone turnover rate and socioeconomic conditions such as low education and low social status. Periodontal health is even worse if these factors are combined.

Permutation entropy: a natural complexity measure for time series.

We introduce complexity parameters for time series based on comparison of neighboring values. The definition directly applies to arbitrary real-world data. For some well-known chaotic dynamical systems it is shown that our complexity behaves similar to Lyapunov exponents, and is particularly useful in the presence of dynamical or observational noise. The advantages of our method are its simplicity, extremely fast calculation, robustness, and invariance with respect to nonlinear monotonous transformations.