Two lessons from fractals and chaos

Liebovitch, Larry S.; Scheurle, Daniela

doi:10.1002/1099-0526(200003/04)5:4<34::aid-cplx5>3.0.co;2-3

Cited by 46 publications

(24 citation statements)

References 3 publications

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“…As a base we use the classical approach to visualization nonlinear system dynamics-the Poincaré plot, which takes a sequence of sample values and plots each sample against the following sample. 12 The significance of this plot is that it is the two-dimensional reconstructed phase space-the projection of the system attractor that describes the dynamics of the time series. 13,14 The geometry of the Poincaré plot reveals properties of the system dynamics.…”

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confidence: 99%

Graphical and Numerical Evaluation of Continuous Glucose Sensing Time Lag

Kovatchev

Shields

Breton

2009

Diabetes Technology & Therapeutics

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Aim: This study introduces a new method for graphical and numerical evaluation of time lags typically associated with subcutaneous glucose sensing, based on Poincaré-type plot and a maximum statistical agreement criterion. Methods:The proposed method is illustrated by retrospective analysis of 56 continuous glucose monitor (CGM) time series collected by the FreeStyle Navigator™ (Abbott Diabetes Care, Alameda, CA) from 28 patients with type 1 diabetes mellitus, each wearing simultaneously two sensors (on arm and abdomen) and parallel reference blood glucose (BG) collected with a reference YSI (Yellow Springs, OH) analyzer every 15 min. The average duration of a time series was 111 h; there were approximately 10,000 sensor-reference data pairs. Results: When sliding in time CGM readings versus BG, the point of minimal spread of a Poincaré-type plot marks visually the time of CGM delay. The same point is numerically estimated by minimizing the distance between BG and CGM readings. The average observed time lag between reference BG and CGM was 12.5 min. Stratified by BG rate of change, the time lag was longer (16.8 min) when BG was falling, compared to steady or rising BG (11.7 min and 9.9 min, respectively) (P Ͻ 0.005). The time lags at the two sensor locations were not significantly different: 12.4 min on the arm, 12.6 min on the abdomen. Conclusions: In this data set, substantial blood-to-sensor time delays were observed, possibly because of both blood-to-interstitial glucose transport and instrumental delay. Analysis of BG-CGM co-dynamics that is free from mathematical approximation of glucose fluctuations resulted in convenient visualization and numerical estimation of these delays. 139

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confidence: 99%

Graphical and Numerical Evaluation of Continuous Glucose Sensing Time Lag

Kovatchev

Shields

Breton

2009

Diabetes Technology & Therapeutics

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“…[5] and [17] already came across the power law in Figure 1 considering one million single trials. However, they did not formulate the law explicitly, as we did.…”

Section: Discussionmentioning

confidence: 99%

The St. Petersburg paradox: An experimental solution

Silva

Matsushita

2016

Physica A: Statistical Mechanics and its Applications

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The St. Petersburg paradox refers to a gamble of infinite expected value, where people are likely to spend only a small entrance fee for it. There is a huge volume of literature that mostly concentrates on the psychophysics of the game; experiments are scant. Here, rather than focusing on the psychophysics, we offer an experimental, "physical" solution as if robots played the game. After examining the time series formed by one billion plays, we: confirm that there is no characteristic scale for this game; explicitly formulate the implied power law; and identify the type of  -stable distribution associated with the game. We find an 1   and, thus, the underlying distribution of the game is a Cauchy flight, as hinted by Paul Samuelson.

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“…These standard descriptors represent the minor axis and the major axis of the ellipse. The description of SD1 and SD2 in terms of linear statistics shows that the standard descriptors guide the visual inspection of the distribution (16).…”

Section: Non-linear Poincaré Analysismentioning

confidence: 99%

Non-linear Poincaré analysis of respiratory efforts in sleep apnea

Behbahani¹,

Mk²

2015

BLL

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OBJECTIVES: Obstructive sleep apnea (OSA) is a risk factor for hypertension, has effects on cardiovascular system and increases the sympathetic activity. The aim of the study was to evaluate the effectiveness of the non-linear Poincaré plot analysis to predict OSA based on polysomnography (PSG). METHODS: The database of this study was collected by the sleep laboratory at the Philipps University in Marburg, Germany. It includes 24 PSG of men and women between 27-63 years old with obstructive and mixed sleep apnea. The start and end of apnea events in PSGs were marked. The Poincaré plots of pre-apneic phase including 4-1 minutes before apnea were evaluated. Wilcoxon test was used for statistical analysis. RESULTS: Poincaré analysis showed that the dynamics of chest and respiratory efforts changed two minutes before the apnea and SD1/SD2 ratios of these parameters signifi cantly increased in the pre-apneic phase (p ≤ 0.01). The SD1/SD2 ratio of nasal airfl ow did not show signifi cant difference even in episodes close to apnea. CONCLUSIONS: Our results suggest that Poincaré plot parameters of PSG have the potential to be considered predictors of apnea with the ability to show the dynamic of changes, which could lead to pre-diagnosis or prediction of apnea about 2-3 minutes before its occurrence (Tab. 2, Fig. 4, Ref. 23). Text in PDF www.elis.sk.

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Two lessons from fractals and chaos

Cited by 46 publications

References 3 publications

Graphical and Numerical Evaluation of Continuous Glucose Sensing Time Lag

Graphical and Numerical Evaluation of Continuous Glucose Sensing Time Lag

The St. Petersburg paradox: An experimental solution

Non-linear Poincaré analysis of respiratory efforts in sleep apnea

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