Daniel T. Kaplan

A single-blind controlled competition among tests for nonlinearity and chaos

Interest has been growing in testing for nonlinearity or chaos in economic data, but much controversy has arisen about the available results. This paper explores the reasons for these empirical difficulties. We designed and ran a single-blind controlled competition among five highly regarded tests for nonlinearity or chaos with ten simulated data series. The data generating mechanisms include linear processes, chaotic recursions, and nonchaotic stochastic processes; and both large and small samples were included in the experiment. The data series were produced in a single blind manner by the competition manager and sent by e-mail, without identifying information, to the experiment participants. Each such participant is an acknowledged expert in one of the tests and has a possible vested interest in producing the best possible results with that one test. The results of this competition provide much surprising information about the power functions of some of the best regarded tests for nonlinearity or noisy chaos.

A comparison of estimators for 1/ f noise

Abstract We use a Monte-Carlo approach to investigate the performance of five different time-series estimators of the exponent α in 1 f α noise. We find that a maximum-likelihood estimator is markedly superior to Fourier regression methods and Hurst exponent methods. The results indicate that useful estimates of α can be made from time series that are much shorter than generally presumed.

Approximate entropy in the electroencephalogram during wake and sleep.

Entropy measurement can discriminate among complex systems, including deterministic, stochastic and composite systems. We evaluated the changes of approximate entropy (ApEn) in signals of the electroencephalogram (EEG) during sleep. EEG signals were recorded from eight healthy volunteers during nightly sleep. We estimated the values of ApEn in EEG signals in each sleep stage. The ApEn values for EEG signals (mean ± SD) were 0.896 ± 0.264 during eyes-closed waking state, 0.738 ± 0.089 during Stage I, 0.615 ± 0.107 during Stage II, 0.487 ± 0.101 during Stage III, 0.397 ± 0.078 during Stage IV and 0.789 ± 0.182 during REM sleep. The ApEn values were found to differ with statistical significance among the six different stages of consciousness (ANOVA, p<0.001). ApEn of EEG was statistically significantly lower during Stage IV and higher during wake and REM sleep. We conclude that ApEn measurement can be useful to estimate sleep stages and the complexity in brain activity.

Relationship between correlation dimension and indices of linear analysis in both respiratory movement and electroencephalogram

OBJECTIVEnWe investigate the relationships between signals from the electroencephalogram (EEG) and those from respiratory movement using the correlation dimension (D(2)).nnnMETHODSnRespiratory movement and EEG were recorded for 7.5h from 7 clinically healthy men. D(2) was calculated by applying an algorithm slightly modified from that proposed by Grassberger and Procaccia (Phys Rev Lett 50 (1983) 346). Non-linearity in respiratory movement and EEG was tested by comparing D(2) for the original data with that for surrogate data.nnnRESULTSnA statistically significant positive correlation between D(2) of the EEG and D(2) of the respiratory movement was observed for the original data, but not for the surrogate data.nnnCONCLUSIONSnA reduced D(2) of the EEG may be associated with an increased regularity of breathing in deep sleep (stage IV). Likewise, the increased D(2) of respiratory movement during rapid eye movement may be associated with increased complexity of the signals. Whether there is a direct coordination between brain and lungs or whether brainstem systems, including that of the cholinergic system, affect both respiration and cortex requires further investigation.

Approximate Entropy of the Electroencephalogram in Healthy Awake Subjects and Absence Epilepsy Patients

The approximate entropy (ApEn) of signals in the electroencephalogram (EEG) was evaluated in 8 healthy volunteers and in 10 patients with absence epilepsy, both during seizure-free and seizure intervals. We estimated the nonlinearity of each 3-sec EEG segment using surrogate data methods. The mean (± SD) ApEn in EEG was 0.83 ± 0.22 in healthy subjects awake with eyes closed. It was significantly lower during epileptic seizures (0.48 ± 0.05) than during seizure-free intervals (0.80 ± 0.13) (P<0.001). Nonlinearity was clearly detected in EEG signals from epileptic patients during seizures but not during seizure-free intervals or in EEG signals from healthy subjects. The ApEn of EEG signals estimated over consecutive intervals could serve to determine pathological brain activity such as that occurring during absence epilepsy.

Nonstationarity and 1/f noise characteristics in heart rate.

Presents several methods for estimating the exponent /spl alpha/ of the 1/f/sup /spl alpha//-type spectral behaviour of heart rate (HR) power spectrum. The authors give a comparison of the statistical properties of various estimators, i.e., maximum likelihood estimation (MLE), Hurst exponent, detrended fluctuation analysis (DFA) and regression of the power spectrum. A method to investigate nonstationarity is explained and applications of these estimators to long term HR recordings are given. The results obtained indicate that the power-law structure of HR is nonstationary and that observed 1/f/sup /spl alpha// patterns may be due to the interaction of the estimators with the nonstationarity in the exponent /spl alpha/.

Markov chain Monte Carlo estimation of nonlinear dynamics from time series

Abstract Much of nonlinear time series analysis is concerned with inferring unmeasured quantities — e.g., system parameters, the shape of attractors in state space — from a noisy measured time series. From a Bayesian perspective, the time series is a vector sample picked at random from a probability density. The density reflects the system dynamics and our subjective uncertainty about system parameters, the measurement function, dynamical noise and measurement noise. The conditional probability density of the system parameters given the measured data is the basis of a Bayesian estimate of the system parameters. Using illustrative chaotic systems with large-amplitude dynamical and measurement noise, we show here that it is feasible to use the Markov chain Monte Carlo (MCMC) technique to generate the Bayesian conditional probabilities. The resulting parameter estimates are markedly superior to those based on conventional least-squares methods: the MCMC-based estimates are unbiased and allow estimates of dynamical parameters on unmeasured components of the state vector. In addition, the MCMC method enables de-noised attractors to be reconstructed, not just in an embedding based on lags of measured variables but in the state space that includes unmeasured components of the dynamics’ state vector. The general purpose MCMC technique effectively combines techniques of nonlinear noise reduction and nonlinear parameter estimation.

Rethinking Assessment of Student Learning in Statistics Courses

Although much attention has been paid to issues around student assessment, for most introductory statistics courses few changes have taken place in the ways students are assessed. The assessment literature describes three foundational elements—cognition, observation, and interpretation—that comprise an “assessment triangle” underlying all assessments. However, most instructors focus primarily on the second component: tasks that are used to produce grades. This article focuses on three sections written by leading statistics educators who describe some innovative and even provocative approaches to rethinking student assessment in statistics classes.

Teaching computation to undergraduate scientists

This paper describes the motivation and design of an introductory computational course for natural, physical, and social scientists.

Depth of Anesthesia Index using Cumulative Power Spectrum

Over the last ten years, monitors of depth of anesthesia have progressively been integrated in the clinical practice. Based on the analysis of the electroencephalogram (EEG), these monitors deliver an index that helps the anesthesiologist to determine the state of the patient during the surgery. Although they employ different kind of algorithms, spectral parameters are always taken into account to achieve the final indexes. In this work, a new spectral parameter based on the cumulative power spectrum is proposed. When compared to the Spectral Edge Frequency (SEF), a classic spectral parameter, the Cumulative Power Spectrum Index (CPSI) presents a higher correlation with reference indexes (AAI, BIS and CePROP) and a higher prediction probability of the state of the patient. Furthermore, when compared to the reference indexes, the CPSI shows similar performances in terms of correlation and presents a higher prediction probability than two of them (BIS and CePROP).