1/f noise in human cognition: Is it ubiquitous, and what does it mean?

Researchers in psychology are paying increasing attention to temporal correlations in performance on cognitive tasks. Recently, Thornton and Gilden (2005) introduced a spectral method for analyzing psychological time series; in particular, this method is tailored to distinguish transient serial correlations from the persistent correlations characterized by 1/f noise. Thornton and Gilden applied their method to word-naming data to support the claimed ubiquity of 1/f noise in psychological time series. We argue that a previously presented method for distinguishing transient and persistent correlations (e.g., Wagenmakers, Farrell, & Ratcliff, 2004) compares favorably with the new method presented by Thornton and Gilden. We apply Thornton and Gilden’s method to time series from a range of cognitive tasks and show that 1/f noise is not a ubiquitous property of psychological time series. Finally, we assess the theoretical developments in this area and argue that the development of well-specified models of the principles or mechanisms of human cognition giving rise to 1/f noise is long overdue.     

Two tools to test time series data for evidence of chaos and/or nonlinearity

Two computer programs are described for evaluating the evidence for chaos and nonlinearity in time series data. “bx” is an efficient algorithm for computing the correlation integral (from which correlation dimension can be estimated); and “surrogat” is a Fourier-transform-based algorithm for generating surrogate data consistent with a null hypothesis that the data arise as a result of a linear stochastic process.     

Approximate entropy and point correlation dimension of heart rate variability in healthy subjects.

The contribution of nonlinear dynamics to heart rate variability in healthy humans was examined using surrogate data analysis. Several measures of heart rate variability were used and compared. Heart rates were recorded for three hours and original data sets of 8192 R-R intervals created. For each original data set (n = 34), three surrogate data sets were made by shuffling the order of the R-R intervals while retaining their linear correlations. The difference in heart rate variability between the original and surrogate data sets reflects the amount of nonlinear structure in the original data set. Heart rate variability was analyzed by two different nonlinear methods, point correlation dimension and approximate entropy. Nonlinearity, though under 10 percent, could be detected with both types of heart rate variability measures. More importantly, not only were the correlations between these measures and the standard deviation of the R-R intervals weak, the correlation among the nonlinear measures themselves was also weak (generally less than 0.6). This suggests that in addition to standard linear measures of heart rate variability, the use of multiple nonlinear measures of heart rate variability might be useful in monitoring heart rate dynamics.     

Approximate entropy and point correlation dimension of heart rate variability in healthy subjects

The contribution of nonlinear dynamics to heart rate variability in healthy humans was examined using surrogate data analysis. Several measures of heart rate variability were used and compared. Heart rates were recorded for three hours and original data sets of 8192 R-R intervals created. For each original data set (n=34), three surrogate data sets were made by shuffling the order of the R-R intervals while retaining their linear correlations. The difference in heart rate variability between the original and surrogate data sets reflects the amount of nonlinear structure in the original data set. Heart rate variability was analyzed by two different nonlinear methods, point correlation dimension and approximate entropy. Nonlinearity, though under 10 percent, could be detected with both types of heart rate variability measures. More importantly, not only were the correlations between these measures and the standard deviation of the R-R intervals weak, the correlation among the nonlinear measures themselves was also weak (generally less than 0.6). This suggests that in addition to standard linear measures of heart rate variability, the use of multiple nonlinear measures of heart rate variability might be useful in monitoring heart rate dynamics.     

On Maraun's Deconstructing of Factor Indeterminancy with Constructed Factors

The nonlinear random coefficient model has become increasingly popular as a method for describing individual differences in longitudinal research. Although promising, the nonlinear model it is not utilized as often as it might be because software options are still somewhat limited. In this article we show that a specialized version of the model can be fit to data using SEM software. The specialization is to a model in which the parameters that enter the function in a linear manner are random, whereas those that are nonlinear are common to all individuals. Although this kind of function is not as general as is the fully nonlinear model, it still is applicable to many different data sets. Two examples are presented to show how the models can be estimated using popular SEM computer programs.     

Latent Growth Modeling for Logistic Response Functions

Throughout much of the social and behavioral sciences, latent growth modeling (latent curve analysis) has become an important tool for understanding individuals' longitudinal change. Although nonlinear variations of latent growth models appear in the methodological and applied literature, a notable exclusion is the treatment of growth following logistic (sigmoidal; S-shape) response functions. Such trajectories are assumed in a variety of psychological and educational settings where learning occurs over time, and yet applications using the logistic model in growth modeling methodology have been sparse. The logistic function, in particular, may not be utilized as often because software options remain limited. In this article we show how a specialized version of the logistic function can be modeled using conventional structural equation modeling software. The specialization is a reparameterization of the logistic function whose new parameters correspond to scientifically interesting characteristics of the growth process. In addition to providing an example using simulated data, we show how this nonlinear functional form can be fit using transformed subject-level data obtained through a learning task from an air traffic controller simulation experiment. LISREL syntax for the empirical example is provided.     

Effects of Data Noise on Statistical Judgement

People made forecasts from graphically presented time series. Series were sinusoids overlaid by a zero or positive linear trend and a zero, low, moderate, or high level of noise. Forecasting performance was affected by both these variables. However, it did not correlate with ability to identify the trend and correlated significantly with ability to detect the sinusoidal pattern only when series were noise-free. A second experiment showed that the effect of data noise was not influenced by the number of forecasts that people made from a series. These findings are consistent with the view that data noise does not affect the way that forecasts are made but that it impairs them for two reasons. First, it renders the anchor-and-adjust heuristics underlying them less effective. Second, it causes people to add more noise to their judgements in their attempts to make them representative of the data.     

Modeling Noisy Data with Differential Equations Using Observed and Expected Matrices

Complex intraindividual variability observed in psychology may be well described using differential equations. It is difficult, however, to apply differential equation models in psychological contexts, as time series are frequently short, poorly sampled, and have large proportions of measurement and dynamic error. Furthermore, current methods for differential equation modeling usually consider data that are atypical of many psychological applications. Using embedded and observed data matrices, a statistical approach to differential equation modeling is presented. This approach appears robust to many characteristics common to psychological time series.     

The point correlation dimension: Performance with nonstationary surrogate data and noise

The dynamics of many biological systems have recently been attributed to low-dimensional chaos instead of high-dimensional noise, as previously thought. Because biological data are invariably nonstationary, especially when recorded over a long interval, the conventional measures of low-dimensional chaos (e.g., the correlation dimension algorithms) cannot be applied. A new algorithm, the point correction dimension (PD2_i) was developed to deal with this fundamental problem. In this article we describe the details of the algorithm and show that the local mean PD2_i will accurately track dimension in nonstationary surrogate data.     

Protective inhibition: A central factor in auditory fatigue

Summary A signal detection experiment was performed to investigate the sensitivity shift after stimulation with discontinuous octave-band noise of 95 dB (A). The influence of psychological factors on the sensitivity shift in the stimulated and the unstimulated ear was investigated in a two-factorial experimental design. In four independent groups of 12 Ss each, the psychological meaning of the noise and the side of sensitivity measurement were varied. Half of the Ss listened passively to the noise stimuli (control groups), the other half was made to believe that the noise signaled errors in a tracking task (experimental groups). After exposure of noise stimuli to the right ear the sensitivity of either the right or the left ear was determined. After exposure to noise the sensitivity of the stimulated ear was shown to be significantly more reduced in the experimental than in the control group. Thus, depending on the meaning of the noise physically identical stimuli produce sensitivity shifts of different amounts. After stimulation of the right ear with noise the sensitivity of the left ear remained completely unaffected. The sensitivity shift of the right ear did not persist as long as a threshold shift would have done. The hypothesis is that intensive auditory stimuli will produce negative feedback which protects — within certain limits — the stimulated ear from the fatiguing after-effects of overstimulation. The effectiveness of this protective inhibition depends on psychological factors.We are grateful to S. Jörg-Wagner for his competent handling of the experiment.     

Exact and resampling probability values for weighted kappa

Permutation procedures to compute exact and resampling probability values for weighted kappa are described. Comparisons with asymptotic probability values demonstrate that exact permutation procedures are advantageous for sparse data sets, whereas resampling permutation procedures are appropriate for both sparse and nonsparse data sets.     

Modeling Individual Damped Linear Oscillator Processes with Differential Equations: Using Surrogate Data Analysis to Estimate the Smoothing Parameter

Among the many methods available for modeling intraindividual time series, differential equation modeling has several advantages that make it promising for applications to psychological data. One interesting differential equation model is that of the damped linear oscillator (DLO), which can be used to model variables that have a tendency to fluctuate around some typical, or equilibrium, value. Methods available for fitting the damped linear oscillator model using differential equation modeling can yield biased parameter estimates when applied to univariate time series. The degree of this bias depends on a smoothing–like parameter, which balances the need for increasing smoothing to minimize error variance but not smoothing so much as to obscure change of interest. This article explores a technique that uses surrogate data analysis to select such a parameter, thereby producing approximately unbiased parameter estimates. Furthermore the smoothing parameter, which is usually researcher-selected, is produced in an automated manner so as to reduce the experience required by researchers to apply these methods. Focus is placed on the damped linear model; however, similar issues are expected with other differential equation models and other techniques in which parameter estimates depend on a smoothing parameter. An example using affect data from the Notre Dame Longitudinal Study of Aging (2004) is presented, which contrasts the use of a single smoothing parameter for all individuals versus use of a smoothing parameter for each individual.     

Extracting thresholds from noisy psychophysical data.

Psychophysical studies with infants or with patients often are unable to use pilot data, training, or large numbers of trials. To evaluate threshold estimates under these conditions, computer simulations of experiments with small numbers of trials were performed by using psychometric functions based on a model of two types of noise: stimulus-related noise (affecting slope) and extraneous noise (affecting upper asymptote). Threshold estimates were biased and imprecise when extraneous noise was high, as were the estimates of extraneous noise. Strategies were developed for rejecting data sets as too noisy for unbiased and precise threshold estimation; these strategies were most successful when extraneous noise was low for most of the data sets. An analysis of 1,026 data sets from visual function tests of infants and toddlers showed that extraneous noise is often considerable, that experimental paradigms can be developed that minimize extraneous noise, and that data analysis that does not consider the effects of extraneous noise may underestimate test-retest reliability and overestimate interocular differences.     

Dynamics of Flow: A Nonlinear Perspective

The aims of this study are to consider the experience of flow from a nonlinear dynamics perspective. The processes and temporal nature of intrinsic motivation and flow, would suggest that flow experiences fluctuate over time in a dynamical fashion. Thus it can be argued that the potential for chaos is strong. The sample was composed of 20 employees (both full and part time) recruited from a number of different organizations and work backgrounds. The Experience Sampling Method (ESM) was used for data collection. Once obtained the temporal series, they were subjected to various analyses proper to the complexity theory (Visual Recurrence Analysis and Surrogate Data Analysis). Results showed that in 80% of the cases, flow presented a chaotic dynamic, in that, flow experiences delineated a complex dynamic whose patterns of change were not easy to predict. Implications of the study, its limitations and future research are discussed.     

A Comparison of Heuristic Procedures for Minimum Within-Cluster Sums of Squares Partitioning

Perhaps the most common criterion for partitioning a data set is the minimization of the within-cluster sums of squared deviation from cluster centroids. Although optimal solution procedures for within-cluster sums of squares (WCSS) partitioning are computationally feasible for small data sets, heuristic procedures are required for most practical applications in the behavioral sciences. We compared the performances of nine prominent heuristic procedures for WCSS partitioning across 324 simulated data sets representative of a broad spectrum of test conditions. Performance comparisons focused on both percentage deviation from the “best-found” WCSS values, as well as recovery of true cluster structure. A real-coded genetic algorithm and variable neighborhood search heuristic were the most effective methods; however, a straightforward two-stage heuristic algorithm, HK-means, also yielded exceptional performance. A follow-up experiment using 13 empirical data sets from the clustering literature generally supported the results of the experiment using simulated data. Our findings have important implications for behavioral science researchers, whose theoretical conclusions could be adversely affected by poor algorithmic performances.     

Psychological experimentation through direct connection of the subject interface to a VAX

We outline the rationale for using a DEC VAX-11/780 for the real-time control of psychological experiments. The properties of VAX hardware and software that make the system well suited to the requirements of many classes of psychological experiments, particularly those that require the concurrent control of several independent experiments or the control of one experiment that must be written as several intercommunicating processes are discussed.     

A simple LSI-11 system for real-time control,interevent time storage,and data analysis

A simple laboratory computer system based on a Digital Equipment Corporation LSI-11, floppy disk, DRV11 parallel input-output board, and the RT-11 operating system is described. Interface to experimental devices is provided through a lab-built relay driver and relay closure sensing interface. An extensive high-level software package provides an easy-to-use control language (e.g., stimuli can be controlled with a simple “TURN ON” or “TURN OFF” instruction) and easy-to-use FORTRAN subroutines for data exploration (e.g., “IFIND” searches a data file for a particular event). The control software automatically generates, codes, and stores a complete log of every input and output event and its time of occurrence in each of five simultaneously running experiments. This provides the capability to reanalyze data in light of hypotheses not available when the experiment was designed. The FORTRAN subroutine library for data exploration provides a conditional and iterative search facility to sift out events or sets of events from the data file for analysis. Standard FORTRAN statements perform arithmetic operations on the resulting data.     

The design and analysis of state-trace experiments

State-trace analysis (Bamber, 1979) addresses a question of interest in many areas of psychological research: Does 1 or more than 1 latent (i.e., not directly observed) variable mediate an interaction between 2 experimental manipulations? There is little guidance available on how to design an experiment suited to state-trace analysis, despite its increasing use, and existing statistical methods for state-trace analysis are problematic. We provide a framework for designing and refining a state-trace experiment and statistical procedures for the analysis of accuracy data using Klugkist, Kato, and Hoijtink's (2005) method of estimating Bayes factors. The statistical procedures provide estimates of the evidence favoring 1 versus more than 1 latent variable, as well as evidence that can be used to refine experimental methodology.     

Executive Function as an Interaction-Dominant Process

Traditional theories of psychology define the cognitive system as composed of insular, encapsulated components, controlled by a central executive. An alternative hypothesis suggests that cognitive control arises from the complex interaction among temporal scales of activity within the system. We examined the hand motions of preschool-age participants gathered during an executive-function task, card sorting, for evidence of multiplicative interactions across temporal scales. The time series of hand motions were submitted to iterated amplitude adjusted Fourier transformation (IAAFT), a surrogate data analysis technique that removes nonlinear, multiscale dependencies while preserving the linear structure of the time series. We found that removing multiscale effects via IAAFT led to a significant change in the width of the multifractal spectrum, an indicator of multiplicative interactions. The results suggest that cognitive control may arise from the interactions among temporal scales of activity within the system rather than as the result of a central executive.     

Effects of federal human subjects regulations on data obtained in environmental stressor research

A series of experiments on environmental noise was performed over a 3-year period. Those experiments in the series that incorporated federal subjects guidelines concerning informed consent failed to replicate the negative aftereffects of noise found earlier by Glass and Singer. The experiments that were performed before implementation of the federal guidelines, however, successfully replicated the Glass and Singer findings. To clarify this pattern of results, a final experiment was performed on two groups of subjects treated identically except that one group gave informed consent while the other did not. The results confirmed that the subject procedures involving informed consent as defined by the federal guidelines prevent the emergence of negative aftereffects of noise, possibly because these procedures give the subjects what amounts to control over the stressor. Similar phenomena may occur in research on other environmental stressors, such as crowding and electric shock, in which subject's perception of control is a critical variable. Possible techniques for performing future research of this type are discussed.