Time domain measures of inter-channel EEG correlations: a comparison of linear, nonparametric and nonlinear measures

Correlations between ten-channel EEGs obtained from thirteen healthy adult participants were investigated. Signals were obtained in two behavioral states: eyes open no task and eyes closed no task. Four time domain measures were compared: Pearson product moment correlation, Spearman rank order correlation, Kendall rank order correlation and mutual information. The psychophysiological utility of each measure was assessed by determining its ability to discriminate between conditions. The sensitivity to epoch length was assessed by repeating calculations with 1, 2, 3, …, 8 s epochs. The robustness to noise was assessed by performing calculations with noise corrupted versions of the original signals (SNRs of 0, 5 and 10 dB). Three results were obtained in these calculations. First, mutual information effectively discriminated between states with less data. Pearson, Spearman and Kendall failed to discriminate between states with a 1 s epoch, while a statistically significant separation was obtained with mutual information. Second, at all epoch durations tested, the measure of between-state discrimination was greater for mutual information. Third, discrimination based on mutual information was more robust to noise. The limitations of this study are discussed. Further comparisons should be made with frequency domain measures, with measures constructed with embedded data and with the maximal information coefficient.     

Noise and complexity in human postural control: interpreting the different estimations of entropy

Background

Over the last two decades, various measures of entropy have been used to examine the complexity of human postural control. In general, entropy measures provide information regarding the health, stability and adaptability of the postural system that is not captured when using more traditional analytical techniques. The purpose of this study was to examine how noise, sampling frequency and time series length influence various measures of entropy when applied to human center of pressure (CoP) data, as well as in synthetic signals with known properties. Such a comparison is necessary to interpret data between and within studies that use different entropy measures, equipment, sampling frequencies or data collection durations.

Methods and Findings

The complexity of synthetic signals with known properties and standing CoP data was calculated using Approximate Entropy (ApEn), Sample Entropy (SampEn) and Recurrence Quantification Analysis Entropy (RQAEn). All signals were examined at varying sampling frequencies and with varying amounts of added noise. Additionally, an increment time series of the original CoP data was examined to remove long-range correlations. Of the three measures examined, ApEn was the least robust to sampling frequency and noise manipulations. Additionally, increased noise led to an increase in SampEn, but a decrease in RQAEn. Thus, noise can yield inconsistent results between the various entropy measures. Finally, the differences between the entropy measures were minimized in the increment CoP data, suggesting that long-range correlations should be removed from CoP data prior to calculating entropy.

Conclusions

The various algorithms typically used to quantify the complexity (entropy) of CoP may yield very different results, particularly when sampling frequency and noise are different. The results of this study are discussed within the context of the neural noise and loss of complexity hypotheses.     

Information and discriminability as measures of reliability of sensory coding

Response variability is a fundamental issue in neural coding because it limits all information processing. The reliability of neuronal coding is quantified by various approaches in different studies. In most cases it is largely unclear to what extent the conclusions depend on the applied reliability measure, making a comparison across studies almost impossible. We demonstrate that different reliability measures can lead to very different conclusions even if applied to the same set of data: in particular, we applied information theoretical measures (Shannon information capacity and Kullback-Leibler divergence) as well as a discrimination measure derived from signal-detection theory to the responses of blowfly photoreceptors which represent a well established model system for sensory information processing. We stimulated the photoreceptors with white noise modulated light intensity fluctuations of different contrasts. Surprisingly, the signal-detection approach leads to a safe discrimination of the photoreceptor response even when the response signal-to-noise ratio (SNR) is well below unity whereas Shannon information capacity and also Kullback-Leibler divergence indicate a very low performance. Applying different measures, can, therefore, lead to very different interpretations concerning the system's coding performance. As a consequence of the lower sensitivity compared to the signal-detection approach, the information theoretical measures overestimate internal noise sources and underestimate the importance of photon shot noise. We stress that none of the used measures and, most likely no other measure alone, allows for an unbiased estimation of a neuron's coding properties. Therefore the applied measure needs to be selected with respect to the scientific question and the analyzed neuron's functional context.     

Comparison of Profile Similarity Measures for Genetic Interaction Networks

Analysis of genetic interaction networks often involves identifying genes with similar profiles, which is typically indicative of a common function. While several profile similarity measures have been applied in this context, they have never been systematically benchmarked. We compared a diverse set of correlation measures, including measures commonly used by the genetic interaction community as well as several other candidate measures, by assessing their utility in extracting functional information from genetic interaction data. We find that the dot product, one of the simplest vector operations, outperforms most other measures over a large range of gene pairs. More generally, linear similarity measures such as the dot product, Pearson correlation or cosine similarity perform better than set overlap measures such as Jaccard coefficient. Similarity measures that involve L₂-normalization of the profiles tend to perform better for the top-most similar pairs but perform less favorably when a larger set of gene pairs is considered or when the genetic interaction data is thresholded. Such measures are also less robust to the presence of noise and batch effects in the genetic interaction data. Overall, the dot product measure performs consistently among the best measures under a variety of different conditions and genetic interaction datasets.     

Information content of molecular structures

For a completely enumerated set of conformers of a macromolecule or for exhaustive lattice walks of model polymers it is straightforward to use Shannon information theory to deduce the information content of the ensemble. It is also practicable to develop numerical measures of the information content of sets of exact distance constraints applied to specific conformational ensembles. We examine the effects of experimental uncertainties by considering "noisy" constraints. The introduction of noise requires additional assumptions about noise distribution and conformational clustering protocols that make the problem of measuring information content more complex. We make use of a standard concept in communication theory, the "noise sphere," to link uncertainty in measurements to information loss. Most of our numerical results are derived from two-dimensional lattice ensembles. Expressing results in terms of information per degree of freedom removes almost all of the chain length dependence. We also explore off-lattice polyalanine chains that yield surprisingly similar results.     

A communication channel model for information transmission in the blowfly photoreceptor

Biological photoreceptors transduce and communicate information about visual stimuli to other neurons through a series of signal transformations among physical states such as concentration of a chemical species, current, or the number of open ion channels. We present a communication channel model to quantify the transmission and degradation of visual information in the blowfly photoreceptor cell. The model is a cascade of linear transfer functions and noise sources that are derived from fundamental principles whenever possible, and whose parameters are estimated from physiological data. We employ the model to calculate the information capacity of blowfly phototransduction; our results compare favorably with estimates of the capacity derived from experimental measurements by de Ruyter van Steveninck and Laughlin (Nature 379 (1996) 642-645) and Juusola (J. Gen. Physiol. 104 (1994) 593-621). The model predicts that photon shot noise and ion channel noise are the dominant noise sources that limits information transmission in the blowfly photoreceptor.     

Enhancing the Functional Content of Eukaryotic Protein Interaction Networks

Protein interaction networks are a promising type of data for studying complex biological systems. However, despite the rich information embedded in these networks, these networks face important data quality challenges of noise and incompleteness that adversely affect the results obtained from their analysis. Here, we apply a robust measure of local network structure called common neighborhood similarity (CNS) to address these challenges. Although several CNS measures have been proposed in the literature, an understanding of their relative efficacies for the analysis of interaction networks has been lacking. We follow the framework of graph transformation to convert the given interaction network into a transformed network corresponding to a variety of CNS measures evaluated. The effectiveness of each measure is then estimated by comparing the quality of protein function predictions obtained from its corresponding transformed network with those from the original network. Using a large set of human and fly protein interactions, and a set of over GO terms for both, we find that several of the transformed networks produce more accurate predictions than those obtained from the original network. In particular, the measure and other continuous CNS measures perform well this task, especially for large networks. Further investigation reveals that the two major factors contributing to this improvement are the abilities of CNS measures to prune out noisy edges and enhance functional coherence in the transformed networks.     

Noise

The proliferation of DNA sequence data has generated a concern about the effects of "noise" on phylogeny reconstruction. This concern has led to various recommendations for weighting schemes and for separating data types prior to analysis. A new technique is explored to examine directly how noise influences the stability of parsimony reconstruction. By appending purely random characters onto a matrix of pure signal, or by replacing characters in a matrix of signal by random states, one can measure the degree to which a matrix is robust against noise. Reconstructions were sensitive to tree topology and clade size when noise was added, but were less so when character states were replaced with noise. When a signal matrix is complemented with a noise matrix of equal size, parsimony will trace the original signal about half the time when there is only one synapomorphy per node, and about 90% of the time when there are three synapomorphies per node. Similar results obtain when 20% of a matrix is replaced by noise. Successive weighting does not improve performance. Adding noise to only some taxa is more damaging, but replacing characters in only some taxa is less so. The bootstrap and g1 (tree skewness) statistics are shown to be uninterpretable measures of noise or departures from randomness. Empirical data sets illustrate that commonly recommended schemes of differential weighting (e.g. downweighting third positions) are not well supported from the point of view of reducing the influence of noise nor are more noisy data sets likely to degrade signal found in less noisy data sets.     

Optimum signal in a simple neuronal model with signal-dependent noise

How does the information about a signal in neural threshold crossings depend on the noise acting upon it? Two models are explored, a binary McCulloch and Pitts (threshold exceedance) model and a model of waiting time to exceedance--a discrete-time version of interspike intervals. If noise grows linearly with the signal, we find the best identification of the signal in terms of the Fisher information is for signals that do not reach the threshold in the absence of noise. Identification attains the same precision under weak and strong signals, but the coding range decreases at both extremes of signal level. We compare the results obtained for Fisher information with those using related first and second moment measures. The maximum obtainable information is plotted as a function of the ratio of noise to signal.     

A synthesis of two decades of research documenting the effects of noise on wildlife

Global increases in environmental noise levels – arising from expansion of human populations, transportation networks, and resource extraction – have catalysed a recent surge of research into the effects of noise on wildlife. Synthesising a coherent understanding of the biological consequences of noise from this literature is challenging. Taxonomic groups vary in auditory capabilities. A wide range of noise sources and exposure levels occur, and many kinds of biological responses have been observed, ranging from individual behaviours to changes in ecological communities. Also, noise is one of several environmental effects generated by human activities, so researchers must contend with potentially confounding explanations for biological responses. Nonetheless, it is clear that noise presents diverse threats to species and ecosystems and salient patterns are emerging to help inform future natural resource‐management decisions. We conducted a systematic and standardised review of the scientific literature published from 1990 to 2013 on the effects of anthropogenic noise on wildlife, including both terrestrial and aquatic studies. Research to date has concentrated predominantly on European and North American species that rely on vocal communication, with approximately two‐thirds of the data set focussing on songbirds and marine mammals. The majority of studies documented effects from noise, including altered vocal behaviour to mitigate masking, reduced abundance in noisy habitats, changes in vigilance and foraging behaviour, and impacts on individual fitness and the structure of ecological communities. This literature survey shows that terrestrial wildlife responses begin at noise levels of approximately 40 dBA, and 20% of papers documented impacts below 50 dBA. Our analysis highlights the utility of existing scientific information concerning the effects of anthropogenic noise on wildlife for predicting potential outcomes of noise exposure and implementing meaningful mitigation measures. Future research directions that would support more comprehensive predictions regarding the magnitude and severity of noise impacts include: broadening taxonomic and geographical scope, exploring interacting stressors, conducting larger‐scale studies, testing mitigation approaches, standardising reporting of acoustic metrics, and assessing the biological response to noise‐source removal or mitigation. The broad volume of existing information concerning the effects of anthropogenic noise on wildlife offers a valuable resource to assist scientists, industry, and natural‐resource managers in predicting potential outcomes of noise exposure.     

Beyond Benford's Law: Distinguishing Noise from Chaos

Determinism and randomness are two inherent aspects of all physical processes. Time series from chaotic systems share several features identical with those generated from stochastic processes, which makes them almost undistinguishable. In this paper, a new method based on Benford''s law is designed in order to distinguish noise from chaos by only information from the first digit of considered series. By applying this method to discrete data, we confirm that chaotic data indeed can be distinguished from noise data, quantitatively and clearly.     

Blind noise reduction for multisensory signals using ICA and subspace filtering, with application to EEG analysis

 In many applications of signal processing, especially in communications and biomedicine, preprocessing is necessary to remove noise from data recorded by multiple sensors. Typically, each sensor or electrode measures the noisy mixture of original source signals. In this paper a noise reduction technique using independent component analysis (ICA) and subspace filtering is presented. In this approach we apply subspace filtering not to the observed raw data but to a demixed version of these data obtained by ICA. Finite impulse response filters are employed whose vectors are parameters estimated based on signal subspace extraction. ICA allows us to filter independent components. After the noise is removed we reconstruct the enhanced independent components to obtain clean original signals; i.e., we project the data to sensor level. Simulations as well as real application results for EEG-signal noise elimination are included to show the validity and effectiveness of the proposed approach. Received: 6 November 2000 / Accepted in revised form: 12 November 2001     

Airport noise exposure and the postnatal growth of children

To determine whether chronic exposure to airport noise affects children, a study was conducted of the physical growth of children, aged 5-13 years, from two communities, one exposed to airport noise (n = 148) and another, not exposed (n = 102). Ten standard anthropometric measurements were made according to U.S. Health Examination Survey guidelines, and information on the social and biological characteristics of each family was collected in interviews. Hotelling's T2-tests were performed comparing the noise-exposed and nonnoise-exposed samples. There was no significant difference between the sample for measurements of social and biological characteristics of the families (including socioeconomic status and maternal reproductive history). Parental anthropometrics differed significantly (T2 = 24.32, P = 0.0001) as did child anthropometrics (T2 = 21.01, P = 0.032). For the child anthropometrics, noise-exposed children's slightly smaller measures of body bulk, together with their larger facial breadths, contributed to the significant T2. When the entire sample (n = 250) was analyzed by multiple linear regression, noise exposure was a significant predictor only of male triceps and subscapular skinfolds. For these two variables, and most other anthropometrics, however, the beta coefficients were negative for both sexes. In order to include information on the covariance structure among all variables, a canonical correlation analysis was performed. Noise exposure loaded negatively on the third canonical variate and was paired with positively loaded measures of body bulk. Taken together, the three analyses suggest that while there is some evidence for a slight effect of airport noise on measures of body bulk for males in this sample, most measures of postnatal growth for both males and females were unaffected by the noise levels experienced.     

Assessing semantic similarity measures for the characterization of human regulatory pathways

MOTIVATION: Pathway modeling requires the integration of multiple data including prior knowledge. In this study, we quantitatively assess the application of Gene Ontology (GO)-derived similarity measures for the characterization of direct and indirect interactions within human regulatory pathways. The characterization would help the integration of prior pathway knowledge for the modeling. RESULTS: Our analysis indicates information content-based measures outperform graph structure-based measures for stratifying protein interactions. Measures in terms of GO biological process and molecular function annotations can be used alone or together for the validation of protein interactions involved in the pathways. However, GO cellular component-derived measures may not have the ability to separate true positives from noise. Furthermore, we demonstrate that the functional similarity of proteins within known regulatory pathways decays rapidly as the path length between two proteins increases. Several logistic regression models are built to estimate the confidence of both direct and indirect interactions within a pathway, which may be used to score putative pathways inferred from a scaffold of molecular interactions.     

Probabilistic analysis of probe reliability in differential gene expression studies with short oligonucleotide arrays

Probe defects are a major source of noise in gene expression studies. While existing approaches detect noisy probes based on external information such as genomic alignments, we introduce and validate a targeted probabilistic method for analyzing probe reliability directly from expression data and independently of the noise source. This provides insights into the various sources of probe-level noise and gives tools to guide probe design.     

Network compression as a quality measure for protein interaction networks

With the advent of large-scale protein interaction studies, there is much debate about data quality. Can different noise levels in the measurements be assessed by analyzing network structure? Because proteomic regulation is inherently co-operative, modular and redundant, it is inherently compressible when represented as a network. Here we propose that network compression can be used to compare false positive and false negative noise levels in protein interaction networks. We validate this hypothesis by first confirming the detrimental effect of false positives and false negatives. Second, we show that gold standard networks are more compressible. Third, we show that compressibility correlates with co-expression, co-localization, and shared function. Fourth, we also observe correlation with better protein tagging methods, physiological expression in contrast to over-expression of tagged proteins, and smart pooling approaches for yeast two-hybrid screens. Overall, this new measure is a proxy for both sensitivity and specificity and gives complementary information to standard measures such as average degree and clustering coefficients.     

Inferences about membrane properties from electrical noise measurements

Four sources of electrical noise in biological membranes, each with a different physical basis, are discussed; the analysis of each type of noise potentially yields a different sort of information about membrane properties. (a) From the thermal noise spectrum, the passive membrane impedance may be obtained, so that thermal noise measurements are essentially equivalent to the type of since wave analysis carried out by Cole and Curtis. (b) If adequately high frequency measurements could be made, the shot noise spectrum should give information about the average motion of a single ion within the membrane. (c) The number of charge carriers and single ion mobilities within the membrane can possibly be inferred from measurements of noise with a 1/f spectrum. Available data indicate, for example, that increases in axon membrane conductance are not achieved by modulations in the mobility of ions within the membrane. (d) Fluctuations arising from the mechanisms normally responsible for membrane conductance changes can produce a type of electrical noise. Analysis of such conductance fluctuations provides a way to assess the validity of various microscopic models for the behavior of individual channels. Two different probabilistic interpretations of the Hodgkin-Huxley equations are investigated here and shown to yield different predictions about the spectrum of conductance fluctuations; thus, appropriate noise measurements may serve to eliminate certain classes of microscopic models for membrane conductance changes. Further, it is shown how the analysis of conductance fluctuations can, in some circumstances, provide an estimate of the conductance of a single channel.     

Haplotype frequencies at the DRD2 locus in populations of the East European Plain

Background

A key to increasing the power of multilocus association tests is to reduce the number of degrees of freedom by suppressing noise from data. One of the difficulties is to decide how much noise to suppress. An often overlooked problem is that commonly used association tests based on genotype data cannot utilize the genetic information contained in spatial ordering of SNPs (see proof in the Appendix), which may prevent them from achieving higher power.

Results

We develop a score test based on wavelet transform with empirical Bayesian thresholding. Extensive simulation studies are carried out under various LD structures as well as using HapMap data from many different chromosomes for both qualitative and quantitative traits. Simulation results show that the proposed test automatically adjusts the level of noise suppression according to LD structures, and it is able to consistently achieve higher or similar powers than many commonly used association tests including the principle component regression method (PCReg).

Conclusion

The wavelet-based score test automatically suppresses the right amount of noise and uses the information contained in spatial ordering of SNPs to achieve higher power.     

When can noise induce chaos and why does it matter: a critique

Noise‐induced chaos illustrates how small amounts of exogenous noise can have disproportionate qualitative impacts on the long term dynamics of a nonlinear system. This property is particularly clear in chaotic systems but is also important for the majority of ecological systems which are nonchaotic, and has direct implications for analyzing ecological time series and testing models against field data. Dennis et al. point out that a definition of chaos which we advocated allows a noise‐dominated system to be classified as chaotic when its Lyapunov exponent λ is positive, which misses what is really going on. As a solution, they propose to eliminate the concept of noise‐induced chaos: chaos “should retain its strictly deterministic definition”, hence “ecological populations cannot be strictly chaotic”. Instead, they suggest that ecologists ask whether ecological systems are strongly influenced by “underlying skeletons with chaotic dynamics or whatever other dynamics”– the skeleton being the hypothetical system that would result if all external and internal noise sources were eliminated. We agree with Dennis et al. about the problem – noise‐dominated systems should not be called chaotic – but not the solution. Even when an estimated skeleton predicts a system's short term dynamics with extremely high accuracy, the skeleton's long term dynamics and attractor may be very different from those of the actual noisy system. Using theoretical models and empirical data on microtine rodent cycles and laboratory populations of Tribolium, we illustrate how data analyses focusing on attributes of the skeleton and its attractor – such as the “deterministic Lyapunov exponent”λ₀ that Dennis et al. have used as their primary indicator of chaos – will frequently give misleading results. In contrast, quantitative measures of the actual noisy system, such as λ, provide useful information for characterizing observed dynamics and for testing proposed mechanistic explanations.