Influence of noise on the sample entropy algorithm

Low EEG Gamma Entropy and Glucose Hypometabolism After Corpus Callosotomy Predicts Seizure Outcome After Subsequent Surgery

Background

Patients with generalized epilepsy who had lateralized EEG abnormalities after corpus callosotomy (CC) occasionally undergo subsequent surgeries to control intractable epilepsy.

Objectives

This study evaluated retrospectively the combination of EEG multiscale entropy (MSE) and FDG-PET for identifying lateralization of the epileptogenic zone after CC.

Methods

This study included 14 patients with pharmacoresistant epilepsy who underwent curative epilepsy surgery after CC. Interictal scalp EEG and FDG-PET obtained after CC were investigated to determine (1) whether the MSE calculated from the EEG and FDG-PET findings was lateralized to the surgical side, and (2) whether the lateralization was associated with seizure outcomes.

Results

Seizure reduction rate was higher in patients with lateralized findings to the surgical side than those without (MSE: p < 0.05, FDG-PET: p < 0.05, both: p < 0.01). Seizure free rate was higher in patients with lateralized findings in both MSE and FDG-PET than in those without (p < 0.05).

Conclusions

This study demonstrated that patients with lateralization of MSE and FDG-PET to the surgical side had better seizure outcomes. The combination of MSE and conventional FDG-PET may help to select surgical candidates for additional surgery after CC with good postoperative seizure outcomes.

EntropyHub: An open-source toolkit for entropic time series analysis

An increasing number of studies across many research fields from biomedical engineering to finance are employing measures of entropy to quantify the regularity, variability or randomness of time series and image data. Entropy, as it relates to information theory and dynamical systems theory, can be estimated in many ways, with newly developed methods being continuously introduced in the scientific literature. Despite the growing interest in entropic time series and image analysis, there is a shortage of validated, open-source software tools that enable researchers to apply these methods. To date, packages for performing entropy analysis are often run using graphical user interfaces, lack the necessary supporting documentation, or do not include functions for more advanced entropy methods, such as cross-entropy, multiscale cross-entropy or bidimensional entropy. In light of this, this paper introduces EntropyHub, an open-source toolkit for performing entropic time series analysis in MATLAB, Python and Julia. EntropyHub (version 0.1) provides an extensive range of more than forty functions for estimating cross-, multiscale, multiscale cross-, and bidimensional entropy, each including a number of keyword arguments that allows the user to specify multiple parameters in the entropy calculation. Instructions for installation, descriptions of function syntax, and examples of use are fully detailed in the supporting documentation, available on the EntropyHub website- www.EntropyHub.xyz. Compatible with Windows, Mac and Linux operating systems, EntropyHub is hosted on GitHub, as well as the native package repository for MATLAB, Python and Julia, respectively. The goal of EntropyHub is to integrate the many established entropy methods into one complete resource, providing tools that make advanced entropic time series analysis straightforward and reproducible.

Effects of ECG Data Length on Heart Rate Variability among Young Healthy Adults

The relationship between the robustness of HRV derived by linear and nonlinear methods to the required minimum data lengths has yet to be well understood. The normal electrocardiography (ECG) data of 14 healthy volunteers were applied to 34 HRV measures using various data lengths, and compared with the most prolonged (2000 R peaks or 750 s) by using the Mann-Whitney U test, to determine the 0.05 level of significance. We found that SDNN, RMSSD, pNN50, normalized LF, the ratio of LF and HF, and SD1 of the Poincaré plot could be adequately computed by small data size (60-100 R peaks). In addition, parameters of RQA did not show any significant differences among 60 and 750 s. However, longer data length (1000 R peaks) is recommended to calculate most other measures. The DFA and Lyapunov exponent might require an even longer data length to show robust results. Conclusions: Our work suggests the optimal minimum data sizes for different HRV measures which can potentially improve the efficiency and save the time and effort for both patients and medical care providers.

A machine learning model for multi-night actigraphic detection of chronic insomnia: development and validation of a pre-screening tool

We propose a novel machine learning-based method for analysing multi-night actigraphy signals to objectively classify and differentiate nocturnal awakenings in individuals with chronic insomnia (CI) and their cohabiting healthy partners. We analysed nocturnal actigraphy signals from 40 cohabiting couples with one partner seeking treatment for insomnia. We extracted 12 time-domain dynamic and nonlinear features from the actigraphy signals to classify nocturnal awakenings in healthy individuals and those with CI. These features were then used to train two machine learning classifiers, random forest (RF) and support vector machine (SVM). An optimization algorithm that incorporated the predicted quality of each night for each individual was used to classify individuals into CI or healthy sleepers. Using the proposed actigraphic signal analysis technique, coupled with a rigorous leave-one-out validation approach, we achieved a classification accuracy of 80% (sensitivity: 76%, specificity: 82%) in classifying CI individuals and their healthy bed partners. The RF classifier (accuracy: 80%) showed a better performance than SVM (accuracy: 75%). Our approach to analysing the multi-night nocturnal actigraphy recordings provides a new method for screening individuals with CI, using wrist-actigraphy devices, facilitating home monitoring.

"Unrest while Resting"? Brain entropy in autism spectrum disorder

Biological systems typically exhibit complex behavior with nonlinear dynamic properties. Nonlinear signal processing techniques such as sample entropy is a novel approach to characterize the temporal dynamics of brain connectivity. Estimating entropy is especially important in clinical populations such as autism spectrum disorder (ASD) as differences in entropy may signal functional alterations in the brain. Considering the models of disrupted brain network connectivity in ASD, sample entropy would provide a novel direction to understand brain organization. Resting state fMRI data from 45 high-functioning children with ASD and 45 age-and-IQ-matched typically developing (TD) children were obtained from the Autism Brain Imaging Data Exchange (ABIDE-II) database. Data were preprocessed using the CONN toolbox. Sample entropy was then calculated using the complexity toolbox, in a whole-brain voxelwise manner as well as in regions of interests (ROIs) based methods. ASD participants demonstrated significantly increased entropy in left angular gyrus, superior parietal lobule, and right inferior temporal gyrus; and reduced sample entropy in superior frontal gyrus compared to TD participants. Positive correlations of average entropy in clusters of significant group differences scores across all subjects were found. Finally, ROI analysis revealed a main effect of lobes. Differences in entropy between the ASD and TD groups suggests that entropy may provide another important index of brain dysfunction in clinical populations like ASD. Further, the relationship between increased entropy and ASD symptoms in our study underscores the role of optimal brain synchronization in cognitive and behavioral functions.

Plastic systemic inhibition controls amplitude while allowing phase pattern in a stochastic neural field model

We investigate oscillatory phase pattern formation and amplitude control for a linearized stochastic neuron field model by simulating Mexican-hat-coupled stochastic processes. We find, for several choices of parameters, that spatial pattern formation in the temporal phases of the coupled processes occurs if and only if their amplitudes are allowed to grow unrealistically large. Stimulated by recent work on homeostatic inhibitory plasticity, we introduce static and plastic (adaptive) systemic inhibitory mechanisms to keep the amplitudes stochastically bounded. We find that systems with static inhibition exhibited bounded amplitudes but no sustained phase patterns. With plastic systemic inhibition, on the other hand, the resulting systems exhibit both bounded amplitudes and sustained phase patterns. These results demonstrate that plastic inhibitory mechanisms in neural field models can dynamically control amplitudes while allowing patterns of phase synchronization to develop. Similar mechanisms of plastic systemic inhibition could play a role in regulating oscillatory functioning in the brain.

Detection of stretch reflex onset based on empirical mode decomposition and modified sample entropy

Background

Accurate spasticity assessment provides an objective evaluation index for the rehabilitation treatment of patients with spasticity, and the key is detecting stretch reflex onset. The surface electromyogram of patients with spasticity is prone to false peaks, and its data length is unstable. These conditions decrease signal differences before and after stretch reflex onset. Therefore, a method for detecting stretch reflex onset based on empirical mode decomposition denoising and modified sample entropy recognition is proposed in this study.

Results

The empirical mode decomposition algorithm is better than the wavelet threshold algorithm in denoising surface electromyogram signal. Without adding Gaussian white noise to the electromyogram signal, the stretch reflex onset recognition rate of the electromyogram signal before and after empirical mode decomposition denoising was increased by 56%. In particular, the recognition rate of stretch reflex onset under the optimal parameter of the modified sample entropy can reach up to 100% and the average recognition rate is 93%.

Conclusions

The empirical mode decomposition algorithm can eliminate the baseline activity of the surface electromyogram signal before stretch reflex onset and effectively remove noise from the signal. The identification of stretch reflex onset using combined empirical mode decomposition and modified sample entropy is better than that via modified sample entropy alone, and stretch reflex onset can be accurately determined.

A method to concatenate multiple short time series for evaluating dynamic behaviour during walking

Gait variability is a sensitive metric for assessing functional deficits in individuals with mobility impairments. To correctly represent the temporal evolution of gait kinematics, nonlinear measures require extended and uninterrupted time series. In this study, we present and validate a novel algorithm for concatenating multiple time-series in order to allow the nonlinear analysis of gait data from standard and unrestricted overground walking protocols. The full-body gait patterns of twenty healthy subjects were captured during five walking trials (at least 5 minutes) on a treadmill under different weight perturbation conditions. The collected time series were cut into multiple shorter time series of varying lengths and subsequently concatenated using a novel algorithm that identifies similar poses in successive time series in order to determine an optimal concatenation time point. After alignment of the datasets, the approach then concatenated the data to provide a smooth transition. Nonlinear measures to assess stability (Largest Lyapunov Exponent, LyE) and regularity (Sample Entropy, SE) were calculated in order to quantify the efficacy of the concatenation approach using intra-class correlation coefficients, standard error of measurement and paired effect sizes. Our results indicate overall good agreement between the full uninterrupted and the concatenated time series for LyE. However, SE was more sensitive to the proposed concatenation algorithm and might lead to false interpretation of physiological gait signals. This approach opens perspectives for analysis of dynamic stability of gait data from physiological overground walking protocols, but also the re-processing and estimation of nonlinear metrics from previously collected datasets.

On the calculation of sample entropy using continuous and discrete human gait data

Sample entropy (SE) has relative consistency using biologically-derived, discrete data >500 data points. For certain populations, collecting this quantity is not feasible and continuous data has been used. The effect of using continuous versus discrete data on SE is unknown, nor are the relative effects of sampling rate and input parameters m (comparison vector length) and r (tolerance). Eleven subjects walked for 10-minutes and continuous joint angles (480Hz) were calculated for each lower-extremity joint. Data were downsampled (240, 120, 60Hz) and discrete range-of-motion was calculated. SE was quantified for angles and range-of-motion at all sampling rates and multiple combinations of parameters. A differential relationship between joints was observed between range-of-motion and joint angles. Range-of-motion SE showed no difference; whereas, joint angle SE significantly decreased from ankle to knee to hip. To confirm findings from biological data, continuous signals with manipulations to frequency, amplitude, and both were generated and underwent similar analysis to the biological data. In general, changes to m, r, and sampling rate had a greater effect on continuous compared to discrete data. Discrete data was robust to sampling rate and m. It is recommended that different data types not be compared and discrete data be used for SE.

Dynamic Cross-Entropy

BACKGROUND

Complexity measures for time series have been used in many applications to quantify the regularity of one dimensional time series, however many dynamical systems are spatially distributed multidimensional systems.

NEW METHOD

We introduced Dynamic Cross-Entropy (DCE) a novel multidimensional complexity measure that quantifies the degree of regularity of EEG signals in selected frequency bands. Time series generated by discrete logistic equations with varying control parameter r are used to test DCE measures.

RESULTS

Sliding window DCE analyses are able to reveal specific period doubling bifurcations that lead to chaos. A similar behavior can be observed in seizures triggered by electroconvulsive therapy (ECT). Sample entropy data show the level of signal complexity in different phases of the ictal ECT. The transition to irregular activity is preceded by the occurrence of cyclic regular behavior. A significant increase of DCE values in successive order from high frequencies in gamma to low frequencies in delta band reveals several phase transitions into less ordered states, possible chaos in the human brain.

COMPARISON WITH EXISTING METHOD

To our knowledge there are no reliable techniques able to reveal the transition to chaos in case of multidimensional times series. In addition, DCE based on sample entropy appears to be robust to EEG artifacts compared to DCE based on Shannon entropy.

CONCLUSIONS

The applied technique may offer new approaches to better understand nonlinear brain activity.

An explorative investigation of functional differences in plantar center of pressure of four foot types using sample entropy method

In the study of biomechanics of different foot types, temporal or spatial parameters derived from plantar pressure are often used. However, there is no comparative study of complexity and regularity of the center of pressure (CoP) during the stance phase among pes valgus, pes cavus, hallux valgus and normal foot. We aim to analyze whether CoP sample entropy characteristics differ among these four foot types. In our experiment participated 40 subjects with normal feet, 40 with pes cavus, 19 with pes valgus and 36 with hallux valgus. A Footscan^® system was used to collect CoP data. We used sample entropy to quantify several parameters of the investigated four foot types. These are the displacement in medial–lateral (M/L) and anterior–posterior (A/P) directions, as well as the vertical ground reaction force of CoP during the stance phase. To fully examine the potential of the sample entropy method for quantification of CoP components, we provide results for two cases: calculating the sample entropy of normalized CoP components, as well as calculating it using the raw data of CoP components. We also explored what are the optimal values of parameters m (the matching length) and r (the tolerance range) when calculating the sample entropy of CoP data obtained during the stance phases. According to statistical results, some factors significantly influenced the sample entropy of CoP components. The sample entropies of non-normalized A/P values for the left foot, as well as for the right foot, were different between the normal foot and pes valgus, and between the normal foot and hallux valgus. The sample entropy of normalized M/L displacement of the right foot was different between the normal foot and pes cavus. The measured variable for A/P and M/L displacements could serve for the study of foot function.

Feature Selection and Predictors of Falls with Foot Force Sensors Using KNN-Based Algorithms

The aging process may lead to the degradation of lower extremity function in the elderly population, which can restrict their daily quality of life and gradually increase the fall risk. We aimed to determine whether objective measures of physical function could predict subsequent falls. Ground reaction force (GRF) data, which was quantified by sample entropy, was collected by foot force sensors. Thirty eight subjects (23 fallers and 15 non-fallers) participated in functional movement tests, including walking and sit-to-stand (STS). A feature selection algorithm was used to select relevant features to classify the elderly into two groups: at risk and not at risk of falling down, for three KNN-based classifiers: local mean-based k-nearest neighbor (LMKNN), pseudo nearest neighbor (PNN), local mean pseudo nearest neighbor (LMPNN) classification. We compared classification performances, and achieved the best results with LMPNN, with sensitivity, specificity and accuracy all 100%. Moreover, a subset of GRFs was significantly different between the two groups via Wilcoxon rank sum test, which is compatible with the classification results. This method could potentially be used by non-experts to monitor balance and the risk of falling down in the elderly population.

Selection of entropy-measure parameters for knowledge discovery in heart rate variability data

Background

Heart rate variability is the variation of the time interval between consecutive heartbeats. Entropy is a commonly used tool to describe the regularity of data sets. Entropy functions are defined using multiple parameters, the selection of which is controversial and depends on the intended purpose. This study describes the results of tests conducted to support parameter selection, towards the goal of enabling further biomarker discovery.

Methods

This study deals with approximate, sample, fuzzy, and fuzzy measure entropies. All data were obtained from PhysioNet, a free-access, on-line archive of physiological signals, and represent various medical conditions. Five tests were defined and conducted to examine the influence of: varying the threshold value r (as multiples of the sample standard deviation σ, or the entropy-maximizing r_Chon), the data length N, the weighting factors n for fuzzy and fuzzy measure entropies, and the thresholds r_F and r_L for fuzzy measure entropy. The results were tested for normality using Lilliefors' composite goodness-of-fit test. Consequently, the p-value was calculated with either a two sample t-test or a Wilcoxon rank sum test.

Results

The first test shows a cross-over of entropy values with regard to a change of r. Thus, a clear statement that a higher entropy corresponds to a high irregularity is not possible, but is rather an indicator of differences in regularity. N should be at least 200 data points for r = 0.2 σ and should even exceed a length of 1000 for r = r_Chon. The results for the weighting parameters n for the fuzzy membership function show different behavior when coupled with different r values, therefore the weighting parameters have been chosen independently for the different threshold values. The tests concerning r_F and r_L showed that there is no optimal choice, but r = r_F = r_L is reasonable with r = r_Chon or r = 0.2σ.

Conclusions

Some of the tests showed a dependency of the test significance on the data at hand. Nevertheless, as the medical conditions are unknown beforehand, compromises had to be made. Optimal parameter combinations are suggested for the methods considered. Yet, due to the high number of potential parameter combinations, further investigations of entropy for heart rate variability data will be necessary.

Sample entropy characteristics of movement for four foot types based on plantar centre of pressure during stance phase

Background

Motion characteristics of CoP (Centre of Pressure, the point of application of the resultant ground reaction force acting on the plate) are useful for foot type characteristics detection. To date, only few studies have investigated the nonlinear characteristics of CoP velocity and acceleration during the stance phase. The aim of this study is to investigate whether CoP regularity is different among four foot types (normal foot, pes valgus, hallux valgus and pes cavus); this might be useful for classification and diagnosis of foot injuries and diseases. To meet this goal, sample entropy, a measure of time-series regularity, was used to quantify the CoP regularity of four foot types.

Methods

One hundred and sixty five subjects that had the same foot type bilaterally (48 subjects with healthy feet, 22 with pes valgus, 47 with hallux valgus, and 48 with pes cavus) were recruited for this study. A Footscan® system was used to collect CoP data when each subject walked at normal and steady speed. The velocity and acceleration in medial-lateral (ML) and anterior-posterior (AP) directions, and resultant velocity and acceleration were derived from CoP. The sample entropy is the negative natural logarithm of the conditional probability that a subseries of length m that matches pointwise within a tolerance r also matches at the next point. This was used to quantify variables of CoP velocity and acceleration of four foot types. The parameters r (the tolerance) and m (the matching length) for sample entropy calculation have been determined by an optimal method.

Results

It has been found that in order to analyze all CoP parameters of velocity and acceleration during the stance phase of walking gait, for each variable there is a different optimal r value. On the contrary, the value m=4 is optimal for all variables.

Sample entropies of both velocity and acceleration in AP direction were highly correlated with their corresponding resultant variables for r>0.91. The sample entropy of the velocity in AP direction was moderately correlated with the one of the acceleration in the same direction (r≥0.673), as well as with the resultant acceleration (r≥0.660). The sample entropy of resultant velocity was moderately correlated with the one of the acceleration in AP direction, as well as with the resultant acceleration (for the both r≥0.689). Moderate correlations were found between variables for the left foot and their corresponding variables for the right foot.

Sample entropies of AP velocity, resultant velocity, AP acceleration, and resultant acceleration of the right foot as well as AP velocity and resultant velocity of the left foot were, respectively, significantly different among the four foot types.

Conclusions

It can be concluded that the sample entropy of AP velocity (or the resultant velocity) of the left foot, ML velocity, resultant velocity, ML acceleration and resultant acceleration could serve for evaluation of foot types or selection of appropriate footwear.

Wearing a safety harness during treadmill walking influences lower extremity kinematics mainly through changes in ankle regularity and local stability

Background

Wearing a harness during treadmill walking ensures the subject's safety and is common practice in biomedical engineering research. However, the extent to which such practice influences gait is unknown. This study investigated harness-related changes in gait patterns, as evaluated from lower extremity kinematics during treadmill walking.

Findings

Healthy subjects (n = 10) walked on a treadmill at their preferred speed for 3 minutes with and without wearing a harness (LiteGait^®, Mobility Research, Inc.). In the former condition, no weight support was provided to the subjects. Lower extremity kinematics was assessed in the sagittal plane from the mean (mean_RoM), standard deviation (SD_RoM) and coefficient of variation (CoV_RoM) of the hip, knee, and ankle ranges of motion (RoM), as well as from the sample entropy (SampEn) and the largest Lyapunov exponent (LyE) of the joints' angles. Wearing the harness increased the mean_RoM of the hip, the SD_RoM and the CoV_RoM of the knee, and the SampEn and the LyE of the ankle. In particular, the harness effect sizes for both the SampEn and the LyE of the ankle were large, likely reflecting a meaningful decline in the neuromuscular stabilizing control of this joint.

Conclusions

Wearing a harness during treadmill walking marginally influences lower extremity kinematics, resulting in more or less subtle changes in certain kinematic variables. However, in cases where differences in gait patterns would be expressed through modifications in these variables, having subjects walk with a harness may mask or reinforce such differences.

Non-invasive atrial fibrillation organization follow-up under successive attempts of electrical cardioversion

The development of non-invasive tools able to provide valuable information about the effectiveness of a shock in external electrical cardioversion (ECV) is clinically relevant to enhance these protocols in the treatment of atrial fibrillation (AF). The present contribution analyzes the ability of a non-linear regularity index, such as sample entropy (SampEn), to follow-up non-invasively AF organization under successive attempts of ECV and to predict the effectiveness of every single shock. To this respect, the atrial activity (AA) preceding each delivered shock was extracted by using a QRST cancellation method. Next, the main atrial wave (MAW), which can be considered as the fundamental waveform associated to the AA, was obtained by applying a selective filtering centered on the dominant atrial frequency (DAF). Finally, the MAW organization was estimated with SampEn and two thresholds (Th1 = 0.1223 and Th2 = 0.0832) were established to predict the ECV outcome. Results indicated that, prior to the first attempt, all the patients who needed only one shock to restore NSR were below Th1. In addition, most of them were above Th2 in case of AF relapsing during the first month. Regarding several shocks, all the patients who maintained NSR more than one month were below Th2 after the first shock. Moreover, all the patients who relapsed to AF during the first month were between Th1 and Th2 and, finally, all the patients with ineffective ECV were above Th1. After each unsuccessful shock, a SampEn relative decrease was observed for the patients who finally reverted to NSR, but the largest variation took place after the first attempt, thus indicating that this shock plays the most important role in the procedure. Indeed, by considering jointly the patients who needed only one shock and the patients who needed several shocks, 91.67% (22 out of 24) of ECVs resulting in NSR, 93.55% (29 out of 31) of ECVs relapsing to AF during the first month and 100% (10 out of 10) of ECVs in which NSR was not restored were correctly classified. As conclusion, the MAW organization analysis via SampEn can provide useful information that could improve the effectiveness of conventional external ECV protocols used in AF treatment.

Characterizing the dynamics of postural sway in humans using smoothness and regularity measures

We investigate human postural sway velocity time series by computing two dynamical statistics quantifying the smoothness (the central tendency measure or CTM) and the regularity (the sample entropy or SampEn) of their underlying dynamics. The purpose of the study is to investigate the effect of aging and vision on the selected measures and to explore the nature of postural dynamics by performing surrogate data tests. A group of 14 young subjects was compared to a group of 11 older healthy subjects in two visual conditions: with eyes open (EO) and with eyes closed (EC). The results suggest that vision and age do not influence the two statistics of the velocity data in the same way. More specifically, the smoothness statistic is able to detect the aging effect. The regularity measure is sensitive to the visual feedback removal. In contrast with some findings in the literature, the results of the surrogate data tests indicate that the center of pressure velocity dynamics are stochastic and are not produced by a purely deterministic behavior. Finally, we discuss some potential implications of our results in terms of postural control mechanisms.

Optimal parameters study for sample entropy-based atrial fibrillation organization analysis

Sample entropy (SampEn) is a nonlinear regularity index that requires the a priori selection of three parameters: the length of the sequences to be compared, m, the patterns similarity tolerance, r, and the number of samples under analysis, N. Appropriate values for m, r and N have been recommended and widely used in the literature for the application of SampEn to some physiological time series, such as heart rate, hormonal data, etc. However, no guidelines exist for the selection of that values in other cases. Therefore, an optimal parameters study should be required for the application of SampEn to not previously analyzed biomedical signals. In the present work, a thorough analysis on the optimal values for m, r and N is presented within the context of atrial fibrillation (AF) organization estimation, computed from surface electrocardiogram recordings. Recently, the evaluation of AF organization through SampEn, has revealed clinically useful information that could be used for a better treatment of this arrhythmia. The present study analyzed optimal SampEn parameter values within two different scenarios of AF organization estimation, such as the prediction of paroxysmal AF termination and the electrical cardioversion outcome in persistent AF. As a result, interesting recommendations about the selection of m, r and N, together with the relationship between N and the sampling rate (f(s)) were obtained. More precisely, (i) the proportion between N and f(s) should be higher than 1s and f(s)>or=256 Hz, (ii) overlapping between adjacent N-length windows does not improve AF organization estimation with respect to the analysis of non-overlapping windows, and (iii) values of m and r maximizing successful classification for the analyzed AF databases should be considered within a range wider than the proposed in the literature for heart rate analysis, i.e. m=1 and m=2 and r between 0.1 and 0.25 times the standard deviation of the data.