Bayesian classifier with multivariate distribution based on D-vine copula model for awake/drowsiness interpretation during power nap

Abstract

In this study, a Bayesian classifier with the multivariate distribution based on the D-vine copula model is developed and evaluated for the awake/drowsiness interpretation during the power nap. The objective is to consider the correlation among the features into the automatic classification algorithm. A power nap is a short sleep process, which is commonly considered as a supplement to the insufficient overnight sleep. It may involve the states of awake and drowsiness. Neurophysiological features are extracted from the EEGs (electroencephalography) and EOGs (electrooculography), which are synchronously recorded during one's short nap after lunch. The multivariate distribution of features is decomposed into independency and dependency products according to the D-vine copula model. The independency product is the marginal probability density function of the features. The dependency product consists of pair-copula functions. The marginal probability density is estimated by the kernel function and k-nearest-neighbor density respectively. The parameters of pair-copula functions are estimated by the maximum likelihood estimation. In total, 8 healthy subjects were involved. The comparison results showed that the Bayesian classifier with the multivariate distribution based on the D-vine copula model obtained quite satisfied classification accuracy. The developed method introduced a feasible way to construct the multivariate distribution, which can enhance the classification performance of Bayesian classifier when dealing with the complex correlation of features in actual cases.

Introduction

Power nap is a short sleep process where the person wakes up before entering the deep sleep [1]. Nap is a supplement to the regular sleep at night. Taking a nap is helpful to repair the damage by sleep deprivation [2]. According to clinical studies, subjects who have a nap have lower coronary mortality than those without napping [3]. Nap is also beneficial for human's long term memory [4], as well as the consolidation of relevant memories [5] etc. In addition, nap is recommended as a good fatigue countermeasure. It makes people quickly recover from the tired state to the normal mental state to maintain a high vigilance level and good working performance [6].

The process of nap may involve the states of being awake and drowsiness. Although the nap is benefit to the mental and physical health, the proper duration is better to be regulated to prevent people from entering the deep sleep [7]. Because waking up from a deep sleep would cause sleep inertia. Sleep inertia is associated with grogginess and disorientation, and people feel more tired than before sleeping [8]. There are several investigation and survey researches on the best sleep timing and length to minimize the sleep inertia effects. However, few have focused on the interpretation of the states from awake to drowsy which are of significance to regulate the nap duration, avoid the fatigue of sleep inertia and maintain the efficient working condition of the body.

The neurophysiological signal contains the characteristic information, which is closely related to the change of the human body state. Physiological states can be diagnosed from neurophysiological signals, e.g. the automatic detection of epilepsy based on EEG (electroencephalography) has the applicant significance for clinical diagnosis [9]. The features extracted from time domain, frequency domain, time-frequency domain of the neurophysiological signals, and the nonlinear dynamic features are feasible for the analysis of physiological states [10]. During the nap, the level of alertness shows a gradual decline process from awake state to the drowsy or sleepy state. The change of the characteristics corresponding to certain frequency activities of EEG and EOG (electrooculography) are typically utilized for analysis [11].

Usually, the extracted features are sent to the classifiers for different interpretation tasks. However, there are complex correlations among the features especially the data from real cases. It would be difficult to obtain desirable performance by sending the features directly to the classifiers. The neurophysiological signals are obtained by placing the electrodes on the scalp. The electrical activities of neurons in the corresponding functional areas are recorded. It is known that the structure made up of millions of neuronal cells is complex. Even for separated recorded positions, there are correlations among the signals at different locations. Therefore, the complex characteristics of neurophysiological signals is worth to be considered to improve the performance of the signal processing or pattern recognition methods for real applications. The methods of feature fusion or feature selection methods can be one of the solution to generate more effective features for classification [12]. On the other hand, with the help of parametric modeling methods, the internal structure of features can be interpreted to improve the classification performance [13]. Comparing the two kinds of solutions, the parametric modeling method is more feasible to make a well relationship between the feature extraction and classification procedures for the neurophysiological signals.

Bayesian classifier is a useful method for various application. The multivariate distribution of parameter is the important prior knowledge for classification. In order to apply the Bayesian classifier for awake/drowsy state interpretation, a well fit of the multivariate distribution is necessary to be developed to deal with the complex characteristics of the neurophysiological signals. The common methods of Bayesian classifier include Gaussian discriminant analysis, naive Bayesian classifier, etc. However, the relationships of multivariable are almost not considered or simplified when implementing those methods. Therefore, how to make accurate estimation of multivariate probability density function for the Bayesian classifier is important to improve its classification performance.

Copula theory is a statistical tool to describe the correlation among random variables. The copula-based model begins with the application in the field of economics [14]. It is also utilized for time series analysis in several fields. Jovanovic et al. proved that copula can capture and visualize the spatial and temporal fluctuations in dependency structures of cardiovascular signals [15]. Ozdemir et al. [16] presented a data-driven classifier fusion method based on the copula statistical theory. Cui et al. [17] introduced a wind power ramp forecast model based on copula theory. Hazarika et al. [18] developed an uncertainty modeling strategy based on multivariate copula. The vine copula is developed based on the theory of copula. Kraus et al. [19] introduced a semi-parametric quantile regression method based on vine copula model. Vernieuwe et al. [20] explored the use of vine copulas in the rainfall field among all storm-related variables. Stübinger et al. [21] developed a statistical arbitrage strategy by using vine copula to interpret the relationships among the random variables. The above presented works showed the application significance and effectiveness of copula and vine copula especially for the data analysis and modeling. Unlike those studies, we focused on the development of classifier which can be adaptive to the real neurophysiological signals for drowsiness detection purpose. The application of parametric model by vine copula for Bayesian classifier is explored and evaluated.

In copula theory, the joint distribution function of multiple random variables is decomposed into the products of several one-dimensional distribution functions and a copula function. The copula function connects several one-dimensional distribution functions to describe the correlation among the random variables. The copula-based Bayesian classification has been realized by some of the authors for drowsiness detection [22]. The constructed multivariate copula model enables the developed Bayesian classifier to obtain better classification performance than the traditional Bayesian classifiers. Due to the structure, the copula function still has limited ability to have a well fit dependency. In terms of optimizing the correlation structure and computational complexity of multi-dimensional variables, the vine copula is better than the traditional multivariate copula. The vine copula builds multi-dimensional dependence structures on the basis of multiple binary copula functions. The realization is more flexible in constructing the dependencies for multivariate copula functions. According to different decomposition strategies, vine copula is mainly divided into C-vine copula and D-vine copula [23]. The D-vine copula is linear structure while the C-vine copula is star structure. The D-vine copula is less complex and more convenient to be optimized than the C-vine copula [24].

In this study, a Bayesian classifier with multivariate distribution is developed based on the D-vine copula model. The multivariate distribution of features is decomposed into independency and dependency products. The marginal probability density of features refers to the independency product that is estimated by the kernel function method and k-nearest-neighbor density estimation method. The pair-copula function is used to construct the correlation among the features, where the parameters are estimated using the maximum likelihood estimation method. The proposed classifier was validated with the actual neurophysiological signals recorded during one's nap after lunch. The obtained results were compared with the Gaussian discriminate analysis, naive Bayesian classifier and supported vector machine model to evaluate the classification performance.