Unbiased estimation of permutation entropy in EEG analysis for Alzheimer's disease classification
Graphical abstract
Introduction
The diagnosis of Alzheimer's disease (AD) is an up-to-date problem which is solved by various techniques. Non-invasive investigation of AD patients is preferred and both magnetic resonance imaging (MRI) and electroencephalography (EEG) are frequently used for image and signal processing related to the diagnosis. Being focused on EEG analysis, spectral analysis is one of the successful tools for AD investigation; however, the analysis of non-linear EEG dynamic offers a more complex signal analysis. Correlation dimension D2 is a good example of complexity measure offering lower values for AD patients [1]. The same trend is seen for the largest Lyapunov exponent λ. Unfortunately, the calculation of D2 and λ1 enforces the analysis of a very long time series [2]. In this study, we prefer entropy estimation from the EEG signal [3], [4] because of the relationship between the complexity of non-linear signals [5], [6] and entropy estimate. Dauwels et al. [7] and many other authors have shown that Alzheimer's disease increases power in the delta and theta-bands in frequency domain, but the power spectrum is a global characteristics of the EEG signal making it impossible to study and localise events in the signal. There are many possibilities how to organize the entropy evaluation. Our approach is based on permutation entropy [8], [9]. There are three reasons why to use permutation entropy for diagnosing AD. Human brain activity can be interpreted as behaviour of a complex chaotic system. Therefore, signals from EEG electrodes carry information about the chaos inside the scalp. The hypothesis of decreasing chaotic behaviour [3] during AD can be tested by using permutation entropy; the second reason for using it is its relative simplicity. This term is exactly defined for the windowed signal and is directly applicable to time series without any methodological difficulties. The time complexity of permutation entropy calculation is acceptable for small windows lengths as discussed in [8], [9]. However, the application to windows of size above 12 is impossible. Permutation entropy has not yet been applied to longer windows. We suppose there is a chance to obtain new and significant results if the window length were prolonged. Using hash table as a special data structure, it is possible to calculate permutation entropy for window lengths up to 30, which is the main novelty of our approach, as will be explained in following sections.
Section snippets
Shannon entropy and its estimation
Definition. Shannon entropy [3], [10] HS of a discrete random variable X with possible values x1, …, xm and probability mass function p(X) is defined aswhere pi = p(xi).
If the probability function is unknown for an experimental data set, and the number of possible values is finite for random variable X, we estimate probability function pi by relative frequency pj,N and number of events kN aswhere nj is the number of occurrences xi of random variable X, and n
Permutation analysis for large samples
The main disadvantage of the original procedure of permutation analysis [8] is in its memory and time complexities. The authors implemented permutation memory as a matrix of columns and ! rows together with counter vector of length !. This enables permutation analysis on a typical computer only for . Traditional applications [8] of permutation entropy use a window of length . The time complexity of single permutation counting is also !, in the worst case. Therefore, for permutation
Application to EEG
Our study provided the EEG data of two patient groups: 10 patients with moderate dementia (MMSE score 10–19) and control group of 10 healthy patients having no memory or other cognitive impairments. The average MMSE was 16.2 (SD of 2.1) for the AD group and 30 for normals, of course. The age profile was similar: 69.4 ± 9.2 for AD and 68.7 ± 7.7 for CN. The AD group consisted of 5 men and 5 women, the CN group of 4 men and 6 women. All subjects were recorded under the same conditions, i.e. lying
Discussion
From the biomedical point of view, the results of statistical testing can be interpreted as follows. The chaotic behaviour of normal human brain is observable by using the standard EEG 10–20 system for sampling frequencies up to 200 Hz at any channel. The differences in permutation entropy between men and women are not statistically significant but the permutation entropy significantly decreases in the case of Alzheimer's disease. Therefore, the permutation entropy on any electrode signal is a
Conclusion
Using Miller approach instead of direct calculation of Shannon entropy from permutation frequencies, we have developed a novel method of EEG analysis via permutation entropy. The advantage of our method is in its very fast permutation analysis and low consumption of computer memory which enables an analysis of large time series with greater length of permutation patterns as well as parametric study of the sampling frequency role. The method was used to diagnose Alzheimer's disease from
Acknowledgement
This work was supported by the Czech Technical University in Prague, grant SGS14/208/OHK4/3T/14.
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