## 1. Introduction

Alzheimer’s disease (AD) is a progressive neurodegenerative disease and the most common form of dementia in the elderly population, affecting intellectual, behavioural, and functional abilities [

1,

2,

3]. A positive diagnosis of AD allows the patient and his/her family time to be informed about the disease, to make life and financial decisions, and to plan for the future. In contrast, a negative diagnosis may reduce worry about memory loss associated with ageing. Moreover, it permits for early treatments of reversible conditions with similar symptoms (like depression and nutrition or medication problems) [

2]. Medical-based diagnosis of AD is not fully reliable and symptoms are frequently dismissed as normal consequences of healthy ageing. Spinal fluid analysis and signal and image processing methods are used to increase the confidence of the diagnosis of AD [

2,

3].

As AD progresses, there are changes in the dynamical brain activity that can be recorded in electroencephalogram (EEG) signals [

1,

2]. The EEG is an affordable, portable, and non-invasive tool to assess brain activity [

4]. In addition, in comparison with other non-invasive brain imaging approaches, EEG has high temporal resolution and includes essential information about abnormal brain dynamics in AD subjects [

2]. The studies show that AD causes a spectral slowdown and alterations in the non-linear dynamics of the brain signal [

5,

6].

A prevailing approach to diagnose of AD is to consider specific frequency bands in EEG, such as

δ (1–4 Hz),

θ (4–8 Hz),

α (8–13 Hz),

β (13–30 Hz), and

γ (30–40 Hz) [

2,

7,

8]. AD affects these different frequency bands in different ways. An increase of power in

δ,

θ, and

γ, and a decrease of power in higher frequencies

α and

β have been reported in AD patients in comparison with healthy age-matched control subjects [

2,

7,

8,

9].

In recent years, because of the non-linearity in the brain, even at the neuronal level [

10], there has been an increasing interest in non-linear techniques for the analysis of EEGs for diagnosis of AD [

1,

2,

11,

12,

13]. One of the most popular non-linear concepts used to assess the dynamical characteristics of signals is that of entropy [

14,

15]. This concept measures the uncertainty and irregularity of a time series [

14,

15]. Higher entropy normally stands for higher uncertainty, whereas lower entropy shows more regularity and certainty in a signal [

14,

16]. Thus, it can be considered as an indicator of dynamical changes along the temporal evolution of EEG signals.

Entropy approaches have been broadly used to characterise different kinds of signals. However, they achieve their maxima for signals with no structure (random) and are defined only for a single temporal scale: the one associated with the original sampling of the time series [

17,

18]. This can be considered as a limitation to investigate dynamics at longer time scales. Accordingly, multiscale entropy (MSE) was proposed to define entropy values for a range of scales to evaluate the complexity of signals at different time scales [

17]. Thus, MSE quantifies signal complexity, which may remain hidden for basic entropy approaches [

19].

Complexity indicates a degree of structural richness [

19]. In fact, neither completely regular (periodic) nor completely irregular (uncorrelated random) time series are truly complex, because none of them is structurally rich at a global level. Thus, the concept of irregularity and complexity are not the same. For example, white Gaussian noise (WGN) is more irregular than 1/

f noise, although the latter is more complex. It is in agreement with this fact that the WGN does not have a rich structure and shows a rapid drop in entropy with an increase in time scale factor [

19,

20,

21].

The MSE algorithm at the temporal scale factor

λ includes two main steps [

17]. First, in the coarse-graining process, the original signal is divided into non-overlapping segments with length

λ, and then the average of each segment is calculated. Second, the sample entropy (SampEn) [

15] of the coarse-grained time series is computed [

17].

For multi-channel signals, the MSE algorithms, though powerful and broadly-used, treat individual time series separately. Therefore, this method is appropriate for the components of multi-channel time series that are statistically independent. However, real multivariate physiological signals are simultaneously recorded and the time series are statistically dependent [

22,

23]. To this end, multivariate MSE using the mean in the coarse-graining process, named mvMSE

_{µ}, has been recently introduced [

22]. The mvMSE

_{µ} algorithm was validated on both illustrative benchmark signals and on real-world multivariate physiological and non-physiological datasets [

22,

24].

However, the dynamics of the volatility (variance) of a time series over multiple temporal scales to extract dynamical properties of spread also need to be inspected. To this end, Costa and Goldberger have recently proposed a modified MSE where the variance is used in the coarse-graining process [

25]. The mean- and variance-based MSE would be referred to as MSE

_{µ} and MSE

_{σ2}, respectively. MSE

_{σ2} was used to analyse heartbeat signals from healthy young and older subjects and patients with congestive heart failure syndrome. It was demonstrated that the dynamics of the volatility of heartbeat signals obtained from healthy young subjects are highly complex. The results also showed that MSE

_{σ2} values decrease with ageing and pathology [

25].

EEG irregularity and complexity analyses have been successfully and widely employed and provide a new view to understand physiological processes in both healthy and pathological conditions in AD [

1,

11,

13,

26,

27,

28]. The MSE- and mvMSE-based methods have been successfully used to characterise biomedical signals to detect different pathological states like epilepsy, schizophrenia, Parkinson’s disease, and AD [

13,

29,

30,

31,

32].

Escudero et al. used multiscale entropy with a coarse-graining process based on the mean to characterise EEGs in AD [

11]. Later, Morabito et al. analysed EEGs in AD patients with multivariate entropy techniques based on the mean [

13]. However, since the dataset included few subjects and channels, the results may not be completely reliable [

13]. Azami and colleagues used only mvMSE

_{µ} for magnetoencephalograms (MEGs) in AD [

32]. They consider five subsets of channels and not all the channels as a whole. Multiscale approaches using the variance in the coarse-graining process have yet to be applied to EEG analysis. Therefore, there is a need to investigate the usefulness of MSE

_{σ2} and mvMSE where the coarse graining process uses variance (mvMSE

_{σ2}) in comparison with the more broadly used methods based on mean (MSE

_{µ} and mvMSE

_{µ}) to characterise EEGs in AD.

The aim of this research is to investigate the first and second moments (mean and variance) for the coarse-graining process of MSE and mvMSE to characterise EEGs to discriminate age-matched control subjects from AD patients. We want to evaluate the differences between results obtained by the multiscale entropy methods and their corresponding multivariate versions. We also test the hypothesis that AD patients’ signals are less complex than controls’ recordings [

13,

19]. In addition, the changes in entropy values for different frequency bands are investigated to understand the effect of AD and entropy-based methods on each frequency band.

The outline of this paper is as follows. The next section describes the EEG data used in this study and explains briefly the MSE

_{µ}, MSE

_{σ2}, mvMSE

_{µ}, and mvMSE

_{σ2} algorithms. Results are presented in

Section 3. The discussions and conclusions are explained in

Section 4 and

Section 5, respectively.