多尺度复杂性分析：粗粒度下高阶矩量揭示睡眠阶段比较研究

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本文是一篇发表于《计算非线性科学与数值模拟》(Communications in Nonlinear Science and Numerical Simulation)第44卷(2017年)的研究论文，标题为“睡眠阶段的比较研究：在粗粒度下使用较高矩量化多尺度复杂度”。该研究关注的是通过粗粒度分析和利用高阶矩来量化不同睡眠阶段的复杂性。在睡眠研究中，理解各个阶段的特性及其内部动态至关重要，因为这有助于评估个体健康状况并开发有效的治疗策略。作者们采用了多尺度复杂度的概念，这是一种数学工具，用于衡量系统中不同尺度下的结构和行为。他们选择在粗粒度水平上进行分析，这是因为粗粒度方法可以提供更宏观的视角，从而揭示出睡眠过程中可能被忽视的模式。研究者们将睡眠信号（可能是脑电图、心率变异等生理指标）视为一个多尺度系统，每个睡眠阶段如浅睡期、慢波睡眠期等具有不同的特征频率和动力学特性。他们通过高阶矩（例如方差、skewness和kurtosis等统计量）来量化这些复杂性，高阶矩不仅能捕捉到数据的分布形状，还能揭示数据中的潜在非线性特征。这种方法允许研究人员区分不同睡眠阶段之间的细微差异，即便是对于那些仅在粗粒度尺度上观察可能难以察觉的差异。在论文中，作者们可能进行了数据采集，对多个受试者的睡眠数据进行了处理，并运用统计和复杂网络分析技术来提取和比较各睡眠阶段的高阶矩特征。他们可能还进行了数据分析，探讨了粗粒度下的高阶矩变化是否能有效反映睡眠阶段的转换，以及这种复杂性变化是否与睡眠质量、健康状况或疾病状态有关。此外，论文可能包含了一些关键发现，如特定睡眠阶段的高阶矩模式的显著性，或者不同人群之间在睡眠复杂性上的差异。同时，这篇研究也可能会提出未来的研究方向，比如改进的量化方法，或者将这种复杂性分析应用到实时睡眠监测和个性化健康干预的可行性。这篇文章通过细致的多尺度复杂度分析，为我们提供了深入理解睡眠阶段之间差异的新视角，为睡眠研究领域提供了重要的理论支持和实践应用潜力。它不仅展示了统计学方法在医疗领域的价值，也为临床医生和睡眠专家提供了新的工具来解读和预测睡眠过程。

294 W. Shi et al. / Commun Nonlinear Sci Numer Simulat 44 (2017) 292–303

Similarly, A

(r) =

N−m −1

and A

(r) =

N−m



N−m

i =1

(r) represent the probability and density for x

m +1

( j) is within r of

m +1

(i ) , respectively. The total number of template matches in a m -dimensional ((m + 1)-dimensional) state space within a

tolerance r can be calculated as

B (r) =

(N − m − 1)(N − m ) B

(r) , (4)

and

A (r) =

(N − m − 1)(N − m ) A

(r) . (5)

Finally, sample entropy is deﬁned by the following formula

SampE n (m, r, N) = − log



A (r)

B (r)



(6)

Hence, the sample entropy is the negative natural logarithm of the conditional probability that a dataset of length N , having

repeated itself for m points within a tolerance r , will also repeat itself for m + 1 points, excluding self-matches [24] .

2.2. Generalized multiscale entropy

We introduce the generalization of multiscale entropy MSE

, where the subscript denotes the moment used to coarse-

grain a time series. The traditional MSE is represented as MSE

, where μ refers to the mean (ﬁrst moment).

For a given time series, x = { x

, x

, ··· , x

} , we divide it into non-overlapping segments with equal length τ . The coarse-

grained time series { y

( τ )

} at scale τ is then derived as

(τ )

jτ



i =( j−1) τ +1

, for the ﬁrst moment (7)

(τ )

τ − 1

jτ



i =( j−1) τ +1

−

)

, for the second moment (8)

(τ )

jτ



i =( j−1) τ +1

−

)

, for the third moment (9)

···

(τ )

jτ



i =( j−1) τ +1

−

)

, for the selected moment n. (10)

In Eq. (10) , τ denotes the scale factor and 1 ≤ j ≤ N / τ . Fig. 1 shows the schematic diagram of the coarse-graining procedure

for MSE

. In our study, we compare the results of the ﬁrst three moments: MSE

( Eq. (7) ), MSE

( Eq. (8) ) and MSE

Skew

( Eq. (9) ), that stand for mean, variance and skewness, respectively. The selections are based on their clear deﬁnition in

mathematics, and hence it will be easier to be comprehended [25,26] . Note that only MSE

can we coarse-grain in τ = 1 ,

which the coarse-graining series is still remained as the original time series. Similarly, the scale is starting from 2 for MSE

and 3 for MSE

Skew

. Besides, the parameters m and r is chosen based on the standard proposed by Lake et al. [27] , a reporting

solution that can calculate the conﬁdence interval of SampEn.

3. Results and discussions

3.1. Numerical simulation

To examine the effectiveness of MSE

method, we use a chaotic deterministic signal (logistic map), a polynomial mapping

that widely cited in the publications related to complex, chaotic and dynamical systems [28–31] , as our simulated signals.

Mathematically, the logistic map is a recurrence relation as

n +1

= Rx

(1 − x

) , (11)

where x

is a number between zero and one that represents the ratio of existing population to the maximum possible

population. The parameter R controls the behaviour of the samples. For example, the samples either converge to a single

ﬁxed point ( R ≤ 3) or oscillate between few values (3 < R < 3.54) for smaller R . For larger R , the variation of the samples

is increasing with more complex behavior emerging. In our study, we use 10 0 0 0 samples of data length to ensure a reliable

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