RCMSE: Composite Multiscale Sample Entropy

Updated 1 August 2025

RCMSE is a complexity measure that refines multiscale entropy by averaging SampEn over shifted coarse-grained sequences to enhance statistical stability.
It is widely applied in biomedical signal analysis, emotion recognition, and financial time series to distinguish between normal and abnormal dynamics.
Despite its higher computational cost compared to dispersion methods, RCMSE offers improved reliability and reduced variance for amplitude-based assessments.

Refined Composite Multiscale Sample Entropy (RCMSE) is an entropy-based complexity measure for time series analysis, designed to improve on standard multiscale entropy (MSE) by increasing statistical reliability and stability across multiple scales, especially for short or noisy signals. RCMSE is widely applied in biomedical signal analysis, emotion recognition from physiological data, financial time series complexity assessment, and stability analysis of dynamical systems.

1. Core Principles and Mathematical Formulation

RCMSE is an extension of multiscale sample entropy (MSE), addressing two major issues in the original methodology: instability for short or noisy signals and increased risk of undefined values at large scales due to reduced data length. Sample entropy (SampEn), which underpins MSE, is defined as the negative natural logarithm of the conditional probability that two sequences similar for $m$ points remain similar at the next point. Letting $A$ and $B$ denote counts of template matches of length $m+1$ and $m$ , respectively,

$\mathrm{SampEn}(m, r) = -\ln\left(\frac{A}{B}\right).$

The standard multiscale process applies coarse-graining via segment averages:

$x_j^{(\tau)} = \frac{1}{\tau} \sum_{b=(j-1)\tau+1}^{j\tau} u_b$

for scale factor $\tau$ , yielding a single coarse-grained sequence per scale.

RCMSE innovates by constructing $\tau$ coarse-grained sequences at each scale by shifting the starting index ( $k = 1,\ldots,\tau$ ):

$x_{k, j}^{(\tau)} = \frac{1}{\tau} \sum_{b = k + \tau(j-1)}^{k + \tau j - 1} u_b.$

SampEn is computed for each sequence. The refined composite entropy is defined as

$\mathrm{RCMSE}(u, m, r, \tau) = -\ln \left( \frac{\frac{1}{\tau}\sum_k A_k}{\frac{1}{\tau}\sum_k B_k} \right),$

where $A_k$ and $B_k$ are the template match counts of length $m+1$ and $m$ on the $k$ th sequence. This averaging reduces variance and the risk of undefined values.

RCMSE shares conceptual motivation with multiscale dispersion entropy (MDE) and its composite version RCMDE, but differs in the underlying entropy computation and computational efficiency (Azami et al., 2016). While RCMSE employs amplitude-based SampEn, MDE and RCMDE use the frequencies of dispersion (pattern) classes; these are mapped from measurements to discrete classes via the normal cumulative distribution function and analyzed in sliding windows. The computational complexity for SampEn (and thus RCMSE) is typically $O(N^2)$ . In contrast, MDE and RCMDE operate in $O(N)$ time, offering significant runtime advantages.

Empirical comparisons across synthetic and real biomedical signals demonstrate that while RCMSE and RCMDE yield similar complexity profiles, the latter are notably faster and more robust for short/noisy data due to the avoidance of undefined values and superior coefficient of variation statistics (Azami et al., 2016).

3. Statistical Properties, Stability, and Computational Considerations

RCMSE exhibits improved statistical reliability in small-sample and high-noise regimes compared to standard MSE, primarily due to:

Averaging across all possible $\tau$ -shifted coarse-grained series, mitigating instability caused by single-sequence under-sampling.
Reduced variance in the entropy estimates, as averaging composite match counts dampens fluctuations associated with particular segmentations.
Lower likelihood of undefined values, as the risk that a coarse-grained sequence is too short for entropy calculation is distributed across $\tau$ independent instances.

Despite these improvements, RCMSE remains significantly slower than pattern-based approaches such as RCMDE. For large-scale physiological time series (e.g., $N \sim 10^6$ ), RCMSE can require computation times of many seconds to days, whereas RCMDE completes in seconds to hours on the same data (Azami et al., 2016).

4. Applications: Biomedical, Physiological, and Financial Time Series

RCMSE has been widely adopted for probing the complexity of physiological recordings (EEG, ECG, blood pressure, galvanic skin response), for emotion recognition, and for differentiating healthy from pathological states (Azami et al., 2016, Tung et al., 2018). For example:

In emotion recognition from GSR signals, RCMSE features at higher scales ( $\tau > 5$ ) produce statistically significant differentiation between low and high arousal states, outperforming both standard MSE and other multivariate entropy measures in specific modalities (Tung et al., 2018).
In physiological studies, RCMSE profiles distinguish between younger and elderly subjects, as well as between patients with neurodegenerative conditions and controls (Azami et al., 2016, Asuke et al., 2023).

In financial research, RCMSE quantifies complexity in markets with nonstationary, noisy log-return series (Masoudi et al., 31 Jul 2025). The method enables:

Robust entropy estimation for relatively short data sets, crucial for markets with high volatility or during regime changes.
Discrimination between assets; for instance, Bitcoin exhibits low entropy at short scales (higher predictability) but high overall summed entropy at longer scales, reflecting pronounced nonlinear correlations and increased overall market complexity.

5. Connections to Information-Theoretical Measures and Variance Statistics

Structural parallels exist between RCMSE (and MSE) and information-variance-based measures such as Allan variance (Asuke et al., 2023). Both classes of metrics analyze the variability of time series across scales via a coarse-graining process:

Both rely on statistical properties of the signal under window averaging (RCMSE: pattern recurrence probabilities; Allan variance: mean differences).
Theoretical links are established through decomposing SampEn as a function of pairwise pattern repetition probabilities, which resemble conditional variances under stationarity and unimodality assumptions.

When the "Neighborhood-Likelihood-to-Variance-Relationship" (NLVR) holds—that is, changes in the conditional probability of recurrence and the variance of the respective block have opposite signs—the scale-dependent trends of RCMSE and Allan variance are aligned (Asuke et al., 2023). This is typically observed in physiological and many natural economic signals.

6. Limitations, Performance Comparison, and Recommendations

While RCMSE substantially enhances reliability over standard MSE, several limitations persist:

Computational cost ( $O(N^2)$ ) is prohibitive for real-time or large-scale deployments, making RCMDE and related dispersion-based approaches the methods of choice for those applications (Azami et al., 2016).
RCMSE may still yield undefined values in extreme cases of very short signals, as pattern count denominators may fall to zero.
Discriminative power is often surpassed by MDE/RCMDE, as evidenced by lower p-values in group separability for both synthetic and real-world physiological data.

A plausible implication is that for most real-world biomedical and complex system applications requiring rapid and robust complexity estimation, dispersion-based refined composite entropy measures (like RCMDE) are preferable, unless precise amplitude-based regularity quantification is required.

Method	Computational Complexity	Robustness for Short/Noisy Data	Discriminative Power	Applicability
MSE	$O(N^2)$	Low (unstable for short data)	Moderate	Biomedical, generic
RCMSE	$O(N^2)$	High (composite averaging)	Moderate–high	Biomed/Finance
MDE/RCMDE	$O(N)$	Very high (no undefined values)	High	Real-time, large-N

RCMSE continues to serve as a statistically robust, scale-sensitive complexity measure for time series, particularly advantageous when amplitude-based dynamics are of specific scientific interest and computational constraints are not prohibitive. However, for most applications demanding efficiency and enhanced robustness against length/noise constraints, dispersion entropy-based refined composite methods are generally superior.