.2024 Feb 10;26(2):155.

doi: 10.3390/e26020155.

A Fast Algorithm for Estimating Two-Dimensional Sample Entropy Based on an Upper Confidence Bound and Monte Carlo Sampling

Zeheng Zhou¹, Ying Jiang¹, Weifeng Liu¹, Ruifan Wu¹, Zerong Li¹, Wenchao Guan¹

Affiliations

PMID:38392410
PMCID: PMC10887568
DOI: 10.3390/e26020155

A Fast Algorithm for Estimating Two-Dimensional Sample Entropy Based on an Upper Confidence Bound and Monte Carlo Sampling

Zeheng Zhou et al. Entropy (Basel).2024.

.2024 Feb 10;26(2):155.

doi: 10.3390/e26020155.

Authors

Zeheng Zhou¹, Ying Jiang¹, Weifeng Liu¹, Ruifan Wu¹, Zerong Li¹, Wenchao Guan¹

Affiliation

¹ School of Computer Science and Engineering, Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China.

PMID:38392410
PMCID: PMC10887568
DOI: 10.3390/e26020155

Abstract

The two-dimensional sample entropy marks a significant advance in evaluating the regularity and predictability of images in the information domain. Unlike the direct computation of sample entropy, which incurs a time complexity of O(N2) for the series withN length, the Monte Carlo-based algorithm for computing one-dimensional sample entropy (MCSampEn) markedly reduces computational costs by minimizing the dependence onN. This paper extends MCSampEn to two dimensions, referred to as MCSampEn2D. This new approach substantially accelerates the estimation of two-dimensional sample entropy, outperforming the direct method by more than a thousand fold. Despite these advancements, MCSampEn2D encounters challenges with significant errors and slow convergence rates. To counter these issues, we have incorporated an upper confidence bound (UCB) strategy in MCSampEn2D. This strategy involves assigning varied upper confidence bounds in each Monte Carlo experiment iteration to enhance the algorithm's speed and accuracy. Our evaluation of this enhanced approach, dubbed UCBMCSampEn2D, involved the use of medical and natural image data sets. The experiments demonstrate that UCBMCSampEn2D achieves a 40% reduction in computational time compared to MCSampEn2D. Furthermore, the errors with UCBMCSampEn2D are only 30% of those observed in MCSampEn2D, highlighting its improved accuracy and efficiency.

Keywords: Monte Carlo algorithm; sample entropy; upper confidence bound strategy.

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Conflict of interest statement

The authors declare no conflicts of interest.

Figures

**Figure 1**
Examples of reference images include: images (a,b) in the Warwick QU Dataset, which represent benign and malignant cases, respectively, each with a size of $580 \times 440$ ; image (c) is a natural image with a size of $775 \times 522$ ; and image (d), called the wallpaper, with a size of $3000 \times 3000$ , is used to verify the method’s performance on large-scale data.

**Figure 2**
(a) depicts the average error variation in MCSampEn2D and UCBMCSampEn2D experiments on the Warwick QU dataset with changing $N_{1}$ , where parameters are set to $m = 2$ and $r = 0.3$ ; (b) depicts the average error variation in MCSampEn2D and UCBMCSampEn2D experiments on natural datasets with changing $N_{1}$ , where parameters are set to $m = 2$ , $r = 0.3$ , $a = 5$ and $b = 1$ . The reward function is set as the cosine function.

**Figure 3**
The average error variation in different reward functionR with changing $N_{1}$ on the Warwick QU dataset, where parameters are set to $m = 2$ and $r = 0.3$ . The parameters for the cosine function in the reward function are set as $a = 8$ and $b = 0.5$ , while for the normal distribution function, the parameters are set as $a = 8$ and $b = 2$ .

**Figure 4**
The error standard deviation variation for the wallpaper, where $N_{0} = 128$ , $N_{1} = 300$ , where the UCB parameters were set at $a = 8$ and $b = 1$ .

**Figure 5**
The mean error variation for the wallpaper, where the UCB parameters were set at $a = 8$ and $b = 1$ .

**Figure 6**
The error standard deviation variation for the wallpaper, where $N_{0} = 128$ and $N_{1} = 300$ .

See this image and copyright information in PMC

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A Fast Algorithm for Estimating Two-Dimensional Sample Entropy Based on an Upper Confidence Bound and Monte Carlo Sampling

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A Fast Algorithm for Estimating Two-Dimensional Sample Entropy Based on an Upper Confidence Bound and Monte Carlo Sampling

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Conflict of interest statement

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