# Fractional Refined Composite Multiscale Fuzzy Entropy of International Stock Indices

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodologies

#### 2.1. Fuzzy Entropy

#### 2.2. Fractional Refined Composite Multiscale Fuzzy Entropy

## 3. Complexity Measure for Synthetic Data

## 4. Complexity Measure for International Stock Indices

#### 4.1. Complexity Measure of Returns

#### 4.2. Complexity Measure of Volatility

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

FRCMFE | Fractional refined composite multiscale fuzzy entropy |

FuzzyEn | Fuzzy entropy |

CMFE | Composite multiscale fuzzy entropy |

RCMFE | Refined composite multiscale fuzzy entropy |

MFE | Multiscale fuzzy entropy |

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Method | Data Length | ||||||
---|---|---|---|---|---|---|---|

1000 | 1500 | 2000 | 2500 | 3000 | 5000 | 10,000 | |

MFE ($\tau $ = 10) | 0.0827 | 0.0794 | 0.0672 | 0.0483 | 0.0470 | 0.0339 | 0.0302 |

CMFE ($\tau $ = 10) | 0.0706 | 0.0538 | 0.0449 | 0.0378 | 0.0331 | 0.0313 | 0.0185 |

RCMFE ($\tau $ = 10) | 0.0684 | 0.0509 | 0.0466 | 0.0366 | 0.0327 | 0.0283 | 0.0180 |

MFE ($\tau $ = 20) | 0.0885 | 0.0823 | 0.0788 | 0.0663 | 0.0717 | 0.0498 | 0.0337 |

CMFE ($\tau $ = 20) | 0.0763 | 0.0681 | 0.0568 | 0.0544 | 0.0397 | 0.0340 | 0.0290 |

RCMFE ($\tau $ = 20) | 0.0716 | 0.0658 | 0.0517 | 0.0397 | 0.0393 | 0.0337 | 0.0228 |

$\mathit{\tau}$ | SSE | S&P500 | SZSE | N225 | HSI |
---|---|---|---|---|---|

1 | 0.9329 | 0.8888 | 0.9533 | 1.0754 | 0.9673 |

3 | 0.5968 | 0.5235 | 0.6029 | 0.6736 | 0.6139 |

5 | 0.4574 | 0.3833 | 0.4637 | 0.5034 | 0.4672 |

7 | 0.4049 | 0.3077 | 0.4059 | 0.4016 | 0.3750 |

9 | 0.3427 | 0.2518 | 0.3411 | 0.3417 | 0.3352 |

12 | 0.3032 | 0.2127 | 0.3050 | 0.2948 | 0.2836 |

16 | 0.2576 | 0.1771 | 0.2621 | 0.2317 | 0.2371 |

20 | 0.2346 | 0.1635 | 0.2493 | 0.2099 | 0.2131 |

$\mathit{\tau}$ | SSE | S&P500 | SZSE | N225 | HSI |
---|---|---|---|---|---|

1 | 0.9329 | 0.8888 | 0.9533 | 1.0754 | 0.9673 |

3 | 0.6022 | 0.5263 | 0.6134 | 0.6550 | 0.6108 |

5 | 0.4715 | 0.3948 | 0.4824 | 0.5058 | 0.4709 |

7 | 0.3965 | 0.3154 | 0.4017 | 0.4070 | 0.3846 |

9 | 0.3414 | 0.2637 | 0.3517 | 0.3418 | 0.3272 |

12 | 0.2984 | 0.2116 | 0.3053 | 0.2826 | 0.2773 |

16 | 0.2615 | 0.1765 | 0.2722 | 0.2332 | 0.2381 |

20 | 0.2319 | 0.1492 | 0.2442 | 0.1975 | 0.2062 |

$\mathit{\tau}$ | SZSE | SSE | HSI | N225 | S&P500 |
---|---|---|---|---|---|

2 | 0.5990 | 0.5972 | 0.5960 | 0.5957 | 0.5943 |

3 | 0.5962 | 0.5951 | 0.5939 | 0.5942 | 0.5927 |

4 | 0.5948 | 0.5939 | 0.5934 | 0.5935 | 0.5927 |

5 | 0.5945 | 0.5939 | 0.5931 | 0.5930 | 0.5924 |

6 | 0.5942 | 0.5935 | 0.5929 | 0.5927 | 0.5920 |

7 | 0.5937 | 0.5930 | 0.5926 | 0.5924 | 0.5920 |

8 | 0.5933 | 0.5928 | 0.5924 | 0.5923 | 0.5919 |

10 | 0.5930 | 0.5924 | 0.5924 | 0.5922 | 0.5918 |

$\mathit{\alpha}$ | SSE | S&P500 | SZSE | N225 | HSI |
---|---|---|---|---|---|

−0.3 | 0.4811 | 0.4795 | 0.4817 | 0.4801 | 0.4803 |

−0.2 | 0.5457 | 0.5439 | 0.5465 | 0.5445 | 0.5448 |

−0.1 | 0.5877 | 0.5856 | 0.5887 | 0.5863 | 0.5867 |

−0.04 | 0.5958 | 0.5934 | 0.5967 | 0.5942 | 0.5946 |

0 | 0.5913 | 0.5889 | 0.5924 | 0.5897 | 0.5901 |

0.1 | 0.5341 | 0.5314 | 0.5353 | 0.5323 | 0.5328 |

0.2 | 0.3822 | 0.3794 | 0.3854 | 0.3803 | 0.3808 |

0.3 | 0.0794 | 0.0767 | 0.0806 | 0.0776 | 0.0780 |

0.4 | −0.4777 | −0.4800 | −0.4767 | −0.4792 | −0.4788 |

0.5 | −1.5028 | −1.5038 | −1.5024 | −1.5034 | −1.5033 |

$\mathit{\alpha}$ | SSE | S&P500 | SZSE | N225 | HSI |
---|---|---|---|---|---|

−0.3 | 0.4798 | 0.4789 | 0.4800 | 0.4793 | 0.4794 |

−0.2 | 0.5442 | 0.5432 | 0.5444 | 0.5435 | 0.5437 |

−0.1 | 0.5860 | 0.5848 | 0.5862 | 0.5852 | 0.5854 |

−0.04 | 0.5938 | 0.5929 | 0.5941 | 0.5930 | 0.5933 |

0 | 0.5893 | 0.5879 | 0.5896 | 0.5884 | 0.5887 |

0.1 | 0.5319 | 0.5304 | 0.5322 | 0.5309 | 0.5312 |

0.2 | 0.3799 | 0.3783 | 0.3802 | 0.3788 | 0.3792 |

0.3 | 0.0771 | 0.0756 | 0.0775 | 0.0761 | 0.0764 |

0.4 | −0.4796 | −0.4809 | −0.4793 | −0.4804 | −0.4802 |

0.5 | −1.5036 | −1.5041 | −1.5035 | −1.5039 | −1.5038 |

$\mathit{\tau}$ | SSE | S&P500 | SZSE | N225 | HSI |
---|---|---|---|---|---|

1 | 0.9614 | 0.7745 | 0.8403 | 0.7986 | 0.8148 |

3 | 0.6337 | 0.4977 | 0.4889 | 0.5158 | 0.5367 |

5 | 0.5148 | 0.4080 | 0.3897 | 0.4275 | 0.4427 |

7 | 0.4538 | 0.3826 | 0.3299 | 0.3783 | 0.3967 |

9 | 0.4301 | 0.3523 | 0.3021 | 0.3633 | 0.3747 |

12 | 0.4057 | 0.3409 | 0.2895 | 0.3496 | 0.3539 |

16 | 0.3661 | 0.3314 | 0.22782 | 0.3505 | 0.3519 |

20 | 0.3909 | 0.3230 | 0.2785 | 0.3615 | 0.3433 |

$\mathit{\tau}$ | SSE | S&P500 | SZSE | N225 | HSI |
---|---|---|---|---|---|

1 | 0.7986 | 0.7745 | 0.8148 | 0.9614 | 0.8403 |

3 | 0.5236 | 0.4868 | 0.5245 | 0.6238 | 0.4876 |

5 | 0.4284 | 0.3985 | 0.4418 | 0.5123 | 0.3783 |

7 | 0.3821 | 0.3668 | 0.3876 | 0.4522 | 0.3261 |

9 | 0.3631 | 0.3519 | 0.3684 | 0.3684 | 0.3056 |

12 | 0.3537 | 0.3410 | 0.3535 | 0.4037 | 0.2954 |

16 | 0.3494 | 0.3280 | 0.3437 | 0.3904 | 0.2851 |

20 | 0.3491 | 0.3197 | 0.3440 | 0.3868 | 0.2791 |

$\mathit{\tau}$ | SSE | S&P500 | SZSE | N225 | HSI |
---|---|---|---|---|---|

2 | 0.5328 | 0.5311 | 0.5335 | 0.5318 | 0.5317 |

3 | 0.5316 | 0.5306 | 0.5321 | 0.5312 | 0.5310 |

4 | 0.5313 | 0.5305 | 0.5316 | 0.5309 | 0.5308 |

5 | 0.5310 | 0.5304 | 0.5314 | 0.5308 | 0.5307 |

6 | 0.5308 | 0.5303 | 0.5312 | 0.5306 | 0.5305 |

7 | 0.5308 | 0.5303 | 0.5311 | 0.5306 | 0.5305 |

8 | 0.5307 | 0.5303 | 0.5309 | 0.5306 | 0.5305 |

10 | 0.5306 | 0.5302 | 0.5308 | 0.5305 | 0.5308 |

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**MDPI and ACS Style**

Wu, Z.; Zhang, W. Fractional Refined Composite Multiscale Fuzzy Entropy of International Stock Indices. *Entropy* **2019**, *21*, 914.
https://doi.org/10.3390/e21090914

**AMA Style**

Wu Z, Zhang W. Fractional Refined Composite Multiscale Fuzzy Entropy of International Stock Indices. *Entropy*. 2019; 21(9):914.
https://doi.org/10.3390/e21090914

**Chicago/Turabian Style**

Wu, Zhiyong, and Wei Zhang. 2019. "Fractional Refined Composite Multiscale Fuzzy Entropy of International Stock Indices" *Entropy* 21, no. 9: 914.
https://doi.org/10.3390/e21090914