# Time Series Analysis Using Composite Multiscale Entropy

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## Abstract

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## 1. Introduction

^{m}to 30

^{m}[16]. As reported in [2,5], in case of m = 2, the SampEn is significantly independent of the time series length when the number of data points is larger than 750. However, for a shorter time series, the variance of the entropy estimator grows very fast as the number of data points is reduced. In the MSE algorithm, for an N points time series, the length of the coarse-grained time series at a scale factor τ is equal to N /τ. The larger the scale factor is, the shorter the coarse-grained time series is. Therefore, the variance of the entropy of the coarse-grained series estimated by SampEn increases as a time scale factor increases. In many practical applications, the data length is often very short and the variance of estimated entropy values at large scale factors would become large. Large variance of estimated entropy values leads to the reduction of reliability in distinguishing time series generated by different systems. In order to reduce the variance of estimated entropy values at large scales, a composite multiscale entropy (CMSE) algorithm is proposed in this paper. The effectiveness of the CMSE algorithm is evaluated through two synthetic noise signals and a real vibration data set provided by Case Western Reserve University (CWRU) [17].

## 2. Methods

#### 2.1. Multiscale Entropy

**Figure 1.**Schematic illustration of the coarse-grained procedure. Modified from reference [3].

**Figure 2.**Brute force method. Modified from reference [18].

#### 2.2. Composite Multiscale Entropy

## 3. Comparative Study of MSE and CMSE

#### 3.1. White Noise and 1/f Noise

Data Length | Signals | Methods | Scales | ||||||||||

1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 20 | |||

2,000 | white noise | MSE | 0.026 | 0.049 | 0.059 | 0.064 | 0.072 | 0.067 | 0.084 | 0.076 | 0.091 | 0.091 | 0.103 |

CMSE | 0.026 | 0.030 | 0.037 | 0.043 | 0.046 | 0.051 | 0.054 | 0.054 | 0.057 | 0.063 | 0.066 | ||

1/f noise | MSE | 0.084 | 0.097 | 0.121 | 0.139 | 0.162 | 0.227 | 0.236 | 0.284 | 0.265 | 0.282 | 0.310 | |

CMSE | 0.084 | 0.088 | 0.092 | 0.099 | 0.108 | 0.126 | 0.122 | 0.148 | 0.153 | 0.155 | 0.163 | ||

10,000 | white noise | MSE | 0.007 | 0.014 | 0.019 | 0.025 | 0.028 | 0.030 | 0.032 | 0.035 | 0.038 | 0.035 | 0.035 |

CMSE | 0.007 | 0.010 | 0.015 | 0.018 | 0.020 | 0.021 | 0.023 | 0.025 | 0.025 | 0.027 | 0.028 | ||

1/f noise | MSE | 0.069 | 0.069 | 0.070 | 0.073 | 0.072 | 0.071 | 0.078 | 0.074 | 0.086 | 0.085 | 0.080 | |

CMSE | 0.069 | 0.068 | 0.068 | 0.067 | 0.069 | 0.069 | 0.068 | 0.069 | 0.072 | 0.072 | 0.070 |

#### 3.2. Real Vibration Data

**Table 2.**Numbers of data sets are corresponding to different faulted classes, defective levels and rotation speeds.

Shaft Speed / Defect Level | Rotation Speed (rpm) | ||||||||

1730 | 1750 | 1772 | |||||||

Fault diameter (mils) | |||||||||

Fault Class | 7 | 14 | 21 | 7 | 14 | 21 | 7 | 14 | 21 |

Normal state | 243 | 242 | 242 | ||||||

Ball | 244 | 243 | 243 | 243 | 243 | 243 | 243 | 243 | 243 |

Inner race fault | 243 | 242 | 244 | 243 | 244 | 245 | 243 | 191 | 242 |

Outer race fault (3) | 243 | 242 | 243 | 243 | 242 | 245 | |||

Outer race fault (6) | 244 | 244 | 244 | 243 | 243 | 244 | 243 | 242 | 244 |

Outer race fault (12) | 242 | 243 | 241 | 243 | 241 | 243 |

**Figure 9.**Measured acceleration signals of vibrations in the time domain of six different bearing conditions (

**a**) normal state, ball fault and inner race fault; (

**b**) outer race faults at 3, 6, and 12 o’clock positions.

**Figure 10.**MSE and CMSE results on bearing vibration data (1,730 rpm, 7 mils). (

**a**) Normal state. (

**b**) Ball fault. (

**c**) Inner race fault. (

**d**) Outer race fault (3 o’clock position). (

**e**) Outer race fault (6 o’clock position). (

**f**) Outer race fault (12 o’clock position)

#### 3.3. Performance Assessment

#### 3.4. Fault Diagnosis Using an Artificial Neural Network

^{−10}and maximum iteration number of 1,000 were used. To improve generalization, the data sets were randomly divided by three parts: (1) training (50%), (2) validation (15%), and (3) testing (35%). The average accuracy of prediction for each experiment was quantified over 200 tests.

Fault Class 1 | Feature Extractor | Fault class 2 | |||||

N | B | I | O3 | O6 | O12 | ||

N | MSE | 25.200 | 25.200 | 20.044 | 25.484 | 5.538 | |

CMSE | 25.898 | 28.069 | 21.614 | 26.115 | 6.992 | ||

B | MSE | 25.200 | 3.675 | 5.312 | 5.515 | 9.657 | |

CMSE | 25.898 | 5.852 | 8.105 | 7.534 | 11.387 | ||

I | MSE | 25.200 | 3.675 | 7.128 | 7.560 | 12.883 | |

CMSE | 28.069 | 5.852 | 8.672 | 9.652 | 16.965 | ||

O3 | MSE | 20.044 | 5.312 | 7.128 | 5.934 | 9.303 | |

CMSE | 21.614 | 8.105 | 8.672 | 6.605 | 13.239 | ||

O6 | MSE | 25.484 | 5.515 | 7.560 | 5.934 | 9.624 | |

CMSE | 26.115 | 7.534 | 9.652 | 6.605 | 12.139 | ||

O12 | MSE | 5.538 | 9.657 | 12.883 | 9.303 | 9.624 | |

CMSE | 6.992 | 11.387 | 16.965 | 13.239 | 12.139 |

Fault Diameter / Feature Extractor | Rotation Speed (rpm) | ||||||||

1730 | 1750 | 1772 | |||||||

7 | 14 | 21 | 7 | 14 | 21 | 7 | 14 | 21 | |

MSE | 97.33% | 98.81% | 96.77% | 99.05% | 98.01% | 95.65% | 99.34% | 96.67% | 95.89% |

CMSE | 99.29% | 99.75% | 98.26% | 99.58% | 99.86% | 98.50% | 99.91% | 99.65% | 98.42% |

## 5. Conclusions

## Acknowledgments

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## Appendix A. The Matlab Code for the Composite Multiscale Entropy Algorithm

function E = CMSE(data,scale) r = 0.15*std(data); for i = 1:scale % i:scale index for j = 1:i % j:croasegrain series index buf = croasegrain(data(j:end),i); E(i) = E(i)+ SampEn(buf,r)/i; end end %Coarse Grain Procedure. See Equation (2) % iSig: input signal ; s : scale numbers ; oSig: output signal function oSig=CoarseGrain(iSig,s) N=length(iSig); %length of input signal for i=1:1:N/s oSig(i)=mean(iSig((i-1)*s+1:i*s)); end %function to calculate sample entropy. See Algorithm 1 function entropy = SampEn(data,r) l = length(data); Nn = 0; Nd = 0; for i = 1:l-2 for j = i+1:l-2 if abs(data(i)-data(j))<r && abs(data(i+1)-data(j+1))<r Nn = Nn+1; if abs(data(i+2)-data(j+2))<r Nd = Nd+1; end end end end entropy = -log(Nd/Nn);

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**MDPI and ACS Style**

Wu, S.-D.; Wu, C.-W.; Lin, S.-G.; Wang, C.-C.; Lee, K.-Y.
Time Series Analysis Using Composite Multiscale Entropy. *Entropy* **2013**, *15*, 1069-1084.
https://doi.org/10.3390/e15031069

**AMA Style**

Wu S-D, Wu C-W, Lin S-G, Wang C-C, Lee K-Y.
Time Series Analysis Using Composite Multiscale Entropy. *Entropy*. 2013; 15(3):1069-1084.
https://doi.org/10.3390/e15031069

**Chicago/Turabian Style**

Wu, Shuen-De, Chiu-Wen Wu, Shiou-Gwo Lin, Chun-Chieh Wang, and Kung-Yen Lee.
2013. "Time Series Analysis Using Composite Multiscale Entropy" *Entropy* 15, no. 3: 1069-1084.
https://doi.org/10.3390/e15031069