To evaluate the ITD–MFFA algorithm's reliability, for verification, white noise and colored noise with an intensity of 1 dB and amplitude of 5% of the sequence standard deviation are added to the original sequence without adjusting the original sequence's attributes (Cheng 2018). The ITD–MFFA and MFDFA methods' calculation results before and after the addition of noise to the original data are compared, and the robustness to noise (i.e., the anti-interference ability) is analyzed. The complexity calculation results obtained using the two methods before and after noise addition are presented in Table 4 and plotted in Figure 8(b), showing that under noise interference, both methods' calculation results exhibit deviations; however, the deviation in the calculation results obtained using the MFDFA method is greater. For the ITD–MFFA method, the correlation coefficients of the sequence complexity before and after the addition of noise are 0.9991 and 0.9993, respectively, and for the MFDFA method, they are smaller at 0.9950 and 0.9355, respectively. Consequently, the ITD–MFFA method displays a stronger anti-interference ability, and the calculation results are more stable. However, due to the detrending process's imprecision, the complexity of the spring EC time series is frequently overestimated using the traditional MFDFA method. Furthermore, the ITD–MFFA method improves the stability and reliability of spring EC measurements.

Table 4

Spring . | Complexity results with ITD–MFFA . | Complexity results with MFDFA (m = 1). | ||||
---|---|---|---|---|---|---|

Original sequence . | Add white noise . | Add colored noise . | Original sequence . | Add white noise . | Add colored noise . | |

Baotu spring | 1.606 | 1.6063 | 1.6061 | 1.8449 | 1.8453 | 1.8103 |

Heihu spring | 1.559 | 1.5671 | 1.5534 | 1.6531 | 1.6326 | 1.6229 |

Tanxi spring | 1.722 | 1.7231 | 1.7275 | 1.8662 | 1.8437 | 1.8891 |

Wangfuchizi | 1.814 | 1.8126 | 1.814 | 1.9226 | 1.9009 | 1.9336 |

Correlation coefficient R with the original sequence complexity | — | 0.9991 | 0.9993 | – | 0.9950 | 0.9355 |

Spring . | Complexity results with ITD–MFFA . | Complexity results with MFDFA (m = 1). | ||||
---|---|---|---|---|---|---|

Original sequence . | Add white noise . | Add colored noise . | Original sequence . | Add white noise . | Add colored noise . | |

Baotu spring | 1.606 | 1.6063 | 1.6061 | 1.8449 | 1.8453 | 1.8103 |

Heihu spring | 1.559 | 1.5671 | 1.5534 | 1.6531 | 1.6326 | 1.6229 |

Tanxi spring | 1.722 | 1.7231 | 1.7275 | 1.8662 | 1.8437 | 1.8891 |

Wangfuchizi | 1.814 | 1.8126 | 1.814 | 1.9226 | 1.9009 | 1.9336 |

Correlation coefficient R with the original sequence complexity | — | 0.9991 | 0.9993 | – | 0.9950 | 0.9355 |

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