# Innovative Variance Corrected Sen’s Trend Test on Persistent Hydrometeorological Data

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## Abstract

**:**

## 1. Introduction

## 2. Sen’s Trend Detection Method

#### 2.1. Sen’s Trend Plot

#### 2.2. Sen’s Trend Test

## 3. The Influence of Persistence on the Original Sen’s Trend Test

#### 3.1. Persistence in Hydrometeorological Data

#### 3.2. Inflation of Trend Slope Variance and Type I Error

## 4. Variance Corrected Sen’s Trend Test

#### 4.1. Theoretical Basis for Correcting Slope Variance

#### 4.2. Practical Procedure of the Method

- Step 1:
- Persistent model specification. The empirical lag-one autocorrelation coefficients of the aggregated time series are plotted versus the time scales. If the lag-one autocorrelations are almost constant or decay slowly for large time scales, the time series is regarded as a representation of an FGN process. Otherwise, the lag-one autocorrelations will drop down to zero after a few time scales; in this case, the time series is classified as an AR(1) process.
- Step 2:
- Persistent parameter estimation. For the AR(1) data, the empirical lag-one autocorrelation coefficient is estimated from the detrended series, and subsequently bias-corrected as recommended by Hamed [39]: $\rho =\left(n\widehat{\rho}+2\right)/\left(n-4\right)$. For the FGN data, the Hurst coefficient $H$ is estimated via the maximum likelihood method, which has been proven to be robust and to present low bias, as compared to several other methods [40].
- Step 3:
- Slope variance correction. The corrected slope variance ${V}_{C}\left(S\right)$ is calculated using Equation (15) for AR(1) data or Equation (16) for FGN data, according to the results of the persistent model specification.
- Step 4:
- Trend significance assessment. The variance-corrected test statistic ${Z}_{SC}$ is compared with the quantiles of the standard Normal distribution at a desired significance level, and the trend significance is quantified.

## 5. Monte-Carlo Simulation

#### 5.1. Simulation Design

#### 5.2. Slope Variance Correction and its Effectiveness on Mitigating Type I Error Inflation

#### 5.3. Power of Trend Detection

## 6. Application to Real-World Data

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Illustration of linear trend identification with Sen’s method: (

**a**) Time series and trend; (

**b**) Sen’s trend plot.

**Figure 2.**Comparison between the simulated distribution of the test statistic ${Z}_{S}$ for stationary independent data of length $n=100$ and the corresponding Normal approximation: (

**a**) original test statistic (Sen, 2017a); (

**b**) bias-corrected test statistic.

**Figure 3.**The effect of data persistence on Type I errors of the original Sen’s trend test: (

**a**) AR(1) data; (

**b**) Fractional Gaussian Noise (FGN) data.

**Figure 4.**The upper tail of the simulated cumulative distribution of the test statistic ${Z}_{SC}$ after variance correction for AR(1) and FGN data as compared to the standard Normal distribution.

**Figure 5.**Effectiveness of variance correction for AR(1) data. Light gray shading indicates acceptable region. White and dark grey shading indicate over-and under-correction, respectively.

**Figure 7.**Type I errors of the variance-corrected Sen’s trend test using a well-specified persistent model: (

**a**) AR(1) data; (

**b**) FGN data. The persistent parameters $\rho $ and $H$ are estimated from the data.

**Figure 8.**Power of the variance-corrected Sen’s trend test with various combinations of data persistence and a dimensionless slope of linear trend (series length: $n=100$) using a well-specified persistent model: (

**a**) AR(1) data; (

**b**) FGN data. The persistent parameters $\rho $ and $H$ are estimated from the data.

**Figure 9.**Power of the innovative variance-corrected Sen’s trend test compared to that of the original test and the Mann–Kendall test, for the case of uncorrelated data (series length: $n=100$).

**Figure 10.**Time series of annual rainy days and frost days of China (

**a1**,

**a3**), and annual total flow from Main River at Wuerzburg in Germany (

**a2**). Trend test results given in (

**a1–a3**) are obtained from variance-corrected Sen’s trend test. Sen’s trend plots for the data sets in a1–3 (

**b1–b3**). Lag-one autocorrelations of the aggregated series versus time scales as for a1–3 (

**c1–c3**).

**Table 1.**Variance inflation factor of Sen’s slope $V\left(S\right)/{V}_{0}\left(S\right)$ calculated for different persistent data sets, given lag-one autocorrelation coefficient $\rho $ and series length $n$.

n | $\mathbf{\rho}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |

AR(1) data | ||||||||||

30 | 1.00 | 1.21 | 1.44 | 1.74 | 2.11 | 2.61 | 3.24 | 4.15 | 5.19 | 5.69 |

50 | 1.00 | 1.21 | 1.46 | 1.79 | 2.20 | 2.77 | 3.54 | 4.73 | 6.63 | 9.21 |

100 | 1.00 | 1.21 | 1.48 | 1.82 | 2.26 | 2.89 | 3.77 | 5.18 | 7.80 | 13.61 |

150 | 1.00 | 1.22 | 1.49 | 1.83 | 2.29 | 2.92 | 3.83 | 5.36 | 8.16 | 15.45 |

200 | 1.00 | 1.22 | 1.49 | 1.84 | 2.31 | 2.94 | 3.89 | 5.44 | 8.38 | 16.31 |

FGN data | ||||||||||

30 | 1.00 | 1.31 | 1.64 | 1.95 | 2.24 | 2.43 | 2.52 | 2.39 | 1.99 | 1.23 |

50 | 1.00 | 1.41 | 1.87 | 2.36 | 2.85 | 3.28 | 3.55 | 3.53 | 3.07 | 1.97 |

100 | 1.00 | 1.54 | 2.24 | 3.06 | 4.02 | 4.91 | 5.69 | 6.03 | 5.55 | 3.74 |

150 | 1.00 | 1.63 | 2.51 | 3.57 | 4.90 | 6.25 | 7.47 | 8.21 | 7.77 | 5.45 |

200 | 1.00 | 1.70 | 2.71 | 3.98 | 5.62 | 7.37 | 9.09 | 10.20 | 9.93 | 7.09 |

**Table 2.**Variance inflation factor of Sen’s slope $V\left(S\right)/{V}_{C}\left(S\right)$ after variance correction for AR(1) data. The underlined values denote that the slope variances are “over-corrected” (<0.8) or “under-corrected” (>1.2).

n | $\mathbf{\rho}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |

Case I: AR(1)-based correction with a known real value of $\rho $ | ||||||||||

30 | 1.00 | 1.01 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.01 | 1.00 | 1.00 |

50 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

100 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

150 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 1.00 | 1.00 | 1.00 |

200 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

Case II: AR(1)-based correction with an estimated value of $\rho $ | ||||||||||

30 | 0.91 | 0.91 | 0.89 | 0.88 | 0.89 | 0.90 | 0.97 | 1.22 | 1.76 | 2.89 |

50 | 0.95 | 0.94 | 0.92 | 0.92 | 0.90 | 0.90 | 0.91 | 0.95 | 1.15 | 2.17 |

100 | 0.97 | 0.96 | 0.96 | 0.96 | 0.95 | 0.95 | 0.93 | 0.91 | 0.89 | 1.19 |

150 | 0.99 | 0.98 | 0.98 | 0.97 | 0.97 | 0.96 | 0.95 | 0.94 | 0.91 | 0.97 |

200 | 0.99 | 0.99 | 0.99 | 0.98 | 0.98 | 0.96 | 0.97 | 0.96 | 0.93 | 0.91 |

Case III: FGN-based correction with a known real value of Hurst coefficient $H$ | ||||||||||

30 | 1.00 | 0.92 | 0.88 | 0.89 | 0.95 | 1.07 | 1.29 | 1.74 | 2.61 | 4.63 |

50 | 1.00 | 0.86 | 0.78 | 0.76 | 0.77 | 0.84 | 1.00 | 1.34 | 2.16 | 4.67 |

100 | 1.00 | 0.79 | 0.66 | 0.59 | 0.56 | 0.59 | 0.66 | 0.86 | 1.41 | 3.64 |

150 | 1.00 | 0.75 | 0.60 | 0.51 | 0.47 | 0.47 | 0.51 | 0.66 | 1.05 | 2.84 |

200 | 1.00 | 0.72 | 0.55 | 0.46 | 0.41 | 0.40 | 0.43 | 0.53 | 0.84 | 2.29 |

**Table 3.**Variance inflation factor of Sen’s slope $V\left(S\right)/{V}_{C}\left(S\right)$ after variance correction for FGN data.

n | $\mathbf{\rho}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0 | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | |

Case I: FGN-based correction with a known real value of Hurst coefficient $H$ | ||||||||||

30 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

50 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

100 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.01 | 1.01 | 1.00 |

150 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

200 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

Case II: FGN-based correction with an estimated value of $H$ | ||||||||||

30 | 0.91 | 1.08 | 1.24 | 1.32 | 1.41 | 1.49 | 1.55 | 1.67 | 1.75 | 1.89 |

50 | 0.90 | 1.08 | 1.16 | 1.24 | 1.26 | 1.32 | 1.37 | 1.47 | 1.56 | 1.67 |

100 | 0.89 | 1.06 | 1.11 | 1.13 | 1.15 | 1.15 | 1.21 | 1.29 | 1.37 | 1.47 |

150 | 0.89 | 1.05 | 1.09 | 1.10 | 1.11 | 1.13 | 1.16 | 1.20 | 1.26 | 1.38 |

200 | 0.89 | 1.05 | 1.07 | 1.08 | 1.08 | 1.10 | 1.12 | 1.16 | 1.21 | 1.30 |

Case III: AR(1)-based correction with a known real value of $\rho $ | ||||||||||

30 | 1.00 | 1.09 | 1.14 | 1.12 | 1.06 | 0.94 | 0.77 | 0.58 | 0.38 | 0.22 |

50 | 1.00 | 1.17 | 1.28 | 1.33 | 1.29 | 1.19 | 1.00 | 0.74 | 0.46 | 0.21 |

100 | 1.00 | 1.27 | 1.51 | 1.68 | 1.77 | 1.71 | 1.51 | 1.16 | 0.71 | 0.27 |

150 | 1.00 | 1.34 | 1.68 | 1.95 | 2.14 | 2.15 | 1.94 | 1.53 | 0.95 | 0.35 |

200 | 1.00 | 1.39 | 1.81 | 2.17 | 2.44 | 2.51 | 2.34 | 1.87 | 1.18 | 0.44 |

Variables | Statistical Features | Data Persistence | Slope Variance | Test Statistic ^{1} | ||||
---|---|---|---|---|---|---|---|---|

Mean $\mathbf{\mu}$ | $\mathbf{Slope}\text{}\mathit{S}/\mathbf{\sigma}$ | Model | Parameter | $\mathbf{Original}\text{}{\mathit{V}}_{0}\left(\mathit{S}\right)$ | $\mathbf{Corrected}\text{}{\mathit{V}}_{\mathit{C}}\left(\mathit{S}\right)$ | $\mathbf{Original}\text{}{\mathit{Z}}_{\mathit{S}}$ | $\mathbf{Corrected}\text{}{\mathit{Z}}_{\mathit{S}\mathit{C}}$ | |

Rainy days | 103 days | 0.008 | AR(1) | $\rho =-0.046$ | 7.9 × 10^{−5} | 7.2 × 10^{−5} | 2.62 ^{++} | 2.74 ^{++} |

Annual total flow | 3.44 Gm^{3} | 0.005 | AR(1) | $\rho =0.380$ | 2.1 × 10^{−6} | 4.5 × 10^{−6} | 3.22 ^{++} | 2.17 ^{+} |

Frost days | 166 days | −0.017 | FGN | $H=0.956$ (0.879)^{b} | 0.2 × 10^{−3} | 0.2 × 10^{−2} (0.9 × 10^{−3}) | −5.18 ^{++} | −1.85 (−2.55 ^{+}) |

^{1}Significant trends at the $5\%$ ($\left|Z\right|>1.96$) level are indicated by “

^{+}”, at the $1\%$ ($\left|Z\right|>2.576$) level by “

^{++}”;

^{2}Values in parentheses are the Hurst coefficients estimated from the detrended series and the corresponding slope variance and trend test statistic.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, W.; Zhu, Y.; Liu, B.; Chen, Y.; Zhao, X.
Innovative Variance Corrected Sen’s Trend Test on Persistent Hydrometeorological Data. *Water* **2019**, *11*, 2119.
https://doi.org/10.3390/w11102119

**AMA Style**

Wang W, Zhu Y, Liu B, Chen Y, Zhao X.
Innovative Variance Corrected Sen’s Trend Test on Persistent Hydrometeorological Data. *Water*. 2019; 11(10):2119.
https://doi.org/10.3390/w11102119

**Chicago/Turabian Style**

Wang, Wenpeng, Yuelong Zhu, Bo Liu, Yuanfang Chen, and Xu Zhao.
2019. "Innovative Variance Corrected Sen’s Trend Test on Persistent Hydrometeorological Data" *Water* 11, no. 10: 2119.
https://doi.org/10.3390/w11102119