3. Research Methods

This study employs four complementary analytical methods to comprehensively characterize rainfall spatiotemporal patterns: (1) Mann-Kendall trend test for significance assessment, (2) Sen's slope estimation for trend magnitude quantification, (3) Sliding window analysis for multi-scale variability characterization, and (4) Hurst exponent analysis for trend persistence evaluation.

3.1 Mann-Kendall (MK) Trend Test

The Mann-Kendall test is a rank-based non-parametric method widely used to detect monotonic trends in hydro-climatic time series. Its primary advantages are its robustness against outliers and its capability to handle non-normally distributed data [4,5].

For a time series $\{x\_{1},x\_{2},...,x\_{n}\}$, the test statistic $S$ is calculated as:

$S=\sum\_{i=1}^{n-1}\sum\_{j=i+1}^{n}sgn(x\_{j}-x\_{i}) (Equation 1)$

where $sgn(θ)$ is the sign function:

$sgn(θ)=\left\{\begin{matrix}+1,&if θ>0\\0,&if θ=0\\-1,&if θ<0\end{matrix}\right.$

For sample sizes $n\geq 8$, the statistic $S$ is approximately normally distributed with mean $E(S)=0$ and variance:$Var(S)=\frac{n(n-1)(2n+5)-\sum\_{p=1}^{g}t\_{p}(t\_{p}-1)(2t\_{p}+5)}{18}$

where $g$ is the number of tied groups and $t\_{p}$ is the number of observations in the $p$-th tied group.

The standardized test statistic $Z$ is computed to assess significance:

$Z=\left\{\begin{matrix}\frac{S-1}{\sqrt{Var(S)}},&if S>0\\0,&if S=0\\\frac{S+1}{\sqrt{Var(S)}},&if S<0\end{matrix}\right. (Equation 2)$

A positive $Z$ indicates an upward trend, while a negative $Z$ indicates a downward trend. The null hypothesis of no trend is rejected at the significance level $α$ if $|Z|>Z\_{1-α/2}$. In this study, significance levels of $α=0.05$ ($|Z|>1.96$) and $α=0.01$ ($|Z|>2.576$) are used.

3.2 Sen's Slope Estimation

Sen's slope estimator [7] is used to estimate the true magnitude of the linear trend. Unlike linear regression, it is insensitive to outliers. For all data pairs $(x\_{i},x\_{j})$ where $j>i$, the slope $Q\_{k}$ is calculated:

$Q\_{k}=\frac{x\_{j}-x\_{i}}{j-i} (Equation 3)$

The Sen's slope $β$ is defined as the median of all $N=n(n-1)/2$ calculated slopes. A positive $β$ signifies an increasing trend (mm/year). The 95% confidence interval for the slope is estimated using a non-parametric procedure based on the normal distribution of the rank statistic.

3.3 Sliding Window Analysis

Table 3a. Sliding Window Configurations

For a window of size $w$, four statistical moments are calculated at each time step $k$. The arithmetic mean ($μ\\_k$) is computed as $μ\\_k=\frac{1}{w}∑\\_i=k^{k+w-1}x\\_i$. The standard deviation ($σ\\_k$) is calculated as $σ\\_k=\sqrt{\frac{1}{w-1}∑\\_i=k^{k+w-1}(x\\_i-μ\\_k)^{2}}$. The coefficient of variation ($CV\\_k$) is derived as $CV\\_k=\frac{σ\\_k}{μ\\_k}$, providing a normalized measure of variability. The trend slope within window ($a\\_k$) represents the instantaneous rate of change within the window.

3.4 Hurst Exponent Analysis

The Hurst exponent ($H$) quantifies the long-term memory or persistence of a time series [8], estimated via Rescaled Range (R/S) analysis. The procedure involves three steps: first, dividing the series into sub-series of length $n$; second, calculating the range of cumulative deviations ($R$) and standard deviation ($S$) for each sub-series; and third, fitting the power law relationship $(R/S)\\_n∝n^{H}$.

The value of $H$ ranges from 0 to 1, with interpretations summarized in Table 3b.

Table 3b. Hurst Exponent Interpretation

3.6 Integration with Digital Twin Water Network Framework

The selection of these statistical methods is not merely for academic characterization but is designed to support the construction of the "Digital Twin Eastern Zhejiang Water Network." Currently, the region is advancing a comprehensive digital reform to build a "Four-Pre" system, which encompasses Forecast, Early Warning, Rehearsal, and Plan. This initiative relies on a robust data infrastructure, including the provincial-wide Integrated Data Resource System (IRS) and the specialized "Data Bottom Board," to digitize physical water conservancy objects.

In this framework, the Mann-Kendall and Sen's slope methods provide the foundational algorithms for the data governance layer. They enable the system to automatically identify long-term climate shifts from the vast accumulation of monitoring data stored in the water conservancy data warehouse. By quantifying trend significance and magnitude, these methods help populate the "Knowledge Platform" with dynamic rules regarding climate change, rather than static historical averages. The Sliding Window Analysis directly supports the "Model Platform" by defining the appropriate temporal scales for hydrological forecasting models. It allows the system to distinguish between short-term tactical scheduling, which requires 3-month scale inputs, and long-term strategic planning, which relies on 12-month scale patterns. Furthermore, the Hurst Exponent serves as a critical parameter for the system's "Intelligent Recognition Models." By quantifying the persistence of historical trends, it provides a mathematical basis for reliability assessments in future scenario rehearsals, enhancing the precision of risk-based joint dispatch decisions. The integration of these mathematical tools transforms raw rainfall data into actionable "knowledge rules," thereby supporting the intelligent generation of spatially balanced water transfer plans and emergency response strategies within the digital twin environment.