Grey Wolf optimization enhanced adaptive decomposition for trend periodic analysis of nonstationary and nonlinear hyrologic series

Abstract

Hydro-meteorological variables are essential drivers of the water cycle, and understanding their dynamics is crucial for effective water resources management and disaster mitigation. Conventional trend analysis methods typically assume sequence normality and independent, lacking flexibility to efficiently analyze non-stationary and nonlinear hydrological series. To overcome these limitations, this study proposes a Grey Wolf Optimization-enhanced Complete Ensemble Empirical Mode Decomposition with Adaptive Noise method for integrated trend-periodic analysis. Validation using synthetic datasets demonstrates superior performance over the traditional methods, achieving accuracy higher than 85% for sequences with complex autocorrelation and heterogeneous distributions, while identifying periodicities undetectable by other methods. Applied to the Yangtze River Basin, the method reveals (1) warming trend in annual mean temperature, (2) spatially heterogeneous precipitation changes i.e., increases in northwest and northeast, decreases in central and southwestern regions. (3) declining runoff trends in the mainstem and most tributaries, with significant periodicities at 2–3-year and 11-year intervals. With the trend-periodic analysis, the runoff extreme can be forecasted. This method achieves an accuracy rate of 79.31%, surpassing the wavelet decomposition method by 20.67% points (58.64%). Extrapolation indicates potential low-value runoff extremes during 2025–2027, suggesting sustained high frequency or intensity of severe droughts over the coming decade.

Introduction

Hydro-meteorological variables are essential drivers of the water cycle^1,2, however, they exhibit significant non-stationarity due to climate change and human activities^3–7. Trend and change detection is the foundation for sustainable water resources management, flood and drought management, hydrological modeling and prediction, and engineering design^8–11.

Trend analysis methods are broadly categorized into parametric and non-parametric approaches^12–15. Parametric methods, such as regression and moving average, assume normal probability distributions and serial independence^16–19. Non-parametric methods, such as the Mann-Kendall(MK) method and innovative trend analysis(ITA), do not require normality probability distribution assumptions^17,20. However, MK assumes that the data are independently and identically distributed^21–25. Modified MK methods, like Mann-Kendall test with trend-free pre-whitening(TFPW-MK) and Modified Mann-Kendall(MMK) Trend Test, can address serial autocorrelation, but their performance depends on sample size and distribution assumptions^4,26. Moreover, the original ITA method can intuitively reveal trend characteristics in data sequences without assuming autocorrelation, normality, or specific sample size requirements. However, it cannot quantitatively assess the significance of trends^27–31. Subsequent improved versions have introduced parametric testing, enabling the quantitative assessment of trend significance^32,33. However, this enhancement comes at the cost of sacrificing the core advantage of the original method—its independence from specific probability distribution types.

The Fast Fourier Transform (FFT), as a classical spectral analysis tool, features high computational efficiency and rapid execution speed. However, its fundamental assumptions require sequences to be stationary and smooth, resulting in poor performance when processing non-stationary and non-smooth sequences. It struggles to capture transient features or local abrupt changes^34–37. In contrast, CWT, leveraging its excellent time-frequency localisation capability, can effectively analyse non-smooth and non-stationary sequences³⁸. Nevertheless, CWT performance heavily depends on the selection of wavelet basis functions, as different wavelets exhibit distinct time-frequency characteristics. This limitation constrains the identifiable frequency range and resolution, potentially causing critical periodic components to be overlooked or misidentified^39,40. Consequently, existing periodicity detection methods still face challenges in adaptability and reliability when confronted with complex and dynamic real-world signals.

Empirical Mode Decomposition (EMD) is an effective analytical method suitable for non-smooth, nonlinear time series⁴¹. This method decomposes the original signal into a series of smooth intrinsic mode functions (IMFs) and a residual term (RES), where the residual typically contains trend information within the sequence. Consequently, EMD has gained widespread application in hydrological time series analysis. However, traditional EMD methods still face challenges such as mode aliasing and reconstruction errors. To overcome these issues, the Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (ICEEMDAN) was proposed by Colominas et al.⁴². This method effectively suppresses modal overlap by adaptively adding finite-order Gaussian white noise and employing ensemble averaging, while significantly reducing reconstruction errors and enhancing decomposition completeness and stability. ICEEMDAN has been progressively applied in non-stationary signal processing domains, demonstrating superior performance compared to traditional EMD and its variants (e.g., EEMD, CEEMDAN). However, the decomposition quality of ICEEMDAN remains susceptible to critical parameters, primarily the number of Gaussian white noise additions and the signal-to-noise ratio settings.

This paper proposes an improved trend-period analysis method named GITPA, which integrates Grey Wolf Optimiser (GWO) with ICEEMDAN. The method enables robust decomposition of non-stationary and nonlinear time series characterised by heterogeneous probability distribution functions (PDFs) and complex autocorrelation structures, thereby overcoming key limitations of traditional approaches. To systematically validate its effectiveness, we generated multiple artificial sequences with controlled variations in key attributes, including time length, trend magnitude, autocorrelation strength, type of probability distribution, and periodic characteristics. The performance of GITPA and its non-optimised baseline version (ITPA) was compared against traditional methods such as MK, ITA, and CWT, demonstrating the superior stability and adaptability of GITPA in decomposition tasks. Furthermore, the proposed method was successfully applied to detect dynamic characteristics of hydrological series in the Yangtze River Basin, providing a reliable basis for predicting future extreme events.

Methods

Figure 1 shows the workflow of the GITPA, which consists of three modules: the GWO-ICEEMDAN decomposition method, the GWO-ICEEMDAN trend analysis(GITA) method, and the GWO-ICEEMDAN periodic analysis(GIPA) method.

Fig. 1.

Flowchart of GWO-ICEEMDAN trend and periodic analysis method.

In the decomposition module, the GWO algorithm is employed to optimize the key parameters of ICEEMDAN—signal-to-noise ratio and average noise order—by minimizing sample entropy. This yields an ICEEMDAN model that enhances the periodic characteristics of decomposed components. This model decomposes the original time series into several IMFs and a RES. The IMF components correspond to sub-components of different frequencies in the data, while the RES, together with some low-frequency IMFs, collectively embody the long-term trend information within the sequence.

In the GITA method, a t-test is first conducted to determine whether the mean of each IMF component significantly deviates from zero. This identifies statistically significant low-frequency IMF components, which are then combined with the RES to construct a trend sequence, from which a trend line is extracted. After sorting this trend sequence, its mean slope serves as the statistical measure of trend significance. Subsequently, a confidence interval for this statistic is generated using the bootstrap method, ultimately yielding a statistically significant trend line along with its corresponding significance assessment results.

In the GIPA method, the FFT is applied to each smoothed IMF component obtained through decomposition to identify its dominant period.

GWO-ICEEMDAN decomposition method

Compared to the traditional method, ICEEMDAN updates the noise addition approach to obtain new IMFs. The improvement of ICEEMDAN greatly suppresses modal aliasing and reconstruction errors of modal decomposition class methods. The specific decomposition procedure of ICEEMDAN is as follows:

(1) Gaussian white noise w⁽ⁱ⁾ is added to the original signal x:

1

(2) Compute and average the local means for each to obtain the first residual component :

2

Where denotes taking the mean sign; is the local mean function.

(3) The 1 st modal component, IMF1 is obtained by subtracting the first residuals from the original sequence:

3

(4) Construct the sequence to obtain the 2nd residual component and compute the 2nd modal component IMF2 :

4

(5) For the kth order residual rk, then it can be obtained by repeating the steps (1) to steps (2):

5

(6) For the kth order component IMFk, the third step can be repeated using the obtained in the previous step to get:

6

Where is the ith constructed signal; x is the original signal; is the standard deviation of the signal’s noise at the ith decomposition; is the first zero-mean unit-variance white noise that was added; and is the computation of the signal’s kth IMF operator.

(7) Return to step (5) and compute until the iteration termination condition is satisfied.

The performance of the ICEEMDAN decomposition method largely depends on two key parameters: signal-to-noise ratio and the number of noise additions. Sample entropy, as a metric for measuring the complexity and regularity of time series, exhibits lower values when the sequence demonstrates stronger regularity and more pronounced periodic components⁴³. The GWO algorithm, proposed by Mirjalili et al., is a meta-heuristic optimization method inspired by the social hierarchy and group hunting behavior of grey wolves⁴⁴. By simulating wolf pack behaviors such as tracking, encircling, and attacking, this algorithm achieves efficient global search. It offers advantages including conceptual simplicity, minimal parameter requirements, ease of implementation, and strong global exploration capabilities. This paper employs the GWO algorithm, with sample entropy minimization as the optimization objective, to perform parameter optimization for ICEEMDAN. This approach effectively enhances its decomposition performance and model adaptability.

GWO-ICEEMDAN trend analysis method

The GWO-ICEEMDAN algorithm decomposes the original data sequence into multiple IMFs components and one residual term, each exhibiting distinct frequency characteristics. Trend information is primarily embedded in the low-frequency IMFs components and the residual term^45,46. Generally, high-frequency IMFs components exhibit approximately symmetric distributions with means close to zero, whereas low-frequency IMFs components often lack symmetry and display means significantly deviating from zero. Although the overall distribution of certain IMFs components may deviate from normality, the central limit theorem ensures that the sampling distribution of their sample means can be approximated as normal when the sample size is sufficiently large (n > 30), thereby satisfying the applicability condition of the t-test^47,48. By performing a t-test to determine whether the sample mean of each IMF component is significantly different from zero, high-frequency and low-frequency IMFs can be effectively distinguished. The identified low-frequency IMFs are then combined with the residual term to construct a trend component that reflects the overall upward or downward movement of the sequence.

To further assess the significance of trends, this study applies sequence ordering to trend series to mitigate the impact of outliers on results. Based on this, the mean slope of the new sequence is calculated as a trend statistic. This statistic exhibits strong robustness and is independent of sequence length. To infer the significance of this statistic, the block bootstrap method is introduced⁴⁹. This method does not rely on the assumption of a Gaussian PDF and preserves the autocorrelation structure within the time series. Through a segmented shifting resampling strategy, the block bootstrap effectively identifies significant trend intervals while suppressing the overestimation of significance levels caused by autocorrelation^50–52. Finally, a confidence interval for the trend statistic is constructed based on bootstrap samples, which is used to determine the statistical significance of the trend.

To further analyze the trend, the following steps were performed:

(1) Sort the trend sequence in descending order by numerical value to obtain an ordered sequence. Use sequential traversal to assign a dense ranking to each element in the ordered sequence. Subsequently, rearrange the assigned ranking values according to the original chronological order to form a ranking sequence consistent with the original sequence.

(2) Let P denote the average gradient of the ranking sequence as a trend statistic, calculated as follows:

7

Where, represents a sequence of data values observed at time points .

(3) This study employs block bootstrap method to construct confidence intervals, following these specific steps. First, the original sequence is converted into an ordinal sequence to eliminate the dimension effect; Subsequently, the slider length is empirically determined based on the sequence length . The sequence is then divided into consecutive, overlapping data blocks using a sliding window to preserve its time-dependent structure. Building upon this, a large number of bootstrap samples are generated by repeatedly sampling these blocks with replacement, and trend statistics are computed for each sample. Finally, corresponding percentiles are extracted from the distribution of these statistics to construct the 95% confidence interval.

GWO-ICEEMDAN periodic analysis method

After decomposing a non-smooth time series into several IMFs, the FFT method can be employed further to analyze the frequency domain characteristics of each IMF. The specific process is as follows: First, perform an FFT calculation on each IMF component to construct the frequency axis and obtain its power spectrum. Identify the dominant frequency by locating the peak positions within the power spectrum. Subsequently, calculate the dominant period corresponding to that IMF based on the inverse relationship between frequency and period. The specific calculation process for the period is as follows:

(1) Perform the FFT calculation.

Perform an FFT on the IMF signal to obtain its discrete spectrum :

8

Where and is a complex number representing the complex amplitude at frequency component k.

(2) Construct the frequency axis and compute the power spectrum.

The frequency corresponding to the frequency index k in the FFT output is calculated using Eq. (8):

9

According to the Nyquist sampling theorem, it is only necessary to analyze the spectral results within the range from to . The latter half is conjugate symmetric and does not provide new frequency information.

To quantify the intensity of each frequency component, calculate the power spectral density (PSD) :

10

Where denotes the magnitude of the complex amplitude . The power spectrum reflects the distribution of signal power across different frequency points.

(3) Identify the dominant frequency.

In the power spectrum locate the frequency index corresponding to the maximum value (excluding the DC component at k = 0).

11

The frequency corresponding to this peak point is the dominant frequency in the signal:

12

(4) Calculate the Dominant Period.

Based on the inverse relationship between frequency and period, the primary period of the signal is calculated from the dominant frequency :

13

Where N is the total number of data in the sequence, is the sampling interval.

Code availability

The custom code supporting the findings of this study has been made publicly available. The source code is deposited in the GitHub repository (https://github.com/zhang1-feng/Trend-and-period) and is released under the MIT license, which permits unrestricted use. The code execution depends on the MATLAB R2022b environment.

Method validation

Validation of GWO-ICEEMDAN trend analysis method

To validate the effectiveness of the GITPA, proposed in this paper, multiple sets of time series are generated with a first-order autoregressive random models (as shown in Eq. (13) were generated using Monte Carlo simulations.

14

Where is the mean value (), is the standard deviation (), is the first-order correlation coefficient, and is the random perturbation following a certain distribution.

A sequence with a linear trend of a specified slope d is generated according to equation . The d is the slope of the trend.

In this study, 700 sequences are generated with varying PDF (Gaussian, Gamma, Exponential), different lengths (25, 50, 100, 200, 500), a range of trend slopes (± 0.001, ± 0.0015, ± 0.005, ± 0.01, ± 0.05, ± 0.1, ± 0.2, ± 0.3, ± 0.4, ± 0.5) and four autocorrelations (−0.4, −0.3, −0.2, −0.1).

1. Validation of GITPA Across Different PDF Sequences

The effectiveness of the proposed GITPA is validated under different PDF, lengths, and the slopes of the sequences compared with ITPA MK and ITA, as shown in Fig. 2.

Fig. 2.

Trend detection results using the GITPA ITPA MK and ITA methods.

Figure 2 displays the results of trend analysis using the GITPA, ITPA, MK, and ITA methods. In each subfigure, the vertical axis represents sequence length, while the horizontal axis denotes trend slope. The terms “ascending” and “descending” in the subfigure captions refer to analyses of upward and downward trends, respectively. Colors in the figure indicate whether a trend was significantly identified: red signifies a significant trend, while blue indicates no significant trend was detected. Comparing these results allows evaluation of the detection performance differences among the methods across varying trend directions and sequence lengths.

Figure 2 results indicate that the GITPA, ITPA and MK methods exhibit low sensitivity to PDF types, demonstrating high robustness. In contrast, the ITA method exhibits pronounced PDF dependency. Specifically, within Gaussian-PDF sequences, ITA can identify significant trends with slopes as low as 0.001 even for sequence lengths below 150. Under non-Gaussian-PDF conditions, however, this method can only detect trends with steeper slopes and more extended sequences. This finding aligns with the conclusions reported by Serinaldi³³.

Furthermore, the detection performance of all four methods improves with increasing series length and trend slope. Under conditions of short series lengths or weak trend strengths, both GITPA and ITPA maintain strong detection capabilities, with their significant recognition areas (marked in red in the figure) being substantially larger than those of the MK and ITA methods, while GITPA slightly outperforms ITPA. Specifically, at a trend slope of 0.001, GITPA can detect trends in series with a minimum length of 150, whereas ITPA requires a series length of at least 200, and the MK method demands a length exceeding 300. Moreover, GITPA accurately identifies trends with slopes ≥ 0.0015 and series lengths ≥ 25, while ITPA and MK require slopes higher than 0.02 and significantly above 0.05, respectively. These results demonstrate that GITPA exhibits higher sensitivity in detecting weak trends and short series.

To quantify the identification performance of the different methods, this study adopted the accuracy as the evaluation metric, with the results shown in Fig. 3. Overall, the GITPA method (red curve) performed the best, with an accuracy consistently exceeding 85%, significantly outperforming the other methods. The ITPA method (blue curve) slightly lagged behind GITPA but still maintained an accuracy above 80%, while the MK method (green curve) remained stable at over 70% accuracy throughout. It is worth noting that GITPA, ITPA, and MK all demonstrated good stability across different PDFs of the input sequences. In contrast, the ITA method (black curve) showed notable sensitivity to distribution type, with its accuracy varying significantly depending on the PDF, reaching as high as 80% in some cases but dropping to just 55% in others.

Fig. 3.

Significant rate changes associated with the PDF of the sequence.

• (2) Validation of GITPA Across Different autocorrelations Sequences

The effectiveness of the proposed GITA is validated under different autocorrelations, lengths, and the slopes of the sequences compared with ITPA, MK and ITA, as shown in Fig. 4.

Fig. 4.

Trend analysis results of different autocorrelation sequences using the GITA MK and ITA methods.

Figure 4 displays the trend identification results of the methods under autocorrelation coefficients of −0.4, −0.3, −0.2, and − 0.1 (arranged from left to right). Since the GITPA, ITPA, and ITA methods do not presuppose independence in the time series, the significant regions they identify show only minor variations across different autocorrelation scenarios. In contrast, the MK method, which relies on the assumption of data independence, exhibits more noticeable fluctuations in its significant regions as the autocorrelation strength changes. In terms of detection capability, both GITPA and ITA identify notably larger significant regions (red areas in the figure) and demonstrate more stable trend detection performance than MK and ITA across all autocorrelation conditions, with GITPA slightly outperforming ITA. It is worth noting that when the series length exceeds 100, GITPA remains effective in detecting trends even with a slope as low as 0.001, whereas ITA requires a minimum series length of 150. For shorter series with a length of 25 and a trend slope no less than 0.005, both GITPA and ITA achieve correct detection. In summary, GITPA exhibits superior overall trend detection performance compared to ITPA, MK, and ITA in the presence of autocorrelated series.

Figure 5 compares the accuracy rates of the GITPA, ITPA, MK, and ITA methods across time series with varying autocorrelation coefficients. Overall, the GITPA method (red curve) demonstrated the best performance, with its accuracy consistently exceeding 85%, significantly outperforming the other methods. The ITPA method (blue curve) performed slightly below GITPA, showing minor fluctuations around 85% accuracy. In contrast, the ITA method (black curve) exhibited the weakest performance, failing to reach 85% accuracy under any condition. Notably, the accuracy of GITPA, ITPA, and ITA remained stable across different autocorrelation coefficients without significant fluctuations. On the other hand, the MK method (green curve) showed strong dependence on autocorrelation strength: its accuracy dropped sharply from approximately 90% to 65% as the autocorrelation coefficient increased. These results indicate that the GITPA method achieves not only the highest accuracy but also the strongest stability when handling autocorrelated sequences.

Fig. 5.

Variation of accuracy with autocorrelation under different methods.

Validation of GWO-ICEEMDAN periodic analysis method

To verify the feasibility of the GWO-ICEEMDAN Periodic analysis(GIPA, the trend analysis module within GITPA) method, the first-order autoregressive stochastic process series were generated using Monte Carlo simulation based on Eq. (15).

15

Where denotes a sequence consisting of trend terms and the random perturbation following different PDF, A and B represent the periods in the sequence.

A total of 72 test sequences were generated in this study, with parameter settings covering different probability density functions (Gaussian, Gamma, and Exponential distributions), sequence lengths (500, 1000, 2000, 2500), and periodic components (100, 50, 20, 10, 10 + 100, 50 + 100). The period detection results are shown in Fig. 6, where the vertical axis represents the number of detected periods, and the horizontal axis indicates the type of probability density function. The subplot titles “GITPA,” “ITPA,” and “CWT” correspond to the period detection methods employed. The legend differentiates the detection results by color: green indicates correct detection, blue denotes partial detection (i.e., not all periods present in the sequence were identified), and red signifies incorrect detection.

Fig. 6.

Recognition results of the GIPA method and CWT method for different periodic sequences.

The titles of each subplot in Fig. 6 indicate the length of the corresponding test sequences. The horizontal axis represents the PDF of the sequences, while the vertical axis denotes the types of periodic components contained in the sequences. The left side of the figure displays the detection results of the GITPA method, the middle-left section shows the results of the ITPA method, and the right side presents the results of the CWT method. As illustrated, GITPA accurately identifies all periodic components in sequences with different PDFs and lengths. In comparison, ITPA fails to fully detect all periodic components in shorter sequences. This issue may stem from mode mixing that occurs during the decomposition of short sequences by ICEEMDAN. The CWT method is only effective in identifying sequences with smaller period values, frequently producing incorrect detections when the period exceeds 100. This discrepancy may originate from the inherent characteristics of the wavelet basis function used in the CWT method, which limit its ability to detect larger periodic components. Once the period exceeds the effective detection range of the wavelet function, misclassification occurs. The experimental results demonstrate that GITPA exhibits higher accuracy and robustness in periodic identification, making it a superior tool for periodic analysis compared to wavelet transform, while also validating the effectiveness of GWO parameter optimization.

In summary, the GITPA method demonstrates excellent robustness and broad adaptability when processing sequences with different probability density functions and autocorrelation structures. For complex sequences with more pronounced periodic characteristics, the GITPA method can effectively and accurately extract the periodic information contained within, significantly enhancing its potential for application in complex time series analysis. Experimental results show that the proposed GITPA method possesses high effectiveness and practical utility, providing a reliable tool for time series analysis in related fields.

Application in the yangtze river basin

The Yangtze River Basin spans approximately 3,219 km from east to west and extends about 966 km from north to south, with a total area of 1.8 million square kilometers. This basin contains 36% of China’s total water resources, supplies drinking water to nearly one-third of the country’s population, and contributes about 40% of the national gross domestic product (GDP), playing a crucial role in supporting China’s economic development and ecological sustainability. However, under the influence of climate change, hydro-meteorological variables within the basin exhibit significant non-stationary characteristics, making accurate identification of their cycles and trends essential for water resource management. There are notable differences in natural and climatic conditions across the basin’s tributaries: the upper reaches of the Jinsha River are characterized by a plateau mountain climate, the Yalong River basin shows transitional features toward a subtropical monsoon climate, while the middle and lower reaches are dominated by a typical subtropical monsoon climate. In addition, the topography transitions from plateaus, basins, and hills in the upper and middle reaches to plains in the lower reaches. These factors collectively lead to significant spatial heterogeneity in the PDF and autocorrelation properties of hydro-meteorological variables across sub-basins, posing major challenges for related research. In previous work, the GITPA method has been validated using artificially generated sequences, demonstrating its applicability to series with different distributions and autocorrelation structures. Based on this, the present study applies the GITPA method to the Yangtze River Basin to systematically reveal the variation patterns of different hydro-meteorological variables in each sub-basin.

This study utilizes a 44-year time series dataset spanning from 1979 to 2022. The natural runoff data were provided by the Changjiang Water Resources Commission, covering nine control hydrological stations along the main stream and major tributaries of the Yangtze River Basin: Panzhihua, Tongzilin, Gaochang, Beiping, Wulong, Chenglingji, Huangzhuang, Hukou, and Datong. This dataset excludes the direct influences of human activities such as reservoir storage, irrigation, and artificial water withdrawals, thus accurately reflecting natural runoff conditions. ENSO and IOBW data were obtained from the National Climate Center of the China Meteorological Administration. Sunspot number data were sourced from the World Data Center for the Sunspot Index and Long-term Solar Observations (WDC-SILSO), Royal Observatory of Belgium, Brussels (DOI: 10.24414/qnza-ac80). Meteorological data, including annual precipitation and mean annual temperature, were obtained from the China Meteorological Science Data Sharing Service Network, comprising observations from 147 meteorological stations within the basin. The spatial distribution of all meteorological stations is shown in Fig. 7.

Fig. 7.

Distribution of hydrological stations and meteorological stations. (the map was generated with data available from the Chinese Geospatial Data Cloud using ArcMap 10.8, URL: https://www.esri.com/zh-cn/arcgis/products/index.).

This study selected natural annual runoff data from the Panzhihua, Tongzilin, Gaochang, Beipei, Wulong, Chenglingji, Huangzhuang, and Hukou hydrological stations to represent the natural annual runoff sequences of the Jinsha River, Yalong River, Min River, Jialing River, Wujiang River, Dongting Lake, Han River, and Poyang Lake, respectively. The natural runoff sequence from Datong Hydrological Station was used to represent the overall natural annual runoff volume of the Yangtze River basin. For meteorological data, based on annual precipitation and average annual temperature observations from meteorological stations within the basin, the Thiessen polygon method was employed using the ArcGIS platform to calculate the area-averaged annual precipitation and area-averaged annual temperature for each tributary’s catchment area. Finally, the resulting area-averaged annual precipitation series and area-averaged annual temperature series were used as the corresponding annual precipitation and annual temperature sequences for each tributary.

To characterize the fundamental properties of meteorological and hydrological sequences across various river basins, this study analyzes their statistical features based on annual runoff data. The indicators employed include mean annual runoff (unit: 10⁶ m³), coefficient of variation (Cv), and skewness coefficient (Cs). Additionally, the Hurst exponent is utilized to assess the long-term persistence in runoff dynamics, the KPSS test is applied to examine sequence stationarity, and the Ljung-Box Q test is employed to analyze autocorrelation. Detailed calculation results for each indicator are presented in Table 1.

Table 1.

Basic statistical information on runoff series for each watershed in the study area.

Watershed	Mean Runoff (106m3)	Cv	Cs	Hurst	KPSS	ADF	Ljung-Box Q
							Q	p
Jinsha River	207,480	0.19	0.80	0.58	1	1	12.62	0.01
Yalong River	95,610	0.18	0.73	0.63	1	1	5.70	0.03
Min River	125,606	0.19	0.90	0.53	1	1	2.87	0.04
Jialing River	127,008	0.23	0.86	0.85	1	1	14.02	0.03
Wu River	94,468	0.18	0.60	0.99	1	1	34.64	0.00
Dongting Lake	89,045	0.27	0.47	0.99	1	1	32.04	0.00
Han River	306,484	0.23	0.35	0.99	1	1	61.07	0.00
Poyang Lake	187,863	0.22	0.38	0.97	1	1	29.41	0.00
Yangtze River	1,456,819	0.15	0.69	0.86	1	1	70.42	0.00

As shown in Table 1, the results indicate that the skewness coefficients of all runoff sequences from the upper Yangtze tributaries exceed 0.5, suggesting a positively skewed distribution that deviates from normality. Furthermore, the Hurst exponents for all sequences are greater than 0.5, confirming significant long-range persistence within the runoff data. Both the KPSS test and ADF tests yielded statistically significant results (statistics both equal to 1), leading to the rejection of their respective null hypotheses and confirming the non-stationarity of the selected runoff sequences. Additionally, the Ljung-Box Q test results across all basins show large Q statistics with associated p-values consistently less than 0.05, indicating strong autocorrelation in the sequences.

Trend analysis of hydrological in the Yangtze river basin

This study employed the GITA method to examine the runoff, annual precipitation, and annual mean temperature time series for the aforementioned river basins. Trend curves were derived for each series, and their statistical P-values were calculated to determine the significance of the observed changes. The trend line is shown in Fig. 8, and the calculated P-values are presented in Table 2.

Fig. 8.

Graphs of temperature, rainfall, and runoff trends in different sub-basins of Yangtze River.

Table 2.

P-values of changes in mean annual temperature, rainfall, and runoff in significant tributaries of the Yangtze River Basin(The P-value represents the significance level of the trend in the sequence.).

Watershed	P
	Temperature	Rainfall	Runoff
Jinsha River	0.88	−0.84	−0.93
Yalong River	0.87	0.26	0.16
Min River	0.90	0.41	0.38
Jialing River	0.91	−0.63	−0.74
Wu River	0.95	−0.5	−0.80
Dongting Lake	−0.75	−0.45	−0.57
Han River	0.89	−0.75	−0.80
Poyang Lake	0.94	−0.32	−0.38
Yangtze River			−0.75

Figure 8 illustrates the trends in precipitation, temperature, and runoff for various river basins between 1979 and 2022, while Table 2 lists the distribution of significance P-values for these trends. During this period, all basins exhibited a consistent upward trend in annual mean temperature, with a warming rate of approximately 0.03 °C per year. This result aligns with the temperature trend identified by Zengchuan Dong et al.⁵³ using the ITA method, which shows that runoff and precipitation generally decreased across most basins, except for the Minjiang and Yalongjiang basins, which exhibited increasing trends. The Dongting Lake basin experienced the most significant reduction in runoff, decreasing by 22,730 m³ per year. This finding aligns with Li Shanshan’s MK-based observation of reduced Dongting Lake runoff and Wenxian Guo’s conclusion on declining Wujiang River runoff^54,55. Notably, both precipitation and runoff in the Minjiang and Yalongjiang basins exhibit significant fluctuations, with runoff variations being more minor than precipitation fluctuations. This phenomenon may be closely related to the warming climate on the Tibetan Plateau. Research by Yongbin et al.⁵⁶ emphasizes the regulatory role of climate change on runoff in this region, noting that while annual precipitation has increased over the past half-century, its growth rate has been significantly lower than that of temperature rise, consistent with the observations in this study. Spatially, precipitation has increased in the northwest and east (primarily covering the Yalong and Min rivers). At the same time, it has generally decreased in the central and southwest regions (mainly including the Jialing, Fu, Han rivers, and the Dongting Lake basin). This spatial distribution pattern is consistent with Pengyang’s research⁵⁷.

Figure 9 illustrates the trend in runoff changes along the main stem of the Yangtze River based on data from the Datong hydrological station. Results indicate an overall fluctuating decline in runoff, with the evolution process divided into two phases: between 1979 and 1999, runoff remained at high levels with a slight increase; since 1995, runoff volume began a sharp decline that persisted until around 2020, after which it has fluctuated at consistently low levels. Concurrently, drought events occurred frequently in the basin, with severe droughts in 2012, 2014, 2019, and 2022, further highlighting the potential link between reduced runoff and extreme hydrological events.

Fig. 9.

Annual runoff trend of the main stream of the yangtze river.

Periodic analysis of runoff in the yangtze river basin

This study employed the GIPA model to analyze annual runoff sequences of major tributaries in the Yangtze River basin, identifying their significant periodic characteristics. The specific results are presented in Table 3.

Table 3.

Annual periodicity of runoff in the mainstream and tributaries of the yangtze river basin.

Watershed	Annual runoff
	Short period	Medium period	Long period
Jinsha River	2.44	11	44
Yalong River	2.75	11	44
Min River	2.75	11	22
Jialing River	3.14	11	22
Wu River	2.10	11	22
Dongting Lake	2.10	11	44
Han River	2.31	11	44
Poyang Lake	2.44	11	44
Yangtze River	2.31	11	44

The periodic characteristics of runoff are shown in Table 3. Since tributary runoff is constrained by precipitation, while mainstem runoff is influenced by tributary inflows, runoff and precipitation exhibit good consistency in their periodic variations. Analysis results indicate that runoff variations primarily exhibit three types of periodic characteristics: a short-term period of 2–3 years, a medium-term period of approximately 11 years, and a long-term period reaching up to 22–44 years. Yang Ruting’s analysis of upper Yangtze River runoff using continuous wavelet transform also identified typical periods of 2–3 years and approximately 11 years, consistent with the findings of this study⁵⁸.

The periodic characteristics of runoff, as shown in Table 3, can be categorized into three types: short-term period of 2–3 years, medium-term period of approximately 11 years, and long-term period of approximately 44 years. Among these, the 2–3-year period aligns closely with the El Niño-Southern Oscillation (ENSO) cycle, which exerts significant influence on interannual climate variability^59,60. The Asian summer monsoon system primarily regulates the interannual variation in China’s summer temperatures. Extensive research indicates that interannual anomalies significantly influence the Asian summer monsoon in Indian Ocean sea surface temperatures, a phenomenon also known as the Indian Ocean Basin Coherent Mode (IOBM)^61,62. Therefore, the 2–3-year period may correspond to the influence of the IOBM, while the 8–11-year and 22–44-year decadal variations may be associated with the Schwabe period and the Hale solar activity period, respectively^63–65.

To clarify the influencing mechanisms of different periods, this study employed the Spearman correlation coefficient to analyse the relationships between various periods and their key potential influencing factors. The results are presented in Table 4. As shown in the table, both ENSO and IOBW exert influence on the short period of natural runoff in all sub-basins of the Yangtze River Basin. From the upstream to the downstream areas, the impact of ENSO on short period shows an increasing trend, whereas the influence of IOBW gradually weakens. For the medium- and Long period, Sunspot Number(SSN) are identified as the predominant influencing factor. These results robustly validate the discussions on the influencing factors for the various periods presented in the preceding section.

Table 4.

Coefficient of correlation between impact factor(ENSO, IOBW and SSN) and regional hyrologic series period.

Short period	ENSO	IOBW	Medium period	SSN	Long period	SSN
Jinsha River	0.43	0.65	Jinsha River	0.66	Jinsha River	0.54
Yalong River	0.48	0.44	Yalong River	0.63	Yalong River	0.51
Min River	0.55	0.42	Min River	0.61	Min River	0.55
Jialing River	0.57	0.43	Jialing River	0.59	Jialing River	0.52
Wu River	0.61	0.32	Wu River	0.66	Wu River	0.56
Dongting Lake	0.67	0.33	Dongting Lake	0.64	Dongting Lake	0.54
Han River	0.65	0.28	Han River	0.67	Han River	0.57
Poyang Lake	0.66	0.22	Poyang Lake	0.61	Poyang Lake	0.51
Yangtze River	0.63	0.42	Yangtze River	0.62	Yangtze River	0.53

Prediction of flood and drought conditions in the Yangtze river basin

This study first employs the GITPA method to analyze the underlying patterns and variation characteristics of the main stream runoff series in the Yangtze River Basin, and then applies the identified patterns to qualitative extrapolation predictions of runoff extreme points (such as high- and low-flow periods). A 44-year runoff series from the Datong Hydrological Station in the Yangtze River Basin was selected as the research subject, and the sliding window method was employed to construct the analysis series. Systematic analysis indicates the presence of three periodic components in the series: a short-term period of 2–3 years, a medium-term period of 11 years, and a long-term period of 44 years, with the 44-year period and the residual term collectively forming the long-term background of the series evolution. During the parameter setting process, after comprehensive consideration, the prediction length was determined to be 5 years, as this duration effectively captures the extreme value characteristics of short- and medium-term periods. At the same time, to meet the basic requirement for sample length in hydrological series analysis (typically no less than 30 years), the analysis window was set to 32 years. Based on this, a total of eight test series were constructed: the first series covers data from 1979 to 2010, the second series (1980–2011) was obtained by sliding the window forward by one year, and so on, until the final series (extended to 2022) was formed. Each series uses the first 32 years of data as the training period. After extracting periodic and trend features using the GWO-ICEEMDAN method, these features were applied to extrapolate runoff changes for the next 5 years. Taking Series 1 as an example, the 32-year dataset from 1979 to 2010 was analyzed. After identifying the variation patterns, each component was extrapolated for 5 years, and the reconstruction process yielded predicted values for flood and drought periods from 2011 to 2015. The specific variation patterns and extrapolation results are shown in Fig. 10.

Fig. 10.

Calculation results based on the first test sequence (1979–2015).

Figure 10 presents the decomposition and prediction results of the runoff sequence: Subfigure (a) shows the trend component sequence. At the same time, subfigures (b) and (c) display the fluctuation curves for IMF1 and IMF2 components, respectively. Subfigure (d) depicts the overall variation curve reconstructed from these three components. The horizontal axis of each subfigure represents the year, while the vertical axis shows the amplitude of fluctuation. Solid lines denote observed values, and dashed lines indicate extrapolated predictions. Subfigure (d) demonstrates that the reconstructed curve effectively identifies extreme points in runoff variation, reflecting the evolving pattern of runoff abundance and scarcity.

To validate the effectiveness of the proposed method, the prediction results of the GITPA method were compared with those of the WPD method under the same testing conditions. As shown in Fig. 11, statistical results indicate that the prediction accuracy of the GITPA method for identifying wet and dry runoff conditions reaches 79.31%, whereas that of the WPD method is 58.64%. The comparison demonstrates that the GITPA method achieves higher accuracy and reliability in recognizing wet and dry runoff characteristics. Furthermore, the performance of the two prediction models was evaluated using the root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R²). The RMSE, MAE, and R² values for GITPA are 755.60 10⁶m³, 582.45 10⁶m³, and 0.52, respectively, while those for WPD are 1169.28 10⁶m³, 1129.38 10⁶m³, and 0.33. Based on these three metrics, the GITPA method outperforms the WPD method in all aspects, demonstrating that GITPA-based extrapolation prediction for runoff wet-dry conditions is an effective approach.

Fig. 11.

Performance evaluation results of GITPA and WPD methods for flood and drought prediction.

Based on the aforementioned research, this paper utilises runoff data from the Datong Hydrological Station spanning 1979–2022. It applies the GITPA method to forecast the wet and dry conditions of runoff in the Yangtze River Basin through 2030. The forecasting process and results are illustrated in Fig. 12.

Fig. 12.

Prediction of annual flood and drought variations in the yangtze river mainstream based on the GITPA method.

Based on the forecast results in Fig. 12, the Yangtze River Basin faces a high probability of experiencing low-flow events between 2026 and 2028, indicating a significant drought risk. It is recommended to strengthen water resource management and allocation during this period and develop drought countermeasures in advance.

Discussion

In the comparison of trend analysis methods, the GITPA method demonstrates exceptional identification performance, consistently achieving accuracy rates above 85%, significantly outperforming the other methods. Both GITPA, ITPA, and MK methods maintain stable accuracy under different probability density function conditions, whereas the ITA method shows significant PDF sensitivity, with its accuracy fluctuating substantially between 80% and 55% depending on the distribution type, as it relies on sequence distribution characteristics when determining trend significance^32,33. In autocorrelated sequences, the significant regions identified by GITPA, ITPA, and ITA methods show only minor differences, and all three maintain high accuracy across different autocorrelation coefficients, demonstrating good robustness. In contrast, the accuracy of the MK method decreases significantly from 90% to 65% as the autocorrelation coefficient increases, due to the failure of its independence assumption in the presence of autocorrelation^21–25. The ITPA and GITPA methods integrate Bootstrap technology and adopt a segment-shifted resampling strategy to preserve the autocorrelation structure of the time series. This approach does not assume a normal distribution and helps mitigate the overestimation of significance levels caused by autocorrelation^50–52. Furthermore, GITPA introduces the Grey Wolf Optimizer for parameter optimization, further enhancing identification accuracy and making it the top-performing method in most scenarios.

For periodic identification, this study further compared the GITPA method with the ITPA method and CWT. The results demonstrate that GITPA accurately identifies all periodic components in sequences with different PDFs and lengths. In contrast, ITPA fails to detect all periodic components in shorter sequences, which may be attributed to mode mixing during the decomposition of short sequences by ICEEMDAN. The CWT method is only effective in identifying sequences with smaller periods and tends to produce misjudgments when the period exceeds 100. This limitation stems from the inherent constraints of the wavelet basis function used in CWT, as its effective detection range is limited, making it difficult to capture large periodic components. In summary, GITPA demonstrates higher accuracy and robustness in periodic identification and, compared to CWT, demonstrates superior performance as a periodic analysis tool.

This study employed the GITPA method to analyze meteorological and hydrological variability characteristics in the Yangtze River Basin. Between 1979 and 2022, all sub-basins exhibited a consistent upward trend in annual mean temperature, with a warming rate of approximately 0.03 °C per year. This result aligns with the temperature change trend identified by Zengchuan Dong et al.⁵³. using the ITA method. Regarding runoff, precipitation generally decreased across most basins, except for the Minjiang and Yalongjiang basins, which exhibited increasing trends. The Dongting Lake basin experienced the most significant reduction in runoff, amounting to 22,730 m³/year. This finding aligns with the conclusions drawn by Li Shanshan⁵⁴ and Wenxian Guo⁵⁵, who independently used the MK method to indicate runoff declines in the Dongting Lake and Wujiang basins. Notably, both precipitation and runoff in the Minjiang and Yalongjiang basins exhibit significant fluctuations, with runoff variability being smaller than that of precipitation. This phenomenon may be closely related to the warming climate on the Tibetan Plateau⁵⁶. Spatially, precipitation trends show increases in the northwest and eastern regions (e.g., Yalong and Min rivers), while decreases prevail in the central and southwestern areas (e.g., Jialing, Han rivers, and Dongting Lake basin), consistent with previous studies⁵⁷.

Regarding the periodic characteristics of runoff in the Yangtze River Basin, this study identified three significant periods: a short-term period of 2–3 years, a medium-term period of approximately 11 years, and a long-term period of 22–44 years. Specifically, the 2–3 year short-term period aligns closely with the interannual climate variability driven by the ENSO phenomenon and is also likely influenced by the IOBM^57–60. The decadal-scale variations of 8–11 years and 22–44 years may be associated with the modulation effects of solar activities, such as the Schwabe cycle and Hale cycle, on the climate system^61–63. To validate the influencing factors of these periods, this study calculated the correlations between the short-term period and ENSO as well as IOBM, and between the medium- and long-term periods and sunspot activity. The results indicate that the short-term natural runoff periods in all sub-basins of the Yangtze River Basin are generally influenced by both ENSO and IOBW. From the upstream to the downstream areas, the impact of ENSO gradually strengthens, while that of IOBW weakens accordingly. In contrast, the medium- and long-term periods are primarily regulated by sunspot activity. Compared with the study by Yang Ruting, which employed CWT and identified periods of 2–3 years and approximately 11 years but failed to detect the 22–44 year long-term signal, this omission likely stems from the limitations in the detection range of the wavelet basis function used in CWT⁵⁸. This further underscores the advantage of the method applied in this study for identifying periodic characteristics across broader temporal scales.

Based on the variation patterns extracted from the runoff series in the Yangtze River Basin, this study further applies an extrapolation method to predict high-flow and low-flow conditions. To objectively evaluate the prediction performance, a systematic comparison was made between the proposed GITPA method and the widely used WPD method. Statistical results show that the prediction accuracy of the GITPA method reaches 79.31%, significantly higher than the 58.64% achieved by the WPD method. In terms of error and fitting performance, the GITPA method also comprehensively outperforms the WPD method, with RMSE, MAE, and R² values of 755.60 10⁶m³, 582.45 10⁶m³, and 0.52, respectively, compared to 1169.28 10⁶m³, 1129.38 10⁶m³, and 0.33 for the WPD method. This performance difference may stem from the high dependency of the WPD method on the wavelet basis function—the choice of basis function directly affects the decomposition effect and prediction accuracy, thereby limiting its adaptability and stability under different hydrological conditions. In contrast, the GITPA method does not rely on the selection of a basis function, thus exhibiting stronger robustness.

In this study, trend extraction is based on the integration of low-frequency IMF components and RES obtained through GWO-ICEEMDAN decomposition. This approach effectively highlights trend characteristics within the sequence, enhancing the stability of trend identification. However, given the strong non-stationarity often present in hydrological sequences, the current trend line obtained may not always represent the globally optimal solution. Future research will focus on optimising trend extraction strategies, including improving modal selection criteria and establishing adaptive reconstruction mechanisms, to further enhance the accuracy and applicability of periodical analysis methods within the GWO-ICEEMDAN framework.

Conclusion

To address the challenges posed by the autocorrelation, non-stationarity, and probabilistic PDF characteristics of hydrological time series that interfere with traditional trend analysis methods, coupled with the limitations of conventional periodical analysis techniques like FFT when handling non-stationary signals, this study combines GWO with ICEEMDAN to propose a novel Trend-Periodical Analysis method (GITPA) and systematically validates its effectiveness.

This method first employs GWO-ICEEMDAN to decompose the original sequence into multiple IMFs and RES, thereby achieving separation of signals at different frequencies. Subsequently, FFT analysis is performed on each IMF component to identify its significant periodicity. Concurrently, the low-frequency IMF and RES are integrated to construct the trend component. By calculating statistical significance metrics and employing the Bootstrap method to establish confidence intervals, the significance of the trend is objectively evaluated.

To comprehensively evaluate the overall performance of the GITPA method, this study designed a systematic numerical simulation experiment. The experiments were conducted on hydrological series with different probability density functions and autocorrelation levels, comparing GITPA with ITPA, traditional trend analysis methods (such as the MK and ITA methods), and continuous wavelet transform. The results indicate that in terms of trend identification, GITPA performed slightly better than ITPA and significantly outperformed the MK and ITA methods. In periodic component detection, GITPA demonstrated greater flexibility and adaptability than both ITPA and CWT, effectively identifying multi-time-scale periodic components within the series. In conclusion, when handling complex hydrological series, the GITPA method not only exhibits superior applicability and reliability but also demonstrates considerable potential for practical application.

This study applied the proposed method to analyse temperature, precipitation, and runoff trends and periodicity in the Yangtze River basin. Results indicate that from 1979 to 2022, the main stem of the Yangtze River and most tributary basins exhibited significant temperature increases, while precipitation and runoff showed overall decreasing trends. Runoff variations exhibited multi-timescale periodic characteristics: short periodical of 2–3 years, medium periodical of approximately 11 years, and long periodical ranging from 22 to 44 years.

This study utilized 44 years of runoff data from the Datong Station in the Yangtze River Basin to construct a test set. The GITPA method and the WPD method were used to extract variation patterns, respectively, and their predictive capabilities for dry-wet cycles were evaluated through extrapolation forecasting. The test results show that the GITPA method achieved an accuracy of 79.31% in predicting dry-wet conditions, significantly higher than the 58.64% accuracy of the WPD method, demonstrating superior identification capability and reliability. Furthermore, the GITPA method also outperformed the WPD method across all error and fitting metrics, with RMSE, MAE, and R² values of 755.60 10⁶m³, 582.45 10⁶m³, and 0.52, respectively, compared to the WPD method’s values of 1169.28 10⁶m³, 1129.38 10⁶m³, and 0.33. In conclusion, the GITPA method offers higher accuracy and reliability in identifying and predicting dry-wet runoff conditions, making it suitable for practical applications in medium- to long-term hydrological forecasting.

Based on the results of periodical analysis of runoff trend, projections for future runoff conditions indicate a potential low-runoff period between 2026 and 2028. This finding suggests that the frequency and intensity of extreme drought events in the Yangtze River Basin may remain at elevated levels over the next decade. This warrants close attention and the timely formulation of corresponding water resource management strategies.

Author contributions

Conceptualization, Hao Wang; methodology, Wei Ding. and Jinbei Li.; software, Jinbei Li.; formal analysis, Wei Ding.; writing—original draft preparation, Jinbei Li; writing—review and editing, Jinbei Li. and, Hao Wang.; visualization, Hao Wang.; funding acquisition, Wei Ding.

Funding

This work was supported by the National Natural Science Foundation of China (no. U2240204).

Data availability

The datasets generated and analysed during the current study are not publicly available due copyright but are available from the corresponding author on reasonable request.

Declarations

Competing interests

The authors declare no competing interests.

Consent for publication

All authors have read and agreed to the published version of the manuscript.