Then et al. (2025) Modified Mann-Kendall with higher-order statistics for trend analysis

Identification

• Journal: Scientific Reports
• Year: 2025
• Date: 2025-12-03
• Authors: Yick Jing Then, Syafrina Abdul Halim
• DOI: 10.1038/s41598-025-30034-0

Research Groups

• Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, Serdang, 43400 UPM, Selangor, Malaysia
• Institute for Mathematical Research (INSPEM), Universiti Putra Malaysia, Serdang, 43400 UPM, Selangor, Malaysia

Short Summary

This study proposes Mann–Kendall with Third-Order Cumulant (MKC3) to improve trend analysis by addressing nonlinearity and autocorrelation, comparing its performance against existing Mann–Kendall variants through simulations and a case study of rainfall trends in Peninsular Malaysia. The findings provide practical guidance for selecting the most suitable trend test based on data characteristics.

Objective

• To propose and evaluate Mann–Kendall with Third-Order Cumulant (MKC3), a modified Mann–Kendall test designed to reduce the impact of both positive serial correlation and nonlinear dependencies in time series data.
• To conduct a Monte Carlo simulation to assess the Type I and Type II error rates of MKC3 in comparison with existing Mann–Kendall variants (MK, TFPW, SMK, MKRD).
• To apply the proposed and existing methods to analyze trends in daily rainfall, monthly maximum 1-day precipitation (Rx1day), and monthly maximum consecutive 5-day precipitation (Rx5day) in Peninsular Malaysia.

Study Configuration

• Spatial Scale: Peninsular Malaysia, focusing on 15 selected rainfall stations located between 1° to 6° North and 100° to 103° East.
• Temporal Scale:
  • Simulation Study: Time series data with sample sizes of 30, 50, 100, and 200, with 10,000 replications for each model.
  • Case Study: Daily rainfall data from 1 January 2019 to 30 July 2024.

Methodology and Data

• Models used:
  • Mann–Kendall (MK) trend test
  • Trend-Free Pre-whitening (TFPW)
  • Seasonal Mann–Kendall (SMK)
  • Mann–Kendall Rank Detrended (MKRD)
  • Mann–Kendall with Third-Order Cumulant (MKC3) - proposed, incorporating a third-order cumulant into MKRD's variance correction formula.
• Data sources:
  • Simulated Data: Nine types of time series models (White Noise, Autoregressive, Nonlinear Autoregressive, Bilinear, Sinusoidal, Chaotic, Deterministic Linear Trend, Autoregressive with Trend, Sinusoidal with Trend) to evaluate Type I and Type II error rates.
  • Observational Data: Daily rainfall data from 15 stations across Peninsular Malaysia, obtained from the Malaysia Drainage and Irrigation Department (DID).
  • Derived Indices: Monthly maximum 1-day precipitation (Rx1day) and monthly maximum consecutive 5-day precipitation (Rx5day), both measured in millimeters (mm).
  • Missing Data Imputation: Inverse Distance Weighting (IDW) interpolation with a power parameter k = 2.
  • Software: Python library pymannkendall for trend tests; QGIS Desktop version 3.44.1 for map generation.

Main Results

• Simulation Study (Type I and Type II Error Rates):
  • No single MK modification is universally robust; performance varies with sample size and time series model type.
  • MK is robust for independent (white noise) data, especially with large sample sizes (100, 200).
  • TFPW is robust in reducing Type II error rates when trends and dependencies coexist, but not outstanding in reducing Type I errors.
  • SMK effectively reduces Type I errors for small sample sizes (30) and moderately autocorrelated, sinusoidal, and chaotic models, but its robustness diminishes with stronger autocorrelation and larger sample sizes, and it tends to overcorrect white noise for small sample sizes.
  • MKRD is robust for strongly autocorrelated data and sinusoidal models with larger sample sizes (50, 100, 200), and robust in strong deterministic trend models (Type II error).
  • MKC3 performs well in bilinear and nonlinear models for most sample sizes (except 30) and is outstanding in detecting weaker trends (reducing Type II error rates in weaker deterministic trend models).
• Case Study (Peninsular Malaysia Rainfall Trends):
  • Daily Rainfall: MK and SMK reported 13 significant increasing trends for the full period, 14 and 13 for the Northeast Monsoon (NEM), and 2 for the Southwest Monsoon (SWM). TFPW, MKRD, and MKC3 identified fewer significant trends, with MKRD and MKC3 reporting no significant trends during SWM.
  • Extreme Rainfall Indices (Rx1day, Rx5day): Generally, fewer stations showed statistically significant trends compared to daily rainfall. During the NEM, some stations exhibited increasing trends, while the SWM showed a higher coverage of negative Z-score statistics, indicating potential decreasing trends, particularly in northern and east coast regions.
  • SMK Limitation: SMK is not recommended for small sample sizes (e.g., Rx1day and Rx5day seasonal data with 22-23 data points) due to insufficient information on ties, leading to unreliable results (high occurrence of zero Z-score values).

Contributions

• Introduces Mann–Kendall with Third-Order Cumulant (MKC3), a novel modification that integrates higher-order statistics (third-order cumulant) into the Mann–Kendall variance correction formula to explicitly address nonlinearity in time series data, a gap not widely covered by existing modifications.
• Provides comprehensive practical guidance for practitioners on selecting the most appropriate Mann–Kendall variant based on data characteristics such as autocorrelation, nonlinearity, and sample size, derived from extensive simulation results.
• Offers a detailed comparative analysis of MKC3 against conventional Mann–Kendall and other modified versions (TFPW, SMK, MKRD) through rigorous simulation studies evaluating Type I and Type II error rates across various time series models.
• Applies the developed methodology to real-world hydro-climatic data from Peninsular Malaysia, providing valuable insights into regional rainfall and extreme precipitation trends, particularly highlighting seasonal discrepancies.

Funding

• Ministry of Higher Education under Fundamental Research Grant Scheme (FRGS/1/2024/STG06/UPM/02/6)

Citation

@article{Then2025Modified,
  author = {Then, Yick Jing and Halim, Syafrina Abdul},
  title = {Modified Mann-Kendall with higher-order statistics for trend analysis},
  journal = {Scientific Reports},
  year = {2025},
  doi = {10.1038/s41598-025-30034-0},
  url = {https://doi.org/10.1038/s41598-025-30034-0}
}