Regional Rainfall Frequency Analysis in Terengganu Using Ward's Clustering, L-Moments, and Trimmed L-Moments

Muhammad Nura¹ and Zahrahtul Amani Zakaria^2,

¹Department of Statistics, Kano State Polytechnic, Kano, Nigeria

²Faculty of Computing and Informatics, Universiti Sultan Zainal Abidin, Kampus Besut, 22200 Besut, Terengganu, Malaysia

Cite this paper:
Muhammad Nura and Zahrahtul Amani Zakaria. Regional Rainfall Frequency Analysis in Terengganu Using Ward's Clustering, L-Moments, and Trimmed L-Moments. American Journal of Water Resources. 2026; 14(2):38-45. doi: 10.12691/ajwr-14-2-2

Abstract

Terengganu, on the northeast coast of Peninsular Malaysia, is a flood-prone region whose annual maximum daily rainfall records are strongly positively skewed owing to the Northeast Monsoon (NEM). This skewness can bias conventional regional frequency analysis (RFA) based on L-moments. Trimmed L-moments (TL-moments) provide a more robust alternative by down-weighting extreme order statistics. Although L-moment RFA is well established for Malaysian rainfall, no study has combined Ward's hierarchical clustering with both L-moments and TL-moments at trimming levels t = 1 and t = 2 within Terengganu, nor systematically evaluated four candidate distributions GEV, GLO, GPA, and the three-parameter Kappa Type-II (K3D-II) under all estimation methods in parallel. Annual maximum daily rainfall records from nine Department of Irrigation and Drainage (DID) stations (1971–2023; 252 station-years) were analysed. Ward's minimum-variance clustering of normalised TL-moment ratio site characteristics, validated by the average silhouette width (ASW) criterion, identified two homogeneous regions. Discordancy screening, heterogeneity testing via 500 Monte Carlo simulations, and Z-statistic goodness-of-fit tests were applied to all four distributions under each estimation method. Normalised growth-factor quantiles were estimated at return periods T = 2–200 years. Ward's clustering yielded an optimal two-region solution (k = 2; ASW = 0.6472): Region R1 (seven stations, low-to-moderate skewness) and Region R2 (two stations most affected by the December 2013 NEM event). Both regions were acceptably homogeneous (H1 < 1) under all methods. TL-moments progressively attenuated regional skewness relative to L-moments, with attenuation most pronounced in R2 (up to 59.7% at t = 2), reflecting reduced influence of the 2013 extreme observations on parameter estimation. The best-fit distribution varied by both region and estimation method: K3D-II performed best for R1 under L-moments and for R2 under TL-moments, while GLO and GEV were preferred for R1 at t = 1 and t = 2, respectively. TL-moment growth factors were 7–59% lower than their L-moment counterparts for T ≥ 20 years. These results suggest that TL-moments combined with the K3D-II distribution provide a more robust framework for regional rainfall frequency analysis in high-skewness, NEM-dominated environments. L-moment and TL-moment estimates are recommended for joint use as upper and lower design bounds, with the appropriate choice guided by the consequence class of the hydraulic structure.

Keywords:
homogeneous region delineation moment-ratio diagram hydro-climatic sub-regions quantile attenuation design rainfall bounds parameter estimation robustness outlier-robust estimation

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References:

[1]	Akasah ZA, Doraisamy SV. 2014 Malaysia flood: Impacts and factors contributing to the worst flooding. J Sci Res Dev. 2015; 2(14): 99–105.
[2]	Suhaila J, Deni SM, Zin WZW, Jemain AA. Trends in Peninsular Malaysia rainfall data during the southwest monsoon and northeast monsoon seasons: 1975–2004. Sains Malaysiana. 2011; 40(6): 533–539.
[3]	Wong CL, Liew J, Yusop Z, Ismail T, Venneker R, Uhlenbrook S. Rainfall characteristics and regionalization in Peninsular Malaysia based on a high-resolution gridded data set. Water. 2016; 8(11): 500.
[4]	Department of Irrigation and Drainage (DID) Malaysia. Laporan Banjir Besar Kelantan dan Terengganu Disember 2013. Kuala Lumpur: Ministry of Natural Resources and Environment Malaysia; 2014.
[5]	Chan NW, Teh GH. The 2013–2014 East Coast Malaysia flood: Assessment and lessons learnt. Int J Disaster Risk Reduct. 2021; 54: 102009.
[6]	Hosking JRM, Wallis JR. Regional Frequency Analysis: An Approach Based on L-Moments. Cambridge University Press; 1997.
[7]	Hosking JRM. L-moments: Analysis and estimation of distributions using linear combinations of order statistics. J R Stat Soc Ser B. 1990; 52(1): 105–124.
[8]	Elamir EAH, Seheult AH. Trimmed L-moments. Comput Stat Data Anal. 2003; 43(3): 299–314.
[9]	Shabri AB, Daud ZM, Ariff NM. Regional analysis of annual maximum rainfall using the TL-moments method. Theor Appl Climatol. 2011; 104(3–4): 561–570.
[10]	Hosking JRM. Some theory and practical uses of trimmed L-moments. J Stat Plan Inference. 2007; 137(9): 3024–3039.
[11]	Noor M, Ismail T, Ul Haq Z, Mostafa SA. Application of four-parameter kappa distribution for regional frequency analysis of annual maximum rainfall. Environments. 2020; 7(2): 37.
[12]	Hosking JRM. The four-parameter kappa distribution. IBM J Res Dev. 1994; 38(3): 251–258.
[13]	Noor M, Ismail T, Ul Haq Z, Mostafa SA. Regional frequency analysis of annual maximum rainfall using the K3D-II distribution. Water. 2022; 14(5): 682.
[14]	Ward JH. Hierarchical grouping to optimize an objective function. J Am Stat Assoc. 1963; 58(301): 236–244.
[15]	Rousseeuw PJ. Silhouettes: A graphical aid to the interpretation and validation of cluster analysis. J Comput Appl Math. 1987; 20: 53–65.
[16]	Sahrin S, Ismail N, Alias NE. Regional frequency analysis on Peninsular Malaysia using L-moments. Far East J Math Sci. 2018; 103(8): 1379–1398.
[17]	Asquith WH. L-moments and TL-moments of the generalized lambda distribution. Comput Stat Data Anal. 2007; 51(9): 4484–4496.
[18]	Milly PCD, Wetherald RT, Dunne KA, Delworth TL. Increasing risk of great floods in a changing climate. Nature. 2002; 415: 514–517.
[19]	Burn DH. Catchment similarity for regional flood frequency analysis using seasonality measures. J Hydrol. 1997; 202(1–4): 212–230.
[20]	Viglione A, Laio F, Claps P. A comparison of homogeneity tests for regional frequency analysis. Water Resour Res. 2007; 43(3): W03428.
[21]	Rao AR, Srinivas VV. Regionalization of watersheds by hybrid-cluster analysis. J Hydrol. 2006; 318(1–4): 37–56.
[22]	Hosking JRM, Wallis JR. Some statistics useful in regional frequency analysis. Water Resour Res. 1993; 29(2): 271–281.
[23]	Zin WZW, Jemain AA, Ibrahim K. The best fitting distribution of annual maximum rainfall in Peninsular Malaysia based on L-moment and LQ-moment. Theor Appl Climatol. 2009; 96: 337–344.
[24]	Han X, Mehrotra R, Sharma A, Rahman A. Incorporating non-stationarity in regional flood frequency analysis procedures. J Hydrol. 2022; 612: 128235.
[25]	Reis DS Jr, Stedinger JR. Bayesian MCMC flood frequency analysis with historical information. J Hydrol. 2005; 313(1–2): 97–116.
[26]	Rahman A, Haddad K, Zaman M, Ishak E, Weinmann PE. Regional flood methods for Australia. Australas J Water Resour. 2020; 24(1): 1–19.
[27]	Kaufman L, Rousseeuw PJ. Finding Groups in Data: An Introduction to Cluster Analysis. Wiley, 1990.
[28]	Dey R, Lewis SC, Arblaster JM, Abram NJ. Projected changes in frequency and intensity of extreme rainfall over South and Southeast Asia. Geophys Res Lett. 2023; 50(7): e2022GL102009.
[29]	Malekinezhad H, Zare-Garizi A. Regional frequency analysis of daily rainfall extremes using L-moments. Atmosfera. 2014; 27(4): 411–422.
[30]	Kumar R, Chatterjee C, Singh RD, Lohani AK. Development of regional flood frequency relationships using L-moments for gauged and ungauged catchments of the north Brahmaputra region of India. Water Resour Manag. 2003; 17: 355–370.
[31]	Donat MG, Lowry AL, Alexander LV, O'Gorman PA, Maher N. More extreme precipitation in the world's dry and wet regions. Nat Clim Chang. 2017; 6: 508–513.
[32]	Peel MC, Wang QJ, Vogel RM, McMahon TA. The utility of L-moments in regional flood frequency analysis. Water Resour Res. 2001; 37(3): 611–622.