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Climate change is intensifying the occurrence of extreme rainfall events, drawing attention to the importance of understanding the return period concept within the realm of extreme weather studies. This study evaluates the stationarity of extreme rainfall series on both monthly and annual series across East Malaysia, employing the Augmented Dickey–Fuller, Phillips Perron, and Kwiatkowski–Phillips–Schmidt–Shin tests. To model these extreme rainfall series, various probability distributions were applied, followed by goodness-of-fit tests to determine their adequacy. The study identified the stationary and non-stationary return values at 25-, 50-, and 100-year return periods. Additionally, maps depicting the spatial distribution for non-stationary increment were generated. The results indicated that extreme monthly rainfall exhibited stationary characteristics, while extreme yearly rainfall displayed non-stationary characteristics. Among the tested probability distributions, the generalised extreme value distribution was found to be superior in representing the characteristics of the extreme rainfall. Furthermore, a significant finding is that the non-stationary rainfall exhibits higher return values than those of stationary rainfall across all return periods. The northeast coast of Sabah highlighted as the most affected area, with notably high return values for extreme rainfall.

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  • The generalized extreme value distribution was the best distribution to represent the characteristics of extreme monthly and yearly rainfall.

  • The northeast coast of Sabah was identified as the most affected area, with high return values.

climate change, East Malaysia, extreme rainfall, non-stationary analysis, return period
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Rainfall is a natural occurrence in which liquid precipitation descends from the sky and plays an indispensable role in the hydrological process. Extreme rainfall is characterised by an unusually larger amount, higher intensity, more frequent occurrence, and longer duration than average rainstorms. Climate change is strongly correlated with extreme rainfall events. As the global temperatures continue to rise, the amount of moisture that can be held in the atmosphere also increases, leading to more intense and frequent rainfall events (Lian et al. 2020; Pandya et al. 2022; Agarwal et al. 2023a). Extreme rainfall can lead to numerous hazards, such as flooding that would eventually cause the loss of life, damage of the infrastructure and economic hardship (Abushandi et al. 2023; Shekar et al. 2023). Moreover, extreme rainfall with strong wind not only puts human lives in jeopardy but also indirectly damages plantation crops (Elahi et al. 2022). It will pose a challenge for agriculture industry to meet the food production requirements of humans. Therefore, study on extreme rainfall continues to gain attention amongst researchers for several decades as the information of extreme rainfall is crucial for agricultural planning, flood mitigation, and risk assessment (Fung et al. 2022; Mohamed et al. 2022; Agarwal et al. 2023b). Recent studies have highlighted a global trend of increasing extreme rainfall events, emphasizing the need for comprehensive understanding and preparedness. For example, Mohamed & Adam (2022) analysed changes in extreme rainfall events in Somalia by comparing data from two distinct periods: 1901–1958 and 1959–2016. Their findings indicate that extreme rainfall events and associated flood risks have been intensifying due to warmer climates, which leads to increased atmospheric moisture. Similarly, Wei et al. (2023) demonstrated that abnormal warming of sea surface temperatures has significantly increased the frequency of extreme precipitation events across Central Asia.

In the context of hydrology, stationarity refers to the statistical properties of hydrological process remaining relatively constant over time. Stationarity is also a flat-looking series without any trend and periodic seasonality (reoccurring patterns at a fixed frequency). However, extreme rainfall events which often show unusual distribution from historical data are categorised as non-stationary rainfall. Non-stationarity in extreme rainfall indicates that the statistical properties, such as mean and variance, change over time. This is often due to factors such as climate change, which makes prediction and modelling more challenging. The Augmented Dickey–Fuller (ADF), Kwiatkowski–Phillips–Schmidt–Shin (KPSS), and Phillips–Perron (PP) test have been widely utilised for stationarity testing of meteorological series. These tests help determine whether a time series, such as rainfall data, is stationary or non-stationary. Understanding the stationarity of extreme rainfall patterns is crucial because it affects the choice of models used for prediction and risk assessment. For instance, Adams et al. (2019) investigated the monthly rainfall frequency in Osun State, Nigeria. The study found that the time series was strong stationary, allowing them to effectively identify periods of maximum and minimum rainfall. Similarly, Repel et al. (2020) conducted a temporal analysis of the rainfall data at Poprad station, Eastern Slovakia. The stationarity of rainfall was tested using the ADF, PP, and KPSS tests and it was found that the daily rainfall data was stationary. This finding suggested that the existing hydrological models can reliably predict future rainfall patterns. In addition, Theng Hue et al. (2022) carried out trend and stationarity analysis of potential evapotranspiration (PET) using KPSS and ADF tests. The authors revealed that non-stationarity was found at certain stations, due to changes in mean and variance over time. This change was attributed to extreme rainfall and meteorological factors, particularly relative humidity, and air temperature. Garba & Abdourahamane (2023) used the peak over threshold method with a generalised Pareto distribution to analyse the extreme rainfall patterns in Niamey, West Africa. The results indicated no significant increasing trend in terms of extreme rainfall magnitude and frequency. Therefore, these tests provide crucial insights into whether extreme rainfall patterns are influenced by long-term trends or are part of a stationary process, which is essential for developing accurate predictive models and effective risk management strategies.

In accordance with the development of stationarity analysis, many attempts have been made to estimate the parameters of the given rainfall distribution. Each method has its advantages and limitations. Maximum Likelihood Estimation (MLE) is an approach that estimates the parameters by maximising the log-likelihood function (Myung 2003). In Nashwan et al. (2019) study on extreme rainfall stationarity, the MLE was used to estimate the probability distribution parameters. It can provide the spatial and temporal information on the non-stationary increment values at various timescales. Seo et al. (2019) modelled the wind turbine power curve using MLE to identify the parameters of Weibull wind speed distribution in the logistic function. The benefit of using MLE is that it provides a uniform approach that can be applied to various estimation scenarios, and the variance of the outcome produced is small. In addition, the variances of the MLE approach can be used to generate confidence limits and test the hypotheses for the parameters. However, if the size of the sample is small, the estimation results obtained by adopting MLE may not be accurate and can be highly biased. Besides, certain specialised software is required to solve the complicated non-linear equations for the MLE process and calculation.

In Malaysia, climate change has resulted in an increase in both the rainfall amount and intensity (Fung et al. 2022). Climate change leads to higher global temperatures, which increases the moisture-holding capacity of the atmosphere. This enhanced atmospheric moisture content can result in heavier and more intense rainfall events (Tamm et al. 2023). The two states of East Malaysia (Sabah and Sarawak) are continuously facing flood issues, exacerbated by these changes in climate .For instance, Panampang, one of the districts of Sabah, suffered from flooding issues due to heavy monsoon rainfall and subsequently, caused massive damage to the area. It affects over 40,000 residents from 70 villages (Roslee et al. 2017). Besides, Sarawak experiences annual floods caused by the extreme and uneven rainfall distribution (Sa'adi et al. 2019). An unwarranted presumption of stationarity can result in the underestimation of severe floods events (Šraj et al. 2016). However, most of the studies in the field of stationarity analysis have only focused on Peninsular Malaysia. There has been a notable lack of focus on studying East Malaysia's situation, which suffers severely from extreme rainfall events. Therefore, the purpose of this study is to conduct an analysis on the non-stationarity of extreme rainfall series across East Malaysia. The stationarity of the extreme rainfall data will be assessed using the ADF, PP, and KPSS tests, classifying data series as stationary or non-stationary based on the results. Various probability distributions, such as the generalised extreme value and generalised Pareto distribution, will be applied to model the extreme rainfall series, and the best-fitting distributions will be identified through goodness-of-fit tests. The return values of both stationary and non-stationary data were estimated. The return period is essential in flood frequency analysis as well as in risk control and risk evaluation. Subsequently, the maps of spatial distribution were interpolated to illustrate the spatial variability in non-stationary increments.

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Study area and description

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East Malaysia (3.7035° N, 114.5243° E) is a component of Malaysia in Southeast Asia, and it is isolated from Peninsular Malaysia by the South China Sea (Roach 2014). The total land area of East Malaysia is around 198,447 km2, accounting for 60% of Malaysia's overall land area. The weather in East Malaysia is tropical, with high temperature from 28 °C–33 °C and high humidity levels all year long. Sabah and Sarawak has a total rainfall amount of 2,630 and 3,830 mm annually. The volume and distribution of rain in East Malaysia are highly influenced by two distinct rainy seasons which are the Northeast monsoon and the Southwest monsoon (Diong et al. 2015).

The historical rainfall data were collected from the Department of Irrigation and Drainage (DID). Figures 1 and 2 illustrate the geographical locations of the rainfall stations in Sabah and Sarawak, correspondingly. Tables 1 and 2 present the detailed information of each rainfall station.
Table 1

The information of the eight rainfall stations in Sabah

Station codeStation nameRecord periodDurationLongitudeLatitude
5558001 Bongawan 1989–2017 29 Years 115° 50′ E 5° 32′ N 
5862002 Ulu Moyog 1989–2017 29 Years 116° 15′ E 5° 51′ N 
4959001 Kemabong 1989–2017 29 Years 115° 55′ E 4° 54′ N 
5269001 Tongod 1989–2017 29 Years 116° 58′ E 5° 16′ N 
6670001 Kobon 1989–2017 29 Years 117° 2′ E 6° 36′ N 
6073001 Basai 1989–2017 29 Years 115° 48′ E 6° 3′ N 
5482001 Bilit 1989–2017 29 Years 118° 11′ E 5° 30′ N 
4486001 Semporna 1989–2017 29 Years 118° 37′ E 4° 27′ N 
Station codeStation nameRecord periodDurationLongitudeLatitude
5558001 Bongawan 1989–2017 29 Years 115° 50′ E 5° 32′ N 
5862002 Ulu Moyog 1989–2017 29 Years 116° 15′ E 5° 51′ N 
4959001 Kemabong 1989–2017 29 Years 115° 55′ E 4° 54′ N 
5269001 Tongod 1989–2017 29 Years 116° 58′ E 5° 16′ N 
6670001 Kobon 1989–2017 29 Years 117° 2′ E 6° 36′ N 
6073001 Basai 1989–2017 29 Years 115° 48′ E 6° 3′ N 
5482001 Bilit 1989–2017 29 Years 118° 11′ E 5° 30′ N 
4486001 Semporna 1989–2017 29 Years 118° 37′ E 4° 27′ N 
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Table 2

The information of the eight rainfall stations in sarawak

Station codeStation nameRecord periodDurationLongitudeLatitude
1111008 Pantu 1989–2018 30 Years 111° 6′ E 1° 8′ N 
1544001 Long Singut 1989–2017 29 Years 114° 25′ E 1° 34′ N 
1996090 Samunsam 1989–2017 29 Years 109° 38′ E 1° 57′ N 
2112027 Rajang 1989–2018 30 Years 111° 15′ E 2° 9′ N 
2325039 Sg. Arau 1989–2018 30 Years 112° 34′ E 2° 18′ N 
3137021 Tubau 1989–2018 30 Years 113° 42′ E 3° 9′ N 
3847035 Long Atip 1989–2018 30 Years 114° 42′ E 3° 49′ N 
4440001 Lutong 1989–2018 30 Years 114° 42′ E 3° 49′ N 
Station codeStation nameRecord periodDurationLongitudeLatitude
1111008 Pantu 1989–2018 30 Years 111° 6′ E 1° 8′ N 
1544001 Long Singut 1989–2017 29 Years 114° 25′ E 1° 34′ N 
1996090 Samunsam 1989–2017 29 Years 109° 38′ E 1° 57′ N 
2112027 Rajang 1989–2018 30 Years 111° 15′ E 2° 9′ N 
2325039 Sg. Arau 1989–2018 30 Years 112° 34′ E 2° 18′ N 
3137021 Tubau 1989–2018 30 Years 113° 42′ E 3° 9′ N 
3847035 Long Atip 1989–2018 30 Years 114° 42′ E 3° 49′ N 
4440001 Lutong 1989–2018 30 Years 114° 42′ E 3° 49′ N 
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Figure 1
The location of meteorological stations in Sabah.
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The location of meteorological stations in Sabah.

Figure 1
The location of meteorological stations in Sabah.
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The location of meteorological stations in Sabah.

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Figure 2
The location of meteorological stations in Sarawak.
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The location of meteorological stations in Sarawak.

Figure 2
The location of meteorological stations in Sarawak.
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The location of meteorological stations in Sarawak.

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Estimation of missing data

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The incomplete of historical rainfall data can often be attributed to both faulty measuring devices and the limited coverage of data collection across certain areas. Therefore, data screening was performed to ensure that the data series is complete. To address the missing data issue, the inverse distance weighting method (IDW) was employed, which assigns weights to nearby stations based on their proximity to the target station. The assumption underlying this method is that closer stations have stronger correlation with each other than more distant ones. The inverse distance method equation is shown below (De Silva et al. 2007):
(1)
Here, represents the estimated value for missing rainfall, represents the rainfall values of various rainfall station, d represents the distance from each location to the estimated target point, N denotes the number of station(s).

Stationarity testing of extreme rainfall

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The concept of stationarity is the extreme rainfall data does not vary over time. The ADF, PP, and KPSS were adopted to determine whether the stationarity of extreme monthly and yearly rainfall data series. When two or more stationarity tests show a stationary result, the data series at the particular station will be classified as stationary. Meanwhile, if less than two stationarity tests show a stationary result, the data series of the particular station will be classified as non-stationary.

Augmented Dickey–Fuller

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The ADF test will be applied to the extreme rainfall data to assess its stationarity at 5% significance level. It is particularly useful for identifying trends and seasonality that might affect the reliability of statistical models. The equation of the ADF test is expressed as (Nashwan et al. 2019):
(2)
where Δx denotes the first difference of the time series, α is the constant term, β represents the coefficient of a linear trend, if present, γ is the coefficient of the lagged dependent variable, δt represents the lag operator and et is the error term.

Phillips–Perron (PP)

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The PP test accounts for the correlation between observations and heteroscedasticity at different points in time. The PP test was employed in this study and expressed as (Adewole & Serifat 2015):
(3)
where represents estimate, represents the t-ratio of α, is the coefficient standard error, s represents the standard error, T represents the test statistics, and are consistent estimates of variance parameter.

Kwiatkowski–Phillips–Schmidt–Shin

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The KPSS test was developed by Kwiatkowski in 1992, which is commonly used to test for stationarity of the extreme rainfall. It provides a comprehensive view of the stationarity properties of the time series The KPSS test is expressed as (Otero & Smith 2007):
(4)
where represents the partial sum of residuals from the regression of the time series and represents the estimated variance of the residuals from a regression of the time series.

Probability distribution fitting

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The GEV distribution

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The GEV distribution is a continuous probability distribution that used to model the distribution of extreme values. The PDF of GEV distribution can be expressed below (Nashwan et al. 2019):
(5)
where σ, μ, and k are the location, scale, and shape parameters.

The GP distribution

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The GP distribution is a two-parameter family of distributions that is used to model exceedances over a threshold (Grimshaw 1993). The PDF can be written as follows (Alam et al. 2018):
(6)
(7)
(8)

The Gumbel (GUM) distribution

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The GUM distribution, also known as Type 1 extreme value distribution, is commonly employed to model the distribution of extreme events such as flood discharge, maximum annual rainfall, and concentration of pollutants. The PDF can be expressed as (Alam et al. 2018):
(9)

The exponential (EXP) distribution

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EXP distribution is a probability distribution that models the time between events in a Poisson process, where events occur at a constant rate and are independent of each other. The PDF can be expressed as (Alam et al. 2018):
(10)

The Log-Pearson type III (LPIII) distribution

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The LPIII distribution is a continuous probability distribution that has three parameters distribution: location, scale and shape. The PDF can be expressed as (Alam et al. 2018):
(11)

The Log-Normal (LNl)

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The LN distribution is probability distribution of a random variable whose natural logarithm is normally distributed. The PDF can be expressed as (Alam et al. 2018):
(12)
where the range of x random variable > 0, represents the mean and represents the standard deviation.

Goodness-of-Fit (GoF) tests

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The GoF tests are statistical procedures used to evaluate how well the extreme rainfall fits into the probability distributions. The distribution models of each time scale will be evaluated using the GoFs, particularly the Kolmogorov–Smirnov (K–S), Chi-square (C-S), and the Anderson–Darling (A–D) tests. All distributions were ranked in ascending order, starting from one (indicating the best fit) and going up to six (indicating the poorest fit). The overall rank will be obtained by summing up the individual scores from each GoF test, and the best distribution with the lowest total score will be selected to fit the extreme rainfall data.

The C-S test

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The C-S test is a statistical method that determine the difference between the observed and expected variables. The equation of the C-S test can be expressed as (Turhan 2020):
(13)
where is the observed frequency and represent the expected frequency.

The K–S test

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The K–S test is a statistical method used to test the GoF of a sample to a continuous probability distribution. The K–S test can be expressed as (Alam et al. 2018):
(14)
(15)
where represents the critical value, Fx(x) and Sn(x) are the empirical and theoretical cumulative distribution function of the tested distribution.

The A–D test

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The A–D test calculates the critical values using the specific distribution. It has the advantage of being more sensitive to the sample data. The A–D statistics can be expressed as (Naumoski et al. 2017):
(16)
where represents the empirical distribution function of the tested distribution.

Estimation of return period

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Stationary rainfall

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The return period is the recurrence interval between the occurrences of specific events such as floods, earthquakes, or volcanic eruptions. The event's return period can be expressed as the percentage of the event that will occur in any given year. For instance, the 100-year return period of an event can be stated as having a 1% probability of occurring in any given year. For stationary rainfall, the probability of occurrence and return period are expressed as (Alam et al. 2018):
(17)
(18)
where represents the probability occurrence, T represents return interval and represents the magnitude of event.

Non-stationary rainfall

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An uneven cycle that alters the means or variance of rainfall over time is known as non-stationary. The rainfall series will be fitted into different distribution models, and the best-fitting distribution based on GoFs will be selected for parameters estimation. The quantile function of the non-stationary return level which derived from the GEV model is shown below (Chikobvu & Chifurira 2015):
(19)
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Stationarity testing of extreme rainfall

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The stationarity of each station was assessed using the ADF, PP and KPSS tests with a 5% significance level. For the ADF and PP test, a p-value greater than 0.05 indicates non-stationarity, while a p-value lower than 0.05 indicates stationarity. In contrast, the KPSS test classifies the data as stationary only when the p-value is greater than 0.05 and its null hypothesis differs from the ADF and PP tests. The overall stationarity result of each station was evaluated based on the results of all three results. A station is classified as stationary if it shows a stationary result in at least two out of three tests.

Table 3 summarised the p-values of extreme monthly and yearly rainfall series obtained from all the tests for all sixteen stations. It was found that all extreme monthly rainfall exhibited stationary characteristics, as the p-values obtained for ADF and PP tests were less than 0.05. There was no trend, constant degree of wiggliness and no heteroscedasticity detected from the ADF and PP tests for each station. Meanwhile, in the KPSS test, non-stationarities of extreme monthly series were found in a few stations, namely the Ulu Moyog, Tongod, Semporna, Long Singut, Rajang, Sg Arau and Lutong. This revealed that data series at these seven stations vary over time, while the remaining nine stations do not change over time. Based on results above, it is interesting to note that all extreme monthly series were labelled as stationary as they acquired stationary results for at least two out of three stationarity tests. Figure 3 displays the trend results, indicating that the extreme monthly rainfall at Station Semporna and Sg. Arau exhibited stationary behaviour. There were no noticeable trend variations over time, and their mean and variance remained constant. Thesefindings are consistent with Hasan et al. (2012) who found that the extreme monthly, biweekly and weekly rainfall in Penang, Peninsular Malaysia showed stationary results. Similar outcomes were observed by Tadesse & Dinka (2017) for monthly series at Waterval River, South Africa.
Table 3

Stationarity test results for the extreme monthly and extreme yearly rainfall series

Station nameExtreme monthly
Extreme yearly
p-value
p-value
ADFPPKPSSADFPPKPSS
Bongawan <0.0001 <0.0001 0.069 0.269* 0.666* 0.369 
Ulu Moyog <0.0001 <0.0001 0.017* 0.075* 0.408* 0.082 
Kemabong <0.0001 <0.0001 0.782 0.136* 0.315* 0.858 
Tongod <0.0001 <0.0001 <0.0001* 0.316* 0.352* 0.000* 
Kobon <0.0001 < 0.0001 0.121 0.220* 0.137* 0.467 
Basai <0.0001 <0.0001 0.871 0.343* 0.282* 0.211 
Bilit <0.0001 <0.000 0.114 0.178* 0.025 0.678 
Semporna <0.0001 <0.0001 0.003* 0.481* 0.398* 0.099 
Pantu <0.0001 <0.0001 0.144 0.632* 0.322* 0.761 
Long Singut 0.004 <0.0001 0.039* 0.029 0.408* 0.614 
Samunsam <0.0001 <0.0001 0.351 0.032 0.300* 0.244 
Rajang <0.0001 <0.0001 0.001* 0.903* 0.456* 0.042* 
Sg. Arau <0.0001 < 0.0001 0.004* 0.166* 0.584* 0.625 
Tubau < 0.0001 <0.0001 0.322 0.432* 0.397* 0.034* 
Long Atip <0.0001 <0.0001 0.298 0.185* 0.742* 0.726 
Lutong <0.0001 <0.0001 0.033* 0.832* 0.416* 0.005* 
Station nameExtreme monthly
Extreme yearly
p-value
p-value
ADFPPKPSSADFPPKPSS
Bongawan <0.0001 <0.0001 0.069 0.269* 0.666* 0.369 
Ulu Moyog <0.0001 <0.0001 0.017* 0.075* 0.408* 0.082 
Kemabong <0.0001 <0.0001 0.782 0.136* 0.315* 0.858 
Tongod <0.0001 <0.0001 <0.0001* 0.316* 0.352* 0.000* 
Kobon <0.0001 < 0.0001 0.121 0.220* 0.137* 0.467 
Basai <0.0001 <0.0001 0.871 0.343* 0.282* 0.211 
Bilit <0.0001 <0.000 0.114 0.178* 0.025 0.678 
Semporna <0.0001 <0.0001 0.003* 0.481* 0.398* 0.099 
Pantu <0.0001 <0.0001 0.144 0.632* 0.322* 0.761 
Long Singut 0.004 <0.0001 0.039* 0.029 0.408* 0.614 
Samunsam <0.0001 <0.0001 0.351 0.032 0.300* 0.244 
Rajang <0.0001 <0.0001 0.001* 0.903* 0.456* 0.042* 
Sg. Arau <0.0001 < 0.0001 0.004* 0.166* 0.584* 0.625 
Tubau < 0.0001 <0.0001 0.322 0.432* 0.397* 0.034* 
Long Atip <0.0001 <0.0001 0.298 0.185* 0.742* 0.726 
Lutong <0.0001 <0.0001 0.033* 0.832* 0.416* 0.005* 

*The bolded values indicate the detection of non-stationarity within the time series.

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Figure 3
Time series of extreme monthly rainfall in (a) Semporna and (b) Sg Arau.
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Time series of extreme monthly rainfall in (a) Semporna and (b) Sg Arau.

Figure 3
Time series of extreme monthly rainfall in (a) Semporna and (b) Sg Arau.
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Time series of extreme monthly rainfall in (a) Semporna and (b) Sg Arau.

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Table 3 reveals that almost all of the yearly series were non-stationary, as evidenced by p-values greater than 0.05 in both the ADF and PP tests. However, the p-values of the ADF test at both Long Singut and Samunsam stations were 0.029 and 0.032, respectively. The p-value of PP test at Bilit station was 0.025. This indicated these three stations were labelled as stationary as their p-values were lesser than 0.05. For the KPSS tests, majority of the stations were found to be stationary except for Tongod, Rajang, Tubau and Lutong staitons. In summary, it can be highlighted that most of the extreme yearly series were non-stationary, with the exception of Bilit, Long Singut and Samunsam stations.

Figure 4 presents a significant downward trend in the extreme yearly series at Tongod and Lutong stations, characterised by frequent fluctuations and drastic mean changes over time. These changes can be attributed to climate change, which has disrupted the distribution of rainfall in Malaysia. This, in turn, has resulted in more frequent and severe climatic extremes (Tarmizi 2019). The great fluctuations in rainfall amount and intensity have caused the extreme yearly rainfall to become non-stationary. Furthermore, a downward trend in extreme yearly rainfall could lead to prolonged dry periods, affecting local ecosystems, agriculture, and infrastructure. Prolonged dry periods could affect local water bodies, wetlands, and aquatic ecosystems, reducing water availability and impacting biodiversity, particularly species dependent on stable water conditions. Crops may suffer from water shortages during dry spells and damage from intense rainfall events, leading to reduced yields and increased vulnerability to pests and diseases. Additionally, the infrastructure in these areas, including roads, bridges, and water supply systems, may be stressed by the variability in rainfall. Prolonged dry periods can lead to ground subsidence and damage to foundations, while intense rainfall can cause flooding and erosion, stressing the resilience of existing infrastructure. (He et al. 2024). These observations highlight the need for adaptive management strategies to mitigate these risks and ensure sustainable development in the face of climate change.
Figure 4
Time series for extreme yearly rainfall in (a) Tongod and (b) Lutong.
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Time series for extreme yearly rainfall in (a) Tongod and (b) Lutong.

Figure 4
Time series for extreme yearly rainfall in (a) Tongod and (b) Lutong.
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Time series for extreme yearly rainfall in (a) Tongod and (b) Lutong.

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Figure 5
Return periods of the extreme monthly rainfall when considered (a) stationary and (b) non-stationary at station Pantu.
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Return periods of the extreme monthly rainfall when considered (a) stationary and (b) non-stationary at station Pantu.

Figure 5
Return periods of the extreme monthly rainfall when considered (a) stationary and (b) non-stationary at station Pantu.
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Return periods of the extreme monthly rainfall when considered (a) stationary and (b) non-stationary at station Pantu.

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In summary, the extreme monthly rainfall series exhibited stationarity, while non-stationary was observed in the extreme yearly rainfall series. It seems possible that these results are due to the sample size of the rainfall. As compared to the extreme yearly series, extreme monthly series used a larger sample size to analyse the stationarity of the rainfall series (Syafrina et al. 2019). As the rainfall sample size increases, the extreme monthly rainfall series (sample mean) will become a better estimate of the population mean. This is because the variability of the extreme monthly rainfall series reduces, and the mean and variance tend to be constant over time (Barri 2019).

Performances of distribution fitting

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The probability distribution that fits the best (best fit) was identified by examining the lowest scores from the GoF tests. Six different probability distribution models were evaluated and ranked from 1 (indicating best fit) to 6 (indicating the poorest fit) in ascending order. In Table 4, the total scores from the 2 GoF tests were combined, and the lowest score distribution was chosen as the best fit.

Table 4

Summary of the best-fit distribution for extreme monthly and extreme yearly series

StationBest- fit probability distribution
Extreme monthlyExtreme yearly
Bongawan GEV GEV 
Ulu Moyog GEV GUM 
Kemabong GEV GEV 
Tongod GEV GP 
Kobon GEV LN 
Basai GEV GEV 
Bilit GEV GEV 
Semporna GEV LPIII 
Pantu GEV GEV 
Long Singut GEV LN 
Samunsam GUM GEV 
Rajang GEV GEV 
Sg. Arau GEV LN 
Tubau GEV GEV 
Long Atip GEV GEV 
Lutong GEV GEV 
StationBest- fit probability distribution
Extreme monthlyExtreme yearly
Bongawan GEV GEV 
Ulu Moyog GEV GUM 
Kemabong GEV GEV 
Tongod GEV GP 
Kobon GEV LN 
Basai GEV GEV 
Bilit GEV GEV 
Semporna GEV LPIII 
Pantu GEV GEV 
Long Singut GEV LN 
Samunsam GUM GEV 
Rajang GEV GEV 
Sg. Arau GEV LN 
Tubau GEV GEV 
Long Atip GEV GEV 
Lutong GEV GEV 
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For the extreme monthly series, the GEV distribution was found to acquire the lowest scores for almost all rainfall stations, except for Samunsam station. The GUM distribution is less favourable as compared to GEV distribution as it has only two parameters (location and scale parameters), which are inadequate for accurately representing the properties of extreme rainfall. Besides, the GEV distribution also performed similarly well for the extreme yearly rainfall series where it obtained the lowest scores for 10 out of 16 stations. The LN distribution was the second favourable distribution and was found to be the best fit for the Kobon, Long Singut and Sg. Arau stations. The LPIII and GP distributions were chosen as the best fits for the Semporna and Tongod stations.

In summary, the GEV distribution showed superior performance in modelling extreme rainfall events at monthly and yearly time scales in East Malaysia. This finding is consistent with previous studies by Alam et al. (2018), Nashwan et al. (2019) and Ng et al. (2021) who demonstrated the effectiveness of the GEV distribution in fitting rainfall at various time scales. The reason for this is that the GEV distribution has a supplementary parameter (shape parameter) that can model the heavy distributions of extreme rainfall events. Moreover, the GEV distribution is a statistical model that unifies the GUM (Type I), Frechet (Type II) and Weibull (Type III) behaviour, making it more flexible than other probability distributions and showed a better fit of the long-tail phenomenon in the histogram of the normalised reflection symmetry metric.

Return values of rainfall

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Table 5 presents the return values of extreme monthly and extreme yearly series, both stationary and non-stationary, for all stations. The GEV distribution was the best fit for extreme monthly and extreme yearly rainfall series, Hence, the non-stationary return values were obtained by fitting the data to the GEV distribution and using MLE for parameters estimation. It can be highlighted that the return value increases with return period, and the non-stationary extreme rainfall had much higher than the stationary extreme rainfall at each return period. For instance, Pantu station had return values of 128.365, 148.276, and 168.187 mm at each return period. On the other hand, the return values of non-stationary extreme monthly rainfall for each return period were 254.564, 281.212, and 305.569 mm. As shown in Figures 5, the 25-year return value of stationary rainfall was 128.365 mm while the non-stationary extreme monthly rainfall was 254.564 mm. In addition, for the 50-year return period, the return values of stationary and non-stationary rainfall were 148.276 m and 281.212 mm, respectively. It was also observed that the return value of non-stationary rainfall at 50-year return period was much higher than that of 100 years return period. These findings mirror those of Nashwan et al. (2019) who observed similar results for Kelantan. Therefore, it is essential to consider non-stationary return levels in analysis to prevent misinterpretation and hydraulic structure design failure caused by the wrong assumptions of stationarity.

Table 5

Results of stationary and non-stationary rainfall return values in extreme monthly and extreme yearly rainfall series

Station nameReturn period (Year)Extreme monthly
Extreme yearly
Stationary return value(mm)Non-stationary return value(mm)Stationary return value(mm)Non-stationary return value(mm)
Bongawan 25 145.909 290.031 212.238 507.072 
50 169.744 323.796 235.285 547.687 
100 193.579 355.433 258.332 582.784 
Ulu Moyog 25 153.805 341.697 233.111 656.542 
50 177.053 382.583 261.728 743.789 
100 200.301 421.227 290.345 828.798 
Kemabong 25 77.199 171.347 116.235 384.398 
50 90.07 196.505 131.921 450.035 
100 102.942 221.486 147.607 518.22 
Tongod 25 121.433 296.574 197.84 513.534 
50 141.928 348.15 225.491 586.93 
100 162.423 401.692 253.142 659.614 
Kobon 25 183.669 452.731 361.536 994.773 
50 220.839 569.717 419.882 1186.698 
100 258.009 704.973 478.227 1392.176 
Basai 25 149.368 348.272 283.783 886.311 
50 175.743 409.877 327.593 1073.108 
100 202.119 474.087 371.404 1278.158 
Bilit 25 529.616 371.719 347.875 1351.568 
50 628.251 455.357 407.467 1790.641 
100 726.886 548.403 467.06 2339.279 
Semporna 25 171.437 398.626 313.935 1132.292 
50 204.705 485.443 366.757 1448.21 
100 237.972 580.98 419.578 1824.124 
Pantu 25 128.365 254.564 182.389 337.354 
50 148.276 281.212 200.739 349.104 
100 168.187 305.569 219.089 357.6 
Long Singut 25 123.233 208.469 161.802 254.292 
50 142.016 222.697 177.84 258.77 
100 160.799 234.484 193.879 261.599 
Samunsam 25 156.62 288.377 245.043 416.018 
50 187.987 338.76 276.344 434.99 
100 219.354 390.78 307.645 449.453 
Rajang 25 132.292 318.327 214.54 537.866 
50 154.36 371.648 245.319 611.733 
100 176.428 426.461 276.097 684.138 
Sg. Arau 25 117.336 259.833 192.714 581.177 
50 135.653 291.913 217.229 670.187 
100 153.97 322.451 241.743 760.102 
Tubau 25 165.8 371.469 293.12 1069.56 
50 197.043 430.892 335.33 1320.506 
100 228.285 491.134 377.539 1604.903 
Long Atip 25 129.119 232.447 170.837 451.03 
50 149.07 251.791 186.967 489.309 
100 169.021 268.548 203.097 522.868 
Lutong 25 165.081 352.726 298.775 969.293 
50 195.165 412.607 343.103 1175.348 
100 225.249 474.234 387.431 1402.176 
Station nameReturn period (Year)Extreme monthly
Extreme yearly
Stationary return value(mm)Non-stationary return value(mm)Stationary return value(mm)Non-stationary return value(mm)
Bongawan 25 145.909 290.031 212.238 507.072 
50 169.744 323.796 235.285 547.687 
100 193.579 355.433 258.332 582.784 
Ulu Moyog 25 153.805 341.697 233.111 656.542 
50 177.053 382.583 261.728 743.789 
100 200.301 421.227 290.345 828.798 
Kemabong 25 77.199 171.347 116.235 384.398 
50 90.07 196.505 131.921 450.035 
100 102.942 221.486 147.607 518.22 
Tongod 25 121.433 296.574 197.84 513.534 
50 141.928 348.15 225.491 586.93 
100 162.423 401.692 253.142 659.614 
Kobon 25 183.669 452.731 361.536 994.773 
50 220.839 569.717 419.882 1186.698 
100 258.009 704.973 478.227 1392.176 
Basai 25 149.368 348.272 283.783 886.311 
50 175.743 409.877 327.593 1073.108 
100 202.119 474.087 371.404 1278.158 
Bilit 25 529.616 371.719 347.875 1351.568 
50 628.251 455.357 407.467 1790.641 
100 726.886 548.403 467.06 2339.279 
Semporna 25 171.437 398.626 313.935 1132.292 
50 204.705 485.443 366.757 1448.21 
100 237.972 580.98 419.578 1824.124 
Pantu 25 128.365 254.564 182.389 337.354 
50 148.276 281.212 200.739 349.104 
100 168.187 305.569 219.089 357.6 
Long Singut 25 123.233 208.469 161.802 254.292 
50 142.016 222.697 177.84 258.77 
100 160.799 234.484 193.879 261.599 
Samunsam 25 156.62 288.377 245.043 416.018 
50 187.987 338.76 276.344 434.99 
100 219.354 390.78 307.645 449.453 
Rajang 25 132.292 318.327 214.54 537.866 
50 154.36 371.648 245.319 611.733 
100 176.428 426.461 276.097 684.138 
Sg. Arau 25 117.336 259.833 192.714 581.177 
50 135.653 291.913 217.229 670.187 
100 153.97 322.451 241.743 760.102 
Tubau 25 165.8 371.469 293.12 1069.56 
50 197.043 430.892 335.33 1320.506 
100 228.285 491.134 377.539 1604.903 
Long Atip 25 129.119 232.447 170.837 451.03 
50 149.07 251.791 186.967 489.309 
100 169.021 268.548 203.097 522.868 
Lutong 25 165.081 352.726 298.775 969.293 
50 195.165 412.607 343.103 1175.348 
100 225.249 474.234 387.431 1402.176 
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The non-stationary increment values for all the return periods of extreme monthly and yearly series were plotted in Figures 6 and 7, respectively. The non-stationary increment values were obtained by calculating the differences between the return values of non-stationary rainfall and stationary rainfall. The positive increment indicates that the return value of non-stationary rainfall is greater than stationary rainfall and vice versa. Different colours were used to represent the non-stationary increment values on the map of East Malaysia. The shaded regions in green, yellow, orange, brown, and pink corresponded to different ranges of non-stationary increment values. For the extreme monthly and extreme yearly series of the 25-year return period, it can be noted that the northeast coastal region of East Malaysia was shaded with pink where that region had the highest non-stationary increment. Similar results were obtained in the spatial map of the 50- and 100- year return periods, where the northeast coastal region had the highest increment. For the 100-year return period, in extreme monthly series, the northeast of Sabah experienced the highest increment, with the range of 321.734–446.743 mm. Meanwhile, in the extreme yearly series, the highest increment was found in the station of Bilit with range of 1511.02–1871.82 mm. The results were in agreement with Jafar et al. (2020) findings, who reported that the northeast region of Sabah would receive a higher rainfall amount due to the occurrence of the northeast monsoon. It is interesting to note that the average rainfall was higher in southwest monsoon. Thus, the flood occurrence frequency is not solely affected by the monsoon season in East Malaysia. In a nutshell, massive flood may occur in East Malaysia due to the northeast monsoon bringing a long duration of extreme rainfall, which will affect the water resources and agriculture in the study area.
Figure 6
The spatial distribution of extreme monthly rainfall amounts of (a) 25-year; (b) 50-year and (c) 100-year return period.
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The spatial distribution of extreme monthly rainfall amounts of (a) 25-year; (b) 50-year and (c) 100-year return period.

Figure 6
The spatial distribution of extreme monthly rainfall amounts of (a) 25-year; (b) 50-year and (c) 100-year return period.
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The spatial distribution of extreme monthly rainfall amounts of (a) 25-year; (b) 50-year and (c) 100-year return period.

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Figure 7
The spatial distribution of extreme yearly rainfall amounts of (a) 25-year; (b) 50-year and (c) 100-year return period.
View largeDownload slide

The spatial distribution of extreme yearly rainfall amounts of (a) 25-year; (b) 50-year and (c) 100-year return period.

Figure 7
The spatial distribution of extreme yearly rainfall amounts of (a) 25-year; (b) 50-year and (c) 100-year return period.
View largeDownload slide

The spatial distribution of extreme yearly rainfall amounts of (a) 25-year; (b) 50-year and (c) 100-year return period.

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A non-stationarity analysis of extreme rainfall for 16 stations in East Malaysia was successfully carried out. The stationarity of the monthly and yearly extreme rainfall was determined using the ADF, PP, and KPSS tests. The extreme rainfall series were then fitted into various distributions and the best distribution based on GoF results was used to estimate parameters. The return periods for stationary and non-stationary rainfall for 25-, 50-, and 100-year extreme monthly and yearly rainfall series were calculated to show the spatial variability in East Malaysia. It was found that the extreme monthly and extreme yearly rainfall series exhibited stationary and non-stationary characteristics, respectively. The GEV distribution was the best distribution to represent the characteristics of extreme monthly and yearly rainfall due to its superior descriptive and predictive capabilities. Additionally, the study showed that the higher the return value correspond to higher return period. Climate change is a major concern as it has affected the extreme rainfall in East Malaysia where the return values of non-stationary rainfall were much higher than stationary rainfall. This study emphasises the need to consider non-stationary for future studies to avoid underestimation of the extreme rainfall because it will cause the failure of hydraulic structure design and frequent occurrences of flood events. The findings in this study will be useful for mitigation planning, hydrological planning, agriculture management, civil engineering design, and climate change assessment in East Malaysia.

It is recommended that future studies should consider using the Peak over Threshold (PoT) technique to extract extreme rainfall, instead of the Block Maxima (BM) method. In this study, the BM method was adopted to extract the extreme rainfall as it is easier to apply since the block periods occur naturally. However, the PoT technique might be preferable in certain situations because it can account for multiple occurrences of extreme events over a particular threshold within a block period. This allows for a more comprehensive analysis of extreme events, as it does not restrict the data to a single maximum value per block, thus providing a richer dataset for statistical modelling and more accurate estimates of extreme rainfall probabilities. Additionally, further studies could explore the impact of urbanisation on rainfall patterns and flood risks in East Malaysia. Implementing real-time monitoring systems and developing comprehensive climate adaptation strategies will be crucial for mitigating the adverse effects of extreme rainfall events.

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The authors would like to express their gratitude to the Ministry of Higher Education Malaysia for funding this research project through Fundamental Research Grant Scheme (FRGS) with project code (FRGS/1/2021/TK0/UCSI/03/3). The authors would like to express the gratitude towards the Malaysian Meteorological Department for providing the meteorological data.

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All relevant data are included in the paper or its Supplementary Information.

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The authors declare there is no conflict.

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