Comparative analysis of ARIMA and Holt's models for predicting water and sanitation access in Bangladesh: toward SDG target Open Access

Access to water and sanitation will be critical to achieving the sustainable development goals (SDGs). Water is essential for addressing our most pressing sustainable development challenges, from eradicating poverty and hunger to combating climate change (Mason et al. 2017). The Water SDG (SDG 6), which consists of eight targets, calls for progress in water supply, sanitation, water quality, water efficiency and scarcity, integrated water resource management, water and the environment, increased international cooperation, and community involvement in water and sanitation management (HLPW 2018). The first two targets of SDG 6 focus on water, sanitation, and hygiene. Water provision targets are also included, which go beyond human use and interaction and address structural, ecosystem, and governance needs in water management (UN 2015).

Access to water and sanitation will play a defining role in achieving the SDGs. From eliminating poverty and hunger to tackling climate change, water is central to addressing our most significant sustainable development challenges (Mason et al. 2017). To address disparities in access to water and sanitation, SDG 6 aims to achieve universal and equitable access to improved drinking water and sanitation for all by 2030. It is crucial to track inequalities in access to drinking water and sanitation to assess progress regarding universal coverage. WHO/UNICEF defines safely managed drinking water as an improved water source that is accessible on premises, available when needed, and free from fecal and priority chemical contamination. Improved water sources include piped water, boreholes or tubewells, protected dug wells, protected springs, and packaged or delivered water. To meet the new SDG criteria for safely managed drinking water services, households must use an improved water source (UNICEF & WHO 2019) that is accessible on premises, available when needed, and free from contamination: compliant with standards for fecal contamination (Escherichia coli) and priority chemical contamination (arsenic and fluoride). Access to safe drinking water is considered to be a human right, not a privilege, for every man, woman, and child. Economic benefits of safe drinking water services include higher economic productivity, more education, and health-care savings. Again, WHO/UNICEF defines safely managed sanitation facilities as improved sanitation facilities that are not shared with other households and where excreta are safely disposed of in situ or transported and treated offsite (UNICEF & WHO 2019). Improved sanitation facilities include flush/pour flush to piped sewer systems, septic tanks or pit latrines: ventilated improved pit latrines, compositing toilets or pit latrines with slabs. Basic and safely managed sanitation services can reduce diarrheal disease, and can significantly lessen the adverse health impacts of other disorders responsible for death and disease among millions of children. The report from WHO and UNICEF indicated that by 2017, 38% of the global population, representing 5.3 billion people, used safely managed drinking water services; an additional 1.4 billion used at least basic services. Furthermore, 206 million people used limited services, 435 million used unimproved sources, and 144 million still relied on surface water (UNICEF & WHO 2019). The report further indicates that eight out of ten people who still lack even basic services live in rural areas, with nearly half living in the least developed countries (LDCs). It is also noted that one out of three people using safely managed drinking water services (1.9 billion) resides in rural areas (UNICEF & WHO 2019).

By 2017, 54% of the global population, or 3.4 billion people, used safely managed sanitation services, while an additional 2.2 billion used at least basic services. Some 627 million people used limited services, 701 million used unimproved facilities, and 673 million still practiced open defecation. Seven out of ten people who still lack even basic services live in rural areas, and one-third of this population resides in LDCs (UNICEF & WHO 2019). In Bangladesh, the proportion of the population using safely managed drinking water services stood at 47.9% (urban: 44.7% and rural: 48.8%) in 2019 (MICS 2019). Additionally, 98.5% of the population (urban: 99.6% and rural: 98.2%) in Bangladesh used improved sources of drinking water in 2019, and 84.6% (urban: 90.6% and rural: 82.9%) had access to improved sanitation (MICS 2019). While globally, the proportion of the population using safely managed drinking water services increased from 61% in 2000 to 71% in 2017, 2.2 billion people worldwide still lack safely managed drinking water, including 785 million without access to basic drinking water. The proportion of the population using safely managed sanitation services increased from 28% in 2000 to 45% in 2017. However, 4.2 billion people worldwide still lack safely managed sanitation, including 2 billion without basic sanitation; of these, 673 million practiced open defecation (UN 2020).

Access to water and sanitation is crucial due to its significant impact on health, dignity, time, and economic losses. Unimproved drinking water and sanitation are the second biggest killers of children globally (Watkins 2006). Every year, 580,000 children die worldwide from waterborne diseases, including diarrhea (UNICEF 2014). Access to improved water sources and sanitation significantly reduces the incidence of waterborne diseases (Armah 2014; Pullan et al. 2014). Due to inadequate sanitation, India loses approximately US$ 53.8 billion annually from decreased productivity and increased health costs (World Bank 2010). Safe water and proper sanitation are essential for health, poverty reduction, and overall prosperity. Water is the common currency linking nearly every SDG, and it will play a critical role in achieving the goals related to energy, cities, health, the environment, disaster risk management, food security, poverty alleviation, and climate change, among others (HLPW 2018). To achieve SDG 6 along with all other SDGs, it is necessary to analyze the trends in access to safe drinking water and improved sanitation. Moreover, projecting future trends regarding access to water and sanitation is essential to meet the targets by 2030.

Analyzing time series data on access to water and sanitation levels and forecasting future trends are required for sustainable development. Time series analysis helps identify trends, behaviors, and the causes of limited access to water and sanitation. Time series forecasting is vital in policy decision-making, particularly concerning water and sanitation management, as it enables the identification of emerging trends and potential future scenarios based on historical data. Accurate forecasting informs policymakers about the availability and demand for water resources, thereby facilitating effective resource allocation (Ghannam & Hussain 2024). It enhances the capability to anticipate crises, such as droughts or floods, thereby allowing for proactive measures. Ultimately, effective time series analysis contributes significantly to achieving the targets set by SDG 6, promoting sustainable development and improving equitable access to water and sanitation services. This study aims to analyze the current trend and forecast the future projection of the proportion of the population using safely managed drinking water services and safely managed sanitation services, contributing to the achievement of the SDGs in Bangladesh.

For analyzing trends and forecasting the people using safely managed drinking water services and sanitation services, the yearly datasets used in this study are collected from the World Bank Databank, which are freely available at https://data.worldbank.org/indicator/SH.H2O.SMDW.ZS and https://data.worldbank.org/indicator/SH.STA.SMSS.ZS for the period 2000–2022 by the indicators ‘Proportion of population using safely managed drinking water services’ and ‘Proportion of population using safely managed sanitation services’, respectively. The datasets used in this study are given in Table 1.

The dataset size significantly influences the accuracy of time series models and the validity of long-term forecasts. With only 23 data points spanning from 2000 to 2022, the limited size may hinder robust parameter estimation, increasing the risk of overfitting the available data and reducing the model's generalizability to future scenarios (Hyndman & Athanasopoulos 2018). Smaller datasets may also result in higher variability in forecast errors, making predictions less reliable, particularly in the context of significant events or trends that may not be adequately captured (Box et al. 2016). Additionally, anomalies or outliers within a sparse dataset can disproportionately skew results, leading to potentially misleading conclusions (Makridakis et al. 2020). Thus, while the dataset allows for some trend analysis, caution is warranted when interpreting forecasts based on such limited historical data.

Multiple statistical methods exist for analyzing trends, spanning from simple linear regression to more sophisticated parametric and non-parametric techniques (Hesel & Hirsch 1992; Chen et al. 2007). It is crucial to choose an appropriate methodology that ensures accurate identification of all factors related to variability and change, as well as their impact on future predictions (Machiwal & Jha 2008). In this study, we employed the Mann–Kendall test, initially introduced by Mann (1945) and expanded upon by Kendall (1975), to perform trend analysis and determine the trend line's slope.

For time series modeling and forecasting, we applied the non-seasonal auto-regressive integrated moving average (ARIMA) model, as recommended by Box & Jenkins (1976). Additionally, we utilized double exponential smoothing, known as Holt's method, developed by Holt in 1957, to enhance predictions and accurately reflect trends within the data series (Holt 1957). The selection of the most appropriate forecasting model involved assessing various error metrics, including root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). The selected model will be used to predict the proportion of people in Bangladesh using safely managed drinking water and sanitation services.

The Mann–Kendall test is a statistical technique frequently used to examine trends in time series data. As a non-parametric method, it does not rely on specific assumptions regarding the distribution of the data (Kendall 1975). One of the key advantages of the Mann–Kendall test is its independence from the statistical distributions required for parametric approaches. In this context, the null hypothesis (H₀) asserts that there is no trend or serial correlation within the population being studied, while the alternative hypothesis (H₁) indicates the existence of a monotonic trend, which can either be increasing or decreasing.

The Mann–Kendall test is based on several assumptions concerning the time series data:

• (1) In the absence of a trend, the data points are independently and identically distributed (iid).

• (2) The measurements accurately reflect the actual states of the variables at the times they were taken.

• (3) The approaches used for sample collection, instrumental measurements, and data management are free from bias.

The Mann–Kendall test offers several advantages:

• (1) It does not require the data to follow any specific distribution, meaning it is not dependent on normal distribution.

• (2) It is not significantly affected by the presence of missing data, except that a reduced number of samples could negatively impact statistical significance.

• (3) It remains unaffected by the irregularity in the spacing of time points.

• (4) The length of the time series does not influence its validity.

However, there are some limitations to consider:

• (1) The Mann–Kendall test is not appropriate for data exhibiting periodicities (such as seasonal effects). To ensure the test's effectiveness, it is advisable to remove any known periodic influences from the data as a preprocessing step before applying the Mann–Kendall test.

• (2) The test generally yields more negative results with shorter datasets, indicating that trend detection is more reliable with longer time series.

The Box–Jenkins (Box & Jenkins 1976) procedure is widely considered the most effective method for selecting models, as it utilizes actual datasets to identify the best-fitting model. This approach offers several advantages, including the ability to minimize the number of estimated parameters and to evaluate seasonality within the data. Several authors (Islam et al. 2019; Gowthaman et al. 2022; Al-Khateeb 2023; Wang et al. 2024; Zafra-Mejía et al. 2024) employed the ARIMA model for forecasting and predicting either water and sanitation status or water quality. The ARIMA model development process consists of four main steps: identification, estimation, diagnostic verification, and forecasting (Box & Pierce 1970). The ARIMA model is one of the most commonly used time series models, characterized by its parameters (p, d, q). Here, ‘p’ represents the autoregressive (AR) coefficient, ‘d’ indicates the degree of differencing needed to achieve stationarity, and ‘q’ denotes the number of moving average (MA) components included in the model.

Initiating with a stationarity test on the time series, if the series does not meet the stationarity condition, methods such as differencing and logarithmic transformations can be applied to induce stationarity. The autocorrelation function (ACF) typically shows a gradual decay over several lags, suggesting minimal correlation, while noticeable spikes in the ACF plot imply the presence of q parameters. In contrast, the partial autocorrelation function (PACF) usually decreases over multiple lags, indicating a lack of significant correlation, with spikes suggesting the presence of p parameters.

Analyzing continuous data on the proportion of people utilizing safely managed drinking water and sanitation services is crucial for understanding variation patterns. The Mann–Kendall test is used to identify trends in the time series data regarding the use of these services. The application of the ARIMA model alongside Holt's method for time series analysis produces results that are then used to identify the most suitable model for forecasting. This selected model will be applied to project the proportion of people in Bangladesh using safely managed drinking water and sanitation services from 2023 to 2041.

The Mann–Kendall test is utilized to analyze trends in time series data related to the proportion of people who have access to safely managed drinking water and sanitation services. To detect seasonal trends in this data, the Mann–Kendall statistical method is employed. The null hypotheses being examined under the Kendall's tau trend test are:

• H₁-1: There is a trend in the proportion of people using safely managed drinking water.

• H₁-2: There is a trend in the proportion of people using safely managed sanitation.

In both instances involving the proportion of people using safely managed drinking water services and safely managed sanitation services, the results of the Mann- Kendall test, as mentioned in Table 2, indicate that the computed p-value falls below the established significance level of α = 0.05. Consequently, the alternative hypothesis, which asserts that there is a trend in the proportion of people utilizing safely managed drinking water and sanitation services in Bangladesh, is accepted. The positive values of Kendall's Tau coefficient (0.459 and 1.000) indicate that there is an upward trend in the proportion of people using safely managed drinking water services and sanitation services over the period analyzed.

Following the identification of the theoretical characteristics of the ACF and PACF as part of an AR process, it becomes clear that the ACF of the differenced series shows a gradual decline toward zero, attributable to the influence of the AR component. This decline effectively diminishes the effect of the MA component, thereby lessening any underlying trend. For the time series concerning the proportion of people using safely managed drinking water services, the appropriate ARIMA models include ARIMA (0,1,0), ARIMA (0,1,1), ARIMA (1,0,0), or ARIMA (1,0,1). In contrast, for the proportion of people utilizing safely managed sanitation services, the suitable ARIMA models can be ARIMA (0,1,0), ARIMA (0,1,1), ARIMA (1,2,0), or ARIMA (2,0,0), depending on the choice to include a lagged error term.

Table 3 presents the suggested models selected based on the principle of parsimony and the discernible patterns observed in the ACF and PACF. The best-fitting model from the ARIMA family is identified by analyzing the values of AIC and BIC.

Figure 6 indicates that the data on the proportion of people using safely managed drinking water and sanitation services in Bangladesh demonstrates a linear trend. Therefore, Holt's method is appropriate for analysis. This method utilizes two smoothing parameters, α and γ, to smooth linear trend data.

The decline in the coverage of safely managed drinking water services in 2022, as indicated by the data showing a drop from 59.13% in 2021 to 59.11%, represents a concerning reversal of the upward trend that had characterized access to safe drinking water in previous years. This decline may be attributed to several factors, including the lingering effects of the COVID-19 pandemic, which disrupted infrastructure projects and shifted governmental priorities. Additionally, environmental challenges such as extreme weather events and climate change could have adversely impacted water source availability and quality. Economic constraints may have further limited funding for water services, and rapid urbanization often strains existing resources, leading to reduced access among underserved populations.

The alpha parameter (α) is utilized to smooth the actual data while continuously adjusting the trend. Its values range from 0 to 1 (Mega 2003), and these values can be determined through trial and error. This parameter indicates the difference between the forecasted values and the actual data. As the alpha value approaches one, the emphasis on the most recent data increases, leading to a minimal smoothing effect. Conversely, when the alpha value approaches zero, the response to the latest data is reduced, resulting in a significant smoothing effect. Although theoretically both alpha (α) and gamma (γ) can take values of 0 and 1, in practice, they are typically determined within a limited range of values. This limitation arises because the choices for alpha (α) and gamma (γ) make the double exponential smoothing method easier to implement (Nurkse 1953). The gamma parameter (γ) adds some flexibility to the forecasted data. The results of the estimated parameters for the exponential smoothing model are presented in Table 4.

Table 4 presents the parameter estimates for the proportion of people utilizing safely managed drinking water and sanitation services in Bangladesh, evaluated using Holt's method. The results show that both models produced estimated values of alpha at 1.000 and 0.999 and gamma at 1.000 for the proportion of those using safely managed drinking water and sanitation services, respectively. The high alpha values (1.000 and 0.999) in both models suggest that the current level estimates heavily rely on the most recent data. Additionally, the gamma values (both 1.000) are also elevated, indicating that the current trend estimates are also based primarily on the latest observations.

Figure 7 indicates that for the proportion of people utilizing safely managed drinking water services, the out-of-sample forecast values generated by the ARIMA (0,1,0) model are closer to the actual values when compared to those from Holt's method. This suggests that the ARIMA (0,1,0) model provides a relatively accurate forecast for the proportion of people using safely managed drinking water services. Conversely, Figure 8 shows that for the proportion of people using safely managed sanitation services, the out-of-sample forecast values from Holt's method are closer to the actual values compared to the ARIMA (0,1,0) model. Therefore, Holt's method delivers a reasonably accurate forecast for the proportion of people using safely managed sanitation services.

In-sample forecasting involves formally assessing the predictive capabilities of developed models using observed data to determine how effectively the algorithms recreate this data. It is comparable to a training set in machine learning, while out-of-sample forecasting resembles a test set. In this study, the dataset spans from 2000 to 2022, with the in-sample forecasting period designated as 2000–2017 and the out-of-sample period set for 2018–2022. To evaluate the performance of the best-fitting models – ARIMA (0,1,0) and Holt's method – the study employed three forecast error statistics: RMSE, MAPE, and MAE. According to Hyndman (2006) and Hastie et al. (2009), the out-of-sample period typically yields smaller errors compared to the in-sample period, as the in-sample includes some relatively large observations. These error statistics were applied to both the in-sample and out-of-sample forecasts, and the results are shown in Table 5. Generally, the method with the lowest forecast error statistic is considered the most effective.

Table 5 shows that for both methods, the in-sample forecasting errors for the proportion of people using safely managed drinking water and sanitation services are greater than the out-of-sample forecasting errors.

The selection of the appropriate forecasting method is determined by the accuracy of the set of measurement values, with the method that yields the lowest forecasting error being deemed the most effective. Table 6 presents the calculated accuracy measures – RMSE, MAE, and MAPE – for the two chosen methods.

Table 6 indicates that for the proportion of people using safely managed drinking water services, the ARIMA (0,1,0) model outperforms Holt's method with the lowest RMSE, MAPE, and MAE values of 0.158, 0.213, and 0.120%, respectively. In contrast, Holt's method has RMSE, MAPE, and MAE values of 0.178, 0.284, and 0.148%. Conversely, when examining the proportion of people using safely managed sanitation services, Holt's method demonstrates superior performance compared to the ARIMA (0,1,0) model, achieving lower RMSE, MAPE, and MAE values of 0.125, 0.219, and 0.052%, while the ARIMA model's values are 0.166, 0.639, and 0.135%. Therefore, based on these results, it is concluded that the ARIMA (0,1,0) model is the most effective forecasting method for the proportion of people using safely managed drinking water services, while Holt's method is best for forecasting the proportion of people using safely managed sanitation services in Bangladesh. Zainun & Majid (2003) also noted that a MAPE value below 10% signals that the model for forecasting the proportion of people using safely managed drinking water and sanitation services performs exceptionally well, whereas a MAPE value between 10 and 20% indicates good performance. As shown in Table 6, the MAPE values of 0.284 and 0.219% for Holt's method regarding the proportions of people using safely managed drinking water and sanitation services, respectively, reflect that these forecasting models perform very well since their MAPE values fall below 10%.

Figure 9 reveals that the proportion of people utilizing safely managed drinking water services is expected to increase steadily from 2023 onward. The projected values for this proportion, along with the corresponding Government of Bangladesh (GoB) targets for the years 2023 to 2041, can be found in Table 7.

From Figure 9 and Table 7, it is apparent that the percentage of people using safely managed drinking water services is projected to rise linearly each year starting in 2023. By 2025, this proportion is expected to reach 60.6%, while the GoB has set a target of 75% for the same year as part of the 8th Five-Year Plan (GoB 2020). Furthermore, the forecast suggests that the proportion will increase to 63.7% by 2030, in contrast to the SDGs target 6.1, which aims for universal and equitable access to safe and affordable drinking water for everyone by 2030. If the current trajectory is maintained, the national targets and the SDG for water (indicator 6.1.1) is unlikely to be achieved by 2030. Therefore, the government, relevant authorities, and other stakeholders need to reassess existing policies and take necessary actions to reach both the national and SDG targets.

Figure 10 illustrates the actual variables, fitted values, forecasts, and a 95% confidence interval using Holt's method. In this graph, the smoothing constants are set at alpha (α) of 0.999 and gamma (γ) of 1.000. The analysis of the predictive models for the percentage of people using safely managed sanitation services in Bangladesh reveals a data pattern with a linear trend, making Holt's method particularly suitable for use. The forecasted values for the proportion of people using safely managed sanitation services in Bangladesh, based on Holt's method for the years 2023 to 2041, are presented in Table 8.

Figure 10 and Table 8 indicate that the percentage of people using safely managed sanitation services is projected to increase each year, following a linear trend. By 2025, this proportion is expected to reach 33.5%, while the GoB has established a target of 80% for safely managed sanitation services as part of the 8th Five-Year Plan (GoB 2020). Additionally, the predicted percentage of users is anticipated to rise to 37.6% by 2030. This contrasts with the SDGs target 6.2, which aims to ensure access to adequate and equitable sanitation and hygiene for all while also striving to eliminate open defecation, particularly for women, girls, and vulnerable populations by 2030. If the current trend persists, the national targets and the SDG for sanitation (indicator 6.2.1) is unlikely to be achieved by 2030. Therefore, the government, relevant authorities, and other stakeholders must reevaluate existing policies and take necessary actions to meet both the national and SDG targets.

Numerous international documents assert that access to improved water and sanitation is a fundamental human right and essential for the health of every individual. This study aims to analyze trends and forecast future projections regarding the proportion of the population utilizing safely managed drinking water and sanitation services in Bangladesh, with the goal of advancing the achievement of SDGs through the use of an ARIMA model. The annual datasets for this analysis were sourced from the World Bank Databank and are freely accessible at https://data.worldbank.org/indicator/SH.H2O.SMDW.ZS and https://data.worldbank.org/indicator/SH.STA.SMSS.ZS, covering the period from 2000 to 2022 for the indicators ‘Proportion of population using safely managed drinking water services’ and ‘Proportion of population using safely managed sanitation services,’ respectively. Initially, the Mann–Kendall test was conducted to identify seasonal trends in the time series data for both drinking water and sanitation services, using the dataset obtained from the World Bank. The ARIMA (0,1,0) model was found to be the best fit for forecasting the proportion of people utilizing safely managed drinking water services, in comparison to Holt's method. Conversely, Holt's method was determined to be the best fit for forecasting the proportion of people using safely managed sanitation services when compared to the ARIMA (0,1,0) model. The ARIMA model indicates a rising trend in the percentage of people with access to safely managed drinking water services, while Holt's method shows an increase in access to safely managed sanitation services in Bangladesh. However, these upward trends do not meet the adequate levels necessary to achieve SDG 6. Predictions suggest that by 2025, the proportion of people using safely managed drinking water services will reach 60.6%, and those using safely managed sanitation services will reach 33.5%. In contrast, the GoB has established targets of 75% for safely managed drinking water and 80% for safely managed sanitation services by 2025, as outlined in the 8th five-year plan (GoB 2020). Additionally, by 2030, the percentage of people using safely managed drinking water services is expected to rise to 63.7%, while those using safely managed sanitation services is projected to reach 37.2%. This analysis clearly indicates that if current trends persist, Bangladesh will not meet the national targets and the SDG targets 6.1 and 6.2 by 2030. Therefore, it is crucial for the government, relevant authorities, and other stakeholders to assess existing policies and implement necessary measures to meet both national and SDG objectives. To speed up the achievement of SDG targets for safe drinking water and sanitation services, the government and relevant stakeholders should emphasize increasing investments in infrastructure development to improve access, especially in underserved rural communities. They should also work on implementing impactful community engagement and education initiatives to promote awareness of hygiene practices. Furthermore, encouraging public-private partnerships can stimulate innovation and secure funding, while ongoing monitoring and assessment of current policies will help ensure that adaptable management strategies are developed to tackle new challenges.

The researchers would like to extend their appreciation to the World Bank for providing the dataset used in this study, which is accessible for free at https://data.worldbank.org/indicator/SH.H2O.SMDW.ZS and https://data.worldbank.org/indicator/SH.STA.SMSS.ZS, covering the period from 2000 to 2022.

This study received no specific support from public, commercial, or non-profit funding entities.

S. M. conceptualized the study, conducted the investigation, developed the methodology, performed data analysis and interpretation, and drafted the manuscript. S.C.B. reviewed and edited the manuscript. All authors have read and approved the final version of the manuscript.

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

The authors declare there is no conflict.