Climate change detection and attribution: Bayesian estimation of abrupt change, seasonality and trend model, and Mann–Kendall trend test approaches Open Access

Climate change is projected to have adverse impacts on environmental sustainability. This research combines statistical analysis and Bayesian modeling for climate change detection and attribution in Kaduna, Northern Nigeria. The study combines the Bayesian estimation of abrupt change, seasonality, and trend model (BEAST) with the Mann–Kendall (M–K) trend test for detection and correlation analysis with optimum fingerprinting for attribution. The study used 122 years of climate data (1901–2022), focusing on average annual rainfall and average annual surface temperature for climate change detection, alongside a 30-year analysis of greenhouse gas (GHG) emissions (1990–2020) for climate change attribution. The result of the M–K test reveals a significant increasing trend in temperature (approximately 0.004 °C/year) and a decreasing trend in rainfall (approximately 0.756 mm/year), indicating a warming climate and potential drought conditions. The Bayesian approach further identified multiple changepoints in temperature data, highlighting years of significant climatic shifts. Correlation analysis demonstrated a weak positive relationship between temperature increases and GHG emissions with a correlation coefficient of 0.27. Optimum fingerprinting results show a non-statistically significant relationship between the variables with an R² value of 0.071, indicating that only 7.1% of the variability in temperature can be explained by the model.

The climate system is a complex, forced, dissipative, nonlinear, and heterogeneous entity that operates out of thermodynamic equilibrium (Ghil & Lucarini 2020). Climate change refers to changes in the statistical values of the climatic variables over the long term (at least decades) (Attah 2013; Mamudu 2021; IPCC 2022). Detection of climate change is a process of demonstrating that climate has changed in a statistically significant manner, without necessarily explaining the cause behind the change (EOS 2022). Attribution involves establishing the most likely causes for the detected change with a defined level of confidence (Zhai et al. 2018). While detection and attribution are interconnected, they serve separate objectives in understanding climate change or variability (Zhai et al. 2018; Mamudu 2021). Scientists use various methods to identify changes in climate observations, like the non-parametric Mann–Kendall (M–K) test (Gocic & Trajkovic 2013; Sharifi et al. 2024), Theil–Sen estimators (Ribes et al. 2017), regression, which also includes optimum fingerprinting (Lewis 2016; Knutson et al. 2017; Ribes et al. 2017; Hammerling et al. 2020), additive decomposition (Ribes et al. 2017), remote sensing (Pilewskie et al. 2015; Tahoun et al. 2016), the Bayesian method (Schnur & Hasselmann 2005; Katzfuss et al. 2017; Shi et al. 2024) and artificial neural network (Bône et al. 2023; Barman et al. 2024; Xalbaevich et al. 2024). These methods rely on expected responses to external forcing (such as greenhouse gas (GHG) emissions) either from physical understanding or as simulated by climate models (Kim et al. 2024). An identified change is ‘detected’ in observations if its likelihood of occurrence by chance due to internal variability alone is determined to be small. However, a failure to detect a particular response might occur for reasons like weak signals relative to internal variability or insensitive metrics. To reduce the risk of spurious detection, scientists look for corroborating lines of evidence that provide a consistent view of the likely cause for the detected changes (Knutson et al. 2017). In this study, the Bayesian estimation of abrupt change, seasonality, and trend model (BEAST) is used to corroborate the M–K trend test for detection, while optimum fingerprinting is used to corroborate correlation analysis for attribution.

The M–K test is a widely used method for detecting trends in time series data, particularly in climate variables. It assesses whether there is a significant monotonic trend, which could be either increasing or decreasing, over time. One of the advantages of the M–K test is that it does not assume any specific distribution for the data. Researchers often apply this test to various climate factors, such as temperature, precipitation, and streamflow. It is a non-parametric statistical method used to identify trends in time series data. In climate science, it is frequently utilized to analyze long-term climate data and determine if there is a noteworthy trend. Kendall's Tau statistic (τ) measures the correlation between data points, quantifying the strength and direction of monotonic trends (Gocic & Trajkovic 2013).

Bayesian networks (BNs), coined by Judea Pearl in 1985 (Aguilera et al. 2011; Holmes 2022), are graphical tools that represent probabilistic relationships among variables using directed acyclic graphs. They stem from Bayesian probability, developed by Thomas Bayes in the 18th century and further refined by Pierre–Simon Laplace (Mamudu et al. 2019; Diniz & Bellhouse 2020). BNs are used to predict the likelihood of causes given an observed event and can model complex relationships, such as between rainfall and streamflow. The BEAST is a Bayesian method used for climate change detection and time series analysis. It focuses on identifying abrupt changes, seasonal variations, and trends in data, making it particularly useful for analyzing complex datasets like those found in climate science (Zhao et al. 2019). It incorporates multiple competing models to improve inference accuracy instead of relying on a single model. It effectively identifies points in time where significant changes occur in the data, which is crucial for understanding climate shifts. The method can decompose time series data into seasonal components and long-term trends, providing a clearer picture of underlying patterns (Zhao et al. 2019).

Correlation measures the strength and direction of a linear relationship between two variables. In the context of climate change and greenhouse gases, it involves examining how changes in climate variables correlate with changes in greenhouse gases. The correlation coefficient, ranging from −1 to 1, indicates the nature of the relationship (Soren et al. 2023): + 1: perfect positive correlation, − 1: perfect negative correlation, 0: no correlation. All detection and attribution methods assume that observed climate change over time can be represented by a linear equation. This method can be applied to single patterns and extended to multiple patterns. Researchers aim to estimate the climate change signal and measure the signal-to-noise ratio by applying a suitable linear filter to the data.

Detection and attribution studies have found that Bayesian hierarchical methods can be used for detecting and attributing climate change (Katzfuss et al. 2017). M–K test, Sen's slope, and kriging were used to detect trends in climate variables (Gwatidaa et al. 2023; Arregocés et al. 2024; Haider et al. 2024). M–K trend test, Sen's estimator, Kendall tau, partial correlation, and participatory surveys were used to examine trends in rainfall, temperature, and runoff (Okafor & Ogbu 2018). It has been observed that greenhouse gases have been causing warming since the beginning of industrialization, with their impact increasing over time (Katzav 2013; Hegerl et al. 2019; Ahmed 2020; Bône et al. 2023).

One of the key novelties of this study is the combination of advanced methods for climate change detection. This study uniquely integrates BEAST with the M–K trend test for detecting climate change. This dual approach enhances the robustness and reliability of the detection process by corroborating results from two different statistical methods. This research provides specific insights into climate impacts in Kaduna, Northern Nigeria, a region that has not been extensively studied in the context of climate change. This regional focus addresses a gap in the literature and contributes valuable data and analysis for local climate policy and adaptation strategies. Another significant aspect of this research is the long-term data analysis. The study utilizes an extensive dataset spanning 122 years (1901–2022) for climate change detection, focusing on average annual rainfall and surface temperature in Kaduna, Northern Nigeria. This long-term analysis provides a detailed historical perspective on climate trends in the region.

Most studies on climate change focused on detection, others on attribution, while the detection studies mostly used trend tests (parametric or non-parametric). However, to fill this gap, this current study aimed to detect climate change by investigating both trend and abrupt changes in climate parameters and combining this with the attribution of the observed change to a possible cause. This dual-method approach enhances the reliability of the detection process, filling a gap where previous studies often focused on either trend detection or abrupt change detection separately. The methods commonly used by climate scientists include climate models, paleoclimate data, correlation analysis, optimal fingerprinting, and remote sensing. However, there is often no focus on identifying abrupt temperature changes. This study fills significant gaps in the literature by integrating advanced detection methods, utilizing long-term data, and focusing on a specific region. It combines detection and attribution, identifies multiple changepoints, provides quantitative trend analysis, demonstrates the complexity of climate change attribution, and offers detailed model diagnostics. These contributions will enhance the understanding of climate change dynamics and provide valuable insights for future research and policy development.

This study sought to answer three main questions and four follow-up questions. The primary questions are: What changes have occurred in the historical climate data of Kaduna, specifically in terms of rainfall and temperature? Are these changes due to natural variability (noise) or external forcing? Is it possible to attribute the change to a factor outside the natural climate system? The secondary questions include: If there are significant changes in temperature, are they indicative of warming (positive) or cooling (negative)? What is the rate of these changes? What specific years mark the points of change, and what are the probabilities associated with these change points? To find answers to these questions, the study employed the M–K trend test, BEAST, correlation analysis, and optimal fingerprinting. While the first two methods will be used in the climate change detection analysis, the last two will be used in the climate change attribution analysis.

The climate change detection study utilized the average annual rainfall (mm) and the average annual surface temperature (°C) for Kaduna Metropolis over 122 years (1901–2022). The climate change attribution study employed 20 years of data (1990–2020) on total GHG emissions, measured in kilotons of CO₂ equivalent (kt of CO₂e), along with the average annual surface temperature (°C) of Kaduna Metropolis. The study extracted 122 years of mean annual rainfall and average annual surface temperature data (1901–2022) for Kaduna from the World Bank Group Climate Change Knowledge Portal (WBG 2021). The climate change attribution study used total annual GHG emissions data for Kaduna from 1990 to 2020 sourced from the same portal (WBG 2023). Python version 3.12.4, along with IPython version 8.27.0, was used in an interactive environment on Spyder version 5.5.1, utilizing Anaconda version 24.7.1 for this analysis. The Python code scripts for each of these methods are shown in the Supplementary files 1–7.

This test was carried out to statistically assess whether there is a monotonic trend, either increasing or decreasing, in temperature or rainfall data. The analysis utilized 122 years of average annual rainfall and temperature records from Kaduna. A two-tailed M–K test was conducted, and the Kendall tau, Sen's slope, and p-value were determined. The hypotheses for the M–K trend test are:

• Null hypothesis (H₀): There is no monotonic trend in the time series data. This means that the data points are randomly ordered over time, and any apparent trend is a result of random variation.

• Alternative hypothesis (H₁): There is a monotonic trend in the time series data.

The Bayesian method was used for climate change detection, and average surface temperature data for the Kaduna metropolis from 1901 to 2022 were used for the analysis. The essence is to identify change points in the temperature of Kaduna. Rbeast, a tool designed for Bayesian evolutionary analysis, was used for this study, and Markov Chain Monte Carlo (MCMC) sampling plays a crucial role. This method was utilized to estimate the posterior distributions of model parameters.

The model coefficient of determination R², the root mean square error (RMSE), the marginal likelihood, and the variance of residuals (σ²) were all examined. While the M–K trend test provides the direction, density, and significance of the trend, the Bayesian method provides the specific change point years.

The climate change attribution addresses the following hypotheses:

• Null hypothesis(^bH₀): The observed changes in temperature are not attributed to greenhouse gases only.

• Alternate hypothesis (^bH₁): The observed changes in temperature are attributed to greenhouse gases only.

The correlation between temperature and GHG concentrations over the 30 years from 1990 to 2020 was analyzed by using Pearson correlation coefficients between temperature and GHG. Correlation measures the strength and direction of a linear relationship between two variables. In the context of climate change and greenhouse gases, it involves examining how changes in climate variables correlate with changes in greenhouse gases. The correlation coefficient, ranging from −1 to 1, indicates the nature of the relationship (Soren et al. 2023): + 1: perfect positive correlation, −1: perfect negative correlation, 0: no correlation. All detection and attribution methods assume that the observed climate change over time can be represented by a linear equation. This method can be applied to single patterns and extended to multiple patterns. Researchers aim to estimate the climate change signal and measure the signal-to-noise ratio by applying a suitable linear filter to the data. This analysis aimed to investigate whether a significant relationship exists between greenhouse gases and temperature in Kaduna.

In this case, n = 1.

As shown in Table 1, the drought aridity trend, showing the results of the M–K test for rainfall, indicates a decreasing trend over the analyzed period. This downward trend in rainfall is statistically significant, as evidenced by a p-value of 0.0164, which is below the 0.05 threshold (alpha value). This suggests that the observed decrease in rainfall is unlikely to be due to random variation. Further supporting this conclusion, Kendall's tau is −0.147, a negative value that confirms the decreasing trend. Additionally, Sen's slope is calculated to be −0.756, indicating that rainfall decreases by approximately 0.756 mm/year; this is an indication of drought in the Kaduna Metropolis.

In contrast, the M–K test for temperature reveals an increasing trend over the same period. The upward trend in temperature is highly significant, with a p-value of 0.0002, well below the 0.05 threshold. This very low p-value indicates that the increase in temperature is statistically significant. Kendall's tau for temperature is 0.227, a positive value that confirms the increasing trend. Sen's slope for temperature is 0.004, suggesting that temperature increases by approximately 0.004 °C/year; this indicates warming in Kaduna. In both cases, the study has sufficient evidence to reject the null hypothesis and accept the alternative hypothesis, which states that there is a monotonic trend in the time series data (both rainfall and temperature). Thus, while Kaduna is experiencing drought, it is simultaneously warming up. Similarly, climate change has been linked to drought in the western region of Bangladesh (Haider et al. 2024) as well as the Gorganrood watershed in northeast Iran (Tavosi et al. 2024).

The Bayesian method was applied to the temperature data to identify specific change points in years, and their respective probabilities and visualize the trend. The length of temperature data points used was 122, regularly spaced with a period of 1 unit.

The graph in Figure 4 illustrates the trend coefficients over the period from 1901 to 2022. The trend coefficient starts slightly above 25.4 °C and remains relatively stable until around 1930. This period of stability suggests that the factors influencing the temperature trend were consistent or balanced during these years. Around 1930 there was a noticeable sharp increase, going above 26 °C, and then followed by a sharp decline around 1944 in the trend, dropping to just below 25.2 °C. This abnormality is obviously a strong signal and could be attributed to external power force, likely significant historical events that may have impacted the variables affecting the temperature trend. World War II took place around this time; however, establishing a definite link will require further studies. After the sharp decline, the trend remains stable at the lower value until about 1970. This suggests a new equilibrium was reached, possibly due to post-war recovery and stabilization. From 1970 onwards, there was a gradual increase in the trend coefficient, becoming more pronounced in the early 2000s and reaching approximately 26 °C by 2022, without reversing, unlike the 1930–1945 era.

The graph in Figure 5 presents a clear visual representation of the rate of change of the trend. The slope shows relative stability until 1970, when the rate of change becomes permanently unstable. The event of 1930 was recovered quickly around 1944. Since the data were annual, the model is of seasonal order ‘1’.

The probabilities associated with these changepoints are derived from the Bayesian model and indicate the confidence level for each detected changepoint. Table 2 shows the years and their corresponding probabilities.

There are some changepoint detection years with high probabilities (P > 0.95), such as 1931 and 1944. There are also years with moderate probabilities (0.45 < P < 0.95) and those with lower probabilities (P < 0.45). The higher probabilities recorded in the earlier years indicate more stability and predictability in the climate. However, the lower probabilities and the frequency of occurrence of the changepoints in recent years show increasing instability and high volatility. Another important fact is the frequency of occurrence of these changepoint years. Between 1901 and 1944, only three changepoints were detected; however, between 1970 and the present, about seven changepoints have been detected, indicating increased climate instability in recent years. Lastly, the error component represents the random noise in the data, which is typically captured by the residuals of the model.

The model fit indicates that the temperature data is modeled as a combination of a trend component, a seasonal component, and a random error. R² is 0.609, indicating that the model explains about 60.90% of the variance in the temperature data. RMSE is 0.277, indicating the average deviation of the predicted temperatures from the actual values. The marginal likelihood and variance of residuals were −214.45 and 0.0903, respectively. A marginal likelihood of −214.45 indicates the logarithm of the probability of the observed data under the model. This value is useful for comparing different models; the model with the highest (least negative) marginal likelihood is generally preferred; however, the MCMC algorithm has already selected the model parameters that give the best fit. Lower values of suggest that the model's predictions are close to the actual data points, indicating a good fit. With an average temperature value of 25.5 °C, a variance of 0.0903 is a small value.

The correlation coefficient of 0.2657 between total GHG emissions and temperature signifies a positive relationship between these two variables. Correlation coefficients range from −1 to 1, where a value of 0 indicates no correlation, values approaching 1 suggest a strong positive correlation, and values nearing −1 indicate a strong negative correlation. In this context, a coefficient of 0.2657, approximately 0.3, suggests a moderate positive correlation, meaning that as GHG emissions increase, temperatures tend to rise as well, albeit the relationship is not particularly strong. This correlation implies that while there is a tendency for temperature to increase with rising GHG emissions, it does not establish GHG emissions as the sole or primary cause of temperature changes. Other factors may also play significant roles in influencing temperature fluctuations. Thus, while the correlation supports the notion that increasing GHG levels contribute to global warming, it also underscores the complexity of climate systems, where multiple variables interact. This finding suggests that efforts to reduce emissions could be beneficial in mitigating temperature increases. However, further analysis is necessary to fully understand the dynamics at play and the various factors influencing climate change.

The regression analysis results, as shown in Table 3, reveal that the model has a low R² value of 0.071, indicating that only 7.1% of the variability in temperature can be explained by total GHG emissions. This suggests a weak explanatory power, as the model does not account for much of the variation in temperature. The adjusted R² value of 0.039 further emphasizes this limitation by adjusting for the number of predictors. In terms of statistical significance, the F-statistic is 2.202 with a corresponding p-value of 0.149, indicating that the overall model is not statistically significant at the conventional alpha level of 0.05. Consequently, there is insufficient evidence to conclude that GHG emissions have a significant effect on temperature rise in Kaduna Metropolis. The constant term is estimated at 24.93 °C, which is statistically significant (p < 0.001), suggesting that when total GHG emissions are zero, the predicted temperature is approximately 24.93 °C. This implies that at zero GHG emission, Kaduna's average surface temperature should be approximately 24.93 °C.

When this is compared to the Kaduna state's average temperature of 2022, which stood at 25.87 °C, a difference of 0.94 °C, approximately a 1 °C rise in average temperature, is observed. The coefficient for total GHG emissions is 3.319 × 10⁻⁶, indicating a negligible increase in temperature per unit increase in emissions, but this coefficient is not statistically significant (p = 0.149), highlighting a weak and unreliable relationship.

A multitude of factors, including solar radiation, volcanic activity, ocean currents, and land use, changes influence climate systems. These factors interact in complex ways, making it difficult for a single-variable model to capture all the variability in temperature. Temperature changes can vary significantly over time and across different regions. A model that does not account for this variability may have a low R² value. For instance, local factors such as urbanization and deforestation can have significant impacts on temperature. Inaccuracies in the measurement of GHG emissions or temperature can lead to errors in the model, reducing its explanatory power. These errors can arise from limitations in data collection methods or inconsistencies in historical records. The relationship between GHG emissions and temperature may not be strictly linear. Nonlinear models or models that include interaction terms might better capture the complexity of the relationship.

Given the weak relationship between GHG emissions and temperature in the current model, it is important to consider other factors that might influence temperature variability. Variations in solar radiation due to changes in the Earth's orbit or solar cycles can significantly impact global temperatures. These variations can cause long-term climate changes independent of GHG emissions. Volcanic eruptions can inject large amounts of aerosols into the atmosphere, reflecting sunlight and causing temporary cooling. The impact of volcanic activity on temperature can be significant, especially in the short term. Ocean currents play a crucial role in distributing heat around the planet. Changes in ocean circulation patterns, such as El Niño and La Niña events, can lead to significant temperature anomalies. Deforestation, urbanization, and changes in land use can alter the Earth's surface properties, affecting local and regional temperatures. These changes can influence the albedo effect, evapotranspiration rates, and heat retention. Besides GHGs, other atmospheric constituents such as aerosols, water vapor, and ozone can influence temperature. The interactions between these components and their combined effect on radiative forcing are complex and can contribute to temperature variability.

The study provides robust statistical evidence of climate change trends and the percentage of those trends attributable to GHG emissions. This information is crucial for policymakers to develop and implement effective climate policies and regulations aimed at mitigating climate change impacts. By identifying significant trends and changepoints in temperature and rainfall, the study helps in formulating adaptation strategies. The Kaduna region is experiencing decreasing rainfall and increasing temperature; hence, it should develop water management plans, invest in rainwater harvesting systems, and promote drought-resistant agricultural practices. The study's focus on Kaduna, Northern Nigeria, provides specific insights into the region's climate dynamics. This localized information is essential for developing tailored solutions that address the unique challenges faced by the region, such as water scarcity and agricultural productivity.

The findings of the study can be used to raise public awareness about the impacts of climate change. Educating communities about the observed trends and their potential consequences can empower individuals to adopt more sustainable practices and advocate for environmental policies. The integration of advanced statistical methods and long-term data analysis contributes to the scientific understanding of climate change. The study's methodology and findings can serve as a reference for future research, promoting further exploration and innovation in climate science. By understanding the specific climate trends and their causes, communities can enhance their resilience to climate change. This includes preparing for extreme weather events, improving infrastructure, and ensuring food and water security. The detailed analysis and robust statistical evidence provided by the study support informed decision-making at various levels, from government agencies to local communities. This leads to more effective and targeted interventions to combat climate change. Implementing adaptation strategies based on the study's findings can lead to economic benefits. For example, improving water management and agricultural practices can enhance crop yields and reduce the economic losses associated with droughts and other climate-related events.

This research reveals significant findings regarding historical climate data of Kaduna from 1901 to 2022. The analysis indicates a statistically significant increasing trend in average annual temperatures, with a notable rise of approximately 0.004 °C/year. This trend suggests a warming climate in Kaduna State, corroborated by the Bayesian method, which identified multiple changepoints, particularly in the years 1931 and 1944, indicating periods of significant temperature shifts and high-temperature volatility in recent years, 1970 onward. Conversely, the study found a significant decreasing trend in annual rainfall, with a decline of about 0.756 mm/year. This trend points toward increasing drought conditions, which could have serious implications for agriculture and water resources in the region. Both the M–K test and Bayesian methods provided robust statistical evidence to reject the null hypothesis, affirming that the observed changes in climate parameters are not merely due to natural variability but are influenced by external factors, including anthropogenic GHG emissions.

In terms of attribution, the research utilized correlation analysis and optimal fingerprinting methods to establish the likelihood that observed temperature changes are attributable to GHG emissions. The correlation coefficient of 0.2657 indicates a moderate positive relationship, suggesting that while there is a tendency for temperature to rise with increasing GHG emissions, this relationship is not strong enough to assert that GHG emissions are the sole cause of temperature changes. Hence, the null hypothesis in this case could not be rejected. However, anthropogenic GHG has caused an increase of 1 °C in the temperature of Kaduna State during the study period (1990–2022). The low R² value in the current climate model highlights the need for a more comprehensive approach to understanding temperature variability. By considering additional factors such as solar radiation, volcanic activity, ocean currents, land use changes, and atmospheric composition, future research can develop more accurate models that better capture the complexity of climate dynamics.

Given the observed decrease in rainfall, it is crucial to develop sustainable water management strategies. This could involve investing in rainwater harvesting systems and promoting drought-resistant agricultural practices. Increasing public awareness about climate change and its impacts can empower communities to adopt more sustainable practices and advocate for environmental policies. Continued research is essential to monitor climate trends and their socio-economic impacts. Future studies should explore the effects of climate change on local ecosystems and vulnerable populations, as well as the effectiveness of adaptation strategies. Climate change trends in Kaduna should be compared with other regions in Nigeria to identify broader patterns and localized effects. Advanced climate modeling techniques should be utilized to predict future climate scenarios and their potential impacts, incorporating variables such as land use changes and population growth.

The authors extend their appreciation to Prince Sattam bin Abdulaziz University for funding this research work through the project number (PSAU/2024/01/29617).

This research was funded by Prince Sattam bin Abdulaziz University (Project number PSAU/2024/01/29617).

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

The authors declare there is no conflict.