Climate change has made rainfall patterns more uneven and unpredictable. Recent advancements in machine learning and deep learning offer the capability to handle the complex, nonlinear nature of weather input parameters, leading to more reliable predictions. Therefore, in this study, trend detection and rainfall prediction using logistic machine learning and deep learning models have been carried out in the Bhopal region of central India. Trend analysis methods such as Mann–Kendall, Sen's slope, and Pettit test methods were applied to detect trends, estimate slope, and change points in weather parameters. The performance assessment of the logistic and deep learning model showed a higher F1 score on classification for the deep learning model (0.93) compared to the logistic model (0.56). The results revealed the greater capability of deep learning models for capturing the variations in rainfall compared to the logistic models. The sensitivity of the deep learning model was studied using gradients of the loss function (mean-square error) with respect to input variables. The gradient-based sensitivity measure revealed that rainfall was highly sensitive to RHmin, BSS, and RHmax. The deep learning model-based rainfall prediction may help with real-time decision-making for irrigation scheduling and others leading to water resource management.

  • Rainfall prediction was conducted using logistic machine learning and deep learning models.

  • Trend analysis used Mann–Kendall, Sen's slope, and Pettit tests.

  • A significant increasing trend was found at maximum temperature, while minimum temperature showed no significant trend.

  • Rainfall was significantly influenced by month, humidity, temperature, and wind velocity.

  • The deep learning model outperformed the logistic model.

Climate change is a complex and far-reaching global issue that affects nations worldwide, including India (IPCC 2021). This evolving scenario is characterized by several notable global trends, such as rising temperatures, shifting rainfall patterns, and increasing frequency of extreme weather events (Pachauri et al. 2014; Gaddikeri et al. 2024). Over the century, global temperatures have steadily increased, with the Earth's average temperature rising by roughly 1.2 °C (2.2 °F) since the late 1800s (Hansen et al. 2012). Recent decades have witnessed an accelerated warming trend, with some of the most recent times ranking among the warmest on record (IPCC 2021). The primary cause of this temperature increase is anthropogenic climate change (Gogoi et al. 2019), caused by conditioning similar to the burning of fossil fuels, leading to an increase in the Earth's surface temperatures. This warming trend has led to an increase in extreme weather events, including storms, droughts, floods, and wildfires (IPCC 2021), causing significant damage to ecosystems, contributing to biodiversity loss, and impacting economies. Furthermore, the melting of ice caps and glaciers, coupled with climate-induced ocean-level rise, poses a severe threat to coastal areas, aggravating coastal flooding and erosion and potentially causing lasting damage to both the environment and human societies (Pachauri et al. 2014).

Climate change has disrupted rainfall patterns, leading to prolonged droughts in some regions and heavy rainfall in others (IPCC 2021; Trenberth 2011). These changes have significant implications for agricultural productivity because rainfall plays a pivotal role in crop cultivation practices. In this context, rainfall is the most important factor because it has the potential to intensify the impact of hydrometeorological disasters (Irwandi et al. 2023). Prediction of rainfall provides valuable awareness to individuals, enabling them to anticipate rainfall in advance and take necessary precautions to safeguard their crops from potential damage (Basha et al. 2020). Moreover, globally, agriculture itself is responsible for a substantial portion (30–40%) of all greenhouse gas emissions (Vermeulen et al. 2012; Abbass et al. 2022), along with other major contributors, including the burning of fossil fuels, deforestation, land-use changes, and industrial processes (IPCC 2014).

The Paris Agreement was adopted in 2015, marking a significant global effort to combat the effects of climate change. The primary objective of the agreement is to limit global warming to well below 2 °C above preindustrial levels, with an even stronger goal of restricting it to 1.5 °C (Rogelj et al. 2016; IPCC 2021). However, recent projections suggest a potential global temperature increase ranging from 1.4 to 5.8 °C by 2100 (Balasubramanian & Birundha 2012; Collins et al. 2013). A climate change assessment report for the Hindu Kush Himalaya region indicates that even if global warming is limited to 1.5 °C, the Himalayan region could still experience a temperature increase of approximately 2 °C by 2100 (Wester et al. 2019; Karki et al. 2022). These trends highlight the urgent need for international cooperation and individual action to mitigate the impacts of climate change, preserve the environment, and secure a sustainable future (IPCC 2018; Rogelj et al. 2018).

India, a country with a substantial population and extensive low-lying coastal areas, is one of the world's most vulnerable nations to the consequences of climate change (Revi et al. 2014; IPCC 2021). Agriculture is a vital pillar of India's economy, and climate-induced changes have profound implications for crop production and food security. India faces significant challenges due to climate change, which necessitates proactive measures to mitigate its impacts on agriculture, water resources, and vulnerable communities. Adaptation strategies and sustainable practices are crucial in this regard. India has experienced increasing weather unpredictability, including temperatures surpassing 50 °C in certain regions and an erratic monsoon season with respect to its timing and rainfall amount. This is highlighted by research conducted by Chateau et al. (2023) and Rohini et al. (2016). The fluctuations in monsoon rainfall have a significant impact on agriculture and the nation's economy, and effective water resource management depends on them, as emphasized by Chakraborty et al. (2023) and Revadekar & Preethi (2012). In addition, studies under various greenhouse gas scenarios show a general decline in runoff availability in different river basins, as detailed by Balasubramanian & Birundha (2012) and Pervez & Henebry (2015). The Indian monsoon, a crucial weather system for agriculture and water resources, is becoming increasingly irregular, which is of great importance (Krishnan et al. 2019). Climate change has interfered with monsoon patterns, resulting in prolonged dry spells interspersed with intense rainfall events, leading to an increase in extreme weather phenomena (Goswami et al. 2006). The notable shifts in India's summer monsoon rainfall from 1951 to 2015 highlighted this vulnerability. During this time, there was a 6% reduction in summer monsoon rainfall, with the Indo-Gangetic Plains and Western Ghats experiencing the most significant declines (Krishnan et al. 2020). Furthermore, the frequency of daily rainfall extremes in central India, characterized by rainfall intensities surpassing 150 mm per day, increased by 75% from 1950 to 2015 (Roxy et al. 2017; Mukherjee et al. 2018). The escalating frequency of droughts in India has been attributable to the overall decrease in seasonal summer monsoon rainfall over the past 6–7 decades. Regions such as central India, the southwest coast, the southern peninsula, and northeastern India have experienced an average of more than two droughts per decade, each with unique consequences due to variations in agricultural geography. Additionally, the area affected by drought has seen a 1.3% increase per decade during the same period (Krishnan et al. 2020).

Therefore, India is facing the urgent need to adapt to climate change, ensure agricultural sustainability, and maintain a resilient food supply for its population (Gupta & Pathak 2016). Studies have examined the impacts of climate change on specific regions in India. For instance, Singh et al. (2015) studied urban flooding caused by heavy rainfall events in Bhopal, finding a decreasing trend in seasonal heavy rainfall events (≥65 mm). In another study, Duhan et al. (2013) examined temperature variations from 1901 to 2002 in Madhya Pradesh, central India, and found that annual mean, maximum, and minimum temperatures increased over the past century, with more significant warming in winter than in summer. Kundu et al. (2015) analyzed monthly rainfall data from 1901 to 2011 for 45 stations in Madhya Pradesh, identifying a significant decrease in annual and monsoon rainfall, with an overall annual decrease of −6.75% in rainfall over the 111-year period, which aligns with the findings.

Climate change impacts vary regionally and are influenced by a wide range of atmospheric factors, leading to uncertainties in rainfall projections (IPCC 2021). Measuring the effects of climate change on temperature and rainfall aims to predict the amount, spatial distribution, and variability of rainfall (Jain & Kumar 2012; Chauhan et al. 2022). Climate change studies continue to advance with the incorporation of several modeling strategies, including machine learning approaches. Machine learning offers the ability to analyze vast datasets rapidly, providing insights into complex climate patterns and trends. This aids in more accurate predictions and proactive mitigation strategies. Weesakul et al. 2018 developed a deep learning neural network (DNN) for forecasting monthly rainfall for optimum reservoir operation and water resources management for the eastern region of Thailand. The application of deep learning models to rainfall prediction signifies a substantial advancement in meteorological modeling, capacity to handle intricate, nonlinear relationships within high-dimensional datasets. Rainfall patterns are influenced by complex, nonlinear interactions among meteorological variables such as temperature, humidity, pressure, and wind speed. Adewoyin et al. 2021 used the deep learning approach to improve the prediction of high-resolution precipitation from coarse-resolution simulation of other weather variables. Praveena et al. 2023 applied logistic regression to accurately predict the rainfall and confirmed better accuracy of support vector machine compared to logistic regression. Analysis using traditional statistical models often struggles to effectively capture these intricate interactions. Deep learning models excel in identifying hidden patterns and relationships within large, multivariate datasets, offering superior predictive capabilities. However, to study the effects of climate change, it is necessary to analyze prevailing trends in climate conditions using long-term meteorological data. Detailed studies using advanced climatic models with machine learning approaches are required to assess the specific effects of climate change on temperature and rainfall. Regional studies can help identify relevant trends to develop strategies for adaptation and mitigation, minimizing the potential negative impacts of climate change.

The objective of this study is to analyze the trend and predict the rainfall using the logistic and deep learning model for understanding the complex dynamics of climate change due to system imbalances such as global warming and witnessing a particularly increase in the average temperature of earth and erratic rainfall.

This study utilized primary weather data to examine the impact of climate change in the study region. Weather data from 1983 to 2023 were collected from the Agro-meteorological Observatory at the ICAR-Central Institute of Agricultural Engineering (CIAE), Bhopal (India). Nonparametric techniques of the trend analysis were employed to estimate the trends and intensities of the weather parameters. Furthermore, machine learning algorithms were applied to capture variations in rainfall patterns. The weather parameters investigated in this study encompassed time-series data on rainfall, relative humidity, temperature, bright sunshine hours, evaporation, and wind velocity.

Site information

This study was carried out in Bhopal, Central India. Bhopal experiences a subtropical climate characterized by significant temperature fluctuations throughout the year. The Agro-meteorological Observatory of the Institute is situated at 23°16′ N latitude and 77°25′ E longitude, at an elevation of 498.7 m above mean sea level (msl), in the Bhopal district of Madhya Pradesh, India (Figure 1). During the summer months, from March to June, high temperatures typically range between 35 and 40 °C (95–104 °F), occasionally exceeding 40 °C owing to intense heatwaves. The winter season, from December to February, brings cool and dry weather conditions, with temperatures ranging from 10 to 25 °C (50–77 °F), making January the coldest month of the year. The region's average annual rainfall varies between 1,076 and 120 mm, based on the long-period rainfall data from 1983 to 2023. Typically, the monsoon arrives in Bhopal between June 10 and June 15, following its normal arrival pattern, and withdraws around the last week of October. However, the arrival and withdrawal dates of the monsoon season can vary slightly from year to year and are influenced by climate patterns and atmospheric conditions. The predominant cropping system in the area is soybean-wheat although rice cultivation has also gained prominence during the Kharif season in recent decades owing to the continuous failure of soybean crops. The soil type in the region is Vertisols, with a pH range of 6.4–9.1 and an electrical conductivity (EC) of 0.87–2.5 mS.
Figure 1

Sitemap of the study area: Bhopal district, Madhya Pradesh, India.

Figure 1

Sitemap of the study area: Bhopal district, Madhya Pradesh, India.

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Data structure and statistical tools and techniques

The primary data on weather parameters, including maximum temperature (Max T), minimum temperature (Min T), maximum relative humidity (RHmax), minimum relative humidity (RHmin), bright sunshine hours (BSS), evaporation (E Pan), and wind velocity (Wind V), were utilized on a date-, month-, and year-wise basis. Trend analysis for the minimum and maximum temperatures was conducted using the Mann–Kendall test (Mann 1945; Kendall 1975; Gilbert 1987). The Pettitt test (Pettitt 1979) was employed to detect change points in the minimum and maximum temperatures, while the magnitude of change in the minimum and maximum temperatures (slope) was estimated using Sen's slope estimator (Sen 1968).

Mann–Kendall test

The Mann–Kendall test is a nonparametric statistical test used to identify trends in a time-series dataset. The test calculates a statistic (S) based on the difference between the data points. Yan et al. 2022 applied the Mann–Kendall test to detect the trend of extreme precipitation indices. The Mann–Kendall test is distribution free and is not affected by outliers and thus making it the most widely used method for trend detection in climatic parameters.

The Mann–Kendall statistic is expressed as follows:
(1)
where sign (xjxk) = 1, if xj > xk, −1 if xj < xk, and 0 otherwise.
(2)
where n is the number of data points, g is the number of tied groups (a tied group is a set of sample data having the same value), and tp is the number of data points in the pth group. In the sequence {2, 3, nondetect, 3, nondetect, 3}, we have n = 6, g = 2, t1 = 2 for the nondetects, and t2 = 3 for the tied value 3. The associated Z value is expressed as follows:
(3)

Sen's slope estimator

Sen's slope estimator, named after Indian statistician M. Sen, is a statistical method used to estimate the slope of a linear trend in time-series data. This is the change in measurements per unit time (Sen 1968). It provides a quantitative measure of the change per unit time. Sen's slope estimator is defined mathematically as follows:
(4)

The confidence interval for Sen's slope is X(N−k)/2 and X(N−k)/2+1, where k = se × z, N is the total number of pairs [n(n − 1)/2], and X represents the slope for a certain pair.

Pettit test

The Pettitt test was named after J. R. Pettitt and is a nonparametric statistical test used to detect abrupt changes or shifts in a time-series dataset. It is a rank-based nonparametric statistical test used to detect change points in a data series (Pettitt 1979). Consider a sequence of random variables X1, X2, … , XT with a change point at time τ that is divided into two sets with distinct distribution functions: F1(Xt), t = 1, 2, … , τ, and F2(Xt), t = τ + 1, … , T. The null hypothesis may be written because there is no change point in the time-series data. The test statistic is given as follows:
(5)
The change point K is given as follows:
(6)
The p value can be evaluated as follows:
(7)
where T is the total number of observations.

Rainfall prediction

Logistic regression machine learning

Logistic regression (LR) is a popular statistical method used for binary classification tasks in machine learning. It is a supervised machine learning model. This type of statistical model (also known as the logit model) is often used in classification and predictive analytics. Logistic regression estimates the probability of the occurrence of an event. In this logistic regression equation, ɳi is the dependent or response variable and x is the vector of the independent variables. The beta parameter, or coefficient, in this model is commonly estimated using an iterative reweighted least-squares algorithm. The probability that an event belongs to a class ‘i’, i = 1, 2 … k for given independent variables x is given as follows:
(8)
(9)

Deep learning architectures

The deep learning model deployed on day-wise data from 2001 to 2022 including day-, month-, and day-wise weather parameters. The datasets contain columns with different types of data, numeric and categorical data. The preprocessing of data was done using one hot encoder to convert categorical variable into numeric variable, which is required for any deep learning model. This also improves the model performance by providing more information to the model about categorical variables. As deep learning models are sensitive to the scale of input variables, the dataset was normalized before model building. The entire datasets are split into training and validation datasets. The training set contained 85% of data points, whereas 15% of data points were used for model validation. A seed value was used to ensure that the same split occurs every time at the end of each epoch. This ensures that the same set of training and validation data will be used in training and validation, respectively, by the model in each and every epochs. The evaluation of model on same validation dataset after each epoch helped in monitoring for overfitting and hyperparameters tuning. A model is overfitting if the validation loss starts to increase while the training loss decreases at a particular epoch. The input layer had 19 input features [12 (for months) + 7 (weather parameters)]. The rectified linear unit (RELU) activation function was used for the neurons of the hidden layer, whereas the output layer had a single neuron with a sigmoid activation function widely used for the classification problem. The loss function used in training was binary cross entropy. Various deep learning architectures with different numbers of layers and neurons were fitted using the ‘Adam’ optimizer. The hyperparameters, including the learning rate, epochs, and batch size, were fine-tuned. The best model was selected based on the performance measures. Several performance metrics, such as accuracy, precision, sensitivity (recall), specificity, and F1 score, were evaluated for both logistic and deep learning models. The performance measures are given by Equations (10)–(14).
(10)
(11)
(12)
(13)
(14)

Statistical analysis and software used

Data entry, organization, and descriptive statistics were performed using Microsoft Excel (2016). The trend analysis was carried out in R 4.3 open-source software using Kendall, trend, and change point packages. The logistic model parameters (coefficients) were estimated using PROC LOGISTIC in SAS 9.3, SAS Institute Inc., Cary, NC, USA. The optimization of the weights for the deep learning model was performed in Python 3.11, an open-source software. The Python 3.11 library included numpy, pandas, tensorflow, sklearn, and matplotlib for analysis.

Trend analysis in rainfall and temperature data

The average annual rainfall of Bhopal was estimated to be 1,076 (±120) mm (Figure 2). There was no trend observed (Table 1) in the annual rainfall (p = 0.47). Sen's slope estimator indicated an insignificant increment in annual rainfall of 1.17 mm/year. An insignificant decrease in rainfall was observed in the months of June and August, estimated at 0.32 mm/year and 0.79 mm/year, respectively. While no trend was detected in the number of rainy days during the Kharif season, a significant increasing trend was observed in the annual series of rainy days and the number of rainy days during the Rabi season (Figure 3 and Table 2). In addition, the monthly rainfall data showed no increasing or decreasing trends over the years (Table 3).
Table 1

Trend analysis for annual rainfall

Mann–Kendall test
Pettit test
Sen's slope (mm/year)
Trendp valueChange pointp value
Total No 0.72 No (2021) 0.73 1.17 (p = 0.73) 
Kharif No 0.91 No (2021) 0.52 −0.41(p = 0.91) 
Rabi No 0.20 No (2018) 0.66 0.87 (p = 0.20) 
Mann–Kendall test
Pettit test
Sen's slope (mm/year)
Trendp valueChange pointp value
Total No 0.72 No (2021) 0.73 1.17 (p = 0.73) 
Kharif No 0.91 No (2021) 0.52 −0.41(p = 0.91) 
Rabi No 0.20 No (2018) 0.66 0.87 (p = 0.20) 
Table 2

Trend analysis for rainy days

SeasonMann–Kendall test
Pettit test
Sen's slope (days/year)
Trendp valueChange pointp value
Total Yes 0.02 Yes (1992) 0.02 0.33 (p = 0.01) 
Kharif No 0.19 – – – 
Rabi Yes 0.04 No (1995) 0.66 0.08 (p = 0.04) 
SeasonMann–Kendall test
Pettit test
Sen's slope (days/year)
Trendp valueChange pointp value
Total Yes 0.02 Yes (1992) 0.02 0.33 (p = 0.01) 
Kharif No 0.19 – – – 
Rabi Yes 0.04 No (1995) 0.66 0.08 (p = 0.04) 
Table 3

Trend analysis in monthly rainfall

MonthMann–Kendall test
Pettit test
Sen's slope (mm/year)
Trendp valueChange pointp value
January No 0.28 No (2013) 0.28 
February No 0.90 No (2003) 0.90 
March No 0.37 No (2006) 0.37 
April No 0.51 No (2006) 0.51 
May No 0.88 No (2007) 0.88 
June No 0.75 No (2019) 0.75 −0.32 
July No 0.36 No (2021) 0.33 1.85 
August No 0.70 No (2021) 0.35 −0.79 
Sep No 0.95 No (2018) 0.81 0.12 
Oct No 0.98 No (2018) 0.56 
Nov No 0.05 Yes (2010) 0.04 
Dec No 0.16 No (1997) 0.16 
MonthMann–Kendall test
Pettit test
Sen's slope (mm/year)
Trendp valueChange pointp value
January No 0.28 No (2013) 0.28 
February No 0.90 No (2003) 0.90 
March No 0.37 No (2006) 0.37 
April No 0.51 No (2006) 0.51 
May No 0.88 No (2007) 0.88 
June No 0.75 No (2019) 0.75 −0.32 
July No 0.36 No (2021) 0.33 1.85 
August No 0.70 No (2021) 0.35 −0.79 
Sep No 0.95 No (2018) 0.81 0.12 
Oct No 0.98 No (2018) 0.56 
Nov No 0.05 Yes (2010) 0.04 
Dec No 0.16 No (1997) 0.16 
Figure 2

Annual rainfall pattern.

Figure 2

Annual rainfall pattern.

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Figure 3

Annual trend in rainy days.

Figure 3

Annual trend in rainy days.

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The variation in minimum and maximum temperature over the years is presented in Figure 4. Based on the results of the trend analysis, it was found that the minimum temperature is increasing at a rate of 0.02 °C/year as estimated by Sen's slope, but the trend was not found significant (tau = 0.15, p = 0.18). A highly significant increasing trend was observed at the maximum temperature (tau = 0.32, p = 0.003). The Pettitt test indicated that the minimum temperature started increasing in 2013. The change in the maximum temperature likely occurred in 2015. It was found that the maximum temperature increased at a rate of 0.05 °C/year, implying a 1 °C rise at the maximum temperature every two decades.
Figure 4

Variation in minimum and maximum temperatures over the years.

Figure 4

Variation in minimum and maximum temperatures over the years.

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Regarding the monthly minimum and maximum temperatures (Tables 4 and 5), a significant increasing trend was observed over the years for all months, except December. However, no significant trend was observed for the maximum temperature for any month during the study period.

Table 4

Trend analysis in minimum temperature over the months

MonthTrendM-KChange pointSen's slope (°C/year)
January 0.01 2013 0.12 
February 0.03 2014 0.13 
March 0.02 2015 0.13 
April 0.002 2015 0.25 
May 0.001 2017 0.22 
June 0.002 1985 0.07 
July 0.001 1988 0.05 
August 0.005 2008 0.03 
September 0.002 2014 0.13 
October 0.002 2012 0.11 
November 0.003 2013 0.14 
December No 0.12 – – 
MonthTrendM-KChange pointSen's slope (°C/year)
January 0.01 2013 0.12 
February 0.03 2014 0.13 
March 0.02 2015 0.13 
April 0.002 2015 0.25 
May 0.001 2017 0.22 
June 0.002 1985 0.07 
July 0.001 1988 0.05 
August 0.005 2008 0.03 
September 0.002 2014 0.13 
October 0.002 2012 0.11 
November 0.003 2013 0.14 
December No 0.12 – – 
Table 5

Trend analysis in maximum temperature over the months

MonthTrendM-KChange pointSen's slope (°C/year)
January No 0.25 – – 
February No 0.32 – – 
March No 0.39 – – 
April No 0.14 – – 
May No 0.44 – – 
June No 0.88 – – 
July No 0.59 – – 
August No 0.22 – – 
September 0.03 1999 0.03 
October 0.01 2012 0.11 
November No 0.52 – – 
December No 0.09 – – 
MonthTrendM-KChange pointSen's slope (°C/year)
January No 0.25 – – 
February No 0.32 – – 
March No 0.39 – – 
April No 0.14 – – 
May No 0.44 – – 
June No 0.88 – – 
July No 0.59 – – 
August No 0.22 – – 
September 0.03 1999 0.03 
October 0.01 2012 0.11 
November No 0.52 – – 
December No 0.09 – – 

Rainfall prediction using logistic regression and deep learning models

Logistic regression

The relationship between the weather parameters and rainfall was developed using a logistic machine learning model and a deep learning algorithm to predict rainfall. The parameter estimates (coefficients) of the logistic model are presented in Table 6. The intercept term was found to be highly significant (p < 0.01) in predicting the phenomenon of rainfall within 1, 3, and 5 days. In predicting rainfall for 1 day, the months of April, February, and July had no significant influence on rainfall, whereas for 3 days, all months except April, August, and February had a significant effect on rainfall. All months except March were found to be significant in predicting the rainfall occurrence for 5 days. Relative humidity and minimum temperature significantly affected rainfall in all categories. The maximum temperature and evaporation had no significant effect on rainfall in any case, but wind velocity contributed significantly in all cases. No lack of fit was found in the fitted logistic models (p > 0.05). The results were also supported by the Wald chi-square test presented in Table 7. All independent variables (month, relative humidity, temperature, and wind velocity) had a significant effect on rainfall, except for BSS and E pan. The association between the predicted probability of rainy days and actual rainy days is presented in Table 8. The measured statistics in Table 8 describe the strength of the association between actual and predicted rainy days. The percent concordant showed the proportion of pairs (actual, predicted) where the predicted rainy day agreed with the actual rainy day in terms of relative ranking. A high percentage indicates the good predictive ability of the model. The measured percentage concordant (90.70%) showed a good agreement between actual and predicted rainy days. Percent discordant showed the proportion of pairs where the predicted rainy day did not agree with the actual rainy day. The lower value of the percentage discordant (9.10%) showed that the performance of the logistic model was not too bad. The percent tied (0.20%) showed the proportion of pairs where the predicted rainy days were identical to the actual rainy day. The ‘Somers’ D,’ ‘Gamma,’ ‘Tau-a,’ and ‘c’ are the indices of rank correlation for assessing the predictive ability of a model. The Somers' D statistics (0.82) showed that there was a strong positive correlation between the actual rainy day and the predicted rainy day. The Gamma statistics (0.82), a nonparametric measure of association measured as the difference between concordant and discordant pairs divided by the total pairs, showed a strong positive association between actual and predicted rainy days. Tau-a (0.20) suggested a moderate association, whereas concordance index c (0.91) indicated a satisfactory predictive ability of the logistic model.

Table 6

Coefficients of logistic model and their significance

ParametersEstimates with p value
Within 1 dayWithin 3 daysWithin 5 days
Intercept −8.7539* −6.0486* −4.7140* 
April 0.3491 0.1412 −0.3641** 
August −0.3074** 0.2442 1.0861* 
December −0.9922* −1.2830* −1.3596* 
February 0.1615 −0.2256 −0.4955* 
January −0.6323** −0.8528* −0.7409* 
July 0.272 0.9185* 1.6341* 
June 1.0872* 1.5182* 1.6857* 
March 0.9699* 0.6038* 0.1051 
May 0.9485* 0.7884* 0.4417* 
November −0.8399* −1.2654* −1.4797* 
October −0.6449* −0.6262* −0.8550* 
RHmax 0.0248* 0.0227* 0.0263* 
RHmin 0.0621* 0.0465* 0.0220* 
Min_T 0.1524* 0.1108* 0.0935* 
Max_T −0.0522** −0.0371 −0.0218 
BSS −0.0198 −0.0369** −0.0369** 
E_Pan 0.0372 0.00941 0.0118 
Wind velocity −0.0572** −0.0530* −0.0592* 
Goodness of fit 0.09 0.80 0.15 
ParametersEstimates with p value
Within 1 dayWithin 3 daysWithin 5 days
Intercept −8.7539* −6.0486* −4.7140* 
April 0.3491 0.1412 −0.3641** 
August −0.3074** 0.2442 1.0861* 
December −0.9922* −1.2830* −1.3596* 
February 0.1615 −0.2256 −0.4955* 
January −0.6323** −0.8528* −0.7409* 
July 0.272 0.9185* 1.6341* 
June 1.0872* 1.5182* 1.6857* 
March 0.9699* 0.6038* 0.1051 
May 0.9485* 0.7884* 0.4417* 
November −0.8399* −1.2654* −1.4797* 
October −0.6449* −0.6262* −0.8550* 
RHmax 0.0248* 0.0227* 0.0263* 
RHmin 0.0621* 0.0465* 0.0220* 
Min_T 0.1524* 0.1108* 0.0935* 
Max_T −0.0522** −0.0371 −0.0218 
BSS −0.0198 −0.0369** −0.0369** 
E_Pan 0.0372 0.00941 0.0118 
Wind velocity −0.0572** −0.0530* −0.0592* 
Goodness of fit 0.09 0.80 0.15 

Note: * and ** indicates the level of significance at 1 and 5%, respectively.

Table 7

Significance of independent variables on rainfall for logistic model

EffectDFWald chi-squarep value
Month 11 117.1486 <0.0001 
RHMax 15.4630 <0.0001 
RHMin 150.9556 <0.0001 
Min_T 39.9901 <0.0001 
Max_T 4.3611 0.0368 
BSS 1.4932 0.2217 
E_Pan 1.5322 0.2158 
Wind_Ve 11.1245 0.0009 
EffectDFWald chi-squarep value
Month 11 117.1486 <0.0001 
RHMax 15.4630 <0.0001 
RHMin 150.9556 <0.0001 
Min_T 39.9901 <0.0001 
Max_T 4.3611 0.0368 
BSS 1.4932 0.2217 
E_Pan 1.5322 0.2158 
Wind_Ve 11.1245 0.0009 
Table 8

Association of predicted probabilities and actual responses (rainy days)

StatisticsMeasurements
Percent concordant 90.70 
Percent discordant 9.10 
Percent tied 0.20 
Somers’ D 0.82 
Gamma 0.82 
Tau-a 0.20 
0.91 
StatisticsMeasurements
Percent concordant 90.70 
Percent discordant 9.10 
Percent tied 0.20 
Somers’ D 0.82 
Gamma 0.82 
Tau-a 0.20 
0.91 

Deep learning models

Numerous deep learning models with different architects were applied in fine-tuning of the hyperparameters. Architecture with one hidden layer containing five neurons was found to be the best, as depicted by the performance measures. The fine-tuned hyperparameters were (learning rate, 0.0001; batch size, 32; and epochs, 50) with the Adam optimizer. The fine-tuned hyperparameters are presented in Table 9. The results for training loss, validation loss, training accuracy, and validation accuracy versus epochs are shown in Figure 5, along with the corresponding confusion matrix. After 40 epochs, there was no further significant decrease in the training or validation loss. In addition, no increase in accuracy (training and validation) was observed after 40 epochs. Thus, the tuned number of epochs was set as 50. The deep learning model achieved an accuracy of 89% in predicting the rainfall for 1 day.
Table 9

Fine-tuned hyperparameters of deployed deep learning architecture

S NoHyperparametersValues
Input features 19 
Hidden layer 
Neurons in hidden layer 
Neurons in output layer 
Batch size 32 
Epochs 50 
Learning rate 0.001 
S NoHyperparametersValues
Input features 19 
Hidden layer 
Neurons in hidden layer 
Neurons in output layer 
Batch size 32 
Epochs 50 
Learning rate 0.001 
Figure 5

Loss and accuracy plots of deep learning models with confusion matrix: (a) for 1 day, (b) for 3 days, and (c) for 5 days.

Figure 5

Loss and accuracy plots of deep learning models with confusion matrix: (a) for 1 day, (b) for 3 days, and (c) for 5 days.

Close modal

Table 10 compares the performance measures of logistic and deep learning models. The accuracy of both models was found to be comparable. However, the correct classification of the rainfall phenomenon requires both precision and sensitivity in the prediction of rainfall. The precision and sensitivity of the deep learning model (0.96 and 0.90) were found to be better than those of the logistic model (0.66 and 0.48). A higher degree of accuracy indicates that there are very few instances of predicting rain that actually did not happen. If instances of predicting no rain when it does rain are less, then sensitivity will be higher. A good measure of performance is the F1 score, which refers to the harmonic mean of precision and sensitivity (recall) that confers greater weight to small items. Overall, based on several performance measures, the deep learning model outperformed the logistic model. In addition, the F1 score was also better for the deep learning model (0.93) compared to the logistic model (0.56). Of all the performance measures, only the specificity of the logistic model (0.96) was better compared to the deep learning model (0.63).

Table 10

Performance measures of logistic and deep learning model

Performance measuresEstimated values
Within 1 dayWithin 3 daysWithin 5 days
Logistic machine learning 
Accuracy 89.11 85.66 84.75 
Sensitivity 0.48 0.69 0.74 
Specificity 0.96 0.91 0.89 
Precision 0.66 0.73 0.77 
F1 score 0.56 0.71 0.76 
Deep learning model 
Accuracy 88.93 84.98 84.39 
Sensitivity 0.90 0.89 0.88 
Specificity 0.63 0.67 0.76 
Precision 0.96 0.88 0.89 
F1 score 0.93 0.89 0.88 
Performance measuresEstimated values
Within 1 dayWithin 3 daysWithin 5 days
Logistic machine learning 
Accuracy 89.11 85.66 84.75 
Sensitivity 0.48 0.69 0.74 
Specificity 0.96 0.91 0.89 
Precision 0.66 0.73 0.77 
F1 score 0.56 0.71 0.76 
Deep learning model 
Accuracy 88.93 84.98 84.39 
Sensitivity 0.90 0.89 0.88 
Specificity 0.63 0.67 0.76 
Precision 0.96 0.88 0.89 
F1 score 0.93 0.89 0.88 

Sensitivity analysis is essential in machine learning and helps in finding how changes in input variables affect the output of the model. It is mostly useful in scenarios where interpretability and model robustness are required. The gradient-based measure of sensitivity has been evaluated by computing the gradients of the output with respect to the inputs to identify most influential input variables. Mean-square error is utilized as the loss function in computing the gradients with respect to the input variables. The measured gradients for all the input variables have been presented in Table 11 and Figure 6. In Table 11, the sensitivity index of a particular input variable has been evaluated by subtracting the minimum of absolute gradients from the absolute gradient of the variable in question and dividing the result by the range of absolute gradients. The rainfall was highly sensitive to RHmin, BSS, RHmax, and the months such as July and August.
Table 11

Gradient-based sensitivity measure of input variables

Input variablesGradientAbsolute gradientSensitivity index (rank)
April −0.0106 0.011 0.048 (14) 
August 0.0349 0.035 0.545 (3
December −0.0107 0.011 0.050 (13) 
February −0.0102 0.010 0.039 (17) 
January −0.0103 0.010 0.042 (16) 
July 0.0329 0.033 0.504 (4
June 0.0092 0.009 0.018 (18) 
March −0.0107 0.011 0.051 (12) 
May −0.0121 0.012 0.079 (10) 
November −0.0106 0.011 0.047 (15) 
October −0.0109 0.011 0.055 (11) 
September 0.0083 0.008 0.000 (19) 
RHMax 0.0275 0.027 0.393 (5) 
RHMin 0.0572 0.057 1.000 (1
Min_T 0.0247 0.025 0.336 (6) 
Max_T −0.0178 0.018 0.196 (9) 
BSS −0.0424 0.042 0.697 (2
E_Pan −0.0210 0.021 0.259 (7) 
Wind_v 0.0185 0.018 0.209 (8) 
Input variablesGradientAbsolute gradientSensitivity index (rank)
April −0.0106 0.011 0.048 (14) 
August 0.0349 0.035 0.545 (3
December −0.0107 0.011 0.050 (13) 
February −0.0102 0.010 0.039 (17) 
January −0.0103 0.010 0.042 (16) 
July 0.0329 0.033 0.504 (4
June 0.0092 0.009 0.018 (18) 
March −0.0107 0.011 0.051 (12) 
May −0.0121 0.012 0.079 (10) 
November −0.0106 0.011 0.047 (15) 
October −0.0109 0.011 0.055 (11) 
September 0.0083 0.008 0.000 (19) 
RHMax 0.0275 0.027 0.393 (5) 
RHMin 0.0572 0.057 1.000 (1
Min_T 0.0247 0.025 0.336 (6) 
Max_T −0.0178 0.018 0.196 (9) 
BSS −0.0424 0.042 0.697 (2
E_Pan −0.0210 0.021 0.259 (7) 
Wind_v 0.0185 0.018 0.209 (8) 

The ranking of sensitivity index has been represented in parenthesis. The top four sensitivity ranks are highlighted with bold font.

Figure 6

Measured gradients of loss function with respect to input variables.

Figure 6

Measured gradients of loss function with respect to input variables.

Close modal

Trend analysis in rainfall and temperature data

The lack of significant trends in rainfall found in annual and monthly rainfall data for Bhopal, as indicated by the p values and Sen's slope estimator, suggests that rainfall patterns have remained relatively stable over the analyzed period. However, the observed increase in the number of rainy days, particularly during the Rabi season, could have implications for crop planning and water management strategies. The increased significance of most months for 3-day and 5-day predictions indicates that short-term weather patterns have more complex interactions that were evident over longer periods.

Moreover, minimum temperatures did not exhibit a significant trend, and the significant increasing trend in maximum temperatures, with an estimated rate of 0.05 °C/year, is concerning. This rise in maximum temperatures could exacerbate heat stress, affecting crop yields, water demand, and human health, necessitating adaptation measures such as heat-tolerant crop varieties, efficient irrigation systems, and improved heat action plans. The identified change point in maximum temperature around 2015, as detected by the Pettitt test, underscores the need for continuous monitoring and assessment of climatic variables to inform timely and appropriate adaptation strategies.

Rainfall prediction using logistic regression and deep learning models

Traditional statistical methods, which rely on linear assumptions, often fall short of accurately predicting rainfall. However, recent developments in machine learning and deep learning provide the solution to the complex, nonlinear nature of weather input parameters, leading to more reliable predictions. The linear model based on historical trends was found less effective in predicting extreme weather events or subtle climatic variations. Endalie et al. 2022 developed the deep learning model for daily rainfall prediction for Jimma, Ethiopia. A long short-term memory (LSTM) model was applied to predict daily rainfall in the Jimma region. The performance assessment revealed that deep learning outperformed the existing statistical methods. By employing logistic machine learning models and deep learning algorithms, the study aimed to create a robust predictive framework for rainfall. The significant intercept term across all prediction periods underscores the strong baseline relationship between the chosen parameters and rainfall occurrence. Relative humidity and minimum temperature consistently emerged as significant predictors of rainfall. This finding aligns with meteorological principles, as higher humidity often precedes rainfall, and minimum temperatures can influence condensation processes. The consistent significance of wind velocity across all prediction periods is noteworthy. Wind patterns can influence moisture transport and cloud formation, directly impacting rainfall likelihood. Finally, the strong association between predicted and actual rainy days, as shown in Table 8, validates the practical applicability of the model. This correlation demonstrates that the model can effectively translate weather parameter data into accurate rainfall predictions, making it a useful tool for weather forecasting and related applications.

In summary, the study's findings highlight the complexity of rainfall prediction and the importance of specific weather parameters. The results not only validate the chosen model but also offer a foundation for future research to refine and enhance rainfall prediction methods. Climate change projections and observed trends highlight the need for proactive adaptation measures to reduce vulnerability and build resilience, especially in sectors such as agriculture that are highly dependent on climatic conditions (Datta et al. 2022; Baraj et al. 2024).

The investigation aimed to quantify the impact of climate change in the Bhopal region of India based on historical time-series weather data spanning from 1983 to 2023. In this study on the effect of climate change on weather parameters, the following conclusions were drawn based on the results of tested hypotheses and the fitted logistic machine learning and deep learning models:

  • 1. The average annual rainfall of the region was estimated to be 1,076 (±120) mm, with no significant increasing or decreasing trend (1.17 mm/year) observed in annual rainfall (p = 0.47). A significant increasing trend in total rainy days and rainy days during the Rabi season was observed. Furthermore, no trend was detected in the number of Kharif rainy days.

  • 2. Minimum temperature showed no significant trend over the years (p = 0.18); however, it is increasing at the rate of 0.02 °C/year. There was a highly significant increasing trend in maximum temperature (p = 0.07). The maximum temperature is increasing at a rate of 0.05 °C/year.

  • 3. A significant increasing trend was observed in monthly minimum temperature over the years, except for December. However, no significant trend was observed in maximum temperature for any month over the years.

  • 4. The Wald chi-square test in the logistic machine learning model detected a significant effect of month, relative humidity, temperature, and wind velocity on rainfall, but BSS and E pan had no effect on rainfall.

  • 5. The deep learning architecture with one hidden layer containing five neurons was found to be the best in predicting rainfall (learning rate, 0.0001; batch size, 32; and epochs, 50).

  • 6. Based on the F1 score performance measure, the deep learning model performed better in predicting rainfall (F1 score: 0.93) compared to the logistic model (0.56). Of the five performance measures, only the specificity of the logistic model (0.96) was better compared to the deep learning model (0.63). Overall, it was concluded that the deep learning model outperformed the logistic model in capturing the variation in rainfall.

Rainfall prediction may play a critical role in crop production. The timely prediction of rainfall could help farmers to make decisions on sowing and harvesting of the crop. In addition, the accurate rainfall prediction is significant for optimizing irrigation schedules, which may result in better crop water productivity. The deep learning model may be further trained for multiple stations from other regions with long-term time-series data. The other influencing input variables (e.g. atmospheric pressure, net radiation, dew point) may be considered for the prediction of rainfall. The findings and methodology adopted in this study may provide valuable insights for researchers and academics interested in exploring climate change through statistical analysis and machine learning approaches.

The authors would like to thank ICAR-Central Institute of Agricultural Engineering, Bhopal, India, for providing Agro-meteorological Observatory data for this study.

Manoj Kumar: Conceptualization, data collection, data analysis, and original draft preparation; Mukesh Kumar: Conceptualization, data collection, and original draft preparation; Ranjay K. Singh: Original draft preparation, manuscript revision; Abhishek M. Waghaye: Conceptualization and interpretation; V. BhushanaBabu: Manuscript revision; Ravindra D. Randhe: Writing – review and editing.

This research received no external funding.

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

The authors declare there is no conflict.

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