## Abstract

The present paper reports a study, where we have developed a methodology to understand the relative uncertainty associated with the rainfall amount corresponding to summer monsoon (JJAS) and post monsoon (OND) for the period 1871–2016 over northeast India. After calculating the partial correlation between two random variables after the removal of the effect of the third one, we have standardized all the realizations of the random variables. Subsequently, after applying the Dempster–Shafer theory, we have obtained joint basic assignments through two judging criteria for the fuzzy sets representing the closeness of the observed values to two measures of central tendency for different window sizes obtained from the original time series. The study revealed a higher rate of increase in the uncertainty with a change in the window size for OND than in the case of JJAS. This study finally concluded that this approach could generate some idea about the most advantageous ratio of training and test cases for predictive models with supervised learning procedures.

## HIGHLIGHTS

Using the Dempster–Shafer theory, we are able to determine joint basic assignments for fuzzy sets.

For OND, the study found a larger increasing pattern in uncertainty with a change in the window size than for JJAS.

We have seen that for OND, the surface lifts significantly as the leading coefficient of the polynomial changes. But, in the case of JJAS, no such significant effect is being noticed.

## INTRODUCTION

The concept of belief functions, also known as the Dempster–Shafer Theory (DST) (Denoeux 2000; Sentz & Ferson 2002; Dezert *et al.* 2012; Kohlas & Monney 2013; Xiao 2020), is a generalization of the Bayesian theory of subjective probability. Instead of requiring probabilities for each question of interest as in the Bayesian theory (Weise & Woger 1993; Rouder & Lu 2005; Ghosh *et al.* 2006; Karni 2007; Bernardo & Smith 2009; Damien *et al.* 2013; Watanabe 2018), belief functions help us build degrees of belief for one question on probabilities for another. These degrees of belief may or may not have the mathematical characteristics of probability, depending on how closely the two problems are related. The DST actually originated from studies by Dempster (1968) and Shafer (1992), but the logic it applies dates back to the seventeenth century. In the early 1980s, when artificial intelligence (AI) researchers (Zadeh 1983; Neapolitan 1990) were seeking to adapt probability theory to expert systems, they came across this theory. Further research has demonstrated that managing uncertainty demands more structure than is accessible in basic rule-based systems, the DST retains its value due to its relative flexibility (Peñafiel *et al.* 2020).The DST (Shafer 1992) is built on two concepts: generating degrees of belief for one issue from subjective probability for a related matter and using Dempster's rule to combine such degrees of belief when they are based on independent pieces of evidence. In general, we derive degrees of belief for one question from probabilities for another. Dempster's rule starts with the concept that the questions for which we have probabilities are independent with respect to our subjective probability judgements but this independence is just a priori; it vanishes once the conflict between the various pieces of evidence is discovered. In order to apply the DST to a specific problem, you must first solve two related problems. First, we must sort the problem's uncertainties into a priori independent pieces of evidence. Second, we must apply Dempster's rule in a computational manner. These two issues, as well as their remedies, are interconnected. Sorting out the uncertainties into individual components yields a structure incorporating elements of evidence that are relevant to various but related questions, which can be used to make computations feasible. Beynon *et al.* (2000) described the DST of evidence's potential as a significant upgrade over ‘traditional’ techniques for decision analysis. The DS approaches are based on Dempster's work on probabilities with upper and lower bounds. They have since gained a lot of grips in the literature on AI and Expert Systems, with a focus on incorporating evidence from many sources. The fundamental ideas of the DST of evidence are introduced in this work, with a brief mention of its origins and similarities to the more conventional Bayesian theory. Following that, they have gone through recent developments in this theory, as well as analytical and application topics of relevance. Finally, they have examined future developments using an example that combines DST with the analytic hierarchy process (AHP).

Back in the 80s, in a pioneering paper, Zadeh (1986) demonstrated how the DST of evidence was having a lot of potentials for the AI field to deal with uncertainty in expert systems. Zadeh (1986) explained how the DST can be regarded in the context of relational databases as the application of conventional retrieval techniques to second-order relations in first normal form. The relational approach clarifies some of DST's controversies and facilitates its implementation in AI-related applications. Nachappa *et al.* (2020) worked on the flood susceptibility maps for the Austrian province of Salzburg by various models supplemented by the DST to optimize the arising flood susceptibility maps based on some flood conditioning factors. In another study, Nachappa *et al.* (2019) demonstrated how the incorporation of DST in the study of landslide susceptibility across Austria could improve the outcomes. In a unified model based on DST, a probabilistic representation of rainfall uncertainty merged with a fuzzy representation of model parameters, Fu & Kapelan (2013) analysed the uncertainty associated with severity probabilities of flood quantities. Al-Abadi (2017) used the DST of evidence in a GIS platform, as an attempt to designate aquifer sensitivity zones for nitrate contamination in the Galal Badra basin, east of Iraq.

In this work, we endeavour to implement DST to understand the intrinsic pattern of rainfall time series over northeast India. At this juncture, let us have a brief overview of the literature works that have reported different climate parameters over northeast India. Jain *et al.* (2013) analysed the rainfall trend in three-time series but did not find any clear trend for the period 1871–2008 over the northeast region also a similar examination of temperature data revealed a growing trend in all four temperature variables over the same region. For observational trend detection of monthly, seasonal, and yearly precipitation in five meteorological subdivisions of Central northeast India during different 30-year normal periods, the Mann–Kendall non-parametric test was used by Subash *et al.* (2011). The maximum and minimum temperature trends were also studied and also the least square linear fitting was employed to obtain the slopes of the trend lines. Mahanta *et al.* (2013) discussed the characteristics of heavy rainfall over the region of northeast India. The seasonal and spatial variations of heavy rainfall occurrences are investigated using daily rainfall data from 15 rainfall stations for a 31-year period (1971–2001) over the same region. Narasimha Murthy *et al.* (2018) investigated and studied the empirical analysis of the time series and used the methodology named the Box-Jenkins Seasonal Autoregressive Integrated Moving Average (SARIMA) for model identification, diagnostic verification, and forecasting of Southwest monsoon rainfall patterns in northeast India. The Mann–Kendall non-parametric test is used by Jhajharia *et al.* (2012) to analyse rainfall trends, rainy days, and 24-h maximum rainfall at 24 sites in subtropical Assam in India's north-eastern region. Both parametric and non-parametric approaches are used to confirm the trends, and the linear regression test is used to determine the magnitudes of significant trends. Mishra *et al.* (2013) considered the possibility of employing short-term weather forecasts to improve rice irrigation efficiency as a possible adaptation to future climate change. To generate alternate irrigation schedules, the rainfall predictions were included in the agro-hydrological model SWAP (Soil Water Plant Atmosphere). To verify the SWAP model, they used data from the field experiments. In a recent study, Varotsos *et al.* (2019) developed an information-modeling tracker to create atmosphere–ocean instability maps in different aquatories based on meteorological stations and meteorological satellites data. Efstathiou & Varotsos (2012) discussed the importance of estimating the prediction of precipitation and rainfall on a short- and long-term basis from the perspective of how substantial changes in rainfall are closely associated with severe socioeconomic and ecological consequences. In a recent study, Varotsos *et al.* (2020) developed a new nowcasting tool to estimate the average waiting time for the extreme values of some climate parameters with a significant accuracy level. This newly introduced tool by Varotsos *et al.* (2020) might help to understand climate variability better to understand the interactions between climatic effects, mitigation, adaptation, and sustainable growth.

The organization of the remaining part of the paper is as follows: In Section 2, we have discussed the details of the data. Afterward, we discussed the theoretical information of belief measures and DST. Section 3 of the paper presents the implementation procedure of the DST to the time series of rainfall amount corresponding to summer monsoon (JJAS) and post monsoon (OND) for the period 1871–2016 over northeast India. We have concluded in Section 4.

## METHODOLOGY

### Data

The observational data on rainfall over northeast India, a meteorological subdivision of India for the period 1871–2016 are collected from the website of Indian Institute of Tropical Meteorology (IITM). Link to the data is https://tropmet.res.in/static_pages.php?page_id=52#data. Parthasarathy *et al.* (1993) developed the homogenization procedure, and IITM documented the methodology through copyrighted material on the IITM website. The data we have utilized in the current study are homogeneous, and the IITM homogenized it using the procedure developed in Parthasarathy *et al.* (1993). Parthasarathy *et al.* (1987, 1995) are two other critical references in this direction. The methodology is available on the IITM in the document copyrighted by the IITM, Homi Bhabha Road, Pune 411008, India. For further details, see https://tropmet.res.in/static_pages.php?page_id=53.

According to the India Meteorological Department (IMD), the meteorological subdivisions that correspond to northeast India are (i) Assam and Meghalaya (A&M); (ii) Arunachal Pradesh (ArP); and (iii) Nagaland, Mizoram, Manipur, and Tripura (NMMT) (Mohapatra *et al.* 2011). Therefore, rainfall over northeast India is not for any single station. In a noteworthy study, Mahanta *et al.* (2013) demonstrated heavy rainfall events over this meteorological subdivision from 1971 to 2001 and identified the most favourable locations between 27.5°N and 28.1°N for serious rainfall occurrence. Kumar & Singh (2021) analyzed the heterogeneity in rainfall patterns over northeast India. They studied the linkages between El Niño and the Indian summer monsoon for the period from 1600 to 2016. Works that have studied the precipitation over the adjacent study zones using various information-theoretic methodologies include Saha & Chattopadhyay (2020), Chakraborty & Chattopadhyay (2021) and Sharma & Chattopadhyay (2021).

### Belief measure

The fuzzy measure theory, which has been thoroughly discussed Wang & Klir (1992) will give us a broad framework within which we may introduce and explore possibility theory, a subject that is closely related to fuzzy set theory and plays a key role in certain of its applications. Wierzcho (1983) connection between fuzzy measures and Lebesgue measures. A noteworthy work in this context is by Leszczyński *et al.* (1985), where they proposed a clustering algorithm using the properties of Sugeno's *g _{λ}* measure. In another work, Grabisch

*et al.*(1992) proposed a general definition of fuzzy measures of fuzzy events compatible with the earlier descriptions of Zadeh. The proposed definition by Grabisch

*et al.*(1992) possesses all the properties of a fuzzy measure, particularly the duality property. A detailed review on the evolution of the theory of fuzzy measure was presented in Garmendia (2005). The fuzzy measure theory allows us to explain how fuzzy set theory and probability theory differ (Wang & Klir 1992). In order to classify fuzzy measure, the function

*g*must meet certain criteria. The fundamental axioms of probability theory were considered to be these necessary features in the past, but that assumption was incorrect. Probability measures are a special form of fuzzy measure since they are defined by weaker assumptions. For the universe of discourse

*X*, the axioms of fuzzy measures are as follows:

For every if then

For every sequence of subsets of *X*, if either or

(i.e., the sequence is monotonic), then

For every and every collection of subsets of *X*.

*X*belongs to the set

*A*but not to any special subset of

*A*, or the degree to which we believe such a claim is justified.

### Dempster–Shafer theory

A generalized approach for representing uncertainty is the DST (Klir & Folger 2015). It consists of sets of prepositions rather than single propositions which assigns to each set an interval where the degree of belief must lie on that set. This theory is described as an extension of probability theory. The DST is especially beneficial when each piece of evidence implicates numerous potential conclusions and the support for each individual conclusion is computed from the overlapping contributions of various pieces of evidence. In target identification applications and tactical inferencing, the DST has been employed to assign a degree of belief (Klir & Folger 2015). The theory of evidence must be presented before the facts may be combined. Because it deals with the weight of evidence, it is termed as a theory of evidence.

*n*mutually exclusive and exhaustive propositions, , where 0 is referred to as a discernment frame. Thus, the Boolean operation OR can be used to build propositions; is the set of all the subsets of 0. To attribute evidence to a proposition, Dempster–Shafer introduced the concept of mass probability, which is denoted by where,

*X*is the universe of discourse.

The methodology presented above would be implemented in Section 3.

## RESULTS AND DISCUSSION

In this section, we are assuming the random variables , , and which represent annual (ANN), post monsoon (OND), summer monsoon (JJAS), and pre monsoon (MAM) rainfall, respectively, in northeast India from 1871 to 2016. We have calculated the partial correlation between all the random variables stated above. For example, we have calculated the partial correlation between and after removal of the effect of . In this way, we have calculated the rest of the pairs. After calculating the partial correlation, we have standardized all the values.

*y*denotes the realization of the time series at some time point, we apply the following transformation to scale the data in such a manner that the values lie between 0 and 1:where the suffix

*i*implies the realization at the th time point. A fuzzy set has been constructed whose elements comes from the time series as elements of a crisp set and the membership function is given bywhere

*a*will be considered as mean and median for the two judging processes for computation of joint belief.

In the final phase, the methodology we discussed in the previous section is implemented. For the first case, we have considered two basic assignments and for summer monsoon (JJAS) and post monsoon (OND) for the time period of 146 years, where and denote the basic assignment about mean and median, respectively. Then, we have considered three focal elements *R*, *D*, and *C* which represent the fuzzy sets ‘very close’ to mean/median, ‘close’ to mean/median, and ‘moderately close’ to mean/median respectively. Furthermore, we have calculated *R*, *D*, *C*, and the unions of all the focal elements, i.e., , , , and which are assigned with the membership grades obtained from different cuts. Applying Dempster's rule to and , we have derived the joint belief, i.e., for each focal element.

Again, we have taken two sets of time period containing the data of 50 years each for summer monsoon (JJAS) and post monsoon (OND). For each case, we have calculated the focal elements *R, D, C*, and the unions of all the focal elements, and assigned each element with the membership grades obtained from the different cuts. Finally, we have calculated the joint belief, i.e., by applying Dempster's rule to the basic assignments and for each focal element.

In the same way, we have calculated the joint belief and basic assignments for all the focal elements of summer monsoon (JJAS) and post monsoon (OND) by taking four sets of time period containing the data of 25 years each.

In Table 1, we have presented a joint belief for JJAS and OND with the basic probability assignments determined by considering the fuzzy sets of rainfall amounts close to mean and median, respectively. We consider the scenario in such a manner that there are two judges to judge the system, one is considering that the rainfall amount is close to the mean and the other is considering that it is close to the median. In Table 1(a) and 1(b), we have presented the joint beliefs for JJAS and OND, respectively. All the elements in the first column are found to have non-zero basic probability assignments and hence all of them are justified as focal elements. In the fourth column of each of the tables, we have presented the joint belief as the combined evidence. It is observed that the joint belief for and are having the maximum joint belief measure for JJAS and OND and hence we understand that for both JJAS and OND most of the monthly rainfall amounts are close to mean and median. However, the combined bodies of evidence show that none of the joint beliefs are greater than or equal to 0.5. Hence, we understand that for both JJAS and OND, the data are close to symmetry but absolute symmetry is not available. This indicates departure from symmetry as well as presence of some degrees of uncertainty within the time series of JJAS and OND rainfall for 146 years over northeast India. It is further observed that the joint belief measure for all the focal elements is higher in the case of JJAS than OND. This can be interpreted as the presence of higher degree of uncertainty in the OND rainfall than Indian summer monsoon, i.e., JJAS rainfall.

. | a . | b . | ||||
---|---|---|---|---|---|---|

About mean (JJAS) . | About median (JJAS) . | Combined evidence . | About mean (OND) . | About median (OND) . | Combined evidence . | |

Focal elements . | . | . | . | . | . | . |

0.390 | 0.397 | 0.465 | 0.384 | 0.370 | 0.426 | |

0.390 | 0.397 | 0.465 | 0.384 | 0.370 | 0.426 | |

0.233 | 0.233 | 0.238 | 0.239 | 0.253 | 0.245 | |

0.390 | 0.397 | 0.155 | 0.384 | 0.370 | 0.142 |

. | a . | b . | ||||
---|---|---|---|---|---|---|

About mean (JJAS) . | About median (JJAS) . | Combined evidence . | About mean (OND) . | About median (OND) . | Combined evidence . | |

Focal elements . | . | . | . | . | . | . |

0.390 | 0.397 | 0.465 | 0.384 | 0.370 | 0.426 | |

0.390 | 0.397 | 0.465 | 0.384 | 0.370 | 0.426 | |

0.233 | 0.233 | 0.238 | 0.239 | 0.253 | 0.245 | |

0.390 | 0.397 | 0.155 | 0.384 | 0.370 | 0.142 |

In the next phase of our study, we have broken the time series considered above into pieces of 50 years. In Table 2(a) and 2(b), we have considered the first and second 50 years for JJAS and Table 2(c) and 2(d) for OND, respectively. If we look at the tables for JJAS, we find that for the first two focal elements the joint belief measures are above 0.50. This indicates a strong body of evidence for which joint belief measures for the first two focal elements display strong evidence favouring to be around mean and median. This implies that mean and median are very close to each other and data to the proximity of mean and median are strongly favoured by the joint body of evidence. This indicates the possibility of the distribution being approximately normal. Similarly for the second 50 years, the joint body of evidence to have belief measure of 0.50 for the first two elements. Hence, here, also the presence of approximately normal distribution is discernible. Next, we look at Table 2(c) and 2(d). For Table 2(c), the first two focal elements are significantly favoured by the joint body of evidence for the first 50 years of OND. However, for the second 50 years of OND, the joint belief measure lies below 0.50. Hence, lack of strength in the joint belief is understandable in this case. From Table 2(c) and 2(d), we can interpret that although for the first 50 years the degree of uncertainty is less, the uncertainty increases for the next 50 years for OND. Combining these two, we observe that there is lack of consistency in the OND time series as far as proximity to mean and median is concerned. However, from Table 2(a) and 2(b), we observe a better consistency in the proximity to mean and median for JJAS. This is in consistency with the observations from Table 1, where we observed less uncertainty in JJAS than OND.

a | b | |||||

About mean (JJAS)
. | About median (JJAS)
. | Combined evidence
. | About mean (JJAS)
. | About median (JJAS)
. | Combined evidence
. | |
---|---|---|---|---|---|---|

Focal elements
. | . | . | . | . | . | . |

0.46 | 0.40 | 0.55 | 0.42 | 0.40 | 0.50 | |

0.46 | 0.40 | 0.55 | 0.42 | 0.40 | 0.50 | |

0.28 | 0.38 | 0.39 | 0.26 | 0.30 | 0.31 | |

0.46 | 0.40 | 0.18 | 0.42 | 0.40 | 0.17 | |

. | c . | d . | ||||

About mean (OND) . | About Median (OND) . | Combined evidence . | About mean (OND) . | About Median (OND) . | Combined evidence . | |

Focal elements
. | . | . | . | . | . | . |

0.42 | 0.46 | 0.57 | 0.40 | 0.38 | 0.46 | |

0.42 | 0.46 | 0.57 | 0.40 | 0.38 | 0.46 | |

0.36 | 0.26 | 0.37 | 0.36 | 0.32 | 0.38 | |

0.42 | 0.46 | 0.19 | 0.40 | 0.38 | 0.15 |

a | b | |||||

About mean (JJAS)
. | About median (JJAS)
. | Combined evidence
. | About mean (JJAS)
. | About median (JJAS)
. | Combined evidence
. | |
---|---|---|---|---|---|---|

Focal elements
. | . | . | . | . | . | . |

0.46 | 0.40 | 0.55 | 0.42 | 0.40 | 0.50 | |

0.46 | 0.40 | 0.55 | 0.42 | 0.40 | 0.50 | |

0.28 | 0.38 | 0.39 | 0.26 | 0.30 | 0.31 | |

0.46 | 0.40 | 0.18 | 0.42 | 0.40 | 0.17 | |

. | c . | d . | ||||

About mean (OND) . | About Median (OND) . | Combined evidence . | About mean (OND) . | About Median (OND) . | Combined evidence . | |

Focal elements
. | . | . | . | . | . | . |

0.42 | 0.46 | 0.57 | 0.40 | 0.38 | 0.46 | |

0.42 | 0.46 | 0.57 | 0.40 | 0.38 | 0.46 | |

0.36 | 0.26 | 0.37 | 0.36 | 0.32 | 0.38 | |

0.42 | 0.46 | 0.19 | 0.40 | 0.38 | 0.15 |

In the next phase of our study, we have broken the time series into pieces of 25 years and we have carried out the computation of joint belief measure for all the windows of length 25 and observed a decline in the joint belief. Although the decline is apparent for JJAS as well as OND, we have specifically observed (Table 3(c) and 3(d)) that for the third and fourth segment of the time series the joint belief is greater than 0.50 for the first two focal elements. However, for OND all the measures of joint belief are far below 0.50. We can interpret from these that the degree of uncertainty increases as we look at the smaller windows than the time series as a whole. However, the increase of uncertainty with change in the window size is more in case of OND than in the case of JJAS.

. | a
. | b
. | ||||
---|---|---|---|---|---|---|

About mean (JJAS)
. | About median (JJAS)
. | Combined evidence
. | About mean (JJAS)
. | About median (JJAS)
. | Combined evidence
. | |

Focal elements
. | . | . | . | . | . | . |

0.28 | 0.36 | 0.30 | 0.36 | 0.32 | 0.35 | |

0.28 | 0.36 | 0.30 | 0.36 | 0.32 | 0.35 | |

0.16 | 0.24 | 0.16 | 0.16 | 0.20 | 0.16 | |

0.28 | 0.36 | 0.10 | 0.36 | 0.32 | 0.12 | |

. | c . | d . | ||||

About mean (JJAS) . | About median (JJAS) . | Combined evidence . | About mean (JJAS) . | About median (JJAS) . | Combined evidence . | |

Focal elements
. | . | . | . | . | . | . |

0.44 | 0.40 | 0.53 | 0.40 | 0.44 | 0.53 | |

0.44 | 0.40 | 0.53 | 0.40 | 0.44 | 0.53 | |

0.20 | 0.28 | 0.26 | 0.32 | 0.36 | 0.40 | |

0.44 | 0.40 | 0.18 | 0.40 | 0.44 | 0.18 | |

. | e . | f . | ||||

About mean (OND) . | About median (OND) . | Combined evidence . | About mean (OND) . | About median (OND) . | Combined evidence . | |

Focal elements
. | . | . | . | . | . | . |

0.32 | 0.24 | 0.23 | 0.24 | 0.24 | 0.17 | |

0.32 | 0.24 | 0.23 | 0.24 | 0.24 | 0.17 | |

0.16 | 0.36 | 0.21 | 0.12 | 0.24 | 0.11 | |

0.32 | 0.24 | 0.08 | 0.24 | 0.24 | 0.06 | |

. | g . | h . | ||||

About mean (OND) . | About median (OND) . | Combined evidence . | About mean (OND) . | About median (OND) . | Combined evidence . | |

Focal elements
. | . | . | . | . | . | . |

0.20 | 0.20 | 0.12 | 0.28 | 0.36 | 0.30 | |

0.20 | 0.20 | 0.12 | 0.28 | 0.36 | 0.30 | |

0.12 | 0.16 | 0.08 | 0.12 | 0.20 | 0.12 | |

0.20 | 0.20 | 0.04 | 0.28 | 0.36 | 0.10 |

. | a
. | b
. | ||||
---|---|---|---|---|---|---|

About mean (JJAS)
. | About median (JJAS)
. | Combined evidence
. | About mean (JJAS)
. | About median (JJAS)
. | Combined evidence
. | |

Focal elements
. | . | . | . | . | . | . |

0.28 | 0.36 | 0.30 | 0.36 | 0.32 | 0.35 | |

0.28 | 0.36 | 0.30 | 0.36 | 0.32 | 0.35 | |

0.16 | 0.24 | 0.16 | 0.16 | 0.20 | 0.16 | |

0.28 | 0.36 | 0.10 | 0.36 | 0.32 | 0.12 | |

. | c . | d . | ||||

About mean (JJAS) . | About median (JJAS) . | Combined evidence . | About mean (JJAS) . | About median (JJAS) . | Combined evidence . | |

Focal elements
. | . | . | . | . | . | . |

0.44 | 0.40 | 0.53 | 0.40 | 0.44 | 0.53 | |

0.44 | 0.40 | 0.53 | 0.40 | 0.44 | 0.53 | |

0.20 | 0.28 | 0.26 | 0.32 | 0.36 | 0.40 | |

0.44 | 0.40 | 0.18 | 0.40 | 0.44 | 0.18 | |

. | e . | f . | ||||

About mean (OND) . | About median (OND) . | Combined evidence . | About mean (OND) . | About median (OND) . | Combined evidence . | |

Focal elements
. | . | . | . | . | . | . |

0.32 | 0.24 | 0.23 | 0.24 | 0.24 | 0.17 | |

0.32 | 0.24 | 0.23 | 0.24 | 0.24 | 0.17 | |

0.16 | 0.36 | 0.21 | 0.12 | 0.24 | 0.11 | |

0.32 | 0.24 | 0.08 | 0.24 | 0.24 | 0.06 | |

. | g . | h . | ||||

About mean (OND) . | About median (OND) . | Combined evidence . | About mean (OND) . | About median (OND) . | Combined evidence . | |

Focal elements
. | . | . | . | . | . | . |

0.20 | 0.20 | 0.12 | 0.28 | 0.36 | 0.30 | |

0.20 | 0.20 | 0.12 | 0.28 | 0.36 | 0.30 | |

0.12 | 0.16 | 0.08 | 0.12 | 0.20 | 0.12 | |

0.20 | 0.20 | 0.04 | 0.28 | 0.36 | 0.10 |

*x*represents timescale and

*y*represents measure of joint belief. In Figures 1 and 2, we have generated three-dimensional (3D) plots to visualize the evolution of joint belief with time scale. To generate the same, we have varied the value of a leading coefficient to understand the impact of change in the belief measure with any possible change in the polynomial. In Figures 1 and 2, we have implemented this technique for JJAS and OND, respectively. The figures show that the surfaces representing the joint belief are almost similar in pattern. In both cases, the joint belief measures have attained their maximum near the time scales that divide the entire window approximately into two halves. It is understandable from this that to develope predictive models for rainfall over northeast India, the relatively suitable option is to divide the entire window 1:1 ratio. Through this observation, we obtain a DST-based approach to have some idea about the most advantageous ratio of training and test cases for predictive method like the Artificial Neural Network with supervised learning procedure.

If we have a deeper insight into the figures, we observed that for OND the surface has a significant lifting with change in leading coefficient of the polynomial. However, in case of JJAS, the surface is not influencing significantly by the change of any leading coefficient of polynomial. This observation is consistent with the earlier outcomes where we identified that OND is characterized by higher degree of uncertainty and JJAS as far as the rainfall amount is concerned.

## CONCLUSIONS

In the study reported in the previous section, we have developed a methodology to understand the relative uncertainty associated with the rainfall amount corresponding to summer monsoon (JJAS) and post monsoon (OND) for the period 1871–2016. We have considered the random variables , , and representing annual (ANN), post monsoon (OND), summer monsoon (JJAS), and pre monsoon (MAM) rainfall, respectively, in northeast India. We have calculated the partial correlation between two random variables after the removal of the effect of the third one. In this way, we have calculated the rest of the pairs. After calculating the partial correlation, we have standardized all the values. In the next phase, we have considered two universes of discourse, one for post monsoon (OND) and other for summer monsoon (JJAS) and calculated the mean and median for both to create a fuzzy set representing the elements of the crisp set to be ‘close to the mean and median respectively’. Then, with the aim of computing joint belief measure through the DST, we have considered two basic assignments and for summer monsoon (JJAS) and post monsoon (OND) for the time period of 146 years, where and denote the basic assignments corresponding to the consideration about mean and median, respectively. Then, we have considered three focal elements R, D, and C which represent the fuzzy sets ‘very close’ to mean/median, ‘close’ to mean/median, and ‘moderately close’ to mean/median, respectively. Also, further we have calculated *R, D, C*, and the unions of all the focal elements, i.e., , , , and which are assigned with the membership grades obtained from different cuts. Applying Dempster's rule to and we have derived the joint belief, i.e., for each focal element. After a rigorous study through the Dempster–Shafer approach, we have observed that the degree of uncertainty increases as we look at the smaller windows of a time series than the time series as a whole. However, the increase of uncertainty with change in the window size is more in case of OND than in the case of JJAS. Finally, we have developed a functional relationship between time scale and measure of joint belief in the form of second-degree polynomial and visualized the association in the form of 3D plots where we have discerned the evolution of the joint belief measure with change in the window size and also with change in the value of a leading coefficient of the polynomial representing a functional relationship between them. We have observed that for OND, the surface has a significant lifting with change in leading coefficient of the polynomial. However, in case of JJAS, the surface is not influencing significantly by the change of any leading coefficient of polynomial. This observation is consistent with the earlier outcomes where we identified that OND is characterised by higher degree of uncertainty and JJAS as far as the rainfall amount is concerned. Figures 1 and 2 have revealed that the surfaces representing the joint belief are almost similar in pattern and in both the cases the joint belief measures have attained their maximum near the time scales that divide the entire window approximately into two halves. It has been interpreted from this that to develop predictive models for rainfall over northeast India, the relatively suitable option is to divide the entire window 1:1 ratio. Through this observation, we obtain a DST-based approach to have some idea about the most advantageous ratio of training and test cases for predictive method like the Artificial Neural Network with supervised learning procedure (Chattopadhyay *et al.* 2010; Acharya *et al.* 2012). As a future study, we propose to extend this approach to a multivariate framework involving other climatological parameters influencing rainfall over the study zone. While concluding, let us comment on the outcomes of the work concerning the need to substantiate the priorities related to the interaction of society and nature, creating a set of Sustainable Development Goals (SDGs), as determined by The World Summit on Sustainable Development ‘Rio + 10’ in Johannesburg in 2002 (Varotsos & Cracknell 2020). The work focuses on precipitation over a meteorological subdivision of India that has immense importance in rainfall. Given the approach and outcomes, we would like to, in a broader sense, this work inclines prediction model development. For a country like India, rainfall prediction is essential to meteorologists because of its role in the economy. Thus, because of sustainable development (Varotsos & Cracknell 2020), this work is consistent with the SDGs of ‘Rio + 10’.

## ACKNOWLEDGEMENT

The authors sincerely acknowledge the supportive and insightful comments from the anonymous reviewers. The data utilized in the work are taken from the Indian Institute of Tropical Meteorology website (IITM), https://tropmet.res.in/static_pages.php?page_id=52#data is the link to the data.

## DATA AVAILABILITY STATEMENT

All relevant data are available from an online repository or repositories (https://tropmet.res.in/static_pages.php?page_id=52#data}).

## CONFLICT OF INTEREST

The authors declare there is no conflict.

Journal Royal Statistical Society: Series B (Methodological)30(2), 205–232.