Over past few decades global warming is assumed to be one of the primary reasons for the irregularities in climatic conditions in the world. The increase and the erratic changes in the temperatures in many parts of the world is because of global warming which is the main reason for disasters such as droughts, heat waves, floods and storms. Prior prediction of weather conditions coupled with adoption of necessary regulations can make lives of mankind better. In climate studies, association between Temperature and Precipitation is of great importance. In order to determine the interdependence
In this direction some studies have been taken place. Choubin et al. ^{1} focused on the application and evaluation of classification and regression trees (CART) model in prediction of seasonal precipitation and compared with adaptive neurofuzzy inference system (ANSIF), auto regressive integrated moving average (ARIMA) models. Choubin et al. ^{2}, have applied Linear regression with two nonlinear models, the adaptive neuro fuzzy inference system (ANSIF) and the multilayer perceptron to forecast an ensemble season precipitation for semi arid catchment in Iron. However, these papers are on precipitation prediction. Pandey et al.
These studies were aimed at predicting the Rainfall concentrated over Humid, Arid and Mediterranean regions. From the review of literature, it is observed that there were hardly any studies on Plateau region and there was no prediction regarding the Temperature which was reported. Our study aims at developing a bivariate Model for monthly mean average Temperature and Precipitation processes which can be used to simulate Temperature of the selected regions (Hyderabad and Medak districts of Telangana State, which is a Plateau region) taking into consideration data pertaining to a closely related variable, namely Precipitation. In this direction Copula analysis is a methodology found to be most appropriate. Objective of the study is to develop a single Copula model to estimate Temperature in selected region, if possible.
Present study is classified into four sections, starting with introduction in section 1. Section 2 consists of materials and methods, deals with the collection of data for selected regions, describing the univariate distribution for both Temperature and Precipitation and identify the bivariate model for Temperature and Precipitation. Section 3 consists of parameter estimation for univariate and bivariate distribution functions using Maximum Likelihood technique followed by prediction. Section 4 includes conclusions on the findings, limitations and future work.
This study is confined to Hyderabad and Medak district.
From the Indian Meteorological Department, Pune month wise Precipitation and Temperature data is collected for Hyderabad and Medak districts for the past 119 years (1901 to 2019).
Hyderabad has a typical dry and wet climate environment surrounding on a hot semidesert climate ^{10}. 26.6^{0}^{ }C is its yearly average Temperature; monthly mean Temperatures are 22.1 – 33.5 ^{0}C. Summer season is during March – June in which the environment is sweltering, with mean Temperature 26 ^{0}C to 35 ^{0}C, most extreme Temperatures frequently surpass 35 ^{0}C among April to June. The coolest Temperatures happen in November, December and January, when the lowest Temperature incidentally plunges to 10^{0}^{ }C. During May the Temperature is the sultriest and the daily maximum Temperature range from 26 to 43^{0}^{ }C; we can observe an oscillating cycle of monthly mean Temperature of Hyderabad from 1901 – 2019.
Monthly average Precipitation of Hyderabad also exhibited similar cyclical behavior from 1901  2019. From June – October, the monthly average Precipitation is found to be maximum every year. The mean Precipitation in the rainy season (JJune – October is 217.56mm.
Medak is situated at a considerable distance from the seacoast, the environment is tropical and is described by very warm summer and dry, except during the SouthWest monsoon season. The yearly mean Temperature is 26.74^{0}^{ }C; monthly mean Temperatures are 22.2 – 32.5^{0}^{ }C. During March – June climate of Medak is sweltering. Its mean Temperature is 28^{0}^{ }C to 33^{0}^{ }C, most extreme Temperature regularly surpass 35^{0}^{ }C between April and June. The hottest Temperature occurs in May and coolest Temperature occurs in December and January. In this region a clear seasonal cycle can be observed in monthly mean Temperature from 1901 – 2019. A clear seasonal cycle is not exhibited by monthly average Precipitation in Medak. During June – October, the monthly average Precipitation is found to be the maximum during the period of study. The mean Precipitation in the rainy season (June to October) is 147.32mm.
The cause of association between Precipitation and Temperature is observed to be influencing soil wetness which in turn influencing Temperature on the surface. The reason for this consequence is indirect control of soil wetness on dividing idle and reasonable warmth situations ^{11}. As the sample data shows a nonGaussian distribution, the Kendall’s tau correlation coefficient is utilized to ascertain relationship between month to month Temperature and Precipitation. A negative association has been observed between Precipitation and Temperature during February – April and June – October (at the 5% significant level), which is given in [
a) Hyderabad 


Jan 
Feb 
Mar 
Apr 
May 
Jun 
July 
Aug 
Sep 
Oct 
Nov 
Dec 
KCC 
0.091 
0.171 
0.276 
0.424 
0.102 
0.308 
0.217 
0.247 
0.429 
0.259 
0.03 
0.118 
pvalue 
0.242 
0.023 
0 
0 
0.16 
0 
0.002 
0.001 
0 
0 
0.67 
0.126 
b) Medak 

KCC 
0.086 
0.121 
0.348 
0.373 
0.548 
0.334 
0.304 
0.166 
0.391 
0.122 
0.015 
0.95 
pvalue 
0.255 
0.11 
0 
0 
0 
0 
0 
0.017 
0 
0.079 
0.824 
0.201 
Copula functions are used to develop a joint probability distribution of Temperature and Precipitation for selected months to represent their association. Let X and Y denote Temperature and Precipitation, which are continuous in nature, with cumulative distribution functions F_{X}(x) = Pr (X ≤ x) and G_{Y}(y) = Pr (Y ≤ y) respectively.
By the definition of Sklar,^{ }
Where C is an unique function and is known as Copula i.e., C(u, v) = Pr(U ≤ u, V ≤ v) is the distribution of (U,V) = (F(X), G(Y)) whose marginal distributions are U[0, 1]. As contended by Joe
Once the parameters of different Copula are estimated, selecting the Copula which can represent the structure of dependency between the interested variables is very important. Few criteria like Aldrian
Here k is the Copula parameters; L is optimized value of the likelihood function of the Copula.
The Bayesian information criterion (BIC) was evolved by Schwarz using Bayesian formalism. It is defined as
In Hyderabad and Medak, Temperature and Precipitation sample data of September during 1901 – 2019 is taken to demonstrate the modeling. A significant negative association can be seen between Temperature and Precipitation in September for Hyderabad and Medak, respectively. (Kendall’s Tau is −0.429 and 0.391, Pvalue=0.000). Temperature has positive skewness (1.01 and 0.99) and Precipitation has a kurtosis (0.93 and 1.26) respectively, which shows that given data follows a normal distribution for Hyderabad and nonnormal distribution for Medak districts.
A forecasted cycle can be observed for Temperature, moves as for the average, impacted because of hazardous atmospheric deviation. Considering all of these characteristics, the Temperature cycle is exhibited as a Normal (Gaussian) and SinhArcsinh (SHASH) distribution for Hyderabad and Medak respectively.
Normal distribution is a two parameter distribution function and the parameterization of the normal distribution given in the function is
Here μ and σ are mean and standard deviation of the distribution respectively.
SinhArcsinh is a four parameter distribution function and the parameterization of the SinhArcsinh distribution given in the function is
Where, r =
c =
and z =
Here μ and σ are the location and scale of the distribution. The parameters
The Precipitation cycle is exhibited as a Skew normal type2 and Reverse Gumbel distributions for Hyderabad and Medak respectively.
Skew Normal type2 distribution is a three parameter distribution function and the parameterization of the Skew Normal type2 distribution given in the function is
Where, z =
Here μ and σ are the location and scale of the distribution. The parameter
Reverse Gumbel distribution is a two parameter distribution function and the parameterization of the Reverse Gumbel distribution given in the function is,
Where, ∞ < x < ∞, here μ  the mean (∞ < μ < ∞) and σ – the standard deviation (σ > 0).
R software has been used to identify the appropriate bivariate Copula distribution. It is found to be the Rotated Gumbel 270 degree distribution and Frank Copula using maximum likelihood estimation having minimum AIC and BIC values for Hyderabad and Medak districts respectively.
The Archimedian Rotated Gumble 270 degree is defined as,
And
Here, the Gumbel parameter θ is given by
The Archimedian Frank Copula is defined as,
And its generator is
Here, the Frank Copula parameter θ is given by
Where,
In this examination to estimate the parameters of the model we utilized the Inference Function for Marginals. In this estimation the marginal distributions likelihood function and the Copula density functions are used.
Given,
Where, F(u) is a Normal & SinhArcsinh distributions and G(v) is a Skew Normal type2 and Reverse Gumbel distributions. Here, t_{1} = (μ_{1}, σ_{1}) & (μ_{2}, σ_{2},
then the bivariate joint probability density function is
where z_{1}, z_{2}, …… z_{n}, is a sample of size ‘n’ and the log likelihood of z_{i} is a univariate function given as
The log likelihood function of the Copula density function reduces to
to get the estimates
The sample data of Hyderabad and Medak shows that there is a negative correlation (0.429 and 0.391) between Temperature and Precipitation
To estimate the parameters of marginal distributions of Temperature and Precipitation in both Hyderabad and Medak districts we used Maximum Likelihood technique and found the following results presented in [
Distibution 
Normal 
SinhArcsinh 

Parameter 
μ 
σ 
μ 
σ 
ϑ 
τ 
Value 
26.49 
0.69 
26.20 
27982.53 
39068.47 
10397.21 
Distibution 
Skew Normal type2 
Reverse Gumbel 

Parameter 
μ 
σ 
ϑ 
μ 
σ 
Value 
82.33 
63.57 
2.26 
141.31 
74.81 
Form the graphs we can notice that fitted marginal distributions of Temperature and Precipitation of both the districts are very close to the observed data.
The parameter estimates of the Bivariate Copula distribution function‘θ’ of Rotated Gumbel 270 and Frank Copula using Maximum Likelihood Estimation and Its minimum AIC and BIC values are presented in [
Bivariate Copula Distribution 
Rotated Gumbel 270 
Frank 
Parameter (θ) 
1.72 
4.09 
AIC 
54.21 
41.62 
BIC 
51.43 
38.84 
Hyderabad and Medak districts Temperature and Precipitation values are simulated using Rotated Gumbel 270 and Frank Copula distributions respectively. These values are compared with the observed data. The pattern exhibited by simulated and observed data is shown in the scatter plots in [
Further, for the identified best Copula model for Temperature, the MAPE is found to be 0.04 for Hyderabad and 0.09 for Medak. It implies that these models can simulate Temperature values with 96% accuracy 91% respectively. Hence, the simulated values of Temperature for testing data of both the districts are computed. The observed and simulated values for testing period are presented in [
Rotated Gumbel 270 Copula for Hyderabad and Frank Copula for Medak are found to be the best models to identify the type of dependency between Temperature and Precipitation. The simulated data for testing period of Temperature has shown good agreement with observed data.
As the variation in Hyderabad data is relatively higher than Medak data. Higher is the variation in the observed data, more is the accuracy in simulating data for future. Hence, for Medak simulated data agreement with observed data is not that perfect.
Though the best joint Copula is identified for both Temperature and Precipitation, only Temperature can be simulated using these models. The associated variable Precipitation cannot be simulated.
In the present study, homoscedasticity is considered. In the literature scan, no such study is made for Hyderabad and Medak. Further, no model is found in the literature for the situation considered in this study for comparison. As the model is able to give prediction with 96% accuracy and a clear agreement is seen for the testing data, we felt that comparing with any other model may be redundant.
The interdependent Temperature and Precipitation can be modeled in the most appropriate way using marginals through Copula method. In our analysis, based on Kendall’s tau the dependency between Temperature and Precipitation is computed. The relation between these two variables is found to be negative in both the districts.
The best joint model of Temperature and Precipitation is found to be the bivariate Rotated Gumbel 270 distribution for Hyderabad and Frank Copula for Medak, based on AIC and BIC criteria. The models are able to explain 96% of the variation hidden in the observed bivariate data. Hence, it is concluded that it can be used for predicting the future Temperature.
In a country like India climatic conditions in southern and northern regions itself vary. So it is not possible to use the same model in both regions. Similarly, as the climate conditions vary at different regions on the globe, it is very difficult to develop a single model which can be used for prediction of temperature for any region on the globe. Though, we are doing analysis of bivariate data and identifying a best Copula bivariate distribution prediction can be carried out only for the variable Temperature which acts independently but not the closely associated variable precipitation.
Similar analysis can be carried out in the geographical regions for the remaining observatory centers of Temperature and Precipitation in Two Telugu States (Telangana and Andhra Pradesh).