The Probability Distribution of Maximum Temperature to Assess the Suitable Statistical Models: Take the North-East and Southern Regions of Pakistan


 Precise maximum temperature probability distribution information is indeed of accurately significance for numerous temperature uses. The purpose of this research to assess the appropriateness of these functions likelihood for evaluating the temperature models at different sites in southern part of Pakistan. The Kumaraswamy distribution function is used initially to approximation the models of maximum temperature. Compare the presentation of the Kumaraswamy distribution with twelve commonly used the probability functions. The consequences obtained show that the more effective functions are not similar across all sites. The maximum temperature features, quality and quantity of the noted temperature observation can be regarded as a factors that affect the presentation of the function. Similarly, the skewness of the noted maximum temperature observations may affect the precision of Kumaraswamy distribution. For the Hyderabad, Lahore and Sialkot sites, the Kumaraswamy distribution obtainable the topmost presentation, however for the Karachi, Multan stations, the generalized extreme value (GEV) distributions provided the best fit, respectively. According to the calculations, the Kumaraswamy distribution usually be regarded as a valid distribution because it runs 3 best fit sites and ranks 2 to 3 among the remaining sites. Though, the tight presentation of the Kumaraswamy and GEV and the flexibility of the Weibull distribution which has been usually verified, more evaluations of the presentation of the Kumaraswamy distribution are needed.

that global warming has had a serious influence on human society (Chen et al., 2019, Zhou et al., 4 2017). As the temperature rises, the ability to retain water in the atmosphere increases, and the 5 temperature of extreme value has changed significantly, leading to forest fires and frequent 6 droughts. At the same time, with temperature changes, the global usual ocean level rise by 19 cm 7 between 1900 and 2011. In other words, changes in extreme temperatures increase the intensity 8 and frequency of extreme weather events (such as extreme temperatures, heavy rains, droughts, 9 and floods. When the temperature rises, the desert will increase, and the infiltration and soil 10 moisture intensity will change. The change in water rotation is caused by an increase in 11 temperature. Rising temperature will also change the restructuring of river excess and the 12 characteristics of water resources in the basin. Extreme temperature events is the main measure of 13 extreme climate events. Therefore, the study of extreme temperature changes under global 14 warming is great significance.

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A current analysis of the detection of extreme temperature trends in Europe (Irannezhad et 16 al., 2019) confirmed that 20th century over time, the warmth of day and night. Also concluded that 17 the frequency and intensity of high temperature (low temperature) (Shaby and Reich, 2012, Parey 18 et al., 2013, Naveau et al., 2014, Huang et al., 2016.Since the temperature probability, it is a 19 feasible method to predict extreme precipitation based on the relationship between temperature 20 and extreme precipitation under weather change. Therefore, in all over the world is studying the 21 basic process of detecting extreme precipitation changes with temperature (Donat et al., 2016, Gao 22 et al., 2018. (Wang et al., 2017) found that, compared with the historical temperature dependency 23 of extreme precipitation, the peak temperature of extreme precipitation will rise with climate 24 warming, which means that the peak arrangement does not mean the probable higher limit of 25 extreme precipitation in the future. As mentioned above, the relationship between extreme 26 precipitations and has been fully studied, but the association among highest extreme rainfall and 27 highest temperature (Teshome and Zhang, 2019).

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All selection periods fit the GEV distribution and estimate the parameter. The likelihood 29 ratio test shows that the best model in which the position parameter increases linearly, and the 30 parameters shape and scale are constant. Model diagnosis including quantile plots, probability 31 plots and density plot showed a good degree of fit. The GOF test Anderson Darling and 1 Kolmogorov Simonov show that modeling can almost gave same fitting result (Hasan et al., 2012, 2 Hughes et al., 2007. A univariate extreme value (EV) model based on the limit temperature data 3 of the block maximum method. The block size selection is important because when size is large, 4 valuable information may be wasted. Using a block length of one year will generate too few 5 maximum sequences (20 data) and result in a higher estimate variance. However, a block length 6 that is too short will not meet the limit approximation of extreme value temperature (EVT). 7 Therefore, the length is half-yearly, quarterly and monthly blocks are still feasible. From the 8 analysis of quantile graph, monthly and annual blocks are more suitable than quarterly and semi-9 annual blocks. Generally, LM is better than MLE in estimating parameters. Numerical results 10 indicate that the maximum temperature will gradually increase (Amin et al., 2018).

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The absolute burden of heat waves that is not conducive to health. This is in the entire 12 community, but workers who work in various hot places are particularly vulnerable. Therefore, 13 the impact is also economical. Since this growing hazard, the health authorities of the Republic of Accurately estimating the long-term trends of global and regional climate change are essential for 28 the impact and prediction attributable to climate change (Li et al., 2020, Li et al.).

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The objective of this study due to the effective application of the Kumaraswamy 30 distribution in different fields, it may be interesting to evaluate its proficiency in the designated 31 case study to estimate the maximum temperature distribution. Thus, the presentation of the 1 Kumaraswamy distribution was tested for certain before used distribution functions (including 2 exponential, normal, invers-Gaussian, logistic, log-logistic, log-normal 3, Gumbel Generalize Generally, understanding the probability distribution of maximum temperature is essential for 10 characterizing temperature behavior, evaluating maximum temperature performance. Therefore, it 11 is important to determine the most suitable function for temperature data. In this study, twelve

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PDFs were used to describe the frequency distribution of temperature. In order to evaluate the effectiveness checked PDF for modeling the probability distribution of (1)

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The KS test the theoretical probability distribution as In equation (2), ( ) is the cumulative distribution function, is the order statistic and n 14 denote the sample size.
Where relates to the test result, X is the variable, ( ) is the distribution function, and n is the 20 sample size and uses the statistical model with the smallest AD score for the data of wind speed 21 as the most-fitting distribution model.

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Pakistan's climate has many unique cyclical changes. The most serious variables affecting the 24 climate are humidity, temperature, wind speed and rainfall. The deserts in some areas are still very 25 hot and dry.

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Karachi is the most populous city in Pakistan and the capital of Sindh Province. Karachi is located 27 in southern Pakistan in Sindh Province. Therefore, the summer is not hot and humid from 28 December to February. Compared with the hot season that started in March and lasted until the 29 June monsoon, it was dry and pleasant. Hyderabad is also located in Sindh province, the desert 30 climate in Hyderabad is hot and warm throughout the year. This city is famous for its tempering 1 the originally hot climate. As a result, houses in Hyderabad have traditionally been equipped with 2 "induction wind" towers that blow breeze in to residential areas to reduce heat. From mid-April to 3 late June is the hottest period of the year, with the highest peak in May at 41.4 o C. The maximum 4 temperature recorded on May was 50 o C, and the lowermost temperature recorded on February was Lahore, the main city and cultural and historical midpoint of Punjab. The weather in Lahore is 7 semi-arid. In June, the rainy season begins. The temperature rises in July.

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Sialkot is located in Panjab in the northwest. It has four sub seasonal humid and subtropical 9 characteristics. The weather in Sialkot is still hot during the day, but cool at night and low 10 humidity. In winter, the climate is a bit warm and there is a lot of precipitation. Multan is located  affection, the Kum distribution is 1st time and previously related with some distribution.

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In Table 2   features must be estimated.

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In order to illustrate that the four most suitable distribution function describe the temperature in 6 different ranges, figure 2(a-e) shows that CDF and PDF and curves fitted by all stations. For pdf 7 and cdf graphs, the horizontal axis is the range of temperature data. For pdf plots, the shows the 8 probability density, which varies between the highest and lowest probable value. For the cdf graph, 9 the perpendicular axis shows the cumulative density, as we move from left to right on the parallel 10 axis, the value increase from 0 to 1. to examine the modulation of the extreme probability of temperature and rainfall, and it was found 18 that the extreme maximum temperature has a statistically significant long-term increase, but has 19 obvious seasonal and regional changes. They used the Wilcoxon test and Boxplots summarized 20 the results of four AEPs (50%, 10%, 5%, and 1%) and fixed and non-fixed generalized extreme and minimum temperature. The bivariate field splits in to two parts "weather" and "local climate".

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The climate factor is spatially related to bivariate simulation. The statistical model adds the  The estimator AIC is the prediction error of sample, and therefore the comparative superiority of 30 the statistical model for data set. (McElreath, 2020, Taddy, 2019 given the set of models used for 31 the data, AIC is the quality of estimations of individually model relative to every model. Therefore, 1 AIC provides a method of model selection. AIC is based on particular theory, when using a 2 statistical model denote the process of generating data, the representation is almost certainly not 3 accurate. Therefore, a model to represent the procedure will lose some information. AIC 4 estimations the comparative amount of data missing a specified model, when assessing the total 5 data lost, AIC will weigh the models goodness-of-fit and model simplicity. For identification of 6 mode the application of BIC widely used in linear regression and time series. However, it can be 7 widely applied to models based on maximum likelihood. Because the interested model is equal to 8 number of parameters.

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The other method is based on the relative measure of information loss when fitting the model to 10 describe the data. This approach includes Akaike information criteria AIC and BIC. However, 11 these two technique is the most popular measure. In the sense of hypothesis testing, AIC is not a 12 model test. Rather, the process and scoring provide a method for comparing data models and a tool  Kum, Wei3, and GEV are more flexible in distribution because they can display better 6 performance. 7 Generally, this research shows that the Kum distribution function is an actual distribution because 8 it runs the most fit in 2 stations, and it ranks 2nd among the remaining 2 stations. However, due to the 9 tightness of the Kum, GEV Wei3 distribution functions and the flexibility of the GEV and wei3 function 10 as its widely proven characteristics in previous studies, the effectiveness of the Kum distribution should be 11 evaluated more in future study. For this reason, more situation study with different temperature features 12 would also be estimated. 13

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We thank our respected reviewers and especially Yejuan Wang for their valuable comments and 15 suggestions that helped us to improve this paper. 17 The authors declare that they have no conflict of interest All the authors of this manuscript confirmed that the data supporting the findings of this study are 27 available in the article. All the required data is available and easily accessible. This study involve no living organisms or their products so don't need any consent of participate    Table. Probability distributions (PDF) and cumulative distribution (CDF) for each distribution.