| Abstract |
Due to the temperature rises global warming has become very popular. Increasing of global temperature will disturb the agronomic sector, intensification some of the contagious diseases that may lead to high mortality rates in humans, high demand for electricity, water and food which ultimately affecting the economy of Pakistan. The current research aims to study the best fitted probability distribution that describes the monthly mean temperature (MMT) of four locations in Pakistan are Islamabad, Lahore, Muzaffarabad and Karachi based on secondary dataset which was collected from Pakistan Meteorological department Lahore for the period 1991 to 2020. The Frechet, Weibull and Log-Logistic distributions are applied and the parameters of these distributions are estimated by maximum likelihood and Bayesian estimation methods. Additionally, Log-normal and Generalized Extreme value distribution are considered using the maximum likelihood estimation method to estimate the parameters. Moreover the graphs of probability density functions also constructed for comparison purposes. The goodness of fit test and model selection criteria such as Kolmogorov-Smirnov test, Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are used to measure the accuracy of the predicted data using theoretical probability distributions. The results show that four stations favor the candidate distributions based on the P-values of Kolmogorov-Smirnov test at 5 percent level of significance. However, Log-Logistic and lognormal distribution are the best fitted as compare to other candidate probability distributions based on AIC and BIC values. Furthermore, quantiles are also calculated using maximum likelihood and Bayesian estimation methods and concluded that Quantile estimates are closed to MMT on the basis of LLD. For trend analysis time series is also conducted. Initially, we have checked the stationarity of the observed data sets by using Augmented Dickey Fuller (ADF) test. Moreover, Auto Regressive Integrated Moving Average (ARIMA) models have been investigated to examine the autocorrelation and seasonality in MMT of different sites in Pakistan. The order of the ARIMA models has determined by using Auto Correlation Function (ACF) and Partial Auto Correlation Function (PACF) plots. Goodness-of-fit criteria such as, Akaike Information Criterion (AIC) have been used for the selection of parsimonious ARMA and ARIMA models. 19 Moreover, Ljung Box test was also used to confirm the best fitted ARMA and ARIMA models. It is concluded that ARMA (6, 2) provides a good fit for Islamabad, Karachi and Lahore stations, while ARIMA (6, 1, 2) is most suitable for Muzaffarabad station. Additionally, these ARIMA and ARMA models are used to make forecast for the next 12 months. |
