| Abstract |
The Universe includes planets, stars, galaxies, dust, clouds, light, rocks and living things. Among these, rock is relatively hard and naturally occurring mineral material. Rocks have wide range of uses that makes them significantly important to human life. In this study, an attempt has been made to fit and identify the most appropriate probability distribution(s) for rocks‟ sample data in Pakistan. The rocks‟ data were collected from different cities of Pakistan and Azad Jammu and Kashmir. The rocks‟ names are Diorite, Gypsum, Marble, Basalt, Sandstone, Limestone, Apatite, Slate, Dolomite, Granite-II, Schist, Gneiss, Uranite, Blochistan, Magnetite, Iron ore, and Granite-I.
Initially, histograms are constructed for the graphical assessment and visualization to test whether the rocks dataset series are positively skewed with heavier tail than normal distribution. Further, coefficient of skewness and kurtosis confirm that positively skewed distributions can be the appropriate candidates for rocks data sets. Therefore, Frechet, Weibull, Log Logistic, Log normal and Generalized extreme value distributions have been used as candidate distributions for analysis. The parameters of candidate probability distributions are estimated by maximum likelihood and Bayesian estimation methods. The Kolmogorov Smirnov test has been adopted as a goodness of fit criteria. Moreover, Akaike information criterion and Bayesian information criterion were used to choose and recommend the most suitable distribution for the selected rocks.
It is worth noting that Frechet, Weibull, and Log logistic distributions are best fitted for the rocks‟ data sets based on the p-values of Kolmogorov Smirnov test at 5% level of significance. Whereas, Log normal is not appropriate for Granite-I and also Generalized extreme value is not best fitted for the two rocks namely Blochistan and Iron Ore. However, Frechet, Weibull, Log Logistic, Log normal and Generlized extreme value distributions models as the first, second, third, fourth and fifth best-fit models respectively for rocks samples due to having the lowest Akaike and Bayesian information criteria values. Moreover, the graphs of probability density functions of candidate probability distributions depict that the appropriateness of candidate distributions.
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Further, Kurskal Wallis test is also applied to assess whether the independent rocks‟ samples having identical population. The results indicated that all the rocks‟ samples do not have identical population.
Additionally, cluster analysis is conducted to classify the rocks into groups that share common characteristics and it is observed that Diorite and Gypsum are placed in one cluster. While, the rocks namely, Slate and Dolomite, Marble and Basalt, Sandstone and Schist, Granite-II and Gneiss, Iron Ore and Granite-I are engaged in different clusters respectively. |
