Determination of Best-Fit Probability Distribution …
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introducing a penalty term for the number of parameters in a probability distribution function. The probability distribution with the minimum BIC and AIC score is
considered the best fit. For further details, readers are referred to [10, 11]. Parameter
estimations of various distributions and the goodness of fit tests mentioned in the
article have been performed using the package ‘fitdistrplus’ in R software [12].
4 Results and Discussions
Mangan, located at the junction of the southeast and southwest quadrants, has a higher
mean monthly rainfall than that of Gangtok in the dry season. This may be attributed
to its location in the mid hills and the orographic effect on northern winds. Given
its high altitude, Thangu experiences orographic precipitation more in the form of
snow than rainfall. This can be validated with the observations of rainfall values that
are consistently low in both wet and dry seasons (refer Fig. 2 and Table 2).
The two-parameter normal, log-normal, gamma, and Weibull distributions are
considered based on the skewness and kurtosis of the sample data points. Figures 3,
4, 5 indicate the Q-Q plots of empirical and theoretical quantiles of the distributions
for annual, wet, and dry seasons, respectively. Assessing the best-fit distribution of
the data may not be decided based only on the Q-Q plots and hence performing the
goodness of fit tests is mandatory.
Four goodness-of-fit tests, namely log-likelihood, AIC, BIC, and chi-square test,
are performed on the empirical data, and the results are tabulated in Tables 3, 4, and
5 for annual, dry, and wet seasons, respectively, for each of the four stations. The
best-fit probability distribution of the rainfall for annual, dry, and wet seasons in each
of the stations can be identified from Tables 3, 4 and 5. The significant values of the
goodness-of-fit tests are marked in bold in the table. For example, from the table wet,
it is obvious that the best-fit probability distribution of rainfall in Gangtok in wet
season is normal distribution with the chi-square p value 0.8886 at 5% significance
level.
The same is validated from the value of log-likelihood and the scores of AIC
and BIC. For Mangan in wet season, the rainfall may follow log-normal distribution
according to the log-likelihood value, AIC and BIC scores, however, it may not
follow any of the tested distributions as the Chi-square p values are not statistically
significant at 5% significance level.
The best-fit distributions have been listed in Table 6. Rainfall in the wet season
is five to six times higher than that in the dry season at Gangtok, Chungthang, and
Mangan, and three times more in Thangu. The Weibull distribution seems to fit best
for the dry season rainfall and the stations receiving less rainfall. However, the rainfall
in Gangtok seems to follow the normal distribution during the dry season. It is also
observed that the best-fit distribution of annual rainfall corresponding to each station
is same as that of the wet season. For Mangan, the log-normal distribution is chosen
as the best fit for annual rainfall over the gamma distribution since the χ
2 p value is
71
introducing a penalty term for the number of parameters in a probability distribution function. The probability distribution with the minimum BIC and AIC score is
considered the best fit. For further details, readers are referred to [10, 11]. Parameter
estimations of various distributions and the goodness of fit tests mentioned in the
article have been performed using the package ‘fitdistrplus’ in R software [12].
4 Results and Discussions
Mangan, located at the junction of the southeast and southwest quadrants, has a higher
mean monthly rainfall than that of Gangtok in the dry season. This may be attributed
to its location in the mid hills and the orographic effect on northern winds. Given
its high altitude, Thangu experiences orographic precipitation more in the form of
snow than rainfall. This can be validated with the observations of rainfall values that
are consistently low in both wet and dry seasons (refer Fig. 2 and Table 2).
The two-parameter normal, log-normal, gamma, and Weibull distributions are
considered based on the skewness and kurtosis of the sample data points. Figures 3,
4, 5 indicate the Q-Q plots of empirical and theoretical quantiles of the distributions
for annual, wet, and dry seasons, respectively. Assessing the best-fit distribution of
the data may not be decided based only on the Q-Q plots and hence performing the
goodness of fit tests is mandatory.
Four goodness-of-fit tests, namely log-likelihood, AIC, BIC, and chi-square test,
are performed on the empirical data, and the results are tabulated in Tables 3, 4, and
5 for annual, dry, and wet seasons, respectively, for each of the four stations. The
best-fit probability distribution of the rainfall for annual, dry, and wet seasons in each
of the stations can be identified from Tables 3, 4 and 5. The significant values of the
goodness-of-fit tests are marked in bold in the table. For example, from the table wet,
it is obvious that the best-fit probability distribution of rainfall in Gangtok in wet
season is normal distribution with the chi-square p value 0.8886 at 5% significance
level.
The same is validated from the value of log-likelihood and the scores of AIC
and BIC. For Mangan in wet season, the rainfall may follow log-normal distribution
according to the log-likelihood value, AIC and BIC scores, however, it may not
follow any of the tested distributions as the Chi-square p values are not statistically
significant at 5% significance level.
The best-fit distributions have been listed in Table 6. Rainfall in the wet season
is five to six times higher than that in the dry season at Gangtok, Chungthang, and
Mangan, and three times more in Thangu. The Weibull distribution seems to fit best
for the dry season rainfall and the stations receiving less rainfall. However, the rainfall
in Gangtok seems to follow the normal distribution during the dry season. It is also
observed that the best-fit distribution of annual rainfall corresponding to each station
is same as that of the wet season. For Mangan, the log-normal distribution is chosen
as the best fit for annual rainfall over the gamma distribution since the χ
2 p value is