It useful information of the long term behavior ofthe

It has been recognized by operational weather forecasters over theglobe that uncertainties associated with the initial conditionsused to initialize the models of numerical weather prediction aswell as errors in those models result in uncertainties in theforecast. It has also been emphasized in the literature that theforecasts might be benefited by the greater amount of informationcontained in a probabilistic forecast than in a deterministicforecast cite{n1,n2}. In view of the above, Roulston and Smithcite{roult} demonstrated how information theory might provide auseful theoretic framework to understand and quantify weather andclimate predictability. The modern age of ergodic theory has begun with thedevelopment of the idea of entropy of a random variable by RudolfClausius  in 1850 and around 1870 it was endowed with astatistical meaning by Ludwig Boltzmann, who established itsrelation with statistical mechanics.  In subsequent times, theconcept of entropy was advanced in the works of J. Willard Gibbsin thermodynamics and Von Neumann in quantum mechanics. ClaudeShannon, in 1948, reintroduced it in information theory. Theentropy provides useful information of the long term behavior ofthe random process and this behavior plays as the key factor indeveloping coding theorems of information theory. For detaileddiscussion on the Shannon entropy please see Gray cite{gray} andreferences therein. The idea of entropy is based on the notion ofmutual information between two processes introduced by Shannon as:egin{equation}I(X,Y)=H(X)+H(Y)-H(X,Y)end{equation}label{1}where, $X$ and $Y$ are two random variables. The mutualinformation is interpreted as the information contained in oneprocess minus the information contained in the process when theother process is known. In spite of existence of another notion ofconditional entropy, the form
ef{1} has been focused by theinformation theorists because it does not require any explanationof what conditional entropy means and has more symmetric form thanthe conditional definition. A detailed account is this regard hasbeen described in Gray cite{gray}.Information theorists, both mathematicians as well as researchersin other areas, have extended Shannon’s basic approach ever moregeneral models of information sources, coding structures, andperformance measures. Xu cite{2} presented a detailed analysis ofthe differences between the relative entropy and Shannon entropyin measuring information content and information loss andconcluded that although the Shannon entropy difference measuresonly the dispersion part, the  relative  entropy  measures  boththe signal and dispersion parts of the information content fromobservations. Brunsell cite{3}, in his study on dailyprecipitation records and of the precipitation event sizedistribution over an area of United States, applied informationtheory metrics are to ascertain the spatial and temporalvariability of precipitation and in course of his study computedboth mutual as well as relative entropy. Kawachi et al cite{4}applied Shannon entropy for daily rainfall observed at a networkof 1107 raingauges in Japan to study the uncertainty of theover-a-year rainfall apportionment. In a remarkable review, Liu etal. cite{5} addressed the entropy measures related tometeorology, including thermodynamic entropy, Boltzmann entropy orGibbs entropy, and Shannon entropy and finally established theintrinsic connection between Gibbs and Shannon entropy. Nebot etal. cite{6} demonstrated a Fuzzy Inductive Reasoning methodologyconsisting of its relevant variables and a set of if-then rulesand they adopted feature selection based on the maximization ofthe models’ forecasting power quantified by a Shannonentropy-based quality measure. They cite{6} used the Shannonentropy measure to determine the uncertainty associated withforecasting a particular output state given any legal input state.The present work is primarily focused on application of Shannonentropy, another important aspect of the methodology is artificialneural network (ANN), which is a very important component of SoftComputing techniques. Recent research activities in ANN have shownthat ANN’s have powerful pattern classification and patternrecognition capacity. They learn from examples and capture subtleabilities. Inspired by biological systems, ANN’s are able tounderlying relationships are unknown or hard to learn from andgeneralize from experience. In a remarkable work, Hsieh and Tangcite{tang} demonstrated that the ANN method with non-linearactivation function could be a versatile and powerful techniquecapable of augmenting traditional linear statistical methods indata analysis and forecasting in meteorology. In anotherfundamental work, Gardner and Dorling cite{Gardner} presented theusefulness of ANN as an alternative to conventional statisticalapproaches in atmospheric modelling, with special emphasis onMultilayer Perceptron (MLP). Applicability of MLP in modellingatmospheric/environmental data was strongly established in variousworks, of which the two works that are very significant, includePerez and Reys cite{perez1} and Perez et al. cite{perez2}. Otherremarkable works in this area include cite{ann1,ann2,ann3,ann4}.In a recent study, Abderrahim et al. cite{ann5} reported efficacyof MLP with backpropagation learning and single hidden layer inair quality forecasting with air temperature, relative humidity,and wind speed as inputs to the model. In another recent work,Gutierrez-Coreaa cite{ann6} applied MLP with differentarchitectures to the short-term prediction of Global SolarIrradiance with temperature, humidity, pressure, wind and otherestimates as inputs. At this juncture, it should be stated thatthe present study differs from an earlier study by Chaudhuri andChattopadhyay cite{solstice} in the sense that instead ofconsidering some discrete values of change $\%$, a range has beenconsidered for maximizing the $H( extbf{p})$ and subsequently anANN has been generated.Thunderstorms result from vigorous convective activity and is oneof the global phenomena that can occur anywhere in the world atany time. Also known as lightning storm or hailstorm, this weatherphenomenon is a form of weather characteristic containing strongwind, lightning, heavy rain, and sometimes snow or hail. Althoughshort-lived phenomena, it has great potential to produce seriousdamage to human life and property. During pre-monsoon months(March–May), many parts over the Indian region are affected bythunderstorms at higher frequency. Highly unstable atmospherebecause of high temperature prevailing at lower levels leads toformation of severe thunderstorms and it moves generally fromnorthwest to southeast over the eastern and northeastern states ofIndia. This weather phenomenon is associated with thunder, squalllines, lightning, torrential rain, and hail. Because of itsdevastating nature, an appropriate prediction with sufficientlead-time has continued to be a challenge to atmosphericscientists. Almost all experiments related to prediction of thesestorms have been based either upon statistical or numericaltechniques. The complexity of the meteorological system andinsufficient data has recurrently led to flawed results.Consequently, no method to date has proved sufficient to predictpre-monsoon thunderstorms over the Northeastern part of India.Considering the potential of ANN in dealing with complex timeseries, Chadhuri and Chattopadhyay cite{sura1} developed afeedforward single hidden layer ANN model to estimate the maximumsurface temperature and relative humidity. Litta et al.cite{litta} attempted ANN with Step, Momentum, ConjugateGradient, Quick Propagation, Levenberg-Marquardt, andDelta-Bar-Delta learning algorithms to predict thunderstormaffected surface parameters over Kolkata, India.The present study has used the methodology based of Shannonentropy to arrange, according to importance, some prominentsurface parameters associated with this kind of thunderstorm. Thepercentage changes in the magnitudes of the correspondingparameters  have been taken as the inputs for the study. Theentropy based methodology has been applied to discern  thevariation in the entropy associated with the probabilitydistributions corresponding to the expected changes $\%$ in themagnitudes of the parameters under study. The parameter withmaximum fluctuation in the entropy with change in the expectedchange in the magnitude $\%$ has been identified as the mostimportant parameter associated with the pre- monsoon thunderstormof the region. Surface parameters tested in this paper are:surface temperature, relative humidity, and air pressure. Based onthe outcomes, ANN in the form of MLP has been attempted to furtherdiscern the impact of pre-monsoon thunderstorms over the surfaceparameters.