Forecasting of Radiation Fog: A Fuzzy Logic Rule Based ApproachRajarshi Bagchi, T. P. Singh Symbiosis Institute of Geoinformatics Symbiosis International University Pune Abstract There are inevitable uncertainties associated with weather. In such a fuzzy environment, weather forecast and its evaluation are very difficult. Weather forecasts will be most significant when conditions are abnormal or chaotic. Under such situations, existing Numerical Weather Prediction (NWP) approach does not produce satisfactory results. To cope up with vague and/or abnormal (chaotic) meteorological information a weather forecasting approach using fuzzy logic based approximate reasoning is considered. Our attempt is to forecast radiation fog with the help of fuzzy logic based approximate reasoning. This process uses the concept of a pure fuzzy logic system where the fuzzy rule base consists of a collection of fuzzy IF-THEN rules. The fuzzy inference engine uses these fuzzy IF-THEN rules to determine a mapping from fuzzy sets in the input universe of discourse to fuzzy sets in the output universe of discourse based on fuzzy logic principles. Effective radiation fog forecast is useful for airlines to keep planes flying under safe schedule and also it may help in avoiding highway accidents. 1. Introduction Fog formation has a range of impacts on aviation, road, rail, operations from a safety perspective and an economic perspective Formation of radiation fog is based on several factors like synoptic scale factors and also on local mesoscale factors that are often not well observed or understood. (Sverre 1940). This introduces uncertainties in the forecasting of fog that can be very difficult to quantify. All these said factors could be modeled by approximate reasoning (generalized modus ponens approach) to forecast radiation fog. The ultimate production will provide multiple possibilities of radiation fog under different climatic variations. As radiation fog causes many problems like highway accidents, maintaining airplanes flying schedule, so prediction of radiation fog is done through conventional thermodynamics models in terms of differential equations (Meyer and Rao, 1999). But due to certain irregularities of weather this model based approach sometimes fails to predict the conditions accurately. To nullify this type of problem the concept of fuzzy rule based is considered. Fuzzy rule based is recommended to developers who are challenged to reduce the knowledge acquisition task, avoid repeating mistakes made in the past, reason in domains that have not been fully understood or modeled, learn over time, reason with incomplete or imprecise data and concepts, provide a means of explanation, and reflect human reasoning (Main et al. 2000), and these are some of the challenges faced by developers of weather forecasting systems (Christopherson 1998).Extensive researches have been carried out upon such weather systems by Roy Bhowmik et al. (2004), Bushan et al. (2003), on fog formation. The fuzzy rule based model uses a combination of observations and forecast weather elements (Puri et al, 1998) and provides a forecast fog risk. In the recent years, fuzzy technique has drawn considerable attention towards handling this kind of complex and non-linear problems. The technique has been widely applied to many meteorological problems such as prediction of ceiling and visibility using fuzzy logic by Hansen and Riordan (1998), marine forecast, Hansen (1997), classification of atmospheric circulation pattern, Bardossy et al. (1995), long term rainfall forecasting, Abraham et al. (2001), climate classification by McBratney et al. (1985), fuzzy logic in operational meteorology by Murtha (1995) and forecasting of temperature-humidity index using fuzzy logic approach by Mitra (2006) Mitra et al. (2008). The aim of fuzzy set theory is to build up a quantitative framework that represents the vagueness of human knowledge as it is expressed through natural languages. The gap, which separates the conventional mathematical models of physical systems from the imprecise mental representation of them, essentially motivated Zadeh (1965) to study how the mental representation could be translated into computable entities so as to overcome some limitations of the conventional models. 2. Methodology The whole processes can be divided into three parts. In the first stage, the parameters responsible for radiation fog represented by appropriate fuzzy sets defined over the different dynamic ranges and corresponding membership functions are assigned to the intervals of the assigned domain. In second step the all possible rules with the possibility occurring of radiation fog was generated. These are according the perception of an expert. At last the generalized form of extended fuzzy reasoning was applied to automate the idea through Visual Basic 6.0. 2.1 Generating Fuzzy Rule Base The parameters under consideration for radiation fog prediction are dew point, dew point spread (difference between air temperature and dew point), rate of change of dew point spread per day, wind speed and sky coverage. Given these parameters our primary task is to predict possibility of occurrence of radiation fog. The occurrence of different parameters responsible for radiation fog represented by appropriate fuzzy sets defined over the different dynamic ranges. Over these ranges primary fuzzy sets are defined over the said domain and the membership functions are attached to the corresponding primary fuzzy sets (figure1). Possibility of fog in terms of (%) is perceptually justified from an expert’s intuition. Here the domain is {0% to 100%}. Primary fuzzy sets defined over the universe are very low, medium, high and very high and the membership functions are also attached as Table 1. 
The general formula of determining the number of possible rules is the product of combinations of the number of primary fuzzy sets taking one at a time from each of them. For example, number of possible rules generated here is equal to 4C1×2C1×2C1×3C1×3C1 = 216 like “If{Dew point is Moist & Dew point spread is Saturated & Rate of change of spread is Saturating & Wind speed is Excellent & Sky condition is Clear} Then {Possibility of Fog is Very High}”. Now, consider the following form of inference in which a fuzzy conditional proposition “if… then…” contains two fuzzy propositions “X is A” and “Y is B” combined using the connective “and”. Premise 1: If X is A and Y is B then Z is C Premise 2: X is A' and Y is B' Consequence: Z is C' where A, A' are fuzzy sets in U; B, B' are fuzzy sets in V and C, C' are fuzzy sets in W. The consequence C' can be deduced from Premise 1 and Premise 2 by taking the max-min composition(o) of a fuzzy set (A' and B') in U×V and a fuzzy relation (A and B) ?C in U×V×W. 2.2 Generating algorithm As a generalized form of fuzzy reasoning with several fuzzy conditional propositions combined with “else”. Premise 1: If X is A1 and Y is B1 then Z is C1 else Premise 2: If X is A2 and Y is B2 then Z is C2 else Premise n: If X is An and Y is Bn then Z is Cn else Premise n+1: If X is A' and Y is B' Consequence: Z is C' If in above discussed fuzzy implications, “else” is interpreted as union ( ) which is valid for the fuzzy implication developed by Zadeh (Ra: 1 [1-µA(uo) + µB(v)]) (Raha and Ray, 1992) in above discussed fuzzy implications, we can deduce the consequences C' as,   “If {Dew point(Td) is ‘moderate’, Dew point spread(?T) is ‘very saturated’, Rate(?T') is ‘saturating’, Wind speed(W) is ‘Excellent’, Sky condition(S) is ‘clear’} Then (Possibility of fog(F) is ‘very high’}”. To test this rule we have take input from Nova data (http://www.ncdc.noaa.gov/pub) as, Td= 25.00C, ?T=-4.00C, ?T'=-2.00C, W=7.0Knts/hr., S=10.1% then by Zadeh’s arithmetic rule, Ra: 1 [1-µA(uo) + µB(v)] as ‘Min’ operator and ‘Max-Min’ composition for relations(R) we get the results for different parameters. (i) For Dew point: (0.7 0.8 0.9 1.0 1.0 1.0 1.0 1.0 1.0 1.0)……………… (a) (ii) For Dew point spread: (0.5 0.5 0.6 0.7 0.9 1.0 1.0 1.0 1.0 1.0)……………… (b) (iii) For Rate of change of Dew point spread: (0.4 0.5 0.5 0.6 0.8 0.9 1.0 1.0 1.0 1.0)……………… (c) (iv) For Wind speed: (0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0)……………… (d) (v) For Sky condition (0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0)……………… (e) Now by fuzzy intersection (?) of (a), (b), (c), (d), (e) the possibility consequence is (0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.0). Now corresponding fuzzy set is, {0.2/0=F < 10, 0.3/10=F < 20, 0.4/20 = F < 30, 0.5/30=F < 40, 0.6/40=F < 50, 0.7/50=F < 60, 0.8/60=F < 70, 0.9/70=F < 80, 1.0/80=F< 90, 1.0/90=F=100} In this way the process was repeated for 216 rules. After fuzzy union of the 216 sets we get the set which is the set of possibility values. The possibility consequence shows that possibility of occurring radiation fog is quiet high and it ranges from 80% to 100%. Now if we compare the computational result with the input value and the corresponding experience (obtained from an expert), it clearly indicates that computational result totally supports the experience of an expert based on the input data. Normally the fuzzy consequence of approximate reasoning is defuzzified by taking the element of the universe of consequence having highest membership value. In case of ‘tie’ situation we can break the tie by taking arbitrary decision or we can take the average of all tie values or we can consider the entire range of tie value. There are many other approaches to defuzzify a fuzzy consequence. The choice of particular process of defuzzification depends on the need of the problem. 4. Conclusion At the time of developing this concept, we review model based numerical approach to prediction of radiation fog. It has observed that instead of using of fuzzy rule based tool to prediction of radiation fog as an independent tool, we should use it as a complement of conventional model based approach to the prediction of radiation fog, so that we can mutually compensate the merits and demerits of both the approaches (i.e. rule based approach and model based approach). The present work addresses the issue of fuzzy rule-based modeling from available data and indicates a solution for predicting the probability of the formation of fog by formulating the problem within a fuzzy framework. The dew point spread and the rate of change of dew point spread have been the most important parameters for the formation of fog. The presented approach aims at complexity reduction using the linguistic fuzzy model, while maintaining the approximation accuracy at a reasonable level. This goal is achieved by modifying the rule antecedents to produce a flexible and interpretable output space. 5. Acknowledgement We express our sincere thanks to Prof. K. S. Ray, Senior Professor and Dr. A. K. Dey of Indian Statistical Institute, Kolkata for their kind advice and guidance time to time. 6. Reference
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