Fog Forecasting using Rule-based Fuzzy Inference ... - Springer Link

Photonirvachak

J. Indian Soc. Remote Sens. (September 2008) 36:243–253

RESEARCH ARTICLE

Fog Forecasting using Rule-based Fuzzy Inference System A.K. Mitra . Sankar Nath . A.K. Sharma

Received: 5 March 2007 / Accepted: 4 April 2008

Keywords Fuzzy logic . Fog . INSAT . Soft computing . FIS . Skill score

Abstract Operational meteorology is perceived as a fuzzy environment in which information is vaguely defined. The mesoscale processes such as fog, stratus and convection are generally dependent on the topography of the place and has always been difficult to forecast for the meteorologists. The main objective of the present study is to introduce the concept of fuzzy inference system (FIS) in the prediction of fog. This approach 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

A.K. Mitra () . S. Nath . A.K. Sharma SATMET Division, India Meteorological Department, New Delhi - 110 003

e-mail: [email protected]

in the input universe of discourse to fuzzy sets in the output universe of discourse based on fuzzy logic principles. Basic weather elements, which affect weather characteristics of fog, are fuzzified. These are then used in fuzzy weather prediction models based on fuzzy inferences. These models are simulated and the crisp results obtained using developed defuzzification strategies are compared with the actual weather data. The basis of methodology is to construct the fuzzy rule base domain from the available daily current weather observations in winter season over New Delhi. The results reveal that dew point spread and rate of change of dew point spread are the most important parameters for the formation of fog. The results further indicate that fog formation over New Delhi are dominant when (i) dew point is greater then 7°C along with dew point spread between 1 and 3°C. (ii) rate of change of dew point spread must be negative and wind speed should be less than 4 knots. This study presents a technique for predicting the probability of fog over New Delhi for 5-6 hours in advance. The skill score indicates that the

123

244

performance of FIS is appreciably good. The method is found to be promising for operational application.

Introduction During the winter season (December to February), fog formation is one of the most important weather events over northern parts of India. The impact and significance of fog ranges from disruption in aviation services, surface transportation and results in serious accidents. Inspite of considerable progress in the field of numerical weather prediction (NWP), current operational NWP models are incapable of simulating the fog progress. This is because of the dependency of fog on mesoscale and synoptic scale processes that act with boundary layer that is influenced by the prevailing synoptic regime. In operational meteorology these events are termed as a complex, ambiguous and vagueness embedded in their nature, therefore such weather hazards are most difficult to forecast precisely. Extensive researches have been carried out upon such weather systems by Roy Bhowmik et al. (2004), Brij Bushan et al. (2003), Thulsidas & Mahapatra (1998), Gupta et al. (1987) on fog formation mostly over airports. In the current scenario, rigorous numerical systems of meteorological processes continue to improve in both spatial and temporal resolution but for analytic solution or those beyond the resolution of systems still need alternative methods for their analysis and subsequent prognosis. With the rapidly evolving technologies in the field of meteorology, it is desirable to evolve such a technique with algorithms, which can provide better problemsolving techniques. One such method is called Fuzzy Inference System (FIS). In the recent years, Soft Computing (SC) 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

123


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). FIS is another Soft Computing technique necessary for analyzing complex systems, especially when the data structure is characterized by several linguistic parameters. These parameters have relationship with fuzzy concepts like fuzzy sets, rule base, linguistic (approximate) variables such as cloudy, foggy, dense, high, low, dry, wet, small, etc., which can account for the effect upon the output of a system due to various input values, without the need for a definite threshold value. The entire FIS framework is based on the concept of fuzzy set theory, fuzzy IF–THEN rules, and fuzzy reasoning (defuzzification). An FIS can utilize human expertise by storing its essential components in rule base and perform fuzzy reasoning to infer the overall output value. For building an FIS, we have to specify the fuzzy sets, fuzzy operators and the knowledge base rules. The concept of fuzzy set theory is applied to find the impact on some of the parameters for the prediction of fog. The theory of fuzzy logic or soft computing was introduced by Lotfi A. Zadeh (1965) to model the uncertainty of the natural language. FIS acquires knowledge from domain experts and that is encoded within the algorithm in terms of the set of IF–THEN rules and employs this rule-based approach and interpolative reasoning to respond to new inputs.

Data Current weather reports of Palam Airport, New Delhi have been used for a period of 6 years (1997–2002) from December to February and validated for the


year (2003–2004 and 2004–2005). VHRR imageries in the Visible channel (0.55 μ – 0.75 μ) from Indian National Satellites (INSAT) from Satellite Meteorological Division, DGM’s Office, Lodi Road were used for validation purpose. In the present study, dew point, dew point spread, wind speed, sky condition, rate of change of the dew point spread have been taken as the basic parameters.

Methodology Fog is generally confined to the horizontal area over the surface. Therefore surface parameters have been considered in this study. Details of the FIS including fuzzy sets, membership functions, fuzzy rule base are as follows: a. Fuzzy Inference Systems Fuzzy Inference Systems (FIS) are popular computing frameworks based on the concept of fuzzy set theory. Their success is mainly due to their closeness to human perception and reasoning, as well as their intuitive handling and simplicity, which are important factors for acceptance and usability of the systems. Nauck et al. (1999). Fuzzy inference is the process of formulating the mapping from a given input to an output using fuzzy logic. The mapping provides a basis from which decisions can be made. The process of fuzzy inference involves formulating Membership Functions, Logical Operations, and IF–THEN rules. Because of its multidisciplinary nature, FIS are associated with a number of names, such as fuzzyrule-based systems, fuzzy expert systems, fuzzy modeling, fuzzy associative memory, fuzzy logic controllers, and fuzzy systems. There are two types of FIS that can be implemented in Fuzzy Logic: Mamdani-type and Sugeno-type. These two types of inference systems vary somewhat in the way outputs are determined, Mamdani et al. (1974) and Takagi–Sugeno (TS), Jang et al. (1997).

245

Fuzzy rule-based model (i.e. FIS), proposed in this paper is Mamdani-type inference. The major difference between Mamdani and Sugeno is the construction of the rule consequent. In the Mamdani, the consequents are linguistic (fuzzy sets) resulting in a transparent model, whereas the Sugeno employs linear combinations of the inputs allowing for approximation. The Mamdani FIS has many attractive features. First, it is suitable for engineering systems because its inputs and outputs are realvalued variables. Second, it provides a natural framework to incorporate fuzzy IF–THEN rules from human experts. Third, there is much freedom in the choices of fuzzifier, fuzzy inference engine, and defuzzifier, so that we may obtain the most suitable fuzzy logic system for a particular problem. Figure 1 illustrates the basic architecture of an FIS. The main components are a fuzzification interface, a fuzzy rule base (knowledge base), an inference engine (decision-making unit), and a defuzzification interface. The input variables are fuzzified whereby the membership functions defined on the input variables are applied to their actual values, to determine the degree of truth for each rule antecedent. Fuzzy if-then rules and fuzzy reasoning are the backbone of FIS. The fuzzy rule base is characterized in the form of if-then rules in which the antecedents and consequents involve linguistic variables. The collection of these fuzzy rules forms the rule base for the fuzzy logic system. Using suitable inference procedure, the truth-value for the antecedent of each rule is computed, and applied to the consequent part of each rule. This results in one fuzzy subset to be assigned to each output variable for each rule. Usually, the rule base and the database are jointly referred to as the knowledge base. Again, by using suitable composition procedure, all the fuzzy subsets assigned to each output variable are combined together to form a single fuzzy subset for each output variable. Finally, defuzzification is applied to convert the fuzzy output set to a crisp output.

123

246


Fig. 1 Fuzzy Inference System.

b. Fuzzy sets and Membership functions The basic idea of fuzzy sets is easy to grasp. Hence an object with membership degree 1 absolutely belongs to the set and those with 0 membership values again absolutely do not belong to the set but objects with intermediate membership degrees belong to the same set partially. The greater is the membership degree the more the object belongs to the set. The parameters that are depicted, as fuzzy sets in this paper are dew point, dew point spread, wind speed, sky condition (Skc), and rate of change of the dew point spread (Rcdps). The basic structure of the fuzzy sets, which has been used in this study, is shown in Table 1. In Table 1, Rcdps (°C /hour) is characterized by two categories, whether the atmosphere is drying (a positive rate) or saturating trend (a negative rate).

The Skc have been classified into either cloudy or clear. Similarly others have been defined. These fuzzy sets have been created from the qualitative study of fog and quantitatively defined by membership functions. These functions contain a specified domain of the value of the system input and have been shown in Fig. 2(a–e) in the form of trapezoids. c. Fuzzy Rule Base The rule base is a set of rules of the If-Then form. The If portion of a rule refers to the degree of membership in one of the fuzzy sets. The Then portion refers to the consequence, or the associated system output fuzzy set. Let a, b, c, d and e be sets of antecedents where a, b, c, d, and e are different

Table 1 Parameter category

Dew-point (°C)

Dew-point spread (°C)

Rate of change of dew point spread (°C /hour)

Wind speed (knots)

Sky condition (octet)

1

Dry

Saturated

Saturating

Light

Cloudy

2

Moderate

Moderate

Drying

Medium

Clear

3

Moist

Unsaturated

-

Strong

-

Dew-point =Dp, Dew-Point Spread=Dps, Rate of change of dew point spread=Rcdps, Wind Speed= Ws, Skc=Sky Condition (0-9): 0 – Clear Sky and (1-9) – Cloudy.

123


247

fuzzy sets, whereas X be sets of consequences. The rules are applied in this manner: Membership

Membership

Membership

Membership

Membership

Fig. 2 Membership functions associated with the (a) dew point, (b) dew point spread, (c) wind speed and (d) sky condition and (e) rate of change of dewpoint spread. (Mod= Moderate, ClrSky= Clear Sky, Bcmg Sat= Becoming saturated).

If (ai & bi & ci & di & ei) Then (Xj)

(1)

where in equation (1), ‘i’ is the variable, which belongs to that fuzzy set, and ‘j’ is the maximum possible consequences in terms of very low, low, medium, high and very high probability of fog. The total number of rules is the product of the number of fuzzy sets in the system. In other words, the number of rules equals all possible permutations of categorized system inputs. From Table 1, there are three sets associated with the Dp, three with the Dps, two with the Rcdps, three with the Ws and two with the Skc. The total number of rules that completely define the set then is 3×3×2×3×2=108. These rules are defined in such a way that if five parameters Dp, Dps, Rcdps, Ws and Skc are moist, saturated, saturating, light and clear respectively then there will be a very high probability of fog formation. In a similar way, other rules have been defined. These rules are shown in tabular form in Table 2. The prediction of fog is based on the degree of the membership of the inputs from the evaluation of a set of predefined rules. The strength of a rule is derived from the corresponding degrees of membership of the inputs. Since an input can be a member of multiple fuzzy sets, then another set of rules involving these sets can be applied. The higher degrees of membership result in corresponding rules, which have more strength in the final computational process. In particular, to be able to deploy fuzzy logic in a rule-based system, one needs to be able to handle the operators ‘AND’ and ‘OR’ and carry out inferencing on the rules. Therefore, we need to be able to perform the intersection and union of two fuzzy sets. To calculate the intersection of a pair of fuzzy sets there are a family of functions, triangular norms or T-norms, that meet certain requirements such as monotonicity, commutativity and associativity and the intersection of a fuzzy set

123

248

J. Indian Soc. Remote Sens. (September 2008) 36:243–253 Table 2

Rules

Dp

Dps

Rcdps

Ws

Skc

Occurrence

1

Moist

Saturated

Saturating

Light

Clear

Very high

2

Moderate

Saturated

Saturating

Light

Clear

High

3

Moderate

Moderate

Saturating

Light

Clear

Medium

4

Moderate

Moderate

Drying

Light

Clear

Low

5

Moderate

Moderate

Drying

Light

Cloudy

Very low

6

———

———

———

———

———

———

——

———

———

———

———

———

———

108

———

———

———

———

———

———

with an ordinary set leads to exclusion of elements or conservation of degrees of membership. d. Mamdani-FIS Mamdani introduced the fuzzification/inference/ defuzzification scheme and used an inference strategy that is generally mentioned as the max-min method. This inference type is a way of linking input linguistic variables to output ones in accordance with the generalized modules, using only the MIN and MAX functions (as T-norm and S-norm (or T-conorm) respectively). It allows achieving approximate reasoning (or interpolative inference). The main feature of such a type of FIS is that both the antecedents and the consequents of the rules are expressed as linguistic constraints. An FIS is simply an expert system that uses a collection of fuzzy membership functions and rules, instead of Boolean logic, to reason about data (Schneider et al., 1996). From a mathematical point of view, let X be a space of objects and x be a generic element of X. A classical set A, A ⊆ X, is defined as a collection of elements or objects x ∈ X, such that x can either belong or not belong to the set A. A fuzzy set A in X is defined as a set of ordered pairs

for the fuzzy set A. The MF maps each element of X to a membership grade (or membership value) between zero and one. The intersection of two fuzzy sets A and B is specified in general by a function T : [0,1] * [0,1] →[0,1], which aggregates two membership grades as follows:

μA1B (x) = T (μA (x), μB (x)) = μA (x) ô μB (x) (2) where ô is a binary operator for the function T. These classes of fuzzy intersection operators are usually referred to as T-norm operators (Jang, Sun and Mizutani, 1997). Four of the most frequently used Tnorm operators are Minimum: Tmin (a, b) = min (a, b) = a Λ b

(3)

Algebraic product: Tap (a, b) = ab

(4)

Bounded product: Tbp (a, b) = 0 V (a + b – 1) (5)

⎛ a, if b = 1 ⎞ ⎜ ⎟ Drastic product : Tdp (a, b) = ⎜ b, if a = 1 ⎟ (6) ⎜ 0, if a, b < 1⎟ ⎝ ⎠

(1)

Like T-norm intersection, the fuzzy union operator is specified in general by a function S : [0,1] * [0,1] →[0,1], which aggregates two membership grades in the following fashion:

where μA (x) is called the membership function (MF)

μAUB (x)= S (μA (x), μB (x))= μA (x) K μB (x) (7)

A = {(x, μA (x)) | x ∈ X}

123


where K is the binary operator for the function S. This class of fuzzy union operators is often referred to as T-conorm (or S-norm) operators (Jang, Sun and Mizutani, 1997). Four of the most frequently used Tconorm operators are Maximum: Smax (a, b) = max (a, b) = a V b

(8)

Algebraic sum: Sas (a, b) = a + b - ab

(9)

Bounded sum: Sbs (a, b) = 1 Λ (a + b)

⎛ a, if b = 0 ⎞ ⎜ ⎟ Drastic sum : Sds (a, b) = ⎜ b, if a = 1 ⎟ ⎜ 0, if a, b < 0 ⎟ ⎝ ⎠

(10)

(11)

Both the intersection and union operators retain some properties of the classical set operation. In particular, they are associative and commutative. According to Mamdani-FIS, the rule antecedents and consequents are defined by fuzzy sets shown in Fig. 3, and has the following structure: if x is A1 and y is B1 then z1 = Fog

(12)

When a crisp output is required, the resulting fuzzy set has been defuzzified by means of several strategies such as centroid, mean of maximum, linear, bisector of area defuzzifier, among which the centroid of area (COA) gave best results and defined as:

249

∫μ Centroid of area ZCOA =

A

( z ).zdz

z

∫μ

A

( z ) dz

(13)

z

Here, μA (z) is the aggregated output membership function.

Results and discussion Based upon the observations collected from the Regional Meteorological Center, New Delhi, validation has been done for the year (2003–2004 and 2004–2005) from December to February. For each season we have taken some of the observations that are presented along with associated probability of fog formation after defuzzification in Table 3. 1. Generally clear skies combined with sufficient moisture and a saturating trend with light winds indicates a higher probability of fog formation. Using rule-based fuzzy theory only one rule would be applicable for (S.N. 1) condition, which is “moist & saturated & medium & clear sky & saturating” according to Table 2. After defuzzification, we have found that probability of fog formation is 74%, which corresponds to high probability of fog

Fig. 3 Mamdani-FIS rule antecedents and the consequent.

123

250


Table 3 Probability of fog formation after defuzzification S.N

1 2 3 4 5 6

Date/ UTC

Dew-point (°C)

Dew-point spread (°C)

Wind speed (knots)

Sky condition (octet)

Rate of change of dew point spread (°C /h)

02/01/04 01Z

4.7

0.3

0

0

0.3

74%

03/01/03 21Z

14.7

3.8

2

2

-0.4

65%

06/12/03 02Z

2.4

9.5

5

0

4.3

15%

28/01/04 00Z

6.2

7.4

0

0

1.3

30%

03/01/05 02Z

9.6

0.8

3

0

-0.1

70%

10/01/05 02Z

7.3

4.1

5

0

0.3

25%

as defined by the current set of the output membership function. The observation at 0300UTC (08:30 hrs IST) of 02/01/04 and satellite imagery of Fig. 4(a) indicated the foggy condition. Applying the same method, as above 16, 2, 1, 4, 2 rules would be applicable for the S.N. 2, 3, 4, 5, and 6 data in Table 3. After defuzzification, it has been found that probability of fog formation is 65%, 15%, 30%, 70%, and 25% respectively. This was verified by the observation and also by satellite imageries shown in Fig. 4(b–f). 2. However, the success of this technique mainly depends on the accurate estimates of the input variables and skill in tuning FIS. Careful construction of the membership functions as well as the rule base is necessary. Though the technique is developed for probability of occurrence of fog, 5–6 hours in advance, the forecast can be updated during the subsequent observation hours. The iterative process of designing the rule base, choosing a defuzzification algorithm, and testing the system performance was repeated

123

Fog probability (%)

several times with different shapes of fuzzy membership function. 3. The code, which accompanies this FIS program in the paper, is built in FORTRAN and JAVA and is easily adaptable to a number of platforms. Very little computing resources are required to implement this type of system. A number of such systems could be developed and inserted into the cron table of any UNIX/LINUX system and could be run on an hourly basis. Alert messages could be displayed if the system recognizes the possibility of development of some meteorological element. Performance of FIS There are several different methods that can be used to forecast. The method a forecaster chooses depends upon the experience of the forecaster, the amount of information available to the forecaster, the level of difficulty that the forecast situation presents, and the degree of accuracy or confidence needed in the forecast.


251

Fig. 4(a–f) INSAT VIS image on (a) 02-01-04 03 Z, (b) 03-01-03 03 Z, (c) 03-01-05 03 Z showing fog and (d) 06-1203 03 Z, (e) 28-01-2004 03 Z, (f) 10-01-05 03 Z without fog (Arrow mark showing fog over Delhi).

The first of these methods is the Persistence Method; the simplest way of producing a forecast. The persistence method assumes that the conditions at the time of the forecast will not change. The persistence method works well when weather patterns change very little and features on the weather maps move very slowly. It may also appear that the persistence method would work only for shorter-term forecasts (e.g., forecast for a day or two). We have tested our FIS model with the developed dataset using persistence method of estimating input values. In other words, we used 21Z or 21UTC (02:30 hrs IST) observations to forecast occurrence of fog for the next 5–6 hours. Here we are presenting categorical statistics, Stanski et al. (1989), such as bias and threat skill scores. A bias score (B) and threat score (T) are defined as B=(F+H)/(M+H) T=H/(F+M+H)

where F = Number of false alarms (occurrence of fog forecast but is not realized). M = Number of missing alarms (nonoccurrence of fog forecast but fog occurred) H = Number of hits (forecast occurrence as well as non-occurrence of fog is realized). B and T becomes 1 in case of F=M=0. When a bias score is less than unity the model indicates underforecast, otherwise over-forecast. The skill score obtained from development data set (1997–2002) are as follows: H 178

F 20

M 30

B 0.95

T 0.78

The result shows that scores are reasonably good and forecasts are slightly under-predicted.

123

252


The skill score obtained from independent data set of January 2004 and January 2005 are as follows: H 41

F 10

M 11

B 0.98

T 0.66

In this case also forecast is slightly under predicted.

Concluding Remarks Fuzzy inference systems can be applied in a vast number of meteorological application areas. An important advantage of the fuzzy expert system is that the knowledge is expressed in the form of easyto-understand linguistic rules. If we have large amount of data, the fuzzy expert system can be taught using neural network or other adaptation techniques. Data collecting and acquisition are initial and one of the most critical parts of expert systems computations. Automated data collecting systems must be available rather than using human manual inputs. Local data collection is very important if we want to make very precise forecasting. This paper addresses the issue of fuzzy rule-based modeling from available data. 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. The present study 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. Fuzzy analysis of fog at Palam Airport, New Delhi reveals that fog occurrence is dominant when dew point and dew point spread is between 6–10°C and 1–3°C respectively. A negative saturating trend and light winds approximately 0–4 knots also play a major role for the formation of fog. The threat score and bias score indicates that the performance of FIS is

123

appreciably good but has a bias slightly towards under-prediction both for development data set as well as independent data set.

Acknowledgements The authors would like to express their sincere gratitude and thanks to the Director General of Meteorology for his encouragement during the course of this study. Authors would like to gratefully acknowledge the facilities provided by Satellite Meteorology Division of the India Meteorological Department, New Delhi. Thanks are also due to Dr. Soma SenRoy, IMD, for providing valuable system and software support.

References Abraham A, Philip NS and Joseph B (2001) “Will we have a wet summer? Long term rain forecasting using Soft Computing models”, Modelling and Simulation 2001, Publication of the Society for Computer Simulation International, Kerchoffs EJH & Snorek M (Editors), Prague, Czech Republic, 1044–1048 Bardossy A, Duckstein L and Bogardi I (1995) “Fuzzy rule-based classification of atmospheric circulation patterns”, International Journal of Climatology 15:1087–1097 Brij Bhusan, Trivedi HK and Bhatia RC (2003) “On the persistence of fog over northern parts of India”, MAUSAM-54, 4,851–860 Gupta RK (1987) “On the technique of forecasting Fog/ Stratus over the Dundigal airfield of Hydrabad”, MAUSAM-38, 4,401-406 Hansen BK (1997) “SIGMAR: A fuzzy expert system for critiquing marine forecasts” AI Applications, Vol. 11(1):59–68 Hansen BK and Riordan D (1998) “Fuzzy case-based prediction of ceiling and visibility”, Preprints of the 1st Conference on Artificial Intelligence, American Meteorological Society, 118–123 Jang JSR, Sun CT and Mizutani E (1997) “Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence”, Prentice Hall Inc, USA

J. Indian Soc. Remote Sens. (September 2008) 36:243–253 Mamdani EH (1974) “Applications of fuzzy algorithms for simple dynamic plant, Proc IEE 121(12):1585– 1588 McBratney AB and Moore AW (1985) “Application of fuzzy sets to climatic classification”, Agricultural and Forest Meteorology 35:165–185 Mitra AK, Meena LR and Giri RK (2006) “Forecasting Of Temperature-Humidity Index using Fuzzy Logic Approach”, National Conference On Advances In Mechanical Engineering (AIME-2006) January 20–21, 2006, Jamia Millia Islamia, New Delhi-110 025 Mohapatra M and Thulsidas A (1998) “Analysis and forecasting of fog over Bangalore airport”, MAUSAM49, 1,135–142 Murtha J (1995) “Applications of fuzzy logic in operational meteorology”, Scientific Services and

253 Professional Development Newsletter, Canadian Forces Weather Service, 42–54 Nauck D and Kruse R (1999) “Obtaining Interpretable Fuzzy Classification Rules from Medical Data”, Artificial Intelligence in Medicine, vol. 16, pp.149–169, Elsevier Roy Bhowmik SK, Sud AM and Singh C (2004) “Forecasting fog over Delhi-Anobjective Method”, MAUSAM-55, 2,313,322 Schneider M, Langholz G, Kandel A and Chew G (1996) “Fuzzy Expert System Tools”, John Wiley & Sons, USA Stanski HR, Wilson LJ and Bossous WR (1989) “Survey of common verification method in Meteorology”, WMO T.NO,358,WMO Geneva Zadeh LA (1965) “Fuzzy Sets”, Information and Control 8:338–353

123