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NUMERICAL EXAMPLE APPENDIX A in “A neuro-fuzzy modeling tool to estimate fluvial nutrient loads in watersheds under time-varying human impact” Rafael Marcé1*, Marta Comerma1, Juan Carlos García2, and Joan Armengol1 1Department of Ecology, University of Barcelona, Diagonal 645, 08028 Barcelona, Spain 2Aigües Ter Llobregat, Sant Martí de l'Erm 30, 08970 Sant Joan Despí, Spain *E-mail: [email protected] April 2004 What is fuzzy logic? Binary logic In binary logic the function that relates the value of a variable with the probability of a judged statement are a ‘rectangular’ one. Taking the seasons as an example... Probability WINTER SPRING SUMMER FALL 1 The result will always be ‘one’ for a season and ‘zero’ for the rest 0 March 7th Winter = 1 WINTER Time (day of the year) SPRING SUMMER FALL Fuzzy logic In fuzzy logic the function can take any shape. The gaussian curve is a common choice... Probability 1 In fuzzy logic, the truth of any statement becomes a matter of degree. 0 March 7th Winter = 0.8 Spring = 0.2 Time (day of the year) Fuzzy reasoning with ANFIS Given an available field database, we define an input-output problem. In this case, the nutrient concentration in a river (output) predicted from daily flow and time (inputs). The first step is to solve the structure identification. We apply the trial-and-error procedure explained in the text with different number of MFs in each input. Suppose that the results were as follows: MFs in input FLOW Error MFs in input TIME Residual Mean Square 1 1 7.52 1 2 5.36 2 1 5.21 2 2 2.95 3 2 2.05 2 3 2.35 3 3 2.04 4 4 2.01 5 5 1.99 This option is considered the optimum trade-off between number of MFs and fit. Fuzzy reasoning with ANFIS Then, the structure identification is automatically solved generating a set of 6 if-and-then rules, i.e. a rule for each possible combination of input MFs. For each rule, an output MF (in this case a constant, because we work with zero-order Sugeno-type FIS) is also generated. and TIME is EARLY ON then CONCENTRATION is C1 Just for If FLOW is LOW and TIME is LATER ON then CONCENTRATION is C2 convenience, we rename the If FLOW is MODERATE and TIME is EARLY ON then CONCENTRATION is C3 different input If FLOW is MODERATE and TIME is LATER ON then CONCENTRATION is C4 MFs with intuitive If FLOW is HIGH and TIME is EARLY ON then CONCENTRATION is C5 linguistic If FLOW is HIGH and TIME is LATER ON then CONCENTRATION is C6 labels, such High or Early on. Rule 1 If FLOW is LOW Rule 2 Rule 3 Rule 4 Rule 5 Rule 6 The next step is to draw the MFs in each input space, an also to assign a value for each output constant. This is the parameter estimation step, which is solved by the Hybrid Learning Algorithm using the available database. Suppose that the algorithm gives the following results: LOW MODERATE HIGH EARLY ON 1 LATER ON 1 0 C3 = 10.58 0 0 10 Flow C1 = 16.23 C2 = 18.56 Probability Probability Remember that a gaussian curve can be defined with two parameters. We give a graphical representation for clarity. 0 10 Time C4 = 16.13 C5 = Now the Fuzzy Inference System is finished. The following slide is a numerical example showing how an output is calculated from an input. Probability 1 Probability If FLOW is LOW 1 Probability p=0 p=0 X 18.56 2 and TIME is LATER ON then CONCENTRATION is C The second step is to combine the p = 0.4 probabilities on the premise part to= get p= 0.1 X 10.58 1.058 p = 0.1 the weight (or probability) of each rule. anis EARLY input, the step toFuzzy FLOW is MODERATEGiven and TIME ON governing then first CONCENTRATION issolve C The six rules the last step is the defuzzyfication It is demonstrable The third step that is1 toapplying calculatethe theand 1 the FIS is the fuzzyfication of inputs, i.e.a Inference System are represented with procedure, when the consequents are p=0 X operator is equivalent to solve consequent of each rule depending 16.13 p = 0.1 logical = 0on p=0 to obtain the probability of each graphical of the MFs thata 0 aggregated (weighted mean) obtain 0 probability) for therepresentation minimum value oftothe their weight (or value in each rule. FLOW is MODERATE and linguistic TIME is LATER ON theneach CONCENTRATION is C apply in rule. crisp output intersection of the MFs p = 0.75 If FLOW is LOW Rule 3 If 1 0 Rule 4 If 1 1 1 0 0 3 1 1 p= 0.4 X MIN = AND 0 0 1 If FLOW is HIGH p = 0.75 0 10 If FLOW is HIGH 8 = 2.636 and TIME is EARLY ON then CONCENTRATION is C5 1 1 p=0 0 0 6.59 0 p=0 INPUT VALUE for FLOW 1.058 + 2.636 0.1 + 0.4 4 p = 0.4 Rule 5 = 0 0 0 1 = 0 1 1 0 Probability 16.23 and TIME is EARLY ON then CONCENTRATION is C1 p=0 Rule 2 X 0 0 0 Probability p=0 p=0 Rule 1 Probability 1 p = 0.4 0 Rule 6 Logical operations 1 X 10.60 = 0 7.388 0 0 10 and TIME is LATER ON then CONCENTRATION is C6 2.5 INPUT VALUE for TIME OUTPUT CONCENTRATION VALUE