Vladimir Dimitrov and Basant Maheshwari
University of Western SydneyHawkesbury,
Richmond 2753, Australia
Fax: +61(245) 701953; +61(245) 885538
Phone: +61(245) 701903; +61(245) 701235
Email:
V.Dimitrov@uws.edu.au
B.Maheshwari@uws.edu.au
Each fuzzy class defines a specific HR fuzzy family. A range of values of Manning roughness coefficient (Manning 1889) corresponds to each of the following five fuzzy families:
Fuzzy families 
Notation used for the fuzzy families 
Range of Manning roughness coefficient 
Very low hydraulic resistance 
b1 
< 0.05 
Low hydraulic resistance

b2 
0.05  <0.10 
Medium hydraulic resistance

b3 
0.10 <0.20 
High hydraulic resistance

b4 
0.20  0.30 
Very high hydraulic resistance 
b5 
>0.30 
The above five families have fuzzy boundaries; that is, they overlap, despite the fact that the shown ranges of Manning roughness coefficient do not overlap. The fuzzy boundaries make the descriptions of HR families used in the present study much more adequate to the continuum of field situations.
First applications of fuzzy genetic algorithms to pattern recognition and classification have started since the late 1960s (Dimitrov 1969, 1970, 1973; Ivakhnenko et al.1976). The algorithm includes the following steps:
The following requirements increase the expected efficiency of the fuzzygenetic algorithm:
The following two hypotheses are used in this application of the fuzzygenetic approach:
Based on this frequency distribution, tables of 'significant gene support' for HR families (see Tables 15) are built in the following way:
For each 4gene combination, occurring in HR families, the degree of its membership to each fuzzy family (equal to its frequency of occurrence in this family) is computed. This degree is expressed through a number in the interval [0,1].
For example, among all 4gene combinations based on genes q, p, r and l occurring in the experimental data, it was found that the combination q1 p1 r1 l1 takes place 17 times in b1 family and 2 times in b2 family; thus, the degree of membership of this gene combination to b1 family is 17/(17+2) = 0.89, and to b2 family 2/(17+2) = 0.11.
Tables 15 contain only significant gene support for HR families that includes combinations whose degree of membership to the corresponding fuzzy family exceeds 0.62  the value of the 'golden section' y = (sqrt(5)  1)/2 = 0.618...
The golden section approach is used in order to avoid the cases of indecisiveness in favour of one or another pattern (class) if the degree of membership to this class is within the 'middle zone' (that is, in the interval between 0.4 and 0.6). The classification decision in favour of certain class is made, if the degree of membership to this class exceeds 0.62; the membership to this class is rejected if the value is less than 0.38 (that is, 1 y).
Only 4gene combinations with degree of membership (to a specific fuzzy family) higher than 0.62 (the golden section value) are considered supportive for this family.
Table1. Significant support of 4gene combinations for b1 family
Gene Combination 
Degree of Membership 
q1p1r1l1 
0.89 
q1p1r2l1 
0.73 
q1p2r1l1 
1.0 
q1p2r2l1 
1.0 
Table 2. Significant support of 4gene combinations for b2 family
Gene Combinations 
Degree of Membersip 
q3p1r2l1 
1.0 
q3p2r1l1 
1.0 
q3p2r1l5 
1.0 
q4p1r2l2 
1.0 
q4p2r1l5 
1.0 
Table 3. Significant support of 4gene combinations for b3 family
Gene Combinations 
Degree of Membership 
q2p1r1l1 
1.0 
q2p2r1l1 
0.8 
q3p1r1l1 
1.0 
q3p1r1l5 
1.0 
q3p1r3l3 
1.0 
q3p2r1l2 
1.0 
q4p1r3l3 
1.0 
q4p2r1l2 
1.0 
q5p2r1l1 
1.0 
Table 4. Significant support of 4gene combinations for b4 family
Gene Combinations 
Degree of Membership 
q2p1r1l2 
0.71 
q3p1r1l3 
1.0 
q4p1r1l3 
1.0 
q5p1r1l3 
0.86 
q5p1r1l4 
1.0 
q5p1r1l5 
1.0 
q5p1r2l5 
1.0 
Table 5. Significant support of 4gene combinations for b5 family
Gene Combinations 
Degree of Membership 
q2p1r1l3 
1.0 
q2p1r2l5 
1.0 
q5p1r3l1 
1.0 
q5p1r3l2 
1.0 
Tables 1 5 can be used for classification of new gene combination. The classification rule is quite simple: if the new gene combination coincides with some of the combinations shown in Tables 1  5, the location of the latter denotes the HR family to which the new combination is classified.
For example, if the new combination is q5p1r1l5, it is classified as belonging to family 4, as q5p1r1l5 is located in Table 4 corresponding to b4 family. The degree of belonging is equal to the degree of membership of q5p1r1l5 to family b4  that is, 1 (see Table 4).
If the new combination is q4p1r2l2, it is classified as belonging to family b2 as q4p1r2l2 is located in Table 2 corresponding to b2 family. The degree of belonging is equal to the degree of membership of q4p1r2l2 to family b2  that is, 1 (see Table 2).
The genes q and l are extremely important as they characterise the flow which is crucial in determining hydraulic resistance; without flow, the study of hydraulic resistance does not make any sense.
From Tables 1  5, we extract all gene combinations which contain both the gene q and the gene l, and thus build the following new table (Table 6); the number adjacent to each combination denotes the degree of membership (belonging) of this combination to a corresponding family.
FLOW l q 
l1  l2  l3  l4  l5 
q1  b1: p1r1(0.9) p1r2(0.7) p2r1(1.0) p2r2(1.0) 

q2  b3: p1r1(1.0) p2r1(0.8) 
b4: p1r1(0.7) 
b5: p1r1(1.0) 
b5: p1r2(1.0) 

q3  b2: p1r2(1.0) p2r1(1.0) b3: 
b3: p2r1(1.0) 
b4: p1r1(1.0) 
b2: p2r1(1.0)


q4  b2: p1r2(1.0) 
b3: p1r3(1.0) 
b2: p2r1(1.0) 

q5  b3: p2r1(1.0) 
b5: p1r3(1.0) 
b4: p1r1(0.9) 
b4: p1r1(1.0) 
b4: p1r2(1.0) 
Is it possible to develop a classification algorithm based only on the flow characteristics (flow rate or/and flow type)?
For example:
An exception is gene q1 which is always associated with q1.
For example:
Although gene l4 is associated only with family b4, it is not decisive in relation to this family as family b4 is significantly supported also by the genes l2, l3 an l5 (as seen from Table 6).
Note:
The reason why classifications based only on the flow rate or only on
the flow type does not work is because the flow is extremely sensitive
to any changes in the other two genes (mode of plantation and
vegetation density) however tiny those changes might appear to the
observer. In this sense, the flow represents an essentially chaotic
parameter  its dynamics are unpredictable and subjected to the
'butterfly effect' considered in Chaos Theory: small changes in the
field situation can result in drastic changes of the flow
characteristics (and hydraulic resistance, respectively).
For example:
The following flow characteristics (extracted from the data in Table 6) could be used for a quick classification (without indispensable participation of the other genes) as they are uniquely associated with a specific family:
Family b1: q1l1
Family b2: q4l5
Family b3: q2l1 q3l2
Family b4: q2l2; q3l3; q5l3, q5l4, q5l5
Family b5: q2l3, q2l5; q5l2.
The idea behind Dissimilarity Measure Algorithm is quite simple:
For each new gene combination, we find how 'distant' is it from the gene combinations included in Tables 15, using a special expression of this distance; the minimal distance determines the family to which the new combination is classified.
The calculation of dissimilarity measure is done with all 4gene patterns included in Tables 1  5 as they are considered as the only available experimental source of information trustworthy enough to be used for the purpose of HR family classification.
The calculation rule consists of the following:
For each 4gene combination q(i)p(j)r(k)l(m) included in Tables 1  5, the dissimilarity measures with any new field situation q(a)p(b)r(c)l(d) are calculated as the following sum (each item of the sum is in absolute value):
ia+jb+kc+md
where i= 1,2,3,4 or 5; j=1 or 2; k= 1,2 or 3; m= 1,2,3,4 or 5, are used to denote the different levels of each of the genes.
Notes:
Example:
Both gene patterns belong to family b4 (see Table 4). Thus, the field situation is classified as belonging to family b4.
A Fuzzy Classification Rule (FCR) is usually of the form IF...THEN..., where both the IF and THEN parts are 'natural language' expressions of some fuzzy classes or their combinations. Fuzzy Logic provides techniques for computing, with these classes aimed at specific problemsolving (classification, pattern recognition, prediction, etc.).
Each HR family is described by a specific FCR. Being different for each family, FCR provides an integral fuzzy description of this family and, therefore, can be used for the purpose of its classification. FCR are entirely contextsensitive; that is, they depend on the available experimental data (as is the case with every fuzzygenetic and fuzzyneuro algorithm).
The data in Table 6 serve as the source for generating FCR . We break this table into five 'subtables' or clusters  each cluster corresponds to one of the five HR famiiies. Here are the clusters extracted from Table 6:
First Cluster  family b1
FLOW l q 
l1 
q1 
p1r1 (0.9) p1r2 (0.7) p2r1 (1.0) p2r2 (1.0) 
Second Cluster  family b2
FLOW l q 
l1  l2  l5 
q3 
p1r2 (1.0) p2r1 (1.0) 
p2r1 (1.0) 

q4 
p1r2 (1.0) 
p2r1 (1.0) 
Third cluster  family b3
FLOW l q 
l1  l2  l3  l5 
q2 
p1r1 (1.0) p2r1 (0.8) 

q3 
p1r1 (1.0) 
p2r1 (1.0) 
p1r1 (1.0) 

q4 
p2r1 (1.0) 
p1r3 (1.0) 

q5 
p1r3 (1.0) 
Fourth Cluster  family b4
FLOW l q 
l2  l3  l4  l5 
q2 
p1r1 (0.7) 

q3 
p1r1 (1.0) 

q4 
p1r1 (1.0) 

q5 
p1r1 (0.9) 
p1r1 (1.0) 
p1r2 (1.0) 
Fifth Cluster  family b5
FLOW l q 
l1  l2  l3  l5 
q2 
p1r1 (1.0) 
p1r2 (1.0) 

q5 
p1r3 (1.0) 
p1r3 (1.0) 
Example:
Let us generate FCR related to the first cluster (family b1) following the steps above. The first cluster contains four gene patterns, including the two major genes q1 and l1, and all possible combinations between levels 1 and 2 of the genes p and r.
 the gene q1 means very low flow rate
 the gene l1 means shallow nonsubmerged flow type
 the genes p1 and p2 mean random pattern of planting and row pattern of planting, respectively
 the genes r1 and r2 mean low density of vegetation and medium vegetation, respectively.
 The description of the second gene pattern is: very low flow rate COMBINED WITH random patterns of planting AND medium density of vegetation WHILE the flow type is shallow nonsubmerged.
 The description of the third gene pattern is: very low flow rate COMBINED WITH row patterns of planting AND low density of vegetation WHILE the flow type is shallow nonsubmerged.
 The description of the fourth gene pattern is: very low flow rate COMBINED WITH law patterns of planting AND medium density of vegetation WHILE the type of flow is shallow nonsubmerged.
IF the flow rate is very low AND the patterns of planting are in raw OR random,
COMBINED WITH a low OR medium density of vegetation,
WHILE the flow type is shallow nonsubmerged,
THEN the hydraulic resistance is very low.
In a strictly logical (computer) form  that is, using IF, THEN, OR, AND and separating brackets  the above rule can be written in the following way:
IF q1 AND [(p1 OR p2) AND (r1 OR r2)] AND l1 THEN b1.
Using the procedure described above, each cluster can be translated into a specific FCR.
FCR related to the second cluster (family b2) is formulated as follows:
IF random patterns of planting
WITH medium density of vegetation are treated
WITH a medium rate flow of shallow nonsubmerged type
OR a high rate flow of medium nonsubmerged type;
OR row patterns of planting WITH a low density of vegetation are treated
EITHER with a medium flow rate of a shallow nonsubmerged
OR submerged type, OR with a high rate flow of a submerged type,
THEN the hydraulic resistance is low.
The logical (computer) form of the above rule is:
IF {(p1 AND r2) AND [(q3 AND l1) OR (q4 AND l2)]}
OR {(p2 AND r1) AND [ [q3 AND (l1 OR l5)] OR (q4 AND l5) ]}THEN b2.
FCR related to the third cluster (family b3) is formulated as follows:
IF random patterns of planting
WITH a low density of vegetation are treated
EITHER with a low rate flow of a shallow nonsubmerged type
OR with a medium rate flow of a shallow nonsubmerged OR submerged type;
OR row patterns of planting WITH a low density of vegatation are treated
EITHER with a low rate flow of a shallow nonsubmerged type,
OR with a medium OR high rate flow of a medium nonsubmerged type;
OR random patterns of planting WITH a high density of vegetation are treated
EITHER with a very high rate flow of a shallow nonsubmerged type,
OR with a high rate flow of a deep nonsubmerged type,
THEN the hydraulic resistance is medium.
The computer form of this FCR is:
IF {(p1 AND r1) AND [ (q2 AND l1) OR [q3 AND (l1 OR l5) ]}
OR {(p2 AND r1) AND [ (q2 AND l1) OR [(q3 OR q4) AND l2] ]}
OR {(p1 AND r3) AND [(q5 AND l1) OR (q4 AND l3)]}
THEN b3.
FCR related to the fourth cluster (family b4) is formulated as follows:
IF random patterns of planting WITH a random density of vegetation are treated
EITHER with a low rate flow of a medium nonsubmerged type,
OR with a medium OR high OR very high rate flow of a deep nonsubmerged type,
OR with a very high rate flow of a just submerged type;
OR random patterns of planting WITH a low density of vegetation are treated
WITH a very high rate flow of a submerged type,
THEN the hydraulic resistance is high.
The computer form of this FCR is:
IF {p1 AND r1 AND [ (q2 AND l2) OR [(q3 OR q4 OR q5) AND l3] OR (q5
AND l4) ]}
OR {p1 AND r2 AND q5 AND l5}
THEN b4.
FCR related to the fifth (family b5) is formulated as follows:
IF random patterns of planting WITH a high density of vegetation are treated
WITH a very high rate flow of shallow nonsubmerged
OR medium nonsubmerged type;
OR random patterns of planting are treated WITH a low rate flow
WHILE this is flow is EITHER deep nonsubmerged AND the density of the plantation is low,
OR submerged AND the density of vegetation is medium,
THEN the hydraulic resistance is very high.
The computer form of this FCR is:
IF {(p1 AND r3) AND (q5 AND (l1 OR l2)}
OR {(q2 AND p1) AND [(r1 AND l3) OR (r2 AND l5)]}
THEN b5.
The Dissimilarity Measurebased classification algorithm, described above, was tested on 70 entirely new field situations which have not been involved in constructing classification Tables 1  5. The right classification occurs in more than 50 cases. This promising result reveals the practical efficiency of the HR family concept in the classification of surface irrigation situations.