ISSN 1320-0682
		Volume 4	1997

Fuzzy-Genetic Study of Hydraulic Resistance Families in Surface Irrigation

Vladimir Dimitrov and Basant Maheshwari
University of Western Sydney-Hawkesbury,
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

Abstract

1 Introduction

2 Hydraulic Resistance Families

• very low HR
• low HR
• medium HR
• high HR
• very high HR

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:

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.

3 Fuzzy-Genetic Approach to Classification

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:

1. A learning sample of objects belonging to different fuzzy classes (families) is given. Each object is described as a sequence of fuzzy (not precisely) defined characteristics expressed in a symbolic form as 'genes'.
2. For all available objects belonging to a specific fuzzy family, a descriptive set of gene sequences is constructed so that each family could be characterised by its own descriptive set (table) of gene sequences.
3. The description set (table) is used for the classification of new objects belonging to a so-called 'control sample' of objects. The result of classification based on the control sample demonstrates the efficiency of the fuzzy-genetic classification algorithm.

The following requirements increase the expected efficiency of the fuzzy-genetic algorithm:

• as small as possible overlaps between the fuzzy families (the ideal situation is a zero overlap between each pair of families);
• the gene sequences in a specific family describe as much as possible objects belonging to this family (the ideal situation is a complete covering of all objects belonging to this family);
• as small as possible overlaps between the descriptive sets (tables) of gene sequences related to different families (the ideal situation is lack of overlap between each pair of sets).

4 Gene Description of Fuzzy Hydraulic Resistance Families

• q : Flow rate, represented by five fuzzy classes: very low (q1), low (q2), medium (q3), high (q4), and very high (q5)

• p : Pattern of Planting, represented by two fuzzy classes: random (p1) and row (p2)

• r : Density of Vegetation, represented by three fuzzy classes: low (r1), medium (r2), and high (r3)

• l : Type of Flow, represented by five fuzzy classes: shallow non-submerged (l1), medium non-submerged (l2), deep non-submerged (l3), just submerged (l4), and submerged (l5).

The following two hypotheses are used in this application of the fuzzy-genetic approach:

1. Typical (characteristic) gene patterns correspond to each HR family.

2. By constructing these patterns, we can use them in solving practical pattern recognition (classification) problems related to the surface irrigation.

5 Search for Significant Gene Support for Hydraulic Resistance Families

Based on this frequency distribution, tables of 'significant gene support' for HR families (see Tables 1-5) are built in the following way:

For each 4-gene 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 4-gene 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 1-5 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 4-gene combinations with degree of membership (to a specific fuzzy family) higher than 0.62 (the golden section value) are considered supportive for this family.

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).

6 Classification Based Only on the Flow Characteristics

• q - the flow rate, represented by five fuzzy classes: q1, q2, q3, q4, q5
•  l - the type of flow, represented by five fuzzy classes: l1, l2, l3, l4 and l5

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.

Is it possible to develop a classification algorithm based only on the flow characteristics (flow rate or/and flow type)?

6.1 Flow Rate

For example:

• both q2 and q5 are associated simultaneously with families b3, b4 and b5

• both q3 and q4 are associated simultaneously with families b2, b3 and b4.

An exception is gene q1 which is always associated with q1.

6.2 Flow Type

For example:

• l1 is associated simultaneously with families b1, b2, b3 and b5
• l2 is associated simultaneously with families b2, b3, b4 and b5
• l3 is associated simultaneously with families b3, b4 and b5
• l5 is associated simultaneously with families b2, b3, b4 and b5

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).

6.3 Both Flow Rate and Flow Type

For example:

• q3 l1, q3l5 and q4l2 are associated simultaneously with families b2 and b3
• q4l3 supports family b3 and b4 at the same time
• q4l1 is simultaneously associated with b4 and b5

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.

7 Classification Algorithm Based on Dissimilarity Measure

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 1-5, using a special expression of this distance; the minimal distance determines the family to which the new combination is classified.

7.1 Main steps of the algorithm

1. For every new field situation described as a gene sequence of the form q(a)p(b)r(c)l(d), find its dissimilarity measure with the all 4-gene patterns already established from the available experimental data and appropriate expert knowledge.

2. Classify the field situation as belonging to the family for which this dissimilarity measure is minimal.

3. If there are two or more families with equal dissimilarity measure, the field situation is classified according to the degree of membership of the corresponding 4-gene pattern: the sequence belongs to the family for which this degree of membership is maximal.

7.2 Calculation of Dissimilarity Measure

The calculation of dissimilarity measure is done with all 4-gene 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 4-gene 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):
|i-a|+|j-b|+|k-c|+|m-d|
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.

7.3 Finding gene patterns with the minimal dissimilarity measure

Notes:

1. If there are more than one 4-gene patterns with equally valued minimal dissimilarity measures, then the gene pattern with a higher degree of membership is usually preferred: its family represents the family to which the field situation q(a)p(b)r(c)l(d) is classified.

2. If the degrees of membership are also equal or very close (indistinguishable) in magnitude, then the final classification is completed by the field expert.

Example:

1. Let the new field situation have the following gene representation q(4)p(1)r(1)l(4). This situation is totally unknown - no such gene pattern exist in Tables 1-5. As a result of the calculation, the minimal dissimilarity measure equal to 1 is found for two gene patterns:
   • q(4)p(1)r(1)l(3) with a dissimilarity measure equal to 1 and weight 1
   • q(5)p(1)r(1)l(4) with a dissimilarity measure equal to 1 and weight 1

   Both gene patterns belong to family b4 (see Table 4). Thus, the field situation is classified as belonging to family b4.

2. Let the new field situation have the following gene representation q(2)p(1)r(2)l(4). This situation is also unknown; no such gene pattern exists in Tables 1- 5. As a result of calculation, the minimal dissimilarity measure equal to 2 is found for the gene pattern q(2)p(1)r(2)l(5) with a dissimilarity measure equal to 2 and weight 1. This gene pattern belongs to family 5 (see Table 5); thus the field situation q(2)p(1)r(2)l(4) is classified as belonging to family b5.

8 Generation of Fuzzy Classification Rules

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 problem-solving (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 context-sensitive; that is, they depend on the available experimental data (as is the case with every fuzzy-genetic and fuzzy-neuro algorithm).

The data in Table 6 serve as the source for generating FCR . We break this table into five 'sub-tables' or clusters - each cluster corresponds to one of the five HR famiiies. Here are the clusters extracted from Table 6:

Procedure for translating a specific cluster into a Fuzzy Classification Rule

• Step 1:
  Describe the meaning of each gene pattern included in the cells of a specific cluster using the gene description provided in section 4.

• Step 2:
  As each gene pattern consists of four genes: q, l, p, and r, after describing each of them, their descriptions must be connected with one another using AND or WHILE or COMBINED WITH or WITH type of connectives, or other connecting word with a meaning of AND (logical intersection); as a result, the description of the gene pattern as a whole is obtained.

• Step 3:
  As every HR family includes a number of gene patterns, after describing each of the patterns, their descriptions must be connected with one another using OR or EITHER...OR or other connecting word with a meaning of OR (logical union); as a result, the description of the whole HR family is obtained. This description is included in the IF part of FCR.

• Step 4:
  Describe the meaning of the family to which the description obtained at the previous step relates; this meaning is included in the THEN part of FCR.

• Step 5:
  The whole FCR is composed as a 'normal' expression (that is, in a natural language form).

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.

• Step 1:
  According to the gene description given in section 4:

  - the gene q1 means very low flow rate

  - the gene l1 means shallow non-submerged 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.

• Step 2:
  - The description of the first gene pattern is: very low flow rate COMBINED WITH a random patterns of planting AND low density of vegetation WHILE the flow type is shallow non-submerged.

  - 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 non-submerged.

  - 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 non-submerged.

  - 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 non-submerged.

• Step 3:
  All four descriptions have to be connected with one another by means of an OR-connective as each of them relates independently to one and the same cluster - family b1. The obtained OR-combined description is simplified and, after avoiding the repetitions of coinciding terms and articulating it in a natural language, is included in the IF part of FCR for the first family b1.

• Step 4:
  According to the family descriptions, b1 means 'very low hydraulic resistance'; this description is included in the THEN part of FCR.

• Step 5:
  The whole FCR is composed in the following (articulated in natural language) way:

  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 non-submerged,
  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 non-submerged type
OR a high rate flow of medium non-submerged type;
OR row patterns of planting WITH a low density of vegetation are treated
EITHER with a medium flow rate of a shallow non-submerged
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 non-submerged type
OR with a medium rate flow of a shallow non-submerged 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 non-submerged type,
OR with a medium OR high rate flow of a medium non-submerged type;
OR random patterns of planting WITH a high density of vegetation are treated
EITHER with a very high rate flow of a shallow non-submerged type,
OR with a high rate flow of a deep non-submerged 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 non-submerged type,
OR with a medium OR high OR very high rate flow of a deep non-submerged 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 non-submerged
OR medium non-submerged type;
OR random patterns of planting are treated WITH a low rate flow
WHILE this is flow is EITHER deep non-submerged 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.

9 Conclusion

The Dissimilarity Measure-based 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.

References

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Dimitrov, V. (1969) Application of Genetic Theory for Selection of Partial Descriptions in GMDH, Avtomatika, 6: 75-80 (in Ukrainian)

2

Dimitrov, V. (1970) Algorithms of GMDH on Zadeh Fuzzy Sets, Avtomatika, 4: 50-56 (in Ukrainian)

3

Dimitrov, V. (1973) Multilayered Stochastic Languages for Pattern Recognition, Proceedings of the First International Joint Conference on Pattern Recognition, (ed.) K.S. Fu, Washington, USA

4

Ivakhnenko, A., Zaichenko, Yu. & Dimitrov, V. (1976) Decision-Making Based on Self-Organisation, Soviet Radio Publ., Moscow (in Russian)

5

Maheshwari, B. (1992) Suitability of Different Flow Equations and Hydraulic Resistance Parameters for Flow in Surface Irrigation, Journal of Water Resources Research, 8: 2059-2066

6

Maheshwari, B. & McMahon, T. (1992) Modelling Shallow Overland Flow in Surface Irrigation, Journal of Irrigation and Drainage Engineering, 118: 201-217

7

Manning R. (1889) On the Flow of Water in Open Channels and Pipes, Trans. Inst. Civ. Eng. (Ireland), 20: 161-207

8

Zadeh, L. (1965) Fuzzy Sets, Information and Control, 1965, 8: 338-353