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Analytic Hierarchy Process (AHP) in ranking watersheds for streamflow data

infilling models

Masengo Ilunga

Department of Civil and Chemical Engineering,

University of South Africa,Pretoria, South Africa.

e-mail: [email protected]

ABSTRACT

The current paper establishes mainly the ranking of watersheds, where streamflow data infilling models, e.g. Artificial neural networks andregression) have been tested previously. Analytic Hierarchy Process (AHP) as decision-making tool is used in the ranking (preference or

suitability) for watersheds when models are applied. To illustrate the power of AHP technique, specific watersheds in Canada, i.e. theEnglish River at Sioux Lookout (ER), the Osilika River (OR), Graham River (GR), the Half River (HR) and the Nagagami River (NR) areobtained from the literature. Streamflow data infilling models tested previously on these watersheds are used in this study as criteria in theAHP formulation. For this study, multi-layed-feedforward autoregressive series model (M-ASM); multi-layed-feedforward bivariate series

model (M-BSM) and multi-dimensional regression (MR) models were used. Based on a consistent judgment from AHP, the weightsobtained after ranking these models were 31 %, 62 % and 7 % respectively. This supports the findings of the previous study that M-BSM performed generally higher as compared to other two models. For each criterion, pairwise comparisons of the watersheds were carried out.The consistency ratio of 0.086 proved that AHP was consistently carried out. The overall preferences (ranking or suitability) for the

watersheds OR, GR, NR, HR and ER, were estimated at 36%, 21%, 18%, 14% and 11% respectively as far as the testing of M-BSM; M-ASM and MR models is concerned. This correlates with the conclusion from the literature that the accuracy of in-filled streamflow dataseries was relatively poor for both ER and HR watersheds.

Keywords: Analytic hierarchical process, infilling streamflow, models

1.

INTRODUCTION

The importance of streamflow data cannot be over-emphasizedin water resources development and management. However,gaps in streamflow data series occur in practice. Infilling data

models are used for the purpose of filling in the gaps in thehydrological time series for given watersheds. Examples of the

application of streamflow data infilling models can be tracedfrom the literature [1], [2], [3], [4]. In hydrological studies, it is

common to assess and rank data infilling models but not payingattention to the ranking or preferences (priorities) that decision-

makers may give to the watersheds where these models have been tested. This situation (i.e. suitability or priority ofwatersheds) can be approached as a multi-criteria decision

problem via AHP where the main goal is the ranking or the

choice of suitable watersheds during streamflow data infilling process. Such a ranking supported by a sound literature is almostinexistent.The choice for suitable watersheds cannot be seen as an easy

task, especially when the selection criteria (i.e. streamflow datainfilling models) are considered. Hence this could possibly leadto subjectivity in the decision-making process. Hence, arelatively high number of streamflow data infilling models may

present a challenge in the ranking process.

In this paper Analytic Hierarchy Process (AHP) isformulated based on a previous study obtained from theliterature [1], where streamflow data infilling techniques wereassessed for specific watersheds in Canada. AHP is proposed forthe ranking (selection) of suitable watersheds with respect to thestreamflow data infilling models. In some way AHP process is

based on subjective considerations that should lead normally to a

consistent judgment. In what follows, the words “model” and“technique” are used interchangeably.

1.

AHP AND STREAMFLOW DATA INFILLINGPROBLEMS

Introduced in the 1980’s by Prof. Saaty, AHP is known as aMulti-Criteria Decision Making (MCDM) method. Later, themethod has been applied to many disciplines including

hydrology and water resources and was proven to be a powerfultool for decision making, e.g. [5]. AHP was recently applied instreamflow prediction, however ranking was done mainly onmodels tested one catchment [5]. However the ranking(suitability or preference) for watersheds with respect to

streamflow data infilling techniques hasn’t been performed based on a consistent through AHP. AHP enables decisionmakers to formulate a complex problem into a hierarchy whichincludes mainly the following three components:

-Goal

-Criteria (attributes)-AlternativesCriteria are so important to be used in the ranking process when

AHP is carried out. The goal is reached by carrying out a seriesof pairwise comparisons based on a scale of 1 to 9. Hence during

pairwise comparison, 1: shows equal importance betweenelements; 3: moderate importance; 5; strong importance; 9:

extreme importance. The values 2, 4, 6 and 8 are intermediatevalues. The computation of the consistency ratio (which should

be less than 0.1) is used to decide on the judgment validity for

AHP. For more details on AHP methodology, the reader can bereferred for example to [5].

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2.

ILLUSTRATION OF AHP FOR WATERSHEDS RANKING

2.1.

DATA AVAILABILITY

As outlined previously, the current study makes use of dataextracted from a previous study found in the literature [1]. This

previous study compared mainly the performance of streamflow

data infilling techniques; i.e. ANNs and regression methods.The streamflow data from watersheds, i.e. the English River at

Sioux Lookout (ER), the Osilika River (OR), Graham River(GR), the Half River (HR) and the Nagagami River (NR) inCanada, were considered in the assessment of the techniques

[1]. Table 1 has been extracted from [1] and constitutes mainly

the data or point of departure for AHP formulation as shown inthe next section. For M-ASM, M-BSM and MR models asdepicted in Table 1, the current study replicates only the values

of root mean square error (RMSE) as model performanceindicator since discussions and conclusions in the previous

study was mainly based on this statistical parameter [1]. Nostatistical analysis for RMSE values is conducted in this studyas it is not within the scope of the current study.

Table 1. Comparison of M-ASM, M-BSM and MR models for watersheds for all seasons as extracted in [1]

M-BSM(RMSE)

M-ASM

(RMSE)

MR

(RMSE)

ER 0.35 4.94 0.36

OR 0.29 6.41 0.15

GR 0.29 8.61 0.24

HR 0.34 7.19 0.26

NR 0.42 2.05 0.22

2.2.

AHP FORMULATION AND

IMPLEMENTATION FOR WATERSHED

RANKING

Based on Table 1, the formulation of AHP for ranking(preference) of watersheds is summarized in the following:

-Goal: Ranking watersheds-Criteria: Streamflow data infilling techniques, i.e. M-ASM, M-BSM and MR. Technique performance is based only on RMSEand used to reach the goal.-Alternatives: Five watersheds are considered: the English River

at Sioux Lookout (ER), the Osilika River (OR), Graham River(GR), the Half River (HR) and the Nagagami River (NR).

The rest of this section shows the implementation of AHP.

2.2.1.

PAIRWISE COMPARISON OF CRITERIA

Table 2 depicts the pairwise comparison of criteria with the

corresponding importance allocation intensity. From Table 1and based on RMSE values, for instance M-BSM can be

considered to be moderately preferred over M-ASM. In fact, M-BSM performed generally better than M-ASM [1]. Hence in

Table 2, M-BSM versus M-ASM will have a value of 3 in termsof priority. The opposite case; i.e. M-ASM versus M-BSM willscore a value of 1/3. A similar reasoning is carried out for therest of values in Table 2, e.g. M-BSM is strongly preferred over

MR, therefore a value of 7 is used), etc. Finally Table 2 presentsthe judgment matrix (3x3) for ranking the watersheds withrespect to the streamflow infilling data models. The subjective

considerations (i.e.….moderately preferred over…;…strongly

preferred over…) are valid only if the consistency ratiocomputed from the judgment matrix is less than 0.1.

Table 2. Pairwise comparison of criteria

M-ASM M-BSM MR

M-ASM 1 0.333 7

M-BSM 3 1 7

MR 0.142 0.142 1

2.2.2. ALTERNATIVES PAIRWISE

COMPARISON

Based on Table 1 and similarly to previous section, pairwisecomparisons among watersheds (alternatives) are summarizedin Tables 3a, 3b and 3c. Pairwise comparisons carried out with

respect to the streamflow data infilling models.

Table 3a. Pairwise comparison between watersheds with respect to M-ASM

ER OR GR HR NR

ER 1 0.333 0.333 1 2

OR 3 1 1 2 3

GR 3 1 1 2 5

HR 1 0.5 0.5 1 3

NR 0.5 0.333 0.2 0.333 1

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Table 3b. Pairwise comparison between watersheds with respect to M-BSM

ER OR GR HR NR

ER 1 0.333 0.333 0.333 0.333

OR 3 1 3 3 3

GR 3 0.333 1 2 0.5

HR 3 0.333 0.5 1 0.5

NR 3 0.333 2 2 1

Table 3c. Pairwise comparison between watersheds with respect to MR

ER OR GR HR NR

ER 1 3 5 5 0.5

OR 0.333 1 3 3 0.142

GR 0.2 0.333 1 0.333 0.142

HR 0.2 0.333 3 1 0.142

NR 3 0.333 7 7 1

3.

RESULTS AND DISCUSSION

From Table 2, the weights of the three criteria can be computedas shown in Table 4. Details of weight computations are givenin Appendix 1. The results depicted in Table 4 (last column)

revealed that approximately 31% of the goal weight is on M-ASM model, 62 % of the goal on M-BSM model and 7 % onMR model. These results suggest a strong preference on M-BSM during streamflow data infilling process. This correlated

generally with the conclusion from the previous study [1].From the judgment matrix, the computation of the consistencyratio was found to be 0.086. Since this value is less than 0.1 asoutlined previously, the judgment is considered to be validduring AHP formulation and implementation.

Table 4. Criterion weights

M-ASM M-BSM MR Average

M-ASM 0.243 0.226 0.467 0.312

M-BSM 0.728 0.678 0.467 0.624

MR 0.034 0.096 0.067 0.066

Table 5a, 5b and 5c depict the weights of the five alternatives

with regards to the performance of each streamflow datainfilling model. Computations are carried out as shown inAppendix 2. From table 5a, it is seen that the preferences forER, OR, GR, HR and NR are estimated at 13%, 31%, 33%,

16% and 7% respectively with respect to M-ASM. Hence, thewatersheds, i.e. OR and GR are ranked relatively higher ascompared to the rest of watersheds. Hence the decision makeror water manager could have more preference for OR and GR

when M-ASM is used alone. Then this is followed by HR, ERand lastly by NR.

From table 5b, the preferences for ER, OR, GR, HR and NR areestimated at 8%, 40%, 17%, 13%, and 22% respectively with

respect to M-BSM. As in the previous case, the decision makercould have more preference for OR and GR when M-BSM isused alone. Then this is followed by HR, NR and lastly by ER.

From table 5c, the preferences for ER, OR, GR, HR and NR areestimated to 33%, 14%, 5%, 8%, and 40% respectively withrespect to M-BSM. As in the previous case, the decision makercould have more preference on NR followed by ER when MR is

used alone. Then this is followed by OR, HR and lastly by GR.

Table 5a. Weights of alternatives with respect to M-ASM

ER OR GR HR NR Average

ER 0.118 0.105 0.110 0.158 0.143 0.127

OR 0.353 0.316 0.330 0.316 0.214 0.306

GR 0.353 0.316 0.330 0.316 0.357 0.334

HR 0.118 0.158 0.165 0.158 0.214 0.163

NR 0.059 0.105 0.066 0.053 0.071 0.071

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Table 5b. Weights of alternatives with respect to M-BSM


ER 0.077 0.143 0.049 0.040 0.062 0.074

OR 0.231 0.429 0.439 0.360 0.563 0.404

GR 0.231 0.143 0.146 0.240 0.094 0.171

HR 0.231 0.143 0.073 0.120 0.094 0.132

NR 0.231 0.143 0.293 0.240 0.188 0.219

Table 5c. Weights of alternatives with respect to MR


ER 0.211 0.600 0.263 0.306 0.260 0.328

OR 0.070 0.200 0.158 0.184 0.074 0.137

GR 0.042 0.067 0.053 0.020 0.074 0.051

HR 0.042 0.067 0.158 0.061 0.074 0.080

NR 0.634 0.067 0.368 0.429 0.519 0.403

Table 6. Computation of overall weights of alternatives, i.e. watersheds

A B C D E F G I J K L M

Criteriaweights

ER OR GR HR NR ER OR GR HR NR

0.311 M-ASM 0.127 0.306 0.334 0.163 0.071 0.039 0.095 0.104 0.051 0.022

0.624 M-BSM 0.074 0.404 0.171 0.132 0.219 0.046 0.252 0.107 0.082 0.137

0.065 MR 0.328 0.137 0.051 0.08 0.403 0.021 0.009 0.003 0.005 0.026

0.107 0.356 0.214 0.138 0.185

Appendix 3 determines the overall preferences for the fivewatersheds. The overall preferences or suitability for ER, OR,GR, HR and NR are 11 %, 36%, 21%, 14% and 18 %

respectively when the three streamflow data infilling models M-ASM, M-BSM and MR are all considered.Hence the overall preferences should be focused on ORwatershed, followed by GR and NR. The watersheds HR and

ER displayed a low preference or priority as far as streamflowdata infilling process is concerned. This is in line with thefindings of the previous study that streamflow data infillingtechniques did not perform well on HR generally [1]. Again, the

results of the current study hold only in the case of considering

RMSE for assessing infilling data techniques. The introductionof other statistics for assessing streamflow data infilling could

possibly impact on the results. In reality, when other factors

(such as economic, social, etc) are taken into consideration, theoverall preference (priority) for the watersheds might change.

Nonetheless, the merit of AHP is to demonstrate the consistency

in the ranking of the watersheds with respect to hydrologicaldata infilling models. In general OR displayed a higher ranking

compared to other watersheds.

4.

CONCLUSIONSThis study has shown the power of AHP in ranking watershedsfor streamflow infilling data techniques. No existing literature

has dealt with such a case while the reversal case has beenstudied before. The preferences (suitability or priority) on thewatersheds were as follows: OR, GR, NR, HR and ER. Thisstudy confirmed that the higher preference is on ANN-based

techniques as opposed to regression techniques. Thetransparency of AHP in this ranking cannot be over-

emphasized. The limitation of the AHP could be that the finalresults depend subjectively on RMSE alone and not on other

statistical criteria. However the computation of the consistency

ratio revealed that the judgment through AHP was consistent.The application of AHP in this study could draw as well thedecision-makers’ attention to the ranking of watersheds in terms

of their suitability for water resources development and planning or suitability as far as hydrological data infilling process is concerned. Further work could be explored forranking watersheds when more than one statistical indicator for

model performance comparison. The current study consideredonly one statistical indicator in the AHP implementation.

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ACKNOWLEGDMENTS

The author acknowledges the use of data extracted from the

publication by U. S. Panu, M Khalil, A. Elshorbagy as adeparture point for AHP formulation.

5. REFERENCES

[1] U. S. Panu, M Khalil, A. Elshorbagy, Streamflow data

Infilling Techniques Based on Concepts of Groups andNeural Networks. Artificial Neural Networks in Hydrology.Kluwer Academic Publishers. Printed in the Netherlands, 2000,

pp.235-258.

[2] S. Starrett, S.K. Starrett, T. Heier, Y. Su, D. Tuan, M.Bandurraga, Filling in missing peak flow data using artificial

neural networks. ARPN Journal of Engineering and AppliedSciences , Vol. 5, No.1, 2010, pp.49-55.

[3] M. Ilunga and E.K. Onyari Infilling Maxima Annual

Monthly Flows Using Feedforward Backpropagation

(BP)Artificial Neural Network (ANNs), Proceedings of the 11th IASTED International Conference, Applied ArtificialIntelligence and Applications, 2011, Innsbruck Austria,February 14 th-16 th, 2011, pp. 69-74

[4] Ilunga, M. and Stephenson, D. Infilling streamflow data

using feedforward backpropagation (BP) artificial neural

networks: Application of the standard BP and pseudo Mac

Laurin power series BP techniques, Water SA Journal , Vol.31, No. 2, 2005, pp. 171-176.

[5] M. Ilunga and E.K. Onyari (2011) Application of

Hierarchical Process (AHP) for ANN model selection in

streamflow prediction, World Multiconference on Systemics,Cybernetics and Informatics (WMSCI 2011), Orlando, USA,July 19 th-22 nd, 2011, pp. 215-219.

[6] E. Onyari, F. Ilunga, Application of MLP neural network

and M5P model tree in predicting streamflow: A case studyof Luvuvhu catchment, South Africa, InternationalConference on Information and Multimedia Technology (ICMT

2010), Hong Kong, China, 2010, pp. V3-156-160.

6.

APPENDICES

Appendix 1: Weights on criteria or attributes (streamflow

data infilling technique)

Table 1. Criteria preference (pairwise comparsion on criteria)

M-ASM M-BSM MR

M-ASM 1 0.333 7

M-BSM 3 1 7

MR 0.142 0.142 1

4.142 1.475 15

Table 2. Weights on criteria or attributes

M-ASM M-BSM MR Average

M-ASM 0.243 0.226 0.467 0.312

M-BSM 0.728 0.678 0.467 0.624

MR 0.034 0.096 0.067 0.066

Each element in Table 2 (weights on criteria) is obtained by

dividing the entry in Table 1 (preference on criteria) by the sumof the column it appears in. Values in the Average column areobtained by averaging values in the different rows. The Averagecolumn represents the weights of criteria.

Appendix 2: Determination of alternative (watershed) weights

Step 1: Weights of watershed (alternatives) with regards to each criterion (streamflow data infilling technique)

Table 1a. Comparison of alternative (watersheds) on data infilling technique-M-ASM

ER OR GR HR NR

ER 1 0.333 0.333 1 2

OR 3 1 1 2 3

GR 3 1 1 2 5

HR 1 0.5 0.5 1 3

NR 0.5 0.333 0.2 0.333 1

8.5 3.166 3.033 6.333 14

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Table 1b. Weights of alternatives with respect to data infilling technique-M-ASM


ER 0.118 0.105 0.110 0.158 0.143 0.127

OR 0.353 0.316 0.330 0.316 0.214 0.306

GR 0.353 0.316 0.330 0.316 0.357 0.334

HR 0.118 0.158 0.165 0.158 0.214 0.163

NR 0.059 0.105 0.066 0.053 0.071 0.071

In a similar way to Table 1a and 1b, comparison of alternatives with respect to data infilling technique-M-BSM and MR were made.Finally the weights with respect to data infilling technique-M-BSMand MR can be obtained, however details are not presented here.

Step 2 Weights of alternatives

Table 5. Weights of Alternatives

ER OR GR HR NR

M-ASM 0.127 0.306 0.334 0.163 0.071

M-BSM 0.074 0.404 0.171 0.132 0.219MR 0.328 0.137 0.051 0.08 0.403

Rows in Table 5 are obtained from weights of alternatives computed in Appendix 2

Appendix 3: Computation of overall Weights of watersheds (alternatives)

Table 1. Overall weights of watershed (alternatives)

A B C D E F G I J K L M

Criteria weights(Appendix 1) Weights of alternatives

(Appendix 2)Overall weights of alternatives

ER OR GR HR NR ER OR GR HR NR

0.311 M-ASM 0.127 0.306 0.334 0.163 0.071 0.039 0.095 0.104 0.051 0.022

0.624 M-BSM 0.074 0.404 0.171 0.132 0.219 0.046 0.252 0.107 0.082 0.137

0.065 MR 0.328 0.137 0.051 0.08 0.403 0.021 0.009 0.003 0.005 0.026

0.107 0.356 0.214 0.138 0.185

Determination of values in column I, J, K,L and M is done by multiplying A by C, A by D, A by E, A by F, and A by G. Overall weights ofER, OR, GR, HR and NR in the last row are obtained by summing values in columns I, J, K, L and M respectively.

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