Comparative analysis of fuzzy inference systems for water consumption time series prediction

Journal of Hydrology 374 (2009) 235–241

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Journal of Hydrology journal homepage: www.elsevier.com/locate/jhydrol

Comparative analysis of fuzzy inference systems for water consumption time series prediction Mahmut Firat a, Mustafa Erkan Turan b, Mehmet Ali Yurdusev b,* a b

Pamukkale University, Civil Engineering Department, Denizli, Turkey Celal Bayar University, Civil Engineering Department, Manisa, Turkey

a r t i c l e

i n f o

Article history: Received 20 March 2009 Received in revised form 16 May 2009 Accepted 7 June 2009

This manuscript was handled by G. Syme, Editor-in-Chief

s u m m a r y Two types of fuzzy inference systems (FIS) are used for predicting municipal water consumption time series. The FISs used include an adaptive neuro-fuzzy inference system (ANFIS) and a Mamdani fuzzy inference systems (MFIS). The prediction models are constructed based on the combination of the antecedent values of water consumptions. The performance of ANFIS and MFIS models in training and testing phases are compared with the observations and the best fit model is identified according to the selected performance criteria. The results demonstrated that the ANFIS model is superior to MFIS models and can be successfully applied for prediction of water consumption time series. Ó 2009 Elsevier B.V. All rights reserved.

Keywords: Water consumption prediction Water management Adaptive neuro-fuzzy inference system Mamdani fuzzy inference systems

Introduction Reliable water demand forecasts are key to planning and management of water resources and also essential for the design and operation of various water infrastructures such as reservoir supply and distribution facilities. Water demand is affected by many factors such as population, rainfall, humidity, temperature, industrial and commercial conditions and a reliable forecast should normally consider the effects of such socio-economic and climatic factors affecting water use. There are different approaches to water demand forecasting, including various statistical or mathematical techniques such as time extrapolation, disaggregated end-uses, single-coefficient method, multiple-coefficient method, probabilistic method, memory based learning technique, Box Jenkins and ARIMA models (Babel et al., 2007). The first four methods are suitable for long term forecasting whereas the others are more appropriate for short term forecasting. The time extrapolation method was used early in the 20th century, but is rarely in use now. In the forecasting of water demand by disaggregated forecast models, a large number of data for different sectors which are normally not available in developing countries are required. Since the water demand depends on socio-economic factors, climatic factors and public water policies and strategies, multiple coefficient demand model can integrate the effects of all or several of these factors. Lahlou and Colyer * Corresponding author. Tel.: +90 5358114054; fax: +90 2362412143. E-mail address: [email protected] (M.A. Yurdusev). 0022-1694/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2009.06.013

(2000) used multiple regression demand models to forecast water demand incorporating conservation activities. Froukh (2001) attempted to integrate both mathematical and heuristic approaches for long-term water-demand forecasts by developing a decisionsupport system that integrates demand forecasting with demand management using decision tools. Wong and Mui (2007) carried out a mathematical model in determining the flushing water demands for high-rise residential buildings in Hong Kong. Babel et al. (2007) developed a multiple coefficient water demand forecast and management model for the domestic sector considering various socio-economic, climatic and policies related factors. Jain et al. (2001) used autoregressive models to forecast peak weekly water demand. Zhou et al. (2000) proposed time series models including trend, seasonality, and climatic correlation and autocorrelation components to forecast the daily water consumption. Artificial intelligence techniques such as artificial neural networks (ANN) and fuzzy logic (FL) have been used as efficient alternative tools for the modeling of complex hydrologic systems and widely used for forecasting (Firat et al., 2009). The FL method was first developed to explain the human thinking and decision system by Zadeh (1965) and used for modeling several engineering systems including water resources (Turan and Yurdusev, 2009). Altunkaynak et al. (2005) predicted future water consumptions from antecedent values using a fuzzy model for modeling monthly water consumption time series for the city of Istanbul. Recently, Adaptive neuro-fuzzy inference system (ANFIS), which consists of the ANN and FL methods, has been used for many application such

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as, database management, system design and forecasting of water resources (Firat and Güngör, 2008). Yurdusev and Firat (2009) proposed an ANFIS approach to model monthly water consumptions based on several socio-economic and climatic factors. The main purpose of this study is to carry out a comparative analysis of two types of fuzzy inference systems, for prediction of water consumption time series. The FISs used include an adaptive neuro-fuzzy inference system (ANFIS) and a Mamdani fuzzy inference systems. The latter is given a specific acronym, MFIS, and used throughout the article. A series of models are constructed based on several combinations of the antecedent values of water consumptions. The performance of the models experimented in training and testing sets are compared with the observations and the best fit model is identified according to the performance criteria including average absolute relative error (AARE), normalized root mean square error (NRMSE) and threshold statistic (Ts). Fuzzy inference systems (FIS) A fuzzy inference system (FIS) is an inference mechanism establishing a relationship between a series of input and output sets. The inference system uses fuzzy sets theory, fuzzy logic principles when establishing such a relationship. Fuzzy inference system (FIS) is a rule based system consisting of three conceptual components. These are: (1) a rule-base, containing fuzzy if–then rules, (2) a data-base, defining the membership functions (MF) and (3) an inference system, combining the fuzzy rules and producing the system results (S ß en, 2001). There are two types of widely used fuzzy inference systems, Takagi–Sugeno FIS and Mamdani FIS (Jang et al., 1997). The most important difference between these systems is the definition of the consequent parameters (Takagi and Sugeno, 1985). The consequent parameters in Takagi–Sugeno FIS are either a linear equation, called ‘‘first-order Takagi–Sugeno FIS”, or constant coefficient, ‘‘zeroorder Takagi–Sugeno FIS (Jang et al., 1997). The ANFIS system used in this study is Takagi–Sugeno type FIS. The other FIS used in this study is a Mamdani type FIS in which the rule base is constructed from the input–output pairs, which is referred to as MFIS in this study. Both are explained in further detail below. Adaptive neuro-fuzzy inference system (ANFIS) FL is employed to describe human thinking and reasoning in a mathematical framework. An ANN has the ability to learn from in-

IF

WDðt  1Þ is A1 and WDðt  2Þ is B1 THEN f 1 ¼ p1 WDðt  1Þ þ q1 WDðt  2Þ þ r 1

IF

WDðt  1Þ is A2 and WDðt  2Þ is B2 THEN f 2 ¼ p2 WDðt  1Þ þ q2 WDðt  2Þ þ r 2

Fig. 1 shows the structure of ANFIS. It is assumed that the ANFIS includes two inputs, WD(t  1) and WD(t  2), and one output, WD(t). Inputnodes (Layer 1): Each node in this layer generates membership grades of the crisp inputs and each node’s output O1i is calculated by:

O1i ¼ lAi ðWDðt  1ÞÞ for i ¼ 1; 2; O1i ¼ lBi2 ðWDðt  2ÞÞ for i ¼ 3; 4

ð1Þ

where WD is the water consumption, WD(t  1) and WD(t  2) are the crisp inputs to the node i, Ai and Bi are the linguistic labels, (pi, qi, ri) are the consequent parameters, lAi and lBi are the MF for Ai and Bi linguistic labels, respectively. In this study, the Gaussian function is used as: ðWDðt1ÞcÞ2 2r2

O1i ¼ lAi ðWDðt  1ÞÞ ¼ e

ð2Þ

Rule nodes (Layer 2): The outputs of this layer, called firing strengths O2i , are the products of the corresponding degrees obtained from the Layer 1.

O2i ¼ wi ¼ lAi ðWDðt  1ÞÞlBi ðWDðt  2ÞÞ;

i ¼ 1; 2

ð3Þ

Average nodes (Layer 3): Main target is to compute the ratio of firing strength of each ith rule to the sum firing strength of all  i Þ as: rules. The firing strength in this layer is normalized ðw

w i ¼ P i O3i ¼ w i wi

i ¼ 1; 2

ð4Þ

Consequent nodes (Layer 4): The contribution of ith rule towards the total output or the model output and/or the function defined is calculated by

 i fi ¼ w  i ðpi WDðt  1Þ þ qi WDðt  2Þ þ ri Þ i ¼ 1; 2 O4i ¼ w

ð5Þ

Output nodes (Layer 5): This layer is called as the output nodes in which the single node computes the overall output by summing all incoming signals:

w1 ðWDðt  1Þ; WDðt  2ÞÞf1 ðWDðt  1Þ; WDðt  2ÞÞ þ w2 ðWDðt  1Þ; WDðt  2ÞÞf2 ðWDðt  1Þ; WDðt  2ÞÞ w1 ðWDðt  1Þ; WDðt  2ÞÞ þ w2 ðWDðt  1Þ; WDðt  2ÞÞ w1 f1 þ w2 f2 ¼ w1 þ w2

f ðWDðt  1Þ; WDðt  2ÞÞ ¼

put and output pairs and adapt to it in an interactive manner. In recent years, the ANFIS method, which integrates ANN and FL methods, has been developed. ANFIS has the potential benefits of both these methods in a single framework. ANFIS eliminates the basic problem in fuzzy system design, defining the membership function parameters and design of fuzzy if–then rules, by effectively using the learning capability of ANN for automatic fuzzy rule generation and parameter optimization (Nayak et al., 2004). For this reason, in this study, the ANFIS methodology is proposed to self-organize model structure and to adapt parameters of the fuzzy system for monthly water consumption prediction. It has the advantage of allowing the extraction of fuzzy rules from numerical data. For the first-order Takagi–Sugeno FIS, two typical rules can be expressed as:

Q 5i ¼ f ðWDðt  1Þ; WDðt  1ÞÞ ¼

X i

ð6Þ

P wf  i f1 þ w  i f2 ¼ Pi i i  i  fi ¼ w w i wi ð7Þ

ANFIS applies the hybrid-learning algorithm, consists of the combination of gradient descent, which is used to assign the nonlinear input parameters, and least-squares methods, which is employed to identify the linear output parameters. The detailed algorithm and mathematical background can be found in Jang et al. (1997). Mamdani type fuzzy inference systems (MFIS) The rule base used in MFIS is constructed by the table look-up method proposed by Wang and Mendel (1991), in which the rule

M. Firat et al. / Journal of Hydrology 374 (2009) 235–241

237

Fig. 1. The structure of ANFIS.

base can be generated from the input–output pairs with no expert knowledge and comprises five steps as follows: 1. Division of input–output spaces into fuzzy regions: Input and output domain intervals are divided into 2N + 1 regions and a fuzzy membership function is assigned for each region. N can be different for each variable. The membership functions of these regions can be named as SN(small N), . . . , S1(Small 1), CE(center), B1(Big 1), . . . , BN(Big N). A triangular membership function is used in this study and the number of membership functions have been increased until the best result is obtained. For example, the best model exercised in this study contains five membership functions as shown in Fig. 2. 2. Generation of fuzzy rules from the data: First, the membership degrees of the data pairs are determined in different regions, then the pairs are assigned with maximum membership degree and then one rule is obtained from one pair of the desired data pairs 3. Assigning a degree to each rule: There are generally many data pairs and one rule can be obtained from each data pair. Most probably, there will be some conflicting rules, having the same IF part but different THEN part. One degree is assigned to each rule to resolve this conflict and only the rule with the highest degree is accepted. The degree of the rule can be defined as:

DðRuleÞ ¼ lðx1 Þlðx2 Þlðx3 Þ

ð8Þ

4. Creation of fuzzy rule base: A fuzzy rule base is created by selecting the rules that have maximum degree. The total number of rule for the best model described in ‘‘Results and discussion” is 62, part of which is shown in Table 1.

5. Determination of a mapping: A defuzzifying procedure is employed to obtain a mapping based on the combined fuzzy rule base (Tayfur et al., 2003). Normalization between 1 and 3 in MFIS models was undertaken using Eq. (2). The aim of such transformation is to be able to compare the different quantities (Tatli and Sßen, 2001). Study area and the data used The performance of ANFIS and MFIS models are tested in the case of municipal water consumptions of the metropolitan area of Izmir, which is the third largest city of Turkey in terms of population and industrial development. Fig. 3 shows the location of the city and the existing water resources. The city is rapidly developing due to the domestic population movement to the city from other parts of the country due especially to its pleasant climatic characteristics. On the other hand, the city is not rich in water resources as it is a coastal city located in western Turkey, where the average temperature is significantly high whereas the rainfall is relatively less. The current water supply system of the city includes several surface water resources and groundwater developments. The most important surface water resource is Tahtalı reservoir, located approximately 60 km away from the city. The reservoir provides around 35% of city supply. The most important groundwater resources of the city is Sarıkız-Göksu wells, which is approximately 50 km away from the city and serve for approximately 40% of the city. A new reservoir development at much further distance is under construction and a long pipe line is required to transfer its water from a neighboring basin to the city. Owing to the complexity of water consumption prediction process, most conventional approaches are often unable to provide

Fig. 2. The membership functions of input–output data.

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Table 1 Part of rule base for the best fit MFIS model. Rule no.

Rule

R1 R2 R3 R4 R5 R58 R59 R60 R61 R62

IF IF IF IF IF IF IF IF IF IF

inp1 = mf2 inp1 = mf3 inp1 = mf4 inp1 = mf3 inp1 = mf3 inp1 = mf2 inp1 = mf3 inp1 = mf3 inp1 = mf5 inp1 = mf3

AND AND AND AND AND AND AND AND AND AND

inp2 = mf3 inp2 = mf4 inp2 = mf3 inp2 = mf3 inp2 = mf3 inp2 = mf3 inp2 = mf3 inp2 = mf5 inp2 = mf3 inp2 = mf3

AND AND AND AND AND AND AND AND AND AND

inp3 = mf4 inp3 = mf3 inp3 = mf3 inp3 = mf3 inp3 = mf3 inp3 = mf3 inp3 = mf5 inp3 = mf3 inp3 = mf3 inp3 = mf2

sufficiently accurate and reliable results. Therefore, in this study, ANFIS and MFIS methods for prediction of water consumption time series are adopted and applied to the city of Izmir. The general structure of model for predicting water consumption can be given as:

WD ¼ f ðWDðt  1Þ; WDðt  2Þ; . . . ; WDðt  nÞÞ

ð9Þ

For the development of the prediction models, the total 108 monthly data records of water consumption were collected in the period 1997–2005 for the city of Izmir, Turkey (Fig. 4). The data

AND AND AND AND AND AND AND AND AND AND

inp4 = mf3 inp4 = mf3 inp4 = mf3 inp4 = mf3 inp4 = mf2 inp4 = mf5 inp4 = mf3 inp4 = mf3 inp4 = mf2 inp4 = mf3

AND AND AND AND AND AND AND AND AND AND

inp5 = mf3 inp5 = mf3 inp5 = mf3 inp5 = mf2 inp5 = mf2 inp5 = mf3 inp5 = mf3 inp5 = mf2 inp5 = mf3 inp5 = mf2

THEN THEN THEN THEN THEN THEN THEN THEN THEN THEN

out1 = mf3 out1 = mf2 out1 = mf3 out1 = mf4 out1 = mf3 out1 = mf3 out1 = mf2 out1 = mf3 out1 = mf3 out1 = mf5

set was divided into two subsets, training and testing data set. The statistical parameters such as minimum value (xmin), maximum x), standard deviation (sx) and skewness value (xmax), mean ( coefficient (csx) for training and testing data sets are calculated and given in Table 2 to see a comparison of the training and testing data. It can be seen from Fig. 4 that there is a trend component in the actual water consumption data. In this study, in order to develop a prediction model the trend in actual records was removed by using the regression line to model the fluctuating part of the data known as residuals as follows:

W ¼ 19117t þ 7  106

ð10Þ

where W is the water consumption fluctuation variable, t is time in months (1 < t < 108). The residuals that are obtained by subtracting the trend from the original data are plotted in Fig. 5 and used for modeling exercises. The data set of residuals given in Fig. 5 is used to train and test the ANFIS and MFIS models. The training data set includes total 84 data records between 1997 and 2004, which is 80% of the total data records. In order to get more reliable evaluation and comparison, models are tested by testing data set which was not used during the training process. The testing data set consists of a total 24 data records, which is 20% of the total data, observed in the last 2 years. Altunkaynak et al. (2005), Jain et al. (2001) and Bougadis et al. (2005) have also used the same strategy for the choice of the training and testing data sets. Fig. 3. Location of the city of Izmir and its water resources.

Fig. 4. General trend of monthly water consumption in Izmir.

Fig. 5. Time series plot of residuals.

Table 2 The statistical parameters for training and testing sets. Water consumption, WD (m3)

xmax

xmin

 x

sx

csx

Training set Testing set

10,933,005 11,328,637

5,887,594 7,279,470

7,953,132 9,147,404

9,918,350 1,176,987

0.42 0.13

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Model development One of the most important steps in developing a satisfactory prediction model is the selection of appropriate input variables as these variables determine the structure of the model and affect the results of the model. Conventionally, the choice of appropriate input variables can be made using the cross-correlations between the variables. In this study, this is achieved by considering the total model performances measured according to several criteria both in training and the testing phases. The model structures are given in Table 3.

Table 3 The structures of water consumption prediction models.

The performances of ANFIS and MFIS models for training and testing data sets are evaluated according to statistical criteria such as, average absolute relative error (AARE), normalized root mean square error (NRMSE) and threshold statistic (Ts). The NRMSE statistic indicates a model’s ability to predict a value away from the mean. The ‘‘threshold statistic” quantifies consistency in the forecasting errors from a particular model. The TSx is calculated for a particular level of relative error (x %) in forecasting. The definition of these performance criteria are given as follows:

hP

N ðWDYi WDDi Þ2 i¼1 N

P  N ð1=NÞ i¼1 ðWDDi Þ   N   1 X WDYi  WDDi   100 AARE ¼   N WD

NRMSE ¼

Input

M1 M2 M3 M4 M5 M6

WD(t  1) WD(t  1)WD(t  2) WD(t  1)WD(t  2)WD(t  3) WD(t  1)WD(t  2)WD(t  3)WD(t  4) WD(t  1)WD(t  2)WD(t  3)WD(t  4)WD(t  5) WD(t  1)WD(t  2)WD(t  3)WD(t  4)WD(t  5)WD(t  6)

where WDYi is the predicted water consumption, WDDi is the observed water consumption, TSx is the threshold statistic for a level of x %, n is the number of data points forecasted having relative error in forecasting less than x % and N is the total number of data points in forecasting. In this study, threshold statistics were calcu-

10

0.12

AARE

0.06

AARE NRMSE

0 2

3

4

5

0.08

6 0.06

0.02

2

0

0

6

0.04 0.02 0 1

2

3

Model

4

5

6

Model

Performances of ANFIS Models Testing Set

Performances of ANFIS Models Testing Set

12

0.14

12

0.14

10

0.12

10

0.12

0.08

6

0.06

4

AARE

8

NRMSE

0.1

0 1

2

3

4

5

0.08

6

0.06

4

0.04

AARE NRMSE

2

0.1

8

0.02

2

0

0

6

AARE NRMSE

0.02 0

1

2

3

Model ANFIS Models Testing Set

5

6

ANFIS Models Testing Set 140

TS5 TS25

TS10

Threshold Statistic (%)

TS1 TS20

100 80 60 40 20 0 M1

4

Model

140 120

0.04

M2

M3

M4

Model

M5

M6

120

TS1 TS20

TS5 TS25

TS10

100 80 60 40 20 0 M1

M2

M3

M4

Model

Fig. 6. Comparison of performances of ANFIS and MFIS models.

M5

M6

NRMSE

1

0.1

4

0.04

2

0.12

8

NRMSE

AARE

6

AARE NRMSE

10

0.1 0.08

0.14

12

NRMSE

0.14

4

ð13Þ

Performances of ANFIS Models Training Set

12

8

AARE

ð12Þ

TSx ¼ ðn=NÞ  100

Performances of ANFIS Models Training Set

Threshold Statistic (%)

ð11Þ

Di

i¼1

Model

i0:5

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lated for level of 1%, 5%, 10%, 20% and 25%. The performances of ANFIS and MFIS models are given in Fig. 6. Results and discussion Firstly, comparing the results of ANFIS models, generally, the training and testing performance of the M5 ANFIS model is better than other ANFIS models. The values of threshold statistics, AARE and NRMSE statistics of the M5 ANFIS model are smaller than those of other ANFIS models. In M5 ANFIS model, 71% of forecasted errors is less than level of 10%. On the other hand, comparison of results of MFIS models, performance of the M5 MFIS model is slightly better than those of other MFIS models. According to these criteria, the M5 ANFIS and MFIS models consisting of five antecedent values of monthly water consumption have shown the best performance for monthly water consumption prediction. Comparison of the performances of M5 ANFIS and MFIS models are given in Table 4. Comparing the performances of the M5 ANFIS and MFIS models, the NRMSE and AARE values of the M5 ANFIS model are lower than those of M5 MFIS model. In addition, the correlation coefficient (CORR) of the M5 ANFIS model is also higher than that of M5 MFIS model. All levels of threshold statistics of M5 ANFIS model are better than those of MFIS model. It can be stated that the performance of the M5 ANFIS model is better than that of M5 MFIS model according to performance criteria previously mentioned. The results demonstrate that the ANFIS method is superior to the MFIS method in predicting monthly water consumption time series. Table 4 Performances of ANFIS and MFIS models. Models

M5 ANFIS M5 MFIS

Testing set NRMSE

AARE (%)

CORR

TS1

TS5

TS10

TS20

TS25

0.076 0.108

6.170 9.129

0.81 0.52

8 8

54 33

71 63

100 92

100 100

Therefore, ANFIS method can be successfully applied to establish accurate and reliable water consumption prediction models. Comparison of training and testing results of M5 ANFIS and MFIS models are shown in Fig. 7. Conclusions In this study, a comparative analysis of ANFIS and MFIS methods, for prediction of water consumption time series, is carried out. For this purpose, six models consisting of the various combinations of antecedent water consumption values are constructed. Comparing the results of ANFIS models, it is observed that the performance of the M5 ANFIS model consisting of five antecedent values of water consumption is better than other ANFIS models. The values of AARE and NRMSE of the M5 ANFIS model are better than those of other ANFIS models. On the other hand, comparison of results of MFIS models, performance of the M5 MFIS model is slightly better than those of other MFIS models. The test statistics of M5 MFIS model are better than those of other MFIS models. According to these criteria, the M5 ANFIS and MFIS models have shown the best performance for monthly water consumption prediction. Comparing the performances of the M5 ANFIS and MFIS models, the NRMSE and AARE values of the M5 ANFIS model are lower than those of M5 MFIS model. In addition, the CORR value of the M5 ANFIS model is also higher than that of M5 MFIS model. The results demonstrate that the ANFIS method is superior to the MFIS method in predicting monthly water consumption time series. Therefore, ANFIS method can be successfully applied to establish accurate and reliable water consumption prediction models and be preferred over MFIS model. References Altunkaynak, A., Özger, M., Çakmakcı, M., 2005. Water consumption prediction of Istanbul city by using fuzzy logic approach. Water Resources Management 19, 641–654.

Fig. 7. Comparison of results of M5 ANFIS and MFIS models.

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