Hydrological Sciences–Journal–des Sciences Hydrologiques, 49(4) August 2004

Open for discussion until 1 February 2005

655

Comparison of grey and phase-space rainfall forecasting models using a fuzzy decision method PAO-SHAN YU1, SHIEN-TSUNG CHEN1, CHE-CHUAN WU1 & SHU-CHEN LIN2

1 Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan, Taiwan yups@mail.ncku.edu.tw

2 Department of Construction Management and Planning, Leader University, Tainan, Taiwan

Abstract This work applies a fuzzy decision method to compare the performance of the grey model with that of the phase-space model, in forecasting rainfall one to three hours ahead. Four indices and two statistical tests are used to evaluate objectively the performance of the forecasting models. However, a trade-off must be made in choosing a suitable model because various indices may lead to different judgements. Therefore, a fuzzy decision model was applied to solve this problem and to make the optimum decision. The results of fuzzy decision making demonstrate that the grey model outperforms the phase-space model for forecasting one hour ahead, but the phase-space model performs better for forecasting two or three hours ahead. Key words fuzzy decision; grey model; phase-space theory; rainfall forecasting

Comparaison grâce à une méthode de décision floue des modèles gris et d’espace des phases pour la prévision de pluie Résumé Ce travail applique une méthode de décision floue pour comparer les performances du modèle gris et du modèle d’espace des phases, pour la prévision de pluie à échéance de une à trois heures. Quatre indices et deux tests statistiques sont utilisés pour évaluer objectivement les performances des modèles de prévision. Cependant, un arbitrage doit être effectué lors du choix du modèle car divers indices peuvent conduire à différents jugements. Par conséquent, un modèle de décision floue a été appliqué pour résoudre ce problème et pour prendre la décision optimale. Les résultats de la prise de décision floue montrent que le modèle gris surpasse le modèle d’espace des phases pour la prévision à une heure, tandis que le modèle d’espace des phases devient plus performant pour la prévision à deux ou trois heures. Mots clefs décision floue; modèle gris; théorie de l’espace des phases; prévision de pluie

INTRODUCTION A flood warning system is a non-structural measure for flood mitigation. In Taiwan, typhoons and southwestern convective storms frequently cause severe floods and disasters. Accordingly, a flood warning system is required to prolong the response time for flood mitigation. However, catchments in Taiwan are usually steep and the hydro-graph rises shortly following the beginning of a rainstorm. Thus, the rainfall fore-casting model is an important part of the flood warning system to extend the lead time of flood forecasting. Hydrological forecasting models (Burlando et al., 1993; Luk et al., 2001) and meteorological forecasting models (Docine et al., 1999) are two approaches to forecasting rainfall. This study focuses on the former models, most of which have been developed for short-duration (1–3 h ahead) rainfall forecasting. Numerous models exist for short-duration rainfall forecasting, of which the grey rainfall forecasting

Pao-Shan Yu et al.

656

model (Chen, 1998; Yu et al., 2000, 2001) and the phase-space model (Lin, 1999) are compared herein. The performance of a rainfall forecasting model is usually evaluated using the fitness of the observed and the forecast rainfalls, as judged by some quantitative criteria. In this work, the rainfalls forecast by both models 1–3 h ahead, are compared using the rainfall hyetograph and the rainfall mass curve, according to four quantitative indices and two statistical tests, to elucidate the models’ performance. However, a troublesome trade-off must still be made in choosing the preferred model, because these criteria may not lead to the same conclusions. Therefore, a fuzzy decision method (Yager, 1977; Hsiao, 1998) is proposed to combine the diverse results of applying these criteria to enable the optimal decision to be made. METHODS GM(1,1) model The grey model, proposed by Deng (1989), provides an effective means of predicting future data using only a few observed data. The GM(1,1) model is one of the most common grey differential equations and has been extensively and successfully used in previous research (for example, Deng, 1989; Xia, 1989; Huang & Huang, 1996; Chen, 1998; Yu et al., 2000, 2001). The GM(1,1) model is a grey model with one variable that is a first-order differential equation and is summarized as follows. A series of observed data are given by X(0) = [x(0)(1), x(0)(2),…, x(0)(k), …, x(0)(n)], where x(0)(k) denotes the kth observed datum. A new time series, X(1) = [x(1)(1), x(1)(2), …, x(1)(k), …, x(1)(n)], is then generated by applying the accumulated generating operation (AGO) to yield a first-order AGO series from the raw series, in which x(1)(k) is defined as:

�=

=k

iixkx

1

)0()1( )()( (1)

The AGO series is then fitted by a first-order differential equation given by:

ba tXt

tX ⊗=⊗+ )(d

)(d )1()1(

(2)

where ⊗a and ⊗b are grey parameters. According to grey system theory, the whitening of grey derivatives for discrete data with a unit time interval (∆t = 1) is given by:

ttkxkx

ttX

tkt ∆

∆−−=→∆

=

)()(limd

)(d )1()1(

1

)1(

(3)

)()1()(d

)(d )0()1()1()1(

kXkxkxt

tX

kt

=−−==

(4)

A new variable Z(1)(k), which is known as the whitening value of ,)()1(

kttX

= is defined as:

[ ] )1()( 5.0)()( )1()1()1()1( −+=≅=

kxkxkZtXkt

(5)

Comparison of grey and phase-space rainfall forecasting models

657

By substituting equations (4) and (5) into equation (2), the differential equation can be rewritten in a discrete difference form as:

ba kZkX ⊗=⊗+ )()( )1()0( (6)

To determine the grey parameters ⊗a and ⊗b, equation (6) is expressed in matrix form as:

θ= UY (7)

where θθθθ is the vector of grey parameters. The matrices can be specified as:

���������

�

�

���������

�

�

⋅⋅⋅=

)(

)4()3()2(

)0(

)0(

)0(

)0(

nx

xxx

Y ,

���������

�

�

���������

�

�

−⋅⋅⋅⋅⋅⋅

−−−

=

1)(

1)4(1)3(1)2(

)1(

)1(

)1(

)1(

nz

zzz

U and ��

���

�

⊗⊗

=θb

a (8)

The parameters Y and U can be obtained from the observed data series, so the vector of grey parameters can then be determined using the least squares method. The least squares method in matrix form is:

YUUU TT

b

a 1)(ˆˆˆ −=�

�

���

�

⊗⊗=θ (9)

where a⊗̂ and b⊗̂ are the identified whitening values of grey parameters ⊗a and ⊗b. After the parameters are determined, equations (2) and (6) together yield:

dk

cakx ⊗+⊗=+ ⊗− ˆe ˆ)1(ˆ ˆ)1( (10)

where dc x ⊗−=⊗ ˆ)1(ˆ )0( ; a

bd ⊗

⊗=⊗ ˆ

ˆˆ , and the solution )1(ˆ )1( +kx indicates the

predicted AGO series. Thus, by applying the inverse AGO approach, the predicted value k time step ahead )1(ˆ )0( +kx is given as:

)(ˆ)1(ˆ)1(ˆ )1()1()0( kxkxkx −+=+ (11)

The parameters of the GM(1,1) model can be updated immediately when new data are obtained. Phase-space reconstruction The rainfall process is a complex dynamic system that can be described using phase-space theory (Packard et al., 1980; Crutchfield & McNamara, 1987; Oiwa & Fiedler-

Pao-Shan Yu et al.

658

Ferrara, 1998). A phase space is an abstract space constructed from all related state variables associated with a dynamic system. However, practically, identifying all related state variables of a dynamic system is difficult. Often, a time series system includes only one variable, which represents the behaviour of the system. Therefore, the phase space of a dynamic system can be reconstructed for a time series using the values of previous time steps as state variables (Packard et al., 1980). Let a time series X = [x1, x2, ..., xN] be the process of rainfall variable x, where N is the number of data values. The time series can be embedded into an m-dimensional Euclidean space Rm as a point set XK = [xk, xk-1, ..., xk-m+1], where k = m, m+1, ..., N. Several algorithms have been proposed to determine the optimal dimension of the phase space (Grassberger & Procaccia, 1983; Sugihara & May, 1990; Kember et al., 1993). In this work, an optimization method is implemented to calibrate the dimensions of the phase space, based on the following objective function:

Maximize �

�−

−− 2

2)(

]')([)](ˆ)([

1xtx

txtx m

(12)

In equation (12), x(t) is the observed series at time t; x′ is a reference value, and )(ˆ )( tx m is the value predicted by the model constructed in m-dimensional phase space, where m = 1, 2, …, N. Nonlinear modelling—local approximation Rodriguez-Iturbe et al. (1989) indicated that rainfall is chaotic. A nonlinear model is required to describe the nonlinear mechanism of a chaotic system. Suppose a dynamic system is reconstructed in a phase space with dimension m, then the state of the system at time t is expressed as Xt = [xt, xt-1, ..., xt-m+1]. At a future time t + T, the state of the system is Xt+T = [xt+T, xt+T-1, ..., xt+T-m+1]. The relationship between such a system at time t and that at time t + T is:

)( tTTt XfX =+ (13)

An appropriate nonlinear model fT must be selected to predict the future behaviour of the system Xt+T at time t + T from the present state Xt. The term fT can be determined by the local approximation method, as suggested by Farmer & Sidorovich (1987). The basic idea that underlies this method is the division of the system into several nearby states in the phase space. Each nearby state is treated separately as a local subspace and exhibits similar behaviour. Only the states near the present state—within the same local subspace—are employed to construct the local model to perform predictions. The extent of the local subspace, or the number of nearby states, is determined by the neighbour number method (Yakowitz, 1987; Kember et al., 1993; Jayawardena & Lai, 1994). The neighbour number method (NNM) is as follows. Assume that the local neighbour region includes K states [

kttt XXX ′′′ ...,,,21

], in which ttk <′ , where k = 1, 2, ..., K. Then, the state Xt+T at time t + T can be estimated using the “future” nearby states [ TtTtTt k

XXX +′+′+′ ...,,,21

] in the neighbour region. Accordingly, equation (13) can be rewritten as:

Comparison of grey and phase-space rainfall forecasting models

659

)...,,,(21 TtTtTtTTt k

XXXfX +′+′+′+ = (14)

The function fT in this study is expressed as a weighted average of nearby states, therefore the forecast ( )Ttx +ˆ with lead time T can be given as:

( ) �=

+′⋅=+K

iTti i

xwTtx1

ˆ (15)

The weight wi of each state is estimated using a Markov process model (Yu & Yang, 1997; Lin, 1999). Sugihara & May (1990) suggested that the number of the nearby states, K, is given by K = m + 1, where m is the embedding dimension of the phase space. The closeness of two states is represented as a Euclidean distance in the phase space:

( ) ( )�=

+−+− −=−=ρn

llnjlnijiij xxXXn

1

2 (16)

where ρij(n) is the Euclidean distance between states Xi and Xj in n-dimensional phase space. CRITERIA FOR DETERMINING MODEL PERFORMANCE Test of rainfall forecasting model Four indices, commonly used to judge the performance of a model, are employed in this study. These indices are (a) percentage error of cumulative rainfall (ECR), (b) root mean square error (RMSE), (c) correlation coefficient (CC) and (d) modified coefficient of efficiency (MCE). The MCE is expressed as follows:

( )�

�

=

=

′−

−−= n

tt

n

ttt

RR

RRMCE

1

2

1

2)ˆ(1 (17)

1−=′ tRR if the lead time is one time step

221 −− +

=′ tt RRR if the lead time is two time steps

3321 −−− ++

=′ ttt RRRR if the lead time is three time steps

where tR̂ is the predicted rainfall; Rt is the observed rainfall; and n is the number of rainfall data. Test of error series Forecasting generally yields an error, which is the difference between the model output and the observed value. However, if a forecasting model accurately describes a real

Pao-Shan Yu et al.

660

system, the values of the error series will be small, random and uncorrelated, and the error series will appear as white noise, with a mean of zero. Therefore, two statistical tests—the mean value test and the white noise test—are applied to investigate the error series. Mean value test The mean value of an error series of a forecasting model is desired to be around zero, if the performance of the model is to be statistically satisfactory. Student’s t-test is usually used to determine these errors (Mujumdar & Kumar, 1990; Meade & Islam, 1998). Consider a variable, T(e), where:

n

eeT2

)(σ

= (18)

where e is the mean of the error series; σ2 is the variance of the series, and n is the number of data in the series. Assume that T(e) follows Student’s t-distribution t(α, n–1). If |T(e)| is less than a critical value K = tα/2(n – 1) with a significance level α, then the mean value of the error series is considered to be around zero. White noise test The Portmanteau test (Mujumdar & Kumar, 1990) is performed herein to test whether the error series is white noise. The variable w(e) is defined as:

�=

−=1

1 01 )()(

n

k

k

RRnnew (19)

where n1 is set to 15% of the number of data, such that n1 = 0.15n; Rk is the covariance with time lagk, and R0 is the covariance with zero time lag. The parameter w(e) is assumed to follow a chi-square distribution ( )1

2 nαχ . If w(e) is less than a critical value of chi-square distribution with a significance level α in the test, then the error series is considered to follow a white noise process. Fuzzy decision model Fuzzy set theory, first proposed by Zadeh (1965), has been applied in a wide variety of disciplines. Fuzzy set theory and its applications are described in the literature (Klir & Yuan, 1995; Zimmerman, 2001). Fuzzy sets are described briefly as follows. Consider n objectives, X{x1, x2, ..., xn}. A fuzzy set ~F of X is characterized by a

fuzzy membership grade µi, such that:

���

��� µµµ

=n

n

xxxF , . . . , ,~

2

2

1

1 (20)

The membership grade µi, ranging from zero to one, can be estimated from the fuzzy membership function µ(xi) that maps the objective xi to the membership grade µi. A fuzzy decision model (Hsiao, 1998) is applied in this work to identify the optimum alternative. The theoretical operations and definitions in the fuzzy decision model are as follows:

Comparison of grey and phase-space rainfall forecasting models

661

Fuzzy relational matrix Consider m influence factors, which influence the decisions, in a fuzzy decision model; these factors construct a factor set U = {u1, u2, ..., ui, ..., um}; n decision-making alternatives, to be determined, construct an alternative set V = {v1, v2, ..., vj, ..., vn}. A fuzzy relational matrix ~R will be defined

to describe the relationship between the influence factor set U and the alternative set V:

�����

�

�

�����

�

�

=

mnmm

n

n

rrr

rrrrrr

R

�

����

�

�

21

22221

11211

~ (21)

where rij is the membership grade that relates the factor ui to the alternative vj. Weighting function set Different influence factors may affect decision making to various extents, so a weighting function set ),,,(~ 21 mwwwW �= is employed to assign

weighting values to capture the relative importance of the influence factors ui. Usually,

the weighting function set is normalized such that �=

=m

iiw

11.

Fuzzy decision making When ~R and ~W are given to suit the problem of interest,

the fuzzy evaluation set ( )neeeE ,,,~ 21 �= is given by:

~~~ RWE �= (22)

or, in detail:

( ) ( )�����

�

�

�����

�

�

==

mnmm

n

n

mn

rrr

rrrrrr

wwweeeE

�

����

�

�

���

21

22221

11211

2121 ,,,,,,~ (23)

The symbol “ � ” in equations (22) and (23) represents the fuzzy composite operation, which is performed by the weighted mean method (Hsiao, 1998) in this work. The weighted mean method is formulated as:

�=

=m

iijij rwe

1

(24)

Each element ej in the fuzzy evaluation set ~E in equation (23) specifies the extent to

which each of the alternatives can be explained in terms of the factor set. The alternative with the highest membership grade in the fuzzy evaluation set is chosen as the optimum alternative, because it explains the factor set to the greatest possible extent.

Pao-Shan Yu et al.

662

STUDY AREA AND DATA SET The case study described herein involves two raingauges, at Hu-Tou-Pei (120°19′44′′E, 23º01′40′′N) and Chi-Ding (120º27′06′′E, 22º57′59′′N), in the Yen-Shui Creek basin as depicted in Fig. 1. The Yen-Shui Creek is in the southwest of Taiwan. It drains 340 km2 into the Taiwan Strait and the main channel is about 41 km long. The mean annual rainfall is around 1650 mm, representing approximately 90% of annual rainfall in the wet season from May to October. Ten rainfall events are collected for the case study; the first six events are for calibration and the last four are for verification. Table 1 lists the attributes of these events at the Hu-Tou-Pei and Chi-Ding raingauges, respectively.

N

Taiwan

Chi-Ding(120º27’06”E,22º57’59”N) Raingauge

Yen-Shui Creek

Hu-Tou-Pei(120º19’44”E,23º01’40”N)

Fig. 1 Location of the study area.

Table 1 Collected events.

Date Rainfall type Hu-Tao-Pei: Chi-Ding: Note No. Duration

(h) Cumulated rainfall (mm)

No. Duration (h)

Cumulated rainfall (mm)

21/06/1991 Storm A1 111 609 B1 94 578 Calibration 03/07/1992 Storm A2 100 571 B2 82 501 Calibration 30/08/1992 Typhoon Polly A3 28 286 B3 34 305 Calibration 04/09/1992 Typhoon Omar A4 32 183 B4 34 236 Calibration 25/05/1993 Storm A5 52 227 B5 48 270 Calibration 11/08/1994 Typhoon Doug (1) A6 34 266 B6 36 336 Calibration 14/08/1994 Typhoon Doug (2) A7 30 163 B7 30 212 Verification 30/06/1997 Storm A8 29 405 B8 30 345 Verification 06/08/1997 Storm A9 46 419 B9 44 650 Verification 05/09/1997 Storm A10 30 249 B10 30 152 Verification APPLICATION AND RESULTS Rainfall forecasting using the GM(1,1) model A set of four chronological observations of rainfall was used dynamically to construct a grey rainfall forecasting model to provide forecasts one to three hours ahead. Yu

Comparison of grey and phase-space rainfall forecasting models

663

et al. (2000) found that using a sequence of data on the rainfall mass curve as the input to the grey rainfall forecasting model leads to better results than obtained using the original rainfall hyetograph. Therefore, the rainfall mass curve was adopted as the input of the grey rainfall forecasting model in this work. A rainfall mass curve

0 10 20 30 40Time (hour)

0

10

20

30

40

Rai

nfal

l Int

ensi

ty (m

m/h

r)

Lead time = 1 hourObservedForecasted

Event A7

0 10 20 30 40Time (hour)

0

50

100

150

200

Cum

ulat

ive

Rai

nfal

l Dep

th (m

m)

Lead time = 1 hourObservedForecasted

Event A7

0 10 20 30 40Time (hour)

0

10

20

30

40

Rai

nfal

l Int

ensi

ty (m

m/h

r)

Lead time = 2 hoursObservedForecasted

0 10 20 30 40Time (hour)

0

50

100

150

200

Cum

ulat

ive

Rai

nfal

l Dep

th (m

m)

Lead time = 2 hoursObservedForecasted

0 10 20 30 40Time (hour)

0

10

20

30

40

Rai

nfal

l Int

ensi

ty (m

m/h

r)

Lead time = 3 hoursObservedForecasted

0 10 20 30 40Time (hour)

0

50

100

150

200

Cum

ulat

ive

Rai

nfal

l Dep

th (m

m)

Lead time = 3 hoursObservedForecasted

Fig. 2 Forecasting results of the grey rainfall forecasting model (Event A7).

Pao-Shan Yu et al.

664

represents the first-order AGO series of a rainfall hyetograph, so the grey rainfall forecasting model herein actually deals with the second-order AGO series of the rainfall hyetograph. The performance of the grey rainfall forecasting model in a single time step is reasonable (Chen, 1998). However, the accuracy of forecasting decreases as the lead time increases, because the forecast errors accumulate. Accordingly, Yu et al. (2000) developed a single time-step forecasting technique to improve forecasting 2 and 3 h ahead. The technique is to transform the forecasts 2 and 3 h ahead into the forecasts over one time step by constructing two grey models using two sets of data at different times. Yu et al. (2000) described the algorithm used to implement this technique. The events in Table 1 were used in the rainfall forecasting. Figure 2 plots the forecasts for Event A7. The results demonstrate that the grey rainfall forecasting model with a single time-step forecasting technique, can reasonably forecast rainfall 1–3 h ahead. However, a phase lag exists between the observed and forecast rainfall hyeto graphs (Fig. 2). This is a common phenomenon in predicting time series. When the trend of the series changes too fast, the inertia of the prediction model causes this phase lag. Rainfall forecasting using the phase-space model The first six events in Table 1 were chosen to calibrate the number of embedding dimensions m of the phase space of the Hu-Tou-Pei and Chi-Ding raingauges, respectively, based on the objective function in equation (12). The reference value x′ in equation (12) was taken as R′ in equation (17); therefore, the objective function was to maximize the MCE. Different embedding dimensions, m = 2, 3, …, 9, were used to construct the phase space and then make predictions for each calibration event. Figure 3 shows the results of the calibration. The curve plots the average calibrated MCE values vs the em-bedding dimensions. Overall, MCE increases with the embedding dimension, but not by much, suggesting that the forecasting performance is slightly improved by increas-ing the embedding dimension. However, the complex structure of the forecasting

2 3 4 5 6 7 8 9Embedding Dimension

-0.4

-0.2

0.0

0.2

0.4

0.6

MC

E

Hu-Tou-Pei 1 hour ahead2 hours ahead3 hours ahead

2 3 4 5 6 7 8 9Embedding Dimension

0.0

0.2

0.4

0.6

MC

E

Chi-Ding1 hour ahead2 hours ahead3 hours ahead

Fig. 3 Calibration results of different embedding dimensions.

Comparison of grey and phase-space rainfall forecasting models

665

model is such that the performance does not improve well with the embedding dimen-sion. Moreover, a model with a lower embedding dimension can begin to predict before a model with a higher embedding dimension. Additionally, for simplicity and consistency, the embedding dimensions of the forecasting model with different lead

0 10 20 30 40Time (hour)

0

10

20

30

40

Rai

nfal

l Int

ensi

ty (m

m/h

r)

Lead time = 1 hourObservedForecasted

Event A7

0 10 20 30 40Time (hour)

0

50

100

150

200

Cum

ulat

ive

Rai

nfal

l Dep

th (m

m)

Lead time = 1 hourObservedForecasted

Event A7

0 10 20 30 40Time (hour)

0

10

20

30

40

Rai

nfal

l Int

ensi

ty (m

m/h

r)

Lead time = 2 hoursObservedForecasted

0 10 20 30 40Time (hour)

0

50

100

150

200

Cum

ulat

ive

Rai

nfal

l Dep

th (m

m)

Lead time = 2 hoursObservedForecasted

0 10 20 30 40Time (hour)

0

10

20

30

40

Rai

nfal

l Int

ensi

ty (m

m/h

r)

Lead time = 3 hoursObservedForecasted

0 10 20 30 40Time (hour)

0

50

100

150

200

Cum

ulat

ive

Rai

nfal

l Dep

th (m

m)

Lead time = 3 hoursObservedForecasted

Fig. 4 Forecasting results of the phase-space model (Event A7).

Pao-Shan Yu et al.

666

times (1–3 h) are set identically for each raingauge. In this study, the embedding dimensions were set to four for Hu-Tou-Pei and three for Chi-Ding. The verification events in Table 1 were then employed in rainfall forecasting using the phase-space model. Figure 4 shows the forecasts of Event A7. The figure demon-strates that the forecasting results are reasonable but somewhat underestimated. However, the performance of a model cannot be directly determined from the graphs. Some quantitative indices are required to evaluate the model more objectively. The following section compares these two models. Comparison of model performance Four indices, mentioned above, were calculated for each event to evaluate the perform-ance of both models. Table 2 presents evaluations for Hu-Tou-Pei raingauge of both forecasting models, according to the stated criteria. The values of the better performing model are presented in bold. The evaluations for both raingauges demonstrate similar results, so that only results for Hu-Tou-Pei raingauge are presented herein. For forecasting 1 h ahead, the grey model outperforms the phase-space model in MCE, RMSE and CC. However for forecasting 2 and 3 h ahead, the phase-space model outperforms in MCE and RMSE. By the ECR criterion, the grey model outperforms the phase-space model in forecasting 2 and 3 h ahead. A significance level of α = 0.95 was adopted in the tests of the error series in Table 3, including the mean value test and the white noise test. The forecasting results by the grey model pass the mean value test for almost all events, but a few forecasts of events by the phase-space model were rejected. The white noise test yield similar results for both models. Forecasts of some events for both raingauges did not pass the white noise test. Although these indices and tests represent objective ways to evaluate the per-formance of forecasting models, identifying the better model is difficult because the various indices do not lead to the same conclusion. Therefore the fuzzy decision model was applied to take into account all of these criteria and yield an optimum decision. The procedures are illustrated as follows. Formulating the factor set and alternative set In this study, the optimum model is either the grey model or the phase-space model so the alternative set is V = {grey model, phase-space model} = {v1, v2}. A set of two-level influence factors was used (as presented in Fig. 5). The rainfall hyetograph, the accumulated rainfall, the model efficiency and the results of the error tests were used as the first-level influence factors in evaluating the performance of the model. Hence, the first-level influence factor set was determined as:

U = {rainfall hyetograph, accumulated rainfall, model efficiency, error test}

= {u1, u2, u3, u4} The indices and tests applied using the criteria in the sections “Test of rainfall forecasting model” and “Test of error series” were used as second-level factors.

u1 = {RMSE, CC} = {c1, c2} u2 = {ECR} = {c3}

Comparison of grey and phase-space rainfall forecasting models

667

Table 2 Results of the indices at Hu-Tou-Pei raingauge.

Grey model: Phase-space model: Event ECR (%) MCE RMSE CC ECR (%) MCE RMSE CC

Lead time = 1 h: A1 15.72 0.27 6.93 0.71 1.05 0.06 7.92 0.47 A2 9.77 0.31 9.32 0.56 –1.83 0.13 10.51 0.26 A3 1.04 0.08 8.61 0.67 –24.00 0.20 8.06 0.65 A4 13.75 0.44 5.16 0.81 –4.73 0.18 6.29 0.59 A5 3.52 0.09 5.16 0.64 10.47 –0.21 6.05 0.44 A6 6.18 0.36 7.44 0.49 –6.84 0.24 8.70 0.21 A7 13.37 0.45 6.57 0.63 –3.64 0.14 8.31 0.29 A8 4.65 0.37 12.01 0.67 –18.32 0.27 13.58 0.40 A9 7.53 0.31 9.91 0.41 –15.82 0.33 10.33 0.17 A10 16.19 0.05 15.35 0.47 3.17 0.10 14.93 0.26 Mean 9.17 0.27 8.65 0.61 –6.05 0.14 9.47 0.37 Lead time = 2 h: A1 24.20 –0.43 12.82 0.17 5.95 0.20 9.65 0.21 A2 11.23 –0.24 13.78 0.07 –6.23 0.14 11.54 0.09 A3 –13.66 –0.38 11.24 0.48 –13.36 –0.07 10.10 0.36 A4 31.74 –0.47 10.78 0.44 –13.16 0.41 6.94 0.46 A5 1.83 –0.38 8.04 0.17 –3.40 0.01 6.92 0.06 A6 –0.79 –0.23 8.11 0.36 4.85 0.06 8.29 0.32 A7 20.38 –0.05 8.85 0.42 –9.00 0.26 7.72 0.33 A8 3.05 –0.16 16.38 0.46 –30.26 0.03 16.41 0.12 A9 –1.41 –0.39 13.02 –0.04 –17.18 0.15 10.61 0.08 A10 –18.43 0.19 14.78 0.28 –17.12 0.28 14.25 0.30 Mean 5.81 –0.25 11.78 0.28 –9.89 0.15 10.23 0.23 Lead time = 3 h: A1 17.25 –0.56 13.88 –0.07 –1.53 0.19 10.11 –0.10 A2 12.77 –0.35 13.67 0.02 –10.22 0.21 10.56 0.07 A3 –27.16 –0.02 9.35 0.59 –28.18 0.01 9.85 0.41 A4 40.12 –1.14 14.96 –0.02 –12.65 0.46 7.76 0.24 A5 4.70 –1.07 8.36 0.20 –0.16 –0.27 6.72 –0.05 A6 –13.23 –0.05 8.04 0.34 –11.76 0.33 7.13 0.39 A7 –5.71 –0.06 9.80 0.02 –27.19 0.39 7.79 0.23 A8 –5.79 –0.15 17.46 0.22 –44.33 0.13 16.00 0.12 A9 –2.81 –0.36 11.82 0.15 –20.79 0.30 9.50 0.29 A10 –7.64 –0.63 19.97 –0.13 –26.65 0.50 11.78 0.61 Mean 1.25 –0.44 12.73 0.13 –18.35 0.23 9.72 0.22 Note: The values in bold indicate the model with better performance.

u3 = {MCE} = {c4} u4 = {Mean value test, White noise test} = {c5, c6}

The second-level influence factors, listed in Tables 2 and 3, were classified as a three-grade decision set, {poor, fair, good}, based on the boundaries of each factor, delineated with reference to the results in Tables 2 and 3, as presented in Table 4. The membership grades of the three-grade decision set were:

���

���

good1,

fair5.0,

poor0

Pao-Shan Yu et al.

668

Table 3 Results of the error tests at Hu-Tou-Pei raingauge.

Mean value test: White noise test: Critical value |T(e)| Critical value w(e)

Event

95.0=αt 90.0=αt Grey model

Phase-space model

295.0χ 2

90.0χ Grey model

Phase-space model

Lead time = 1 h: A1 1.66 1.29 1.32 0.08 26.30 23.54 35.49 21.17 A2 1.66 1.29 0.60 0.10 23.68 21.06 12.17 6.54 A3 1.72 1.32 0.06 1.67 7.81 6.25 12.43 1.72 A4 1.71 1.32 0.86 0.24 9.49 7.78 1.73 1.67 A5 1.68 1.30 0.21 0.54 14.07 12.02 23.68 14.76 A6 1.70 1.31 0.37 0.35 9.49 7.78 5.10 1.85 A7 1.71 1.32 0.60 0.13 7.81 6.25 3.61 7.89 A8 1.72 1.32 0.29 1.02 7.81 6.25 3.68 1.07 A9 1.68 1.30 0.46 0.94 12.59 10.64 13.29 7.27 A10 1.71 1.32 0.47 0.09 7.81 6.25 11.02 5.85 Mean 1.70 1.31 0.52 0.52 12.69 10.78 12.22 6.98 Lead time = 2 h: A1 1.66 1.29 1.09 0.36 26.30 23.54 79.40 38.66 A2 1.66 1.29 0.46 0.30 23.68 21.06 43.85 13.73 A3 1.72 1.32 0.65 0.72 7.81 6.25 5.31 4.06 A4 1.71 1.32 0.95 0.61 9.49 7.78 14.06 4.50 A5 1.68 1.30 0.07 0.15 14.07 12.02 36.87 32.34 A6 1.70 1.31 0.04 0.25 9.49 7.78 3.47 1.39 A7 1.71 1.32 0.68 0.33 7.81 6.25 4.52 0.74 A8 1.72 1.32 0.14 1.42 7.81 6.25 2.79 1.39 A9 1.68 1.30 0.07 1.01 12.59 10.64 22.42 6.53 A10 1.71 1.32 0.56 0.53 7.81 6.25 3.88 5.64 Mean 1.70 1.31 0.47 0.57 12.69 10.78 21.66 10.90 Lead time = 3 h: A1 1.66 1.29 0.72 0.09 26.30 23.54 109.99 55.72 A2 1.66 1.29 0.53 0.55 23.68 21.06 47.19 24.84 A3 1.72 1.32 1.63 1.66 7.81 6.25 5.70 11.49 A4 1.71 1.32 0.86 0.53 9.49 7.78 15.17 8.38 A5 1.68 1.30 0.17 0.01 14.07 12.02 22.60 13.68 A6 1.70 1.31 0.74 0.70 9.49 7.78 4.14 4.38 A7 1.71 1.32 0.17 0.99 7.81 6.25 7.02 2.04 A8 1.72 1.32 0.24 2.12 7.81 6.25 6.08 10.37 A9 1.68 1.30 0.15 1.40 12.59 10.64 19.05 10.07 A10 1.71 1.32 0.17 1.04 7.81 6.25 3.04 3.01 Mean 1.70 1.31 0.54 0.91 12.69 10.78 24.00 14.40 Note: The values in bold indicate the model with better performance and that passing the test with α = 0.95. Constructing the fuzzy relational matrix Consider the forecast 1 h ahead at Hu-Tou-Pei (Table 2); the number of times the RMSE (c1) of the grey model in Table 2 fell into the classes of the decision set, {poor, fair, good}, were zero, two and eight, respectively. Thus, its fuzzy grade was 9 (= 0 × 0 + 0.5 × 2 +1 × 8)), which was then divided by the number of total events, for normalization. Finally, the fuzzy member-ship grade of c1 was 0.9. The fuzzy membership grades of the elements in the second-level factor set were estimated using the same procedure. They are u1 = {0.9, 0.85}, u2 = {0.8}, u3 = {0.6} and u4 = {0.95, 0.5}. Averaging the elements of the second-level

Comparison of grey and phase-space rainfall forecasting models

669

RainfallModel

AccumulatedRainfall (u2)

Rainfall Hyetograph (u1)

Model Efficiency (u3)

Error Test (u4)

RMSE (c1)

CC (c2)

ECR (c3)

MCE (c4)

Mean Value Test (c5)

White Noise Test (c6)

1st Level 2nd Level

Fig. 5 Diagram of the two-level factor set.

Table 4 Boundaries of each factor.

Factor Good Fair Poor RMSE ≤ 10 >10 and ≤ 20 >20 CC ≥ 0.5 <0.5 and ≥ 0 <0 ECR ≤ 10% >10% and ≤ 25% >25% MCE ≥ 0.4 <0.4 and ≥ 0 <0 Mean value test pass, α = 0.90 pass, α = 0.95 fail, α = 0.95 White noise test pass, α = 0.90 pass, α = 0.95 fail, α = 0.95 Table 5 Fuzzy relational matrices.

Hu-Tou-Pei: Chi-Ding: Factor Grey model Phase-space

model Grey model Phase-space

model Lead time = 1 h: Rainfall hyetograph 0.56 0.44 0.57 0.43 Accumulated rainfall 0.50 0.50 0.52 0.48 Model efficiency 0.57 0.43 0.60 0.40 Error test 0.45 0.55 0.47 0.53 Lead time = 2 h: Rainfall hyetograph 0.47 0.53 0.44 0.56 Accumulated rainfall 0.48 0.52 0.74 0.26 Model efficiency 0.09 0.91 0.18 0.82 Error test 0.46 0.54 0.57 0.43 Lead time = 3 h: Rainfall hyetograph 0.47 0.53 0.42 0.58 Accumulated rainfall 0.62 0.38 0.68 0.32 Model efficiency 0.00 1.00 0.09 0.91 Error test 0.54 0.46 0.50 0.50 influence factors yields the fuzzy membership grades associated with the first-level factor set. The membership grades with the grey model (alternative 1) were given as {r11, r21, r31, r41} = {0.875, 0.8, 0.6, 0.725}. Likewise, the fuzzy membership grades

Pao-Shan Yu et al.

670

that relate to the first-level factor set with the phase-space model (alternative 2) were {r12, r22, r32, r42} = {0.7, 0.8, 0.45, 0.875}. The fuzzy relational matrix for forecasting 1 h ahead at Hu-Tou-Pei was formulated and normalized as:

�����

�

�

�����

�

�

=�

�����

�

�

�����

�

�

=

55.045.043.057.050.050.044.056.0

~

875.0725.045.060.080.080.070.0875.0

~ RR

Based on the above procedures, the fuzzy membership grades that relate the first-level factor set to alternatives 1 and 2 can be calculated for forecasts 1–3 h ahead, and the fuzzy relational matrices can be constructed for two gauges, as listed in Table 5. Determining the weighting function set Next, the weighting function set for the first-level factors must be determined. The rainfall hyetograph and MCE are two factors that are considered to be significant in forecasting rainfall. Thus, the weighting function set for the first-level factors is:

���

���

=4321

1.0,4.0,1.0,4.0~ wwwwW

Fuzzy decision making Accordingly, for forecasting 1 h ahead at Hu-Tou-Pei, the fuzzy evaluation set is finally determined to be:

( )�����

�

�

�����

�

�

=

42

32

22

12

41

31

21

11

4321 ,,,~

rrrr

rrrr

wwwwE �

( )�����

�

�

�����

�

�

=

55.045.043.057.050.050.044.056.0

1.0 ,4.0 ,1.0 ,4.0 �

)45.0 ,55.0(=

In the fuzzy evaluation set, the membership grade, 0.55, of alternative 1 (grey model) exceeds that, 0.45, of alternative 2 (phase-space model), indicating that the grey model outperforms the phase-space model.

Comparison of grey and phase-space rainfall forecasting models

671

Table 6 Results of fuzzy decision making.

Hu-Tou-Pei: Chi-Ding: Lead time Result Better model Result Better model

1 h (0.55, 0.45) Grey model (0.57, 0.43) Grey model 2 h (0.32, 0.68) Phase-space model (0.38, 0.62) Phase-space model 3 h (0.30, 0.70) Phase-space model (0.33, 0.67) Phase-space model The proposed fuzzy decision-making model was applied to determine the optimum alternative, and the results are listed in Table 6. These results reveal that the grey model outperforms the phase-space model in forecasting 1 h ahead, but the phase-space model outperforms in forecasting 2 or 3 h ahead. DISCUSSION AND CONCLUSION Short-duration rainfall forecasting is important for forecasting floods, especially in the catchment, where the runoff responds very quickly to the rainfall. Two rainfall forecasting models—the grey model and the phase-space model—were applied to forecast rainfall 1–3 h ahead in the study area. The performance of a model is usually evaluated objectively using quantitative indices. However, a trade-off must always be made in determining a suitable model because various indices may suggest different models. Thus, the fuzzy decision model was proposed herein to solve this problem and determine a single optimum alternative. The results of fuzzy decision making indicate that the grey model outperforms the phase-space model in forecasting 1 h ahead, but the phase-space model performs better in forecasting 2 or 3 h ahead. Rainfall is a volatile phenomenon. However, trends apply over short periods. The grey rainfall forecasting model is used essentially to forecast trends in rainfall. The grey outperforms the phase-space model in forecasting 1 h ahead. When the rainfall changes too fast, the grey model cannot effectively predict rainfall over longer lead time, so a phase lag exists between the observed and the forecast rainfall hyetographs. The phase-space model outperforms the grey model in forecasting 2 and 3 h ahead. The phase-space model is constructed using phase-space theory and nonlinear modelling. Rainfall is a complex dynamic system. The information in a rainfall time series is used to reconstruct the phase space of a dynamic system. It can be used to predict the trajectory of a dynamic system in the phase space from historical informa-tion about the system. Although the phase-space model does not predict very accurately in this case study, it does not yield the phase lag common to predictions based on time series. The performance of the phase-space model strongly relies on historical rainfall data. If sufficient data are available to enable the model to be calibrated and to learn, then the phase-space model will be expected to forecast better. Although the fuzzy decision model can be used directly and objectively to evaluate the performance of models, the choice of factors and their weights is rather subjective. The selection of the factors and their weights may considerably affect the results. The fuzzy decision method should be appropriately applied according to the characteristics of the models to be evaluated and the significance of the factors chosen by the decision maker.

Pao-Shan Yu et al.

672

Acknowledgement The authors would like to thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract no. NSC 87-2621-P006-027. REFERENCES Burlando, P., Rosso, R., Cadavid, L. G. & Salas, J. D. (1993) Forecasting of short-term rainfall using ARMA models.

J. Hydrol. 144, 193–211. Chen, S. T. (1998) Application of grey prediction model on runoff forecasting. MS Thesis, National Cheng Kung

University, Tainan, Taiwan (in Chinese). Crutchfield, J. P. & McNamara, B. S. (1987). Equations of motion from a data series. Complex Systems 1, 417–452. Deng, J. L. (1989) Introduction to grey system theory. J. Grey Systems 1(1), 1–24. Docine, L., Andrieu, H. & Creutin, J. D. (1999) Evaluation of a simplified dynamical rainfall forecasting model from rain

events simulated using a meteorological model. Physics and Chemistry of the Earth, Part B: Hydrology, Oceans and Atmosphere 24(8), 883–887.

Farmer, J. D. & Sidorovich, J. J. (1987) Predicting chaotic time series. Phys. Rev. Lett. 59(8), 845–848. Grassberger, P. & Procaccia, I. (1983) Characterization of strange attractors. Phys. Rev. Lett. 50(5), 346–349. Hsiao, S. W. (1998) Fuzzy logic based decision model for product design. Int. J. Industr. Ergonomics 21, 103–116. Huang, Y. P. & Huang, C. C. (1996) The integration and application of fuzzy and grey modeling methods. Fuzzy Sets and

Systems 78, 107–119. Jayawardena, A. W. & Lai, F. (1994) Analysis and prediction of chaos in rainfall and stream flow time series. J. Hydrol.

153(1–4), 23–52. Kember, G., Flower, A. C. & Holubeshen, J. (1993) Forecasting river flow using nonlinear dynamics. Stochast. Hydrol.

Hydraul. 7, 205–212. Klir, G. J. & Yuan, B. (1995). Fuzzy Sets and Fuzzy Logic: Theory and Applications. Prentice Hall, Englewood Cliffs,

New Jersey, USA. Lin, S. C. (1999). Fractals and chaos in the analysis and prediction of nonlinear hydrologic systems. PhD dissertation,

National Cheng Kung University, Tainan, Taiwan (in Chinese) Luk, K. C., Ball, J. E. & Sharma, A. (2001) An application of artificial neural networks for rainfall forecasting. Math.

Computer Modelling 33, 683–693. Meade, N. & Islam, T. (1998) Technological forecasting-model selection, model stability, and combining model.

Management Sci. 44(8), 1115–1130. Mujumdar, P. P. & Kumar, D. N. (1990) Stochastic models of streamflow: some case studies. Hydrol. Sci. J. 35(4), 395–

410. Oiwa N. N. & Fiedler-Ferrara, N. (1998) A fast algorithm for estimating Lyapunov exponents from time series. Phys. Lett.

A 246(1–2), 117–121. Packard, N. H., Crutchfield, J. P., Farmer, J. D. & Shaw, R. S. (1980) Geometry from a time series. Phys. Rev. Lett. 45(9),

712–716. Rodriguez-Iturbe, I., de Power, B. F., Sharifi, M. B. & Georgakakos, K. P. (1989) Chaos in rainfall. Water Resour. Res.

25(7), 1667–1675. Sugihara, G. & May, R. M. (1990) Nonlinear forecasting as a way of distinguishing chaos from measurement error in time

series. Nature 344, 734–741. Yakowitz, S. (1987) Nearest-neighbour methods for time series analysis. J. Time Series Analysis 8(2), 235–247. Yager, R. R. (1977) Multiple objective decision-making using fuzzy set. Int. J. Man-Machine Studies 9, 375–382. Yu, P. S. & Yang T. C. (1997) A probability-based renewal rainfall model for flow forecasting. Natural Hazards 15,

51–70. Yu, P. S., Chen, C. J. & Chen, S. J. (2000) Application of grey and fuzzy methods for rainfall forecasting. J. Hydrol.

Engng 5(4), 339–345. Yu, P. S., Chen, C. J., Chen, S. J. & Lin, S. C. (2001) Application of grey model toward runoff forecasting. J. Am. Water

Resour. Assoc. 37(1), 151–166. Xia, J. (1989) Research and application of grey system theorem to hydrology. J. Grey Systems 1(1), 43–52. Zadeh, L. A. (1965) Fuzzy sets. Information Control 8, 338–353. Zimmermann, H. J. (2001) Fuzzy Sets Theory—and its Applications, fourth edn. Kluwer Academic Publishers, Dordrecht,

The Netherlands. Received 30 June 2003; accepted 1 April 2004