Wavelet Support Vector Machines for Forecasting Precipitation in Tropical Cyclones: Comparisons with GSVM, Regression, and MM5

438 WEATHER AND FORECASTING VOLUME 27

CHIH-CHIANG WEI Department of Information Management, Toko University, Pu-Tzu City, Taiwan

(Manuscript received 5 January 2011, in ﬁnal form 23 October 2011)

ABSTRACT

This study presents two support vector machine (SVM) based models for forecasting hourly precipitation during tropical cyclone (typhoon) events. The two SVM-based models are the traditional Gaussian kernel SVMs (GSVMs) and the advanced wavelet kernel SVMs (WSVMs). A comparison between the fifth- generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5) and statistical models, including SVM-based models and linear regressions (regression), was made in terms of performance of rainfall prediction at the Shihmen Reservoir watershed in Taiwan. Data from 73 typhoons affecting the Shihmen Reservoir watershed were included in the analysis. This study designed six attribute combinations with different lag times for the forecast target. The modified RMSE, bias, and estimated threat score (ETS) results were employed to assess the predicted outcomes. Results show that better attribute combinations for typhoon climatologic characteristics and typhoon precipitation predictions occurred at 0-h lag time with modified RMSE values of 0.288, 0.257, and 0.296 in GSVM, WSVM, and the regression, respectively. Moreover, WSVM having average bias and ETS values close to 1.0 gave better predictions than did the GSVM and regression models. In addition, Typhoons Zeb (1998) and Nari (2001) were selected for comparison between the MM5 model output and the developed statistical models. Results showed that the MM5 tended to overestimate the peak and cumulative rainfall amounts while the statistical models were inclined to yield underestimations.

1. Introduction forecasts (Jian et al. 2003). Microphysical schemes, used only in research before the 1990s, are now being adopted In meteorology and the atmospheric sciences, the by operational numerical weather prediction models. prediction of rainfall at landfall during tropical cyclones Furthermore, tremendous growth in computer capabil- (TCs) is an important research topic that has attracted ity in recent years has led to significant improvements in much interest. TCs, also known as typhoons, often de- forecasts of cloud condition and precipitation. To un- velop in the western North Pacific region. As soon as a typhoon makes landfall, the upstream watershed re- derstand the complicated physical mechanisms that oc- ceives voluminous rainfall within a short time, which cur during typhoon attacks in Taiwan, some insightful quickly converges downstream. The heavy precipitation real-case numerical studies (see section 2) have been un- can easily lead to floodwater exceeding the downstream dertaken (Businger et al. 1990; Jian et al. 2003). However, embankments, causing considerable economic losses and the physically based model is mathematically a highly casualties (Wei and Hsu 2009). Therefore, an accurate complicated, nonlinear numerical model in space and quantitative precipitation forecast for TC events is one time, and an accurate quantitative precipitation forecast of the most difficult challenges in meteorology (Businger remains one of the most difficult tasks (Lee et al. 2006). et al. 1990; Lonfat et al. 2004). Forecasting the behavior of complex systems has been High-resolution mesoscale numerical models hold a broad application domain for artificial intelligence or promise for enhancing the accuracy of precipitation machine learning. In weather forecasting, the prediction of rainfall during typhoons by statistical approaches has received much attention in recent years (see section 2). Corresponding author address: Chih-Chiang Wei, Dept. of In- formation Management, Toko University, No. 51, Sec. 2, Univer- As is well known, support vector machines (SVMs) pro- sity Rd., Pu-Tzu City, Chia-Yi County 61363, Taiwan. posed by V. Vapnik and his group at AT&T Bell Lab- E-mail: [email protected] oratories offer new and promising classification and

DOI: 10.1175/WAF-D-11-00004.1

Ó 2012 American Meteorological Society Unauthenticated | Downloaded 09/28/21 07:32 PM UTC APRIL 2012 W E I 439 regression techniques (Cortes and Vapnik 1995). SVMs Li et al. 2005). Comparison of performance between the are developed from the idea of structural risk minimiza- above-mentioned statistical models and MM5 (Li et al. tion, which shows that the generalization error is bounded 2005) was made in terms of the prediction of typhoon by the sum of training errors (Hao 2008). SVMs learn floods (Typhoons Zeb in 1998 and Nari in 2001) at the from a separating hyperplane to maximize the margin Shihmen Reservoir watershed. and to produce good generalization ability (Burges 1998). Recent theoretical research has solved the existing diffi- 2. Statistical and numerical approaches culties in practical applications of SVMs (Joachims 1999; Platt 1999). In recent years, statistical and numerical approaches On the other hand, wavelet transforms have proven have been successfully applied to typhoon precipitation to be very popular and effective in regression and pat- prediction. For the statistical approaches, Yeh (2002) tern recognition applications and the wavelet technique used empirical orthogonal function analysis including shows promise for both nonstationary signal approxi- the climatology average method, deviation persistence mations and classifications (Zhang and Benveniste 1992; method, and regression equations to forecast the 6-h ac- Szu et al. 1992; Chen and Xie 2007). Hence, it is of value cumulated typhoon rainfall over Taiwan. Lee et al. (2006) to explore whether better performance could be ob- employed a climatology model for forecasting typhoon tained if the wavelet technique is combined with SVMs rainfall in Taiwan. The proposed model used a simple (Zhang et al. 2004). In recent years, the combination of statistical approach to estimate reasonable cumulative wavelet theories and SVMs, called wavelet support vec- rainfall for each river basin. Lonfat et al. (2007) devel- tor machines (WSVMs; see section 3) has drawn con- oped a parametric hurricane rainfall prediction scheme, siderable attention owing to its high predictive ability which applied the rainfall climatology and persistence for a wide range of applications and its better perfor- model to forecasting rainfall accumulations. Using the mance compared with other traditional leaning machines skills of the artificial intelligence/machine learning, Lin (Kivanc et al. 2003). For example, Chen and Xie (2007) and Chen (2005) developed a neural network with two used the dual-tree complex wavelet features and SVMs hidden layers for typhoon rainfall forecasting. The model for pattern recognition. Yang and Wang (2008) applied configuration is evaluated using typhoon characteristics. the WSVM to distributed denial of service intrusion Fan and Lee (2007) developed a Bayesian mixture re- detections. Widodo and Yang (2008) established an gression model where the data of the nonnegative re- intelligent system for fault detection and classification sponse variable contain many zero measurements. The of induction motors using WSVM. Wu (2009) proposed model was applied to typhoon rainfall predictions in new WSVMs for setting up a nonlinear system of Taipei, Taiwan. Sheng et al. (2008) used TC data (po- product sale series. Yao et al. (2010) presented a method sition, pressure, and wind) to establish the distribution that used WSVM for chatter identification. Chen et al. functions of TC rainfall and ran the SVM regression (2010) proposed an improved voice activity detection models for TC rainfall forecast in Jiaxing, in eastern algorithm using WSVM. At present, the WSVM tech- China. nique has not been applied to typhoon precipitation For the numerical approaches, Yang and Houze (1995) forecast. indicated that the simulated rainfall amount, distribu- This study aims to develop SVM-based models, which tion, and internal mesoscale structure were highly sensi- include the traditional Gaussian radial basis function tive to the hydrometeor types and microphysical schemes kernel SVMs (GSVM) and the advanced WSVM, for implemented in the model. Liu et al. (1997) successfully forecasting hourly precipitation during typhoon events. simulated the track, storm intensity, and detailed inner- Comparison of performance between these developed core structure of Hurricane Andrew in 1992, using MM5 models and linear regressions (regression) was made in with a grid nesting down to a 6-km grid size and a so- terms of precipitation forecasts at the Shihmen Reservoir phisticated explicit-scale cloud microphysics scheme. watershed in Taiwan. Wang (2002) further indicated that the detailed cloud Moreover, this study also compares the above statis- structures of an idealized TC showed various cloud mi- tical models with the fifth-generation Pennsylvania State crophysics schemes. Jian et al. (2003) simulated precipi- University–National Center for Atmospheric Research tation associated with Typhoon Sinlaku over Taiwan (PSU–NCAR) Mesoscale Model (MM5; Grell et al. 1994). using MM5 initialized diabatically with the Local Anal- MM5 has been widely used by the mesoscale and mi- ysis and Prediction System (LAPS). Li et al. (2005) used croscale meteorology communities. It has also been em- a physically based distributed hydrological model to ployed to investigate rainfall characteristics and typhoon simulate typhoon floods over a mountainous water- tracks in Taiwan (Wu et al. 2002; Yang and Tung 2003; shed in Taiwan. The meteorological forcings include

Unauthenticated | Downloaded 09/28/21 07:32 PM UTC 440 WEATHER AND FORECASTING VOLUME 27 the observed gauge rainfall data and the predicted rain- where (a*i , ai) are coefﬁcients determined by training fall data from a mesoscale meteorological MM5 model. and K(xi, xj) is the support vector (SV) kernel. The SV kernel formed is a kernel of dot-product type in some feature space and it should satisfy Mercer’s condition. 3. Theory of SVM-based algorithms That is to say, the Mercer theorem gives the conditions The principles of the SVM and wavelet SVM ap- that a dot-product kernel must be satisﬁed by K(x, x9) 5 proaches are described below. K(hx x9i). More details can be found in Mercer (1909). Then, the decision function takes the form a. Support vector machines SVMs use a device called kernel mapping to map the l data in input space to a high-dimension feature space in f (x) 5 å (a*i 2 ai)K(x, x9) 1 b. (5) which the problem becomes linearly separable (Burges i51 1998). Assume that there is a training dataset f(x, y), x 2 The selection of an appropriate kernel function plays < n, y 2

n ! l xj 2 xj9 a* 2 a 5 a* C K(x, x9) 5 P c . (9) å ( i i) 0 i 2 [0, ], (4) j51 a i51

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It is a multipurpose reservoir for irrigation, hydroelectric energy generation, public water supply, ﬂood control, and tourism. The watershed covers an area of 763.4 km2. It is currently managed by the Water Resources Agency (WRA). This study included a total of 73 typhoon events affecting the study site over the past 18 yr (1990– 2007), as shown in Table 1. Complete data of hourly typhoon characteristics can be obtained from the Cen- tral Weather Bureau (CWB), while historical hydrological data from the reservoir watershed are available from the WRA. FIG. 1. Location of study site and an example of Typhoon Wipha (2007) approaching the study site. 5. Statistical models and applications

A translation invariant wavelet kernel, the so-called In this section, statistical models, namely SVM-based Morlet wavelet kernel, is (Szu et al. 1992; Yang and models and regression, are implemented for forecasting Wang 2008; Tolambiya and Kalra 2010) hourly precipitation estimates during typhoon attacks. x2 a. Data preprocessing and statistics c(x) 5 cos(1:75x) exp 2 . (10) 2 The collected data include climatologic characteristics of typhoons and the corresponding hourly precipitations Given the above mother wavelet, the corresponding in the reservoir watershed. First, the attributes of ty- wavelet kernel function is phoon climatologic characteristics, namely pressure at " ! !# 2 n 2 typhoon center at time t, denoted A1(t)(10 Pa); lati- xj 2 xj9 kxj 2 xj9k K(x, x9) 5 P cos 1:75 exp 2 . tude of typhoon center at time t, A2(t)(8N); longitude j51 a 2a2 of typhoon center at time t, A3(t)(8E); radius of ty- (11) phoon at time t, A4(t) (km); speed of typhoon at time 21 t, A5(t)(kmh ); and maximum wind speed of ty- 21 phoon center at time t, A6(t)(kmh ), are collected for analysis. 4. Study area and data The watershed area studied is relatively small in com- The location of Shihmen Reservoir watershed can be parison with the region affected by the typhoons; hence, seen in Fig. 1. The Shihmen Reservoir, one of the largest the average rainfalls at the watershed are assumed to water reservoirs in Taiwan, was completed in 1964 and is represent the precipitation characteristics. Take for ex- located along the upstream reaches of the Tahan River. ample Typhoon Wipha of 2007. As seen in Fig. 1, it was

TABLE 1. Typhoon events studied.

Year No. Typhoons Year No. Typhoons 1990 1 Yancy 1991 1 Ellie 1992 1 Polly 1993 0 — 1994 4 Doug, Fred, Gladys, Seth 1995 0 — 1996 1 Herb 1997 3 Winnie, Amber, Ivan 1998 5 Nichole, Otto, Yanni, Zeb, Babs 1999 3 Maggie, Sam, Dan 2000 7 Kai-Tak, Billis, Prapiroon, Bopha, Yagi, 2001 9 Cimaron, Chebi, Utor, Trami, Yutu, Toraji, Xangsane, Bebinca Nari, Lekima, Haiyan 2002 3 Rammasun, Nakri, Sinlaku 2003 9 Kujira, Nangka, Soudelor, Imbudo, Morakot, Vamco, Krovanh, Dujuan, Melor 2004 9 Conson, Mindulle, Kompasu, Rananim, Aere, 2005 7 Haitang, Matsa, Sanvu, Talim, Khanun, Haima, Meari, Nock-Ten, Nanmadol Damrey, Longwang 2006 6 Chanchu, Ewiniar, Billis, Kaemi, Saomai, Shanshan 2007 4 Pabuk, Sepat, Wipha, Krosa

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TABLE 2. List of attributes and their statistics.

Pressure at Lat (8N) of Lon (8E) of Radius of Speed of Max wind speed Hourly Statistics typhoon center typhoon center typhoon center typhoon typhoon at typhoon center rainfall Unit 102 Pa 88km km h21 km h21 mm h21

Denotation A1 A2 A3 A4 A5 A6 A7 Mean 966.3 22.7 122.4 214.3 16.9 126.1 2.6 Std dev 19.7 2.6 2.9 73.0 6.9 41.1 5.5 Max 1000.0 29.5 132.5 400.0 65.0 270.0 55.0 Min 912.0 15.0 114.2 0 0 0 0

approaching the study site on 18 September and its ra- say 5%. With a 5% significance level, the critical value for dius was 200 km. The percentage of the watershed area R is close to 0.038. Hence, choosing ‘‘jRj . 0.1’’ as a to the area invaded by Typhoon Wipha was 0.61%. In threshold is reasonable. That is, if jRj . 0.1, the cor- 21 view of this, for simplicity, A7(t) (mm h ) denotes the responding attribute is selected; otherwise, the attribute average precipitation of all rain gauges in the watershed will be omitted. According to this criterion, attributes A1, at each time period. A2, A4, A6,andA7 are selected for model analysis. A total of 2780 hourly records was available. Table 2 To identify the suitable model inputs to T, six attribute shows the seven attributes selected and their statistics combinations are designed. They are as follows. with the mean, standard deviation, and maximum and Case 1 has attributes including typhoon climatologic minimum values. All variables in the database were characteristics and precipitation. The forecast func- measured on an hourly basis. tion can be expressed as b. Feature selection and cases designed T(t) 5 fCase1[A1(t), A2(t), A4(t), A6(t), A7(t)]. Table 3 lists the correlation coefficient (R) between (13) the specific attribute A(t)(i.e.,A1–A7) and the target T(t) (i.e., the 1-h ahead precipitation). Here, R is defined as Cases 2–6 show that, owing to the strong correlation

between A7(t 2Dt)Dt50,5 and the target, the lag time N of attribute A7 should be considered. Assume that å [A(t) 2 A][T(t) 2 T ] Dt of cases 2–6 vary from 1 to 5 h (i.e., case 1 at Dt 5 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit51 R 5 u , (12) 0), then case 3, for example, can be expressed as uN N t 2 2 å [A(t) 2 A] å [T(t) 2 T ] T(t) 5 f fA (t), A (t), A (t), A (t), t51 t51 Case3 1 2 4 6

[A7(t 2Dt)]Dt50,1,2g. (14) where A is the average of the specific attribute, T is the average of the target, and N is the number of hourly records. We refer to T(t)asA7(t 1 1). As can be seen in c. Construction of models Table 3, A –A show weak correlations, while A shows 1 6 7 In the following analysis, the statistical models were a strong correlation with T. Because of the high corre- cross validated. The entire dataset excluding Typhoons lation (autocorrelation) in A , the time delay of rainfalls 7 Zeb (1998) and Nari (2001) (these two typhoons will be was examined. The temporal lag of A is denoted as 7 discussed in section 6) was partitioned into five equal- A (t 2Dt), where Dt refers to the lag time. The R values 7 sized subsets (i.e., fivefold cross validation). During each between the target T(t) and attribute A (t 2Dt)atDt 7 run, one of the subsets was chosen for validation, while from 0 to 5 h are 0.803, 0.663, 0.609, 0.569, 0.500, and the other subsets were used for training. 0.441, respectively, indicating a decrease in R value with an increase in lag time. With reference to the above analysis of the correlations, this study set the threshold of R value for selecting TABLE 3. Correlation coefficients (R) of attribute vs target. the attributes. For simplicity, the dependence among the Attribute attributes is not considered when determining the critical Target A1 A2 A3 A4 A5 A6 A7 value for R. There is a well-established test procedure for examining the correlation at certain significance levels, T 20.170 0.193 20.020 0.121 20.064 0.140 0.803

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1) SVM-BASED MODEL the output field according to the input fields. The regression equation represents a straight line or plane For constructing the SVM-based models, the penalty that minimizes the squared differences between pre- value C is set to be 1000 and the small positive number « dicted and actual output values (SPSS Inc. 2006). The is 0.05 for both GSVM and WSVM. The dilation fac- general form of linear multiple regression in matrix tor a in WSVM is 1. The values of the model weighting notation is (Blaker 2000) (w, b) are trained. During the training, the weights grad- ually converge to values in which input vectors produce output values as close as possible to the target output Y 5 Xb 1 E, (17) desired. Here, for example, GSVM and WSVM in case 1 can where Y is the (p 3 1) vector of observations, X is the be expressed, respectively, as (p 3 q) matrix containing q input factors, b is the (q 3 1) vector containing the regression coefficients (model T(t) 5 GSVM [A (t), A (t), A (t), A (t), A (t)] Case1 1 2 4 6 7 parameters), and E is the (p 3 1) vector containing the and (15) noise terms. Here, for comparison purposes, the selected attribute T(t) 5 WSVMCase1[A1(t), A2(t), A4(t), A6(t), A7(t)]. combinations were the same as the six cases in the SVM- (16) based model. For example, the general form of regression in case 1, REGCase1(), can be expressed as 2) LINEAR REGRESSION MODEL

Regression is one of the most important statistical T(t) 5 REGCase1[A1(t), A2(t), A4(t), A6(t), A7(t)], methods applied to science, engineering, economics, and (18) management (Wang 1999). The linear regression model estimates the best-ﬁtting linear equation for predicting or in matrix form as

2 3 2 3 2 3 2 3 T(1) A (1) A (1) A (1) A (1) A (1) b1 e 1 2 4 6 7 6 7 1 6 7 6 A (2) A (2) A (2) A (2) A (2) 7 6 b 7 6 7 6 T(2) 7 6 1 2 4 6 7 7 6 2 7 6 e2 7 6 7 5 6 7 6 b 7 1 6 7 , (19) 6 . 7 6 . . . . . 7 6 3 7 6 . 7 4 . 5 4 . . . . . 5 4 5 4 . 5 b4 T(N) N31 A (N) A (N) A (N) A (N) A (N) e 1 2 4 6 7 N35 b5 531 N N31

where b1–b5 are the regression coefficients and e1–eN are T(t) 5 29:496A1(t) 1 1:898A2(t) 1 3:365A4(t) the noise terms. 1 2:933A (t) 1 1:825A (t) 2 33:466. (20) This study adopted a stepwise regression method and 6 7 specified selection criteria according to the statistical probability (the p value) associated with each field. d. Performance definitions Thesecriteriawereemployedforaddingandremoving For comparison purposes, three measures of rainfall fields with the stepwise estimation methods (SPSS Inc. are employed: the root-mean-square error (RMSE), bias 2006). The classical p value of 0.05 is historically ac- score (BIAS), and equitable threat score (ETS). The cepted and is a convention in the scientific community RMSE is often employed to verify the amount of error (Genell et al. 2010). Therefore, as a first step, the best in the rain forecasted. RMSE is commonly defined as remaining variable is added, provided that it passes the vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi significant at the 5% criterion. Then, all variables cur- u u N rently in the regression are checked to see if any one of t 1 pre obs 2 them can be removed, using the criterion of signifi- RMSE 5 å [P (t) 2 P (t)] , (21) N t51 cance . 10% (or 5%, according to model run; both can obtain the same regression equations). The process con- where Ppre(t) is the predicted precipitation at record t tinues until no more variables are added or removed. and Pobs(t) is the observed precipitation at record t. After analysis, the linear regression equation for case 1 Generally, the smaller the RMSE criteria, the better is was the performance of the predicted outcomes. RMSE

Unauthenticated | Downloaded 09/28/21 07:32 PM UTC 444 WEATHER AND FORECASTING VOLUME 27 measures assume implicitly that the precipitation is nor- mally distributed; hence, this study modified the RMSE for handling zero precipitations: Modified RMSE vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u N t1 5 å flog[1 1 Ppre(t)] 2 log[1 1 Pobs(t)]g2 N t51 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u " # u N 2 t1 1 1 Ppre(t) 5 å log obs . (22) N t51 1 1 P (t) FIG. 2. Modified RMSE measures of precipitation predictions by GSVM, WSVM, and regression. BIAS measures the ratio of the predicted rain frequency to the observed frequency, regardless of forecast accuracy (McBride and Ebert 2000; Ebert et al. 2003). indicating that WSVM is superior to GSVM and re- That is, BIAS can be employed to assess the tendency gression in precipitation prediction. Moreover, of the six of the model to under- or overpredict rain occurrence. cases, the lowest average RMSE value of 0.280 appears BIAS is defined as in case 1, while the highest value of 0.330 is found in case 6, revealing that case 1 simulations give better per- H 1 F BIAS 5 , (23) formance than those of cases 2–6. H 1 M Further, the scatterplots of the observed versus the where H is the frequency of correct predictions of rain predicted precipitations are shown in Fig. 3 with out- occurrence, F is the frequency of incorrect predictions comes derived from case 1. In addition, the linear re- 2 of rain occurrence, and M is the frequency of rain oc- gression equation and squared correlation (R )aremade currences that are not predicted. Here, the frequency in Figs. 3a–c. As can be seen, all three models have of correct forecasts of no rain can be denoted by Z. slopes of less than 1.0 in the order of (SlopRegression 5 Therefore, the total number of forecasts (or observa- 0.5803) , (SlopGSVM 5 0.6997) , (SlopWSVM 5 0.7009), tions) is (H 1 M 1 F 1 Z). If BIAS is equal to 1.0, then indicating that WSVM gives the best estimation clos- 2 the predicted rainfall frequency is the same as that ob- est to the observed results. Furthermore, the R value served (Ebert et al. 2003). of WSVM is also better than that of GSVM and re- ETS has been used for several decades to measure the gression. correspondence between the forecasted and observed To evaluate the model capability in estimating light, rain occurrences (Schaefer 1990; Ebert et al. 2003). ETS moderate, and heavy rains during the typhoon period, is defined as both BIAS and ETS scores were computed to give a correct estimate of the observed rainfall over a certain

H 2 Hrandom threshold. As seen in Figs. 4 and 5, the thresholds range ETS 5 . (24) 2 from 0 to 40 mm h 1. As mentioned, if BIAS . 1.0, the H 1 F 1 M 2 Hrandom model overpredicts rain occurrence; otherwise, the model The ‘‘equitable’’ measure accounts for the random underpredicts rain occurrence. As seen in Fig. 4, the chance that both forecasted and observed rain occur- straight line BIAS 5 1 divided the figure into two re- 21 rences meet the criteria Hrandom 5 (H 1 M)(H 1 F)/ gions. For thresholds , 2mmh (i.e., the BIAS . 1 (H 1 M 1 F 1 Z). ETS ranges from 21/3 to 1, with a region), regression has the highest bias among these value of 1 indicating perfect correspondence between models, while both GSVM and WSVM have BIAS values predicted and observed rain occurrences. ETS is now close to 1.0. Moreover, for thresholds . 5mmh21,all used for rainfall verification at most operational centers threemodelshaveBIASvalues, 1.0. ETS measures (Ebert et al. 2003). the number of forecast fields that match the observed threshold. Figure 5 illustrates the ETS as a measure of e. Model evaluation and discussion the relative performance for different thresholds of rain- In this section, to identify the suitable model and case, fall. As can be seen, the ETS scores of GSVM, WSVM, the modified RMSE was employed. Figure 2 shows the and regression decrease from 1.0 to 0.20, 0.26, and 0.19 results of cases 1–6 obtained using the GSVM, WSVM, with increasing thresholds of rainfall until 19, 22, and and regression approaches. As can be seen, WSVM is 18 mm h21, respectively; then, they rise to 35 mm h21 the most precise of these three models for each case, and, finally, fall again to 40 mm h21.

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FIG. 4. BIAS scores at various thresholds for SVM-based and regression models.

The increase in BIAS and ETS values observed in Figs. 4 and 5, respectively, at thresholds . 22 mm h21 for both GSVM and WSVM can be attributed to the following reason. First, this study lists the frequencies of fH, M, F, Zg, which appear in the BIAS and ETS for- mulas [i.e., Eqs. (23) and (24), respectively]. As can be seen in Table 4, at thresholds of 25, 30, and 35 mm h21 for GSVM, these frequencies are the same (i.e., 10 times) for H, (24, 14, 4) for M, and all zero values for F.There- fore, when calculating the BIAS measures at thresholds of 25, 30, and 35 mm h21, the numerator of (H 1 F)in Eq. (23) will be ﬁxed at 10, while the denominators of (H 1 M) will become 34, 24, and 14, respectively. There- fore, BIAS will increase at thresholds . 22 mm h21. The same reason applies to the WSVM case. Similarly, the reason why ETS measures increases at thresholds . 22 mm h21 canalsobeexplained. According to Figs. 4 and 5, the BIAS and ETS scores of GSVM and WSVM are better than those of regression at thresholds . 35 mm h21, implying good prediction capacity of GSVM and WSVM for the heavy rain situations. Moreover, to evaluate the average performance

FIG. 3. Scatterplots showing the observed vs the predicted rainfall FIG. 5. ETS scores at various thresholds for SVM-based and for GSVM, WSVM, and regression. regression models.

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TABLE 4. Rain contingency table for the specific threshold and TABLE 5. BIAS and ETS scores for three models. measures of BIAS and ETS. Performance score GSVM WSVM Regression Frequency Measure Threshold BIAS Avg 0.883 0.892 0.884 21 Model (mm h ) HMF Z BIAS ETS Min, max 0.250, 1.047 0.250, 1.030 0, 1.224 GSVM 25 10 24 0 2626 0.294 0.291 ETS Avg 0.705 0.715 0.615 30 10 14 0 2636 0.417 0.414 Min, max 0.211, 1.000 0.188, 1.000 0.000, 1.000 35 10 4 0 2646 0.714 0.713 40 4 4 4 2648 1.000 0.332 WSVM 25 10 24 2 2624 0.353 0.275 30 10 14 2 2634 0.500 0.382 s (nondimensional pressure) vertical coordinate. The 35 10 5 0 2645 0.667 0.665 physical parameterizations include the Grell (1993) 40 6 4 3 2648 0.900 0.460 subgrid-scale cumulus parameterization scheme, the Regression 25 12 22 2 2624 0.412 0.330 Blackadar (1979) planetary boundary layer scheme, a 30 10 14 0 2636 0.417 0.414 35 8 6 0 2646 0.571 0.570 radiation scheme with interaction between clear sky 40 0 8 0 2652 0.000 0.000 and clouds (Dudhia 1989), and a simple ice microphysics scheme (Dudhia 1989). Details of the MM5 model can be found in Grell (1993) and Grell et al. (1994). of all models, both BIAS and ETS scores are estimated b. Experimental design and simulations of for all thresholds as follows: two typhoons Typhoons Zeb (1998) and Nari (2001) possessed unique Average BIAS 5 å BIASk Ok å Ok and meteorological features and both caused severe flood- k k ing. Table 6 lists the characteristics of the two typhoons, (25) and their historical tracks are shown in Figs. 6 and 7 respectively. Average ETS 5 å ETSk Ok å Ok , (26) The initial and boundary conditions of MM5 are taken k k from the European Centre for Medium-Range Weather where k is the threshold index (from 0 to 40 mm h21); Forecasts (ECMWF) analyses with 1.258 latitude 3 1.258 longitude horizontal resolution. The MM5 horizontal BIASk and ETSk are the scores of BIAS and ETS at the configuration includes three nested grids (with grid sizes assigned threshold k, respectively; and Ok is the number of observations at assigned threshold k. Table 5 lists the of 60, 20, and 6.67 km) for Typhoon Zeb (1998) and four average BIAS and ETS scores of these models. As can nested grids (with grid sizes of 60, 20, 6.67, and 2.22 km) be seen, WSVM having average BIAS and ETS scores for Typhoon Nari (2001). There are 31 s levels in the close to 1.0 gives better a estimation than do GSVM and vertical for all MM5 runs of both typhoons. Sea surface regression, meaning that WSVM achieves better pre- temperature is kept constant during the period of inte- diction performance. gration. Further details of the model analysis procedure can be found in Li et al. (2005). Both typhoons have two protruding peak episodes (see 6. Numerical model and comparisons Fig. 8). The lifespans of the typhoons, basin-averaged In this section, the MM5 results from Li et al. (2005) rainfall hyetographs, and MM5 simulation results are were compared with those obtained using the above described below. statistical models. Typhoons Zeb (1998) and Nari (2001) were selected for comparison. First, a brief description of the MM5 model is given in the following subsection. TABLE 6. Statistics of two typhoon characteristics. a. Description of MM5 Zeb Nari Characteristics (1998) (2001) The MM5 is used as a common model framework for investigating the sensitivity of simulated track, central Duration of impact periods (h) 50 72 Avg of pressure at typhoon center (Pa) 943.9 982.3 pressure, maximum wind, and accumulated rainfall of Maximal radius during typhoon periods (km) 350 150 typhoon to physical parameterizations, as well as the Avg moving speed of typhoon (km h21) 19.3 5.6 topographic effects (Yang and Ching 2005). The MM5 Maximal rainfall rate (mm h21)4238 model is a three-dimensional, limited-area, primitive Total precipitation during typhoon 503.5 703.5 equation, nested-grid model with a terrain-following periods (mm)

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FIG. 6. Historical track of Typhoon Zeb in 1998 (the map is taken FIG. 7. Historical track of Typhoon Nari in 2001 (the map is taken from the CWB web site: http://www.cwb.gov.tw/eng/index.htm). from the CWB web site: http://www.cwb.gov.tw/eng/index.htm).

d Typhoon Zeb (1998) moved along the east coast of 25.1 mm h21 (at 35 h), with relative peak errors of 26.4% Taiwan without making landfall (see Fig. 6). The and 40.4%, respectively. interaction between typhoon circulation and moun- As shown in Fig. 8b, the observed rainfall peaks tainous terrain brought heavy rainfall over eastern and of Typhoon Nari (2001) are 37.6 mm h21 (at 28 h) and northern Taiwan. The rain periods in the watershed 35.4 mm h21 (at 46 h), while the MM5-predicted rainfall were from 0100 LT 15 October to 0200 LT 17 Octo- peaks are 58.8 mm h21 (at 28 h, the same as the observed ber 1998. The basin-averaged rainfall hyetographs peak time) and 34.0 mm h21 (at 48 h), with relative peak estimated from the rain gauge data and the MM5- errors of 56.4% and 0.4%, respectively. Moreover, the predicted rainfall are given in Fig. 8a. Note that the WSVM-predicted rainfall peaks are 31.5 mm h21 (at basin-averaged rainfall amounts will be regarded as 27 h) and 32.5 mm h21 (at 45 h), with relative peak errors the observed rainfall totals for the watershed in the of 16.1% and 8.1%, respectively. following analysis. d Typhoon Nari (2001) broke several hydrometeoro- logical records and produced the most severe damage over Taiwan. The rain periods in the watershed were from 2000 LT 15 September to 1900 LT 18 September 2001 (see Fig. 7). The basin-averaged rainfall hyetographs and MM5-predicted rainfall are given in Fig. 8b. c. Comparisons between numerical and statistical models

Here, the results obtained by statistical models in case 1 were chosen to compare with those of MM5.

1) RAINFALL HYETOGRAPHS As shown in Fig. 8a, the observed rainfall peaks of Typhoon Zeb (1998) are 27.5 mm h21 (at the time period of 21 h) and 42.1 mm h21 (at 37 h), while the MM5-predicted rainfall peaks are 33.7 mm h21 (at 17 h) and 49.2 mm h21 (at 38 h), with relative peak errors of

22.5% and 16.8%, respectively. Furthermore, the WSVM- FIG. 8. Basin-averaged rainfall hyetographs and rainfalls predicted 2 predicted rainfall peaks are 20.2 mm h 1 (at 20 h) and by MM5 and WSVM.

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compared with the observations, while statistical models, such as WSVM, give underestimations. The following reasons are suggested to account for the above phe- nomena.

d For statistical models, as indicated in Tuleya et al. (2007), the underestimation of large rain frequency is not surprising because the climatological rain rates represent averages over rainfall totals of many typhoons that average out to a relatively low value. Therefore, it might cause the predicted values to be relatively underestimated for a heavy typhoon (Hsu and Wei 2007). d For numerical models, as mentioned in Jian et al. (2003), overestimation might be attributed to the complicated and ill-understood precipitation physics. Another important reason might be that most model initialization routines provide adiabatic initial conditions, leading to the infamous spinup problem (Heckley 1985; Donner 1988). The associated lack of condensa- tion and latent heat release during the early part of model integration restricts the short-range (0–12 h) forecasting accuracy of the mesoscale models.

FIG. 9. The observed and model-estimated hourly accumulated Although it might be more desirable to forecast peak hyetographs for Typhoons Zeb (1998) and Nari (2001). typhoon rainfalls with smaller relative peak errors according to the numerical model outputs (such as MM5), the statistical models (such as WSVM) could provide 2) ACCUMULATED PRECIPITATIONS reasonable cumulative rainfall estimates for river basins. Figure 9 shows the observed and the model-estimated Emergency management personnel can use this infor- cumulative amounts of rainfall for the two typhoons. mation in their hazard mitigation decision making be- In the case of Typhoon Zeb (Fig. 9a), the results show fore a typhoon hits (Fread et al. 1995; Lee et al. 2006). that MM5 (698.9 mm) yields an overestimation com- On the other hand, MM5 might not be more accurate in pared with the observed rainfall (504.1 mm), while predicting accumulated rainfall than statistical models. GSVM (352.2 mm), WSVM (377.8 mm), and regression However, the insightful numerical MM5 model has dem- (375.9 mm) give underestimated predictions. In the case onstrated that the high-resolution nonhydrostatic model is of Typhoon Nari (Fig. 9b), the model-estimated rainfall capable of simulating detailed local circulation variations amounts for MM5 (804.8 mm), WSVM (759.8 mm), and (Jian et al. 2006). That is to say, MM5 could be employed GSVM (746.6 mm) are close to the observed rainfall to understand the complicated physical mechanisms oc- (763.5 mm). However, it should be noted that although curring during typhoons. MM5 gives a reasonable estimate for the total cumulative rainfall, the difference between the observed and model- 7. Conclusions estimated rainfall is still large. As see in Fig. 9, the maximal difference in accumulated precipitation at 27 h is In meteorology and engineering ﬁelds, the prediction an underestimation of 147 mm while that at 58 h is an of quantitative precipitation during tropical cyclone overestimation of 102.3 mm. Based on Fig. 9, WSVM (typhoon) events is an important research topic that has gives the best estimation among the four models in the attracted much interest. This study developed the SVM- case of Typhoon Nari. based models for forecasting the hourly precipitation amounts during typhoons. SVMs include the traditional d. Discussion Gaussian radial basis function kernel SVMs and the As seen in Fig. 8, the MM5 model tends to over- advanced wavelet kernel SVMs. Furthermore, the nu- estimate the peak rainfall amounts for these typhoons, merical model (MM5) was compared with the statistical while WSVM tends to underestimate the larger rainfall models, including GSVM, WSVM, and regression. frequency. In addition, resultsshowninFig.9reveal The developed models were applied to rainfall pre- that MM5 overestimates the cumulative rainfall totals dictions at the Shihmen Reservoir watershed in Taiwan.

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This study analyzed data related to climatologic charac- Chen, S.-H., R. C. Guido, T.-K. Truong, and Y. Chang, 2010: teristics, typhoon tracks, and hourly precipitation amounts Improved voice activity detection algorithm using wavelet at the study site. A total of 73 typhoon events affecting and support vector machine. Comput. Speech Lang., 24, 531– 543. the Shihmen Reservoir watershed from 1990 to 2007 Chen, X., Y. Li, R. Harrison, and Y.-Q. Zhang, 2008: Type-2 fuzzy were selected. The attributes of typhoon climatologic logic-based classifier fusion for support vector machines. Appl. characteristics include the pressure at the typhoon Soft Comput., 8, 1222–1231. center, latitude of the typhoon center, longitude of the Cortes, C., and V. Vapnik, 1995: Support-vector networks. Mach. typhoon center, radius of the typhoon, speed of the ty- Learn., 20, 273–297. Donner, L. J., 1988: An initialization for cumulus convection in phoon, maximum wind speed at the typhoon center, and numerical weather prediction models. Mon. Wea. Rev., 116, average precipitation at the reservoir watershed. 377–385. This study designed six attribute combinations with Dudhia, J., 1989: Numerical study of convection observed dur- different lag times. The predictions obtained by the ing the Winter Monsoon Experiment using a mesoscale two- GSVM, WSVM, and regression approaches for the de- dimensional model. J. Atmos. Sci., 46, 3077–3107. Ebert, E. E., U. Damrath, W. Wergen, and M. E. Baldwin, 2003: signed cases were compared. Meanwhile, the modified The WGNE assessment of short-term quantitative precipita- RMSE, BIAS, and ETS were employed to assess the tion forecasts (QPFs) from operational numerical weather forecasted results. Results show that the optimal at- prediction models. Bull. Amer. Meteor. 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