A Bayesian Regression Approach to Seasonal Prediction of Tropical Cyclones Affecting the Fiji Region

Savin S. Chand School of Earth Sciences, University of Melbourne, Melbourne, Victoria, Australia

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Kevin J. E. Walsh School of Earth Sciences, University of Melbourne, Melbourne, Victoria, Australia

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Johnny C. L. Chan Guy Carpenter Asia-Pacific Climate Impact Centre, City University of Hong Kong, Hong Kong, China

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Abstract

This study presents seasonal prediction schemes for tropical cyclones (TCs) affecting the Fiji, Samoa, and Tonga (FST) region. Two separate Bayesian regression models are developed: (i) for cyclones forming within the FST region (FORM) and (ii) for cyclones entering the FST region (ENT). Predictors examined include various El Niño–Southern Oscillation (ENSO) indices and large-scale environmental parameters. Only those predictors that showed significant correlations with FORM and ENT are retained. Significant preseason correlations are found as early as May–July (approximately three months in advance). Therefore, May–July predictors are used to make initial predictions, and updated predictions are issued later using October–December early-cyclone-season predictors. A number of predictor combinations are evaluated through a cross-validation technique. Results suggest that a model based on relative vorticity and the Niño-4 index is optimal to predict the annual number of TCs associated with FORM, as it has the smallest RMSE associated with its hindcasts (RMSE = 1.63). Similarly, the all-parameter-combined model, which includes the Niño-4 index and some large-scale environmental fields over the East China Sea, appears appropriate to predict the annual number of TCs associated with ENT (RMSE = 0.98). While the all-parameter-combined ENT model appears to have good skill over all years, the May–July prediction of the annual number of TCs associated with FORM has two limitations. First, it underestimates (overestimates) the formation for years where the onset of El Niño (La Niña) events is after the May–July preseason or where a previous La Niña (El Niño) event continued through May–July during its decay phase. Second, its performance in neutral conditions is quite variable. Overall, no significant skill can be achieved for neutral conditions even after an October–December update. This is contrary to the performance during El Niño or La Niña events, where model performance is improved substantially after an October–December early-cyclone-season update.

Corresponding author address: Savin S. Chand, School of Earth Sciences, University of Melbourne, Melbourne, VIC 3010, Australia. Email: schand@pgrad.unimelb.edu.au

Abstract

This study presents seasonal prediction schemes for tropical cyclones (TCs) affecting the Fiji, Samoa, and Tonga (FST) region. Two separate Bayesian regression models are developed: (i) for cyclones forming within the FST region (FORM) and (ii) for cyclones entering the FST region (ENT). Predictors examined include various El Niño–Southern Oscillation (ENSO) indices and large-scale environmental parameters. Only those predictors that showed significant correlations with FORM and ENT are retained. Significant preseason correlations are found as early as May–July (approximately three months in advance). Therefore, May–July predictors are used to make initial predictions, and updated predictions are issued later using October–December early-cyclone-season predictors. A number of predictor combinations are evaluated through a cross-validation technique. Results suggest that a model based on relative vorticity and the Niño-4 index is optimal to predict the annual number of TCs associated with FORM, as it has the smallest RMSE associated with its hindcasts (RMSE = 1.63). Similarly, the all-parameter-combined model, which includes the Niño-4 index and some large-scale environmental fields over the East China Sea, appears appropriate to predict the annual number of TCs associated with ENT (RMSE = 0.98). While the all-parameter-combined ENT model appears to have good skill over all years, the May–July prediction of the annual number of TCs associated with FORM has two limitations. First, it underestimates (overestimates) the formation for years where the onset of El Niño (La Niña) events is after the May–July preseason or where a previous La Niña (El Niño) event continued through May–July during its decay phase. Second, its performance in neutral conditions is quite variable. Overall, no significant skill can be achieved for neutral conditions even after an October–December update. This is contrary to the performance during El Niño or La Niña events, where model performance is improved substantially after an October–December early-cyclone-season update.

Corresponding author address: Savin S. Chand, School of Earth Sciences, University of Melbourne, Melbourne, VIC 3010, Australia. Email: schand@pgrad.unimelb.edu.au

1. Introduction

Tropical cyclones (TCs) are some of the most destructive naturally occurring phenomena; strong winds coupled with heavy rainfall often have devastating consequences for life and property. Advance warning of a TC strike can therefore have great socioeconomic benefits in terms of more effective and coordinated preparation for cyclone events. A number of seasonal forecast schemes have been developed and implemented over the past three decades, using both dynamical and statistical modeling techniques to predict TC activity in several cyclone basins and subbasins (Camargo et al. 2007). In the southwest tropical Pacific region, small islands such as Fiji, Samoa, and Tonga [the “FST region” (Fig. 1), defined as the area between 5° and 25°S and 170°E and 170°W] are very vulnerable to catastrophic effects of TC activity (Mimura et al. 2007). However, no seasonal prediction scheme yet exists for TCs over the FST region. Therefore, the present study seeks to develop an appropriate statistical model to make predictive inferences for TCs affecting the FST region.

A recent study by Chand and Walsh (2009) showed that TC genesis positions and tracks in the FST region exhibit marked interannual variability due to the El Niño–Southern Oscillation (ENSO) phenomenon. During the El Niño phase, for example, TC genesis is enhanced east of the date line, extending from north of Fiji to over Samoa. TCs formed during El Niño years take three characteristic paths depending on their mean genesis locations. In the La Niña phase, fewer TCs are observed compared to the El Niño phase and genesis is more common in the west of the region to about 170°E. The TCs formed during La Niña years are often steered over the Fiji islands and Tonga from the Coral Sea region with relatively little or no threat to Samoa. Such variations seen in the ENSO–TC relationship over the FST region are linked to concomitant changes in the large-scale environmental conditions such as low-level relative vorticity (VORT), upper-level divergence (DIV), and environmental vertical wind shear (see Chand and Walsh 2009 for details).

Considerable effort has also been directed toward understanding the ENSO–TC relationship in other ocean basins, for example, in the Coral Sea and broader southwest Pacific region (e.g., Revell and Goulter 1986a,b; Hastings 1990; Basher and Zheng 1995), in the Australian region (e.g., Nicholls 1979, 1984, 1985, 1992; Solow and Nicholls 1990; Evans and Allan 1992; McDonnell and Holbrook 2004; Ramsay et al. 2008), and in the western North Pacific basin (e.g., Chan 1985, 2000, 2007; Chan et al. 1998, 2001; Chia and Ropelewski 2002; Camargo and Sobel 2005). These investigations not only established a strong ENSO–TC relationship but also confirmed that this relationship is identifiable prior to a cyclone season through the use of appropriate ENSO indices, thus making it possible to develop suitable statistical forecast schemes to predict TC activity for a particular year.

Pioneering work on the development of seasonal cyclone forecasting schemes was made by Nicholls (1979) for the Australian region and Gray (1984) for the North Atlantic basin using regression-based linear statistical models. Subsequent studies led to the development of prediction schemes for different cyclone basins with an improved methodological framework for statistical cyclone modeling. Chan et al. (1998, 2001) and Liu and Chan (2003), for example, used the projection pursuit regression technique of Friedman and Stuetzle (1981) to develop seasonal forecasting schemes for the western North Pacific and the South China Sea. This method projects high-dimensional data into a low-dimensional subspace with an ability to retain dominant linear and nonlinear features of the original high-dimensional data. Elsner and Schmertmann (1993) considered a different approach to predict seasonal numbers of intense Atlantic hurricanes. They showed that a nonlinear Poisson model is superior to a linear statistical model in terms of improvements in the hindcast skill. From a classical (or frequentist) perspective, parameter values for Poisson regression models are assumed fixed and therefore estimated by a maximum likelihood procedure (e.g., Solow and Nicholls 1990; Elsner and Schmertmann 1993; McDonnell and Holbrook 2004). Recent work—for example, Elsner and Jagger (2004, 2006), Chu and Zhao (2007), and Flay and Nott (2007)—used a Bayesian approach to the Poisson regression model. Here, the parameter values are assumed to have a distribution and therefore inference is made by computing the posterior probability density estimates of parameters conditioned on the observed data. This approach of estimating parameters has obvious advantage over a linear or a classical Poisson method in that the posterior distributions can be used to make probability statements regarding the different number of cyclones in a given year.

A review of literature reveals that the Australian–southwest Pacific region (105°–170°E) has garnered more attention than the FST region. The FST region spans the approximate location of the main center of action of ENSO (Trenberth and Shea 1987), thus making it possible for TCs to occur in both El Niño and La Niña conditions. This gives rise to a somewhat nonlinear pattern of the ENSO–TC relationship. As a result, earlier work by Basher and Zheng (1995) could only detect a very weak correlation between the FST region cyclone activity and the Southern Oscillation index (SOI), which is often used as one of the indicators of ENSO events. In this paper, we reexamine the contemporaneous relationship between various ENSO indicators and TC activity over the FST region with the inclusion of some large-scale environmental parameters that are well documented in the literature for their role in TC formation (e.g., Gray 1968, 1975, 1979) and track variations (e.g., Chan and Gray 1982; Gray 1994). This forms the first objective of our present work. The other objective is to develop appropriate statistical (Poisson regression) models using the Bayesian approach to forecast cyclones in the FST region. Because TCs formed in the Coral Sea region are more often steered into the FST region during La Niña years than during El Niño years, and in contrast more TCs are formed in the FST region during El Niño years than during La Niña years (Chand and Walsh 2009), this adds to a rather weaker statistical relationship between the total annual number of TCs affecting the FST region and ENSO indices. Accordingly, it is essential to develop separate Poisson models for TCs that form within the FST region and for those that enter the domain from the Coral Sea region to obtain more rigorous and stable statistical relations with appropriate ENSO predictors.

The Poisson regression model using the Bayesian approach is considered optimal for the development of the cyclone prediction scheme in the FST region primarily for two reasons. First, the observed cyclones in the FST region are a small number of counts from a relatively large sample size (i.e., only a total of 123 cyclones are observed from 39 yr of data). Such rare events are often modeled using the Poisson process, for example, in a study by McDonnell and Holbrook (2004) for the Australian–southwest Pacific region (6°–20°S, 105°–170°E). Second, the Bayesian approach (as opposed to the classical approach) permits availability of model parameters in terms of their posterior distribution. This readily facilitates predictive inferences on future cyclone occurrences within a probabilistic framework.

The paper is structured as follows. Section 2 describes various datasets used in this study, while section 3 outlines the procedure for predictor selection. Section 4 explains the formulation of predictor models using the Bayesian approach. Results are presented in section 5. Lastly, a discussion and summary are given in section 6.

2. Data

a. TC data and definitions

A TC in the FST region is defined as a nonfrontal, synoptic-scale disturbance in which 10-min sustained winds reach at least gale strength (a minimum of 17.5 m s−1). This definition is same as one proposed by Revell (1981) and adopted by, for example, Thompson et al. (1992) and Sinclair (2002) for their studies in the broader southwest tropical Pacific region. Consistent with Chand and Walsh (2009), we have included all TCs forming in the FST region, as well as those crossing this defined boundary during some part of their lifetime. TCs that are formed within the FST region (hereafter “FORM”) and those that are formed outside but eventually entered the FST region (hereafter “ENT”) are distinguished according to the following two criteria.

  1. Any cyclone that first attains a minimum of 17.5 m s−1 wind speed within the defined FST region or along its boundary is considered a FORM.

  2. Any cyclone that formed outside the defined region but registered its 6-hourly track within the defined region during some part of its lifetime is considered an ENT.

The TC data used here are archived by Joint Typhoon Warning Center (JTWC) at 6-h intervals (available online at http://www.usno.navy.mil/NOOC/nmfc-ph/RSS/jtwc/best_tracks/). Only TCs in the austral summer season (i.e., November–April) are included. Altogether, 123 TCs from the period beginning November 1970 and ending April 2009 are considered in the analysis. The period chosen for the present investigation is consistent with the era after which routine satellite observations became available. To ensure maximum accuracy, observations within the JTWC database were cross-referenced with information from the Fiji Meteorological Service (FMS). Note that the maximum sustained wind in the JTWC dataset is the 1-min-averaged wind speed, while a 10-min average is used in the FMS dataset. The strength of 10-min sustained wind is statistically 88% of the 1-min sustained wind (e.g., Courtney and Knaff 2009). Therefore, the JTWC data are scaled to 10-min averages before making a comparison with the FMS dataset.

b. Sea surface temperature data and ENSO indices

The monthly values of standardized anomalies of various ENSO indices [such as Darwin and Tahiti sea level pressures (SLPs), the SOI, and Niño-3.4 and Niño-4 region SST anomalies] used in this study were obtained online from the Climate Prediction Center’s Web site (http://www.cpc.ncep.noaa.gov/data/indices/) for the periods beginning January 1970 and ending July 2009. The SST data for the same period were obtained from the National Oceanic and Atmospheric Administration (NOAA) extended reconstructed SST (ERSST) version 2 dataset of Smith and Reynolds (2004). The ERSST data have a horizontal resolution of 2° × 2° available globally for individual months.

c. Atmospheric data

The atmospheric data required for computation of the large-scale environmental fields were extracted from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis products (Kalnay et al. 1996) for the period January 1970–July 2009. The data have a horizontal resolution of 2.5° × 2.5° available globally for individual months. Environmental fields calculated from the atmospheric data include (i) environmental vertical wind shear (EVWS), defined as the magnitude of the vector difference of winds between the 200- and 850-hPa pressure levels; (ii) relative vorticity at the 850-hPa level computed using a centered finite-differencing scheme; (iii) divergence at the 200-hPa level also computed using a centered finite-differencing scheme; (iv) midlevel relative humidity (Rhum; between 500- and 700-hPa levels), and 5) vertical gradient of equivalent potential temperature (ThetaE) calculated using temperature fields at 1000- and 500-hPa levels following the procedure described by Bolton (1980).

3. Potential predictors

The selection of appropriate predictors is one of the most important steps for the development of a statistical forecast model. The choice of predictors could vary from region to region depending on their physical or statistical link with cyclones. Often, two factors are considered important when choosing a potential predictor for the development of a cyclone prediction scheme: (i) the predictor should have a strong and stable physical link with TCs and (ii) it must be readily available prior to the onset of a cyclone season (at least by October in our case for November–April predictions to be useful).

A review of the literature on statistical cyclone prediction schemes for various ocean basins, particularly those in the adjacent Australian–southwest Pacific region (105°–170°E), provided insights into various potential predictors that can also be applied to the FST region. As in other cyclone basins, it is well established that the ENSO phenomenon also influences the annual number of TCs in the FST region (Chand and Walsh 2009). Therefore, indices that are used as proxies of ENSO can be considered potential predictors. Since environmental influences are also crucial for TC genesis and movement in the FST region (Chand and Walsh 2009), parameters associated with large-scale environmental conditions are also considered. In this section, statistical links of various potential predictors with the annual number of cyclones associated with FORM and ENT are determined. We begin by first exploring the periodicity of TCs associated with FORM and ENT using a wavelet analysis to acquire a better understanding of their temporal variability.

a. Climatology and periodicity of TCs associated with FORM and ENT

The climatological average number of cyclones that form within the FST region in a season is 2.54 compared to 0.62 that enter. Most cyclones that enter the FST region are formed in the Coral Sea region (Fig. 2a), where TC activity is relatively high in La Niña years as opposed to El Niño years (e.g., Revell and Goulter 1986a,b; Basher and Zheng 1995). The annual number of cyclones associated with FORM and ENT in the FST region exhibits a large variation (Fig. 2b). Here, we examine the extent to which this variation can be accounted for by the variation in ENSO events.

A wavelet analysis using the Morlet basis function is employed to determine the dominant modes of variability associated with FORM and ENT and how these modes vary in time. The procedure, detailed in Torrence and Compo (1998), is implemented using a MATLAB toolbox (available online at http://atoc.colorado.edu/research/wavelets/software.html). The Morlet wavelet is a complex function, as opposed to the real-valued Mexican hat wavelet (Addison 2002), and is therefore more appropriate to capture various oscillatory behaviors.

Distinct features are evident in the power spectrum plots of the annual number of cyclones associated with FORM (Fig. 3a) and ENT (Fig. 3b). Shaded contours enclose variances that are statistically significant at 90% and 95% significance levels—the null hypothesis being that the power spectrum peaks are not significantly different from the red-noise background spectrum determined using the one-lag autocorrelation technique (Torrence and Compo 1998). Clearly, the annual number of TCs in the FST region associated with FORM exhibits a 4- to 8-yr periodicity, which lies well within the 2- to 8-yr ENSO band (e.g., Trenberth 1976; Torrence and Webster 1998). The associated 2- to 8-yr scale-averaged time series shows that a significantly high variance of the power spectrum associated with FORM occurred from around 1980 to 2000. This implies the presence of several El Niño (warm) and La Niña (cold) events of large amplitude during 1980–2000. Elsewhere, the edge effect becomes dominant (as indicated by the cone of influence) and therefore variance becomes significantly weaker due to zero padding (Torrence and Compo 1998). Because our time series of FORM and ENT have finite lengths, errors will occur at the beginning and end of the wavelet power spectrum. Padding the end of time series with sufficient zeroes is necessary here to bring the total length up to the next higher power of 2.

The annual number of TCs in the FST region associated with ENT also exhibits a periodicity in the 2- to 8-yr ENSO band (Fig. 3b). However, the variance within this band is much weaker compared to that with FORM. A period of more than 16 yr, which may be related to the Pacific decadal oscillation (PDO; e.g., Mantua and Hare 2002), is also noted. However, this periodicity lies outside the cone of influence and because only 39 yr of data are considered here, this is insufficient to resolve conclusively the effect of the PDO.

b. Predictors associated with FORM

1) ENSO indices

The potential ENSO indices as candidate predictors of the annual number of TCs in the FST region associated with FORM are identified using the following procedure. A correlation analysis is performed for the 1970–2009 period between the annual number of TCs forming in the FST region for the November–April season and 3-monthly running mean values of various ENSO indices starting January–March preceding the TC season and ending October–December (OND) following the TC season. The statistical significance of correlation is determined using the Pearson correlation technique (e.g., Chu and Zhao 2007). The critical value of the Pearson correlation coefficient r for a sample size of 39, using two-tailed test at the 99% significance level, is 0.38 (Sheskin 2007). Thus, any predictor with a correlation of magnitude below 0.38 is not considered.

It is found that 3-monthly running mean values of the Niño-3.4 index and the Niño-4 index (Niño-4) show substantially strong positive correlation with the annual number of cyclones associated with FORM at various time leads (Fig. 4a). The maximum preseason correlation can be identified during May–July (MJJ) periods, with the correlation coefficient associated with Niño-4 index (r = 0.58) having a slight edge over the Niño-3.4 index (r = 0.56). The OND early-cyclone-season correlation coefficients of these two indices are also large, with Niño-3.4 index (r = 0.58) now having a slight edge over the Niño-4 index (r = 0.55). Other preseason ENSO indices (such as Darwin and Tahiti SLPs and the SOI) show very weak correlation with the annual number of TCs associated with FORM (see also Basher and Zheng 1995). Therefore, only Niño-3.4 and Niño-4 indices are retained for further analysis.

2) Large-scale environmental parameters

The large-scale environmental parameters that are known to be crucial for TC formations in the FST region (Chand and Walsh 2009), including the component of Gray’s genesis parameters (Gray 1975), are also considered. Here, the annual number of cyclones in the FST region associated with FORM is correlated with various environmental variables for the May–July preseason period (Fig. 5), the October–December early cyclone season (Fig. 6), and the November–April cyclone season (Fig. 7), over the Pacific domain between 40°N and 40°S and 120°E and 120°W. Dynamical variables, particularly EVWS and relative vorticity, are found to have strong local correlation with the annual number cyclones associated with FORM for both the October–December early cyclone season (Figs. 6a,b) and the November–April cyclone season (Figs. 7a,b). In contrast, thermodynamical variables (SST, equivalent potential temperature, and relative humidity) exhibit weak local correlation with the annual number cyclones associated with FORM for all seasons. Nevertheless, their correlation is strong in the near-equatorial region, particularly in the Niño-3.4 (5°N–5°S, 170°–120°W) and the Niño-4 (5°N–5°S, 160°E–150°W) regions.

Thermodynamical conditions are usually satisfied in the FST region for all cyclone seasons (e.g., Gouriou and Delcroix 2002; Harr 2004) and, with exception of equivalent potential temperature, they show relatively little local variability because of the ENSO phenomenon, as implied in the primary-mode empirical orthogonal function (EOF) spatial loading patterns (Figs. 8d–f). Here, we use the standard EOF technique (e.g., Preisendorfer 1988) to visualize the dominant spatial variability present in the large-scale environmental conditions. The primary EOF modes identified here are well separated from possible perturbations by their subsequent modes, as determined by the North et al. (1982) criteria (not shown), and are reflective of tropical and equatorial Pacific variability patterns associated with ENSO. The ENSO pattern of variability associated with dynamical variables, particularly relative vorticity, is reasonably large in the FST region (Figs. 8a–c). As expected, these regions of high variability are consistent with relatively high local correlations seen during the November–April cyclone season.

Since large-scale environmental parameters occur simultaneously with TC seasons, they cannot be used as predictors. Instead, predictors during the May–July winter are used based on the assumption that changes in the winter conditions are representative of summer environment in which TCs usually form. The appropriate winter predictors are identified as follows. For each of the environmental variables, a grid point with a Pearson correlation of magnitude above 0.38 is deemed significant at the 99% level and therefore selected as a critical region. To ensure the stability of the chosen statistical relationship, a simple average within a 5° square box centered over the critical region of the predictor variable is taken.

c. Predictors associated with ENT

The November–April Niño-3.4 and Niño-4 indices show weak negative correlation with the annual number of TCs entering the FST region (Fig. 4b). This negative correlation is an indication that more cyclones enter the FST region during La Niña years than during El Niño years. However, the reverse is true for FORM, in which TC formation is characteristically higher in El Niño years than in La Niña years (as per Fig. 4a). Interestingly, the correlation between the annual number of cyclones associated with ENT and 3-monthly running mean values of Niño-4 index is slightly larger and more statistically significant for the preceding May–July winter than for the November–April cyclone season. This is somewhat similar to an earlier study by Nicholls (1984) in the Australian region, where a significantly higher correlation between the annual number of TCs and the east Pacific SST was found during the preceding May–July season than that during the cyclone season (see Fig. 2 of Nicholls 1984). Determining reasons for this vanishing correlation from winter season to summer cyclone season is not within the scope of the present investigation. Because the May–July Niño-4 index is significantly related to the annual number of TCs associated with ENT, it is considered a candidate predictor.

Large-scale environmental parameters that are known to affect TC movement are also considered. These include 500–700-hPa mean steering flow (e.g., Chan and Gray 1982; Chand and Walsh 2009) and 500-hPa geopotential height (e.g., Goh and Chan 2010). It appears that the May–July variations of these large-scale environmental fields, including SST, in the East China Sea around 15°–30°N, 120°–150°E have effects on the annual number of TCs entering the FST region (Figs. 9a,c,e). Significantly high correlation remains here even after removing the effect of ENSO through partial correlation analysis (not shown). This implies that correlations between these environmental fields and TCs entering the FST region are, to some extent, independent of the ENSO phenomenon. A similar result was found by Ramsay et al. (2008) for Australian region cyclones but for the November–April cyclone season. Here, we consider these predictors on the basis of the assumption that changes in these large-scale environmental fields during the May–July preseason in the East China Sea are statistically related to changes in conditions that influence the annual number of TCs entering the FST region. As evident later, including these large-scale environmental fields from the East China Sea as predictors of the annual number of TCs entering the FST region substantially improves the hindcast skill of our model. Further investigation is necessary to identify the physical basis of the relationship between the number of TCs entering the FST region and large-scale environmental fields in the East China Sea. However, it is hypothesized that the westerly wind bursts associated with the monsoon surges (e.g., Briegel and Frank 1997) and the MJO (e.g., Zhang 2005) traveling south from the northwest Pacific may be causing TC formation, particularly in the Coral Sea region, where conditions are more favorable during La Niña conditions (e.g., Nicholls 1984; Basher and Zheng 1995). Since TCs entering the FST region are predominantly formed in the Coral Sea region, it is also reasonable to examine genesis parameters (such as those discussed earlier for TCs associated with FORM) as likely predictors. However, no potential statistical relationship could be found between preseason genesis parameters and the annual number of cyclones entering the FST region. Hence, they are not considered.

4. Model formulation

a. FORM and ENT model structures

The two separate Poisson models—the FORM model and the ENT model—are developed for the FST region using the procedure described in the appendix. As shown later in section 5, a number of predictor combinations are assessed for each of these models through a cross-validation technique. Only those combinations that yielded significant hindcast skill for both the May–July preseason and the October–December early cyclone season are retained. The October–December early-season update is necessary here to reevaluate predictions for seasons where the onset of ENSO events is after the May–July period or where the decay phase of the previous ENSO event extends through the May–July season, thus contaminating the signal. The climatological models of FORM and ENT are also constructed. From Eq. (A1) in the appendix, the overall structures of FORM and ENT models for various predictor combinations can be specified as follows:
i1520-0442-23-13-3425-e1
where ES-SST, ES-GHT and ES-UWNDS represent East China Sea SST, East China Sea geopotential height, and East China Sea zonal winds, respectively.

Implementation of these models using the Bayesian approach requires that prior distributions be specified for model parameters. Here, we take the standard route and apply noninformative priors to each component of our model parameters (e.g., Elsner and Jagger 2006; Chu and Zhao 2007). The noninformative priors used are the mean of 0 and the variance of 106. This very large variance indicates very small precision (i.e., 10−6) and thus contributes little information about the data.

b. Posterior distributions of model parameters

The Gibbs sampling scheme outlined in the appendix is applied to various models of FORM and ENT for a total of 10 000 iterations. To ensure stability of the models, we discard the first 3000 iterations as a burn-in and use the subsequent 7000 as an output of the Gibbs sampler to obtain posterior distributions of regression parameters. The model convergence within the first 3000 simulations is diagnosed through repeated samplings with different values of initial conditions. The posterior distributions of respective parameters, after kernel smoothing, are nearly identical for different initial conditions (not shown). This implies that models converge to a posterior distribution of interest within the first 3000 simulations.

Posterior density distributions of parameter coefficients associated with different models in Eq. (1) are obtained using the 7000 updates of the Gibbs sampler [see Fig. 10 for examples of posterior density plots of May–July preseason all-parameter-combined FORM and ENT models and the climatology models, which contain only the regression constant (i.e., intercept) and the indicator variable]. Distributions are smoothed using a kernel density estimator to remove insignificant fluctuations. The autocorrelation values associated with each parameter reach zero fairly quickly. This indicates that the output of the Gibbs sampler is independently drawn from their joint posterior distribution.

The density of posterior distributions on either side of the zero line provides an insight into the relative contribution of each parameter, and therefore its associated predictor, to the regression model. In the May–July preseason FORM model, for example, a large proportion of samples associated with VORT and, to lesser extent, the Niño-4 index lie on the left side (negatively oriented) and on the right side (positively oriented), respectively (as per Fig. 10). In fact, the 95% credible intervals (−0.61 and −0.021) associated with the parameter values of VORT lie on the left side of the zero reference line. Similarly, the 90% credible intervals on values of Niño-4 (0.00 and 0.61) lie on the right side of the zero reference line. This shows that these predictors play a key role in the prediction equation as opposed to other predictors where the associated credible intervals enclose zero reference lines. Indeed, the combination of VORT and Niño-4 predictors alone appears to have greater hindcast skill to predict the annual number of TCs associated with FORM than other combinations of the predictors evaluated here (see section 5). On the other hand, no clear distinction on predictor importance can be seen for the May–July preseason ENT model. As evident later, using the all-parameter-combined model appears to have relatively good hindcast skill to predict the annual number of TCs associated with ENT than other possible combinations of predictors.

To ensure credible results, mean Bayesian regression coefficients and associated standard deviations (SDs) are cross-referenced with maximum likelihood estimates (MLEs) and associated standard errors (SEs) from the classical (or frequentist) approach. The statistics of the latter are obtained using a MATLAB script developed by Smyth (2009). As expected, mean Bayesian coefficients and standard deviations compare favorably with the MLEs and standard errors, respectively (see examples of this comparison in Table 1).

c. Cross validation

Validation of a forecast model is necessary to assess its performance in practice. Here, we use a leave-one-out cross-validation (LOOCV; e.g., Elsner and Schmertmann 1994) technique to assess the skill of various FORM and ENT models and how they compare with each other and with the overall climatology model. The LOOCV technique works by successively omitting an observation from the dataset and repeating the modeling procedure to predict the omitted observation, with the resulting “prediction” often referred to as the “hindcast.” Because our TC data (both FORM and ENT) exhibit no serial correlations (as per the autocorrelations associated with climatology models shown in Fig. 10, top and bottom right), the use of the LOOCV procedure is justified, as the presence of serial correlation will introduce bias in the estimation of forecast skill (Elsner and Schmertmann 1994).

The hindcast skill of different models is compared using the RMSE. The RMSE is the square root of the mean-square error (MSE), which, as defined by Elsner and Jagger (2006), is the squared difference between the posterior predicted probability and the observation, such that
i1520-0442-23-13-3425-e2
Here, pi(k) is the posterior predicted probability of observing each cyclone (k = 0, … , 10) and oi(k) is the observed count for respective seasons (i = 1, 2, … , 39). The closer the hindcasts are to the observations, the lower the RMSE and hence the better the model.

5. Results

a. Final prediction models for the FST region

The LOOCV procedure described earlier is applied to various modeling strategies of FORM and ENT with different predictor combinations (Table 2). As evident, all models appear to have substantial skill in forecasting the annual number of TCs affecting the FST region. However, we need to select one model of each from this large pool of potential FORM and ENT models. Selecting an appropriate Bayesian model from a large class of potential models is discussed by various authors (e.g., Bernardo and Smith 2000; Marriott et al. 2001). In the present investigation, we consider the general model choice problem examined by Bernardo and Smith (2000) based on the assumption that no true Bayesian model exists, but the most appropriate model is the one in which the expected loss is minimized. Thus, the Niño-4 simple FORM model (RMSE = 1.62) and the all-parameter ENT model (RMSE = 0.98) are the two leading contenders, as they have the smallest RMSE. However, because relative vorticity also plays an important role in the overall prediction equation associated with the FORM model (see section 4b), we choose the relative vorticity and Niño-4 combined FORM model (RMSE = 1.63) based on the rationale that inclusion of relative vorticity may capture the precursor signal of TC numbers in a neutral season where Niño-4 index alone may not be sufficient. For ease of comparison, the same model is applied to update FORM using October–December early-cyclone-season predictors. For TCs entering the FST region, the May–July preseason prediction using the all-parameter-combined model appears appropriate.

b. Predictive distribution of TC probabilities

Because the Bayesian approach provides a unified probabilistic framework for inferences, it is interesting to examine the predictive probability of observing different number of cyclones for a particular year. The rationale for doing this is to understand the limitations of our models when applied in practice and to evaluate the effects of different ENSO conditions on the model performance. For ease of interpretation, we present two cases each of below-normal, normal, and above-normal years where models performed the best and two cases where models performed the worst. A year is considered above (below) normal if TCs observed in that year are half a standard deviation above (below) the mean value. The best and the worst cases associated with each category are evaluated by first ranking the RMSE in ascending order and then selecting years with the two smallest (largest) values as the best (worst) cases.

The predictive distributions of cyclones for years when the FORM and ENT models performed the best (left panel) and the worst (right panel) for the three observation categories are shown in Fig. 11. For years with below-normal observations, the best hindcasts associated with FORM (ENT) are made in 1973/74 and 1974/75 (2004/05 and 2006/07) seasons, where the modeled number of TCs with the maximum probability clearly coincided with the observations. The worst hindcasts for the same category associated with FORM (ENT) are made in 1970/71 and 1990/91 (1978/79 and 1997/98) seasons, where the predictor model indicated normal seasons when the actual observations are below normal. Similarly, the best hindcasts for years of normal observations associated with FORM (ENT) are made in 1981/82 and 2004/05 (1996/97 and 2000/01) seasons. The worst hindcasts associated with FORM are made in 1983/84 and 1993/94 seasons, where the model predicted slightly above-normal activity. However, the ENT model has yielded good hindcasts even for the worst cases of 1972/73 and 1976/77 seasons. For years with above-normal observations, the best hindcasts associated with FORM (ENT) are made in 1972/73 and 1997/98 (1973/74 and 1974/75) seasons, where the modeled number of TCs with the maximum probability clearly coincided with the observations. The worst hindcasts for the same category associated with FORM (ENT) are made in 1986/87 and 1996/97 (1984/85 and 1999/00) seasons, where the predictor model indicated below-normal seasons when actual observations are above normal.

Careful examination of the worst case hindcasts associated with FORM clearly reveals model limitations for seasons where the onset of El Niño or La Niña events is after the May–July preseason (e.g., 1970/71 and 1986/87) or where the previous event continued through the May–July season during its decay phase (e.g., 1983/84). Years corresponding to ENSO events are obtained from the Climate Prediction Center (available online at http://www.cpc.noaa.gov/products/analysis_monitoring/ensostuff/ensoyears.shtml). Accordingly, issuing updates for these seasons using the October–December early-season predictors shows substantial improvement in the hindcast skill (Fig. 12). In the 1970/71 season, for example, the probability of observing zero cyclones increases from approximately 2% to 32% after making the October–December prediction update [Figs. 12a(1),a(2)]. Similarly, the probability of observing two cyclones in the 1983/84 season increases by approximately 50% [Figs. 12b(1),b(2)]. In the 1986/87 season, the above-normal observation becomes consistent with the associated probability distribution after the October–December early-cyclone-season update [Figs. 12c(1),c(2)]. These results imply that predictions using our FORM model need to be reassessed for cases where the onset of ENSO events is after the May–July preseason or where the previous event is continued through the May–July season during its decay phase.

The other limitation is that the model performance in neutral conditions is quite variable, ranging from the best performance—for example, in the 1981/82 season—to the worst performances—the 1990/91 and 1993/94 seasons. Overall, no significant skill can be achieved for neutral conditions—even after an October–December update (Table 3). This is contrary to what occurs during El Niño or La Niña events when model performance is improved as expected after an October–December update. No update is made for TCs entering the FST region because of the lack of statistically significant early-cyclone-season predictors (as per section 3c).

6. Discussion and summary

This paper represents the first comprehensive study on the development of a seasonal prediction scheme for cyclones affecting the Fiji, Samoa, and Tonga (FST) region. Because the FST region spans the approximate location of the main center of action of ENSO (Trenberth and Shea 1987), cyclones can occur here in both El Niño and La Niña conditions (Chand and Walsh 2009). In El Niño years, more cyclones are observed to form within the FST region than in La Niña years. On the other hand, the annual number of cyclones entering the FST region during La Niña years is greater than those entering the region during El Niño years. Accordingly, two separate regression models are developed for the FST region: (i) for TCs that form within the FST region (type “FORM”) and (ii) for TCs that enter the FST region (type “ENT”). Cyclones generally enter the FST region from the Coral Sea, where formation is more common during La Niña years compared to that in El Niño years.

Climatologically, the average number of cyclones that forms within the FST region in a season is approximately 2.54 compared to only 0.62 that enter. Moreover, it is apparent from the wavelet analysis that TCs forming within the FST region exhibit a 4- to 8-yr periodicity, which lies well within the 2- to 8-yr ENSO band. As a result, the annual number of TCs forming within the FST region has high correlation with several of the ENSO indices examined in this study. The correlation is also high with various large-scale environmental parameters that are widely known in the literature to affect cyclone genesis. Of all the large-scale environmental parameters examined here, the dynamical parameters such as 850-hPa relative vorticity, 200-hPa divergence and environmental vertical wind shear show strong local correlation with the annual number of TCs associated with FORM. The correlation disappears after removing the effect of ENSO through the partial correlation technique, implying that changes in the annual number of TCs forming within the FST region are related to changes in the dynamical parameters arising because of ENSO activity. This observation is consistent with earlier studies in other basins (e.g., Wang and Chan 2002; Chia and Ropelewski 2002). In contrast, thermodynamical variables (SST, equivalent potential temperature, and relative humidity) exhibit very weak local correlation with the annual number of TCs associated with FORM for all seasons. Regardless, strong correlations can be identified with the May–July preseason ENSO indices and near-equatorial region large-scale environmental parameters, enabling the prediction of the annual number of TCs associated with FORM to be made at least three months before the November–April cyclone season.

The periodicity of TCs associated with ENT, as evident by the wavelet analysis, is only weakly associated with the ENSO variability. As a result, various ENSO indices and large-scale environmental parameters (such as 500–700-hPa mean steering flow and 500-hPa geopotential height) examined here for November–April season show only weak correlations, with the annual number of TCs entering the FST region. Nevertheless, their correlations with the May–July preseason ENSO indices (particularly with the Niño-4 index) and with the May–July preseason large-scale environmental conditions in the East China Sea are found to be statistically significant. As with FORM, this makes possible the prediction of the annual number of TCs associated with ENT at least three months before the cyclone season. The statistically significant relationship between the annual number of TCs associated with ENT and large-scale environmental parameters, including SST, in the East China Sea remains strong even after removing the effect of ENSO. This implies that TCs entering the FST region may also be influenced by some phenomenon other than ENSO alone.

After exploring a suite of predictors associated with FORM and ENT, two separate Poisson regression models—that is, the FORM model and the ENT model—are then developed for the FST region. The parameter values of these Poisson models are determined using the Bayesian approach. The Bayesian approach is considered optimal here, as it permits the availability of model parameters in terms of their posterior distribution that readily facilitates predictive inferences on future cyclone occurrences within a probabilistic framework. A number of predictor combinations are evaluated for FORM and ENT models through a cross-validation technique. Our findings suggest that the relative vorticity and Niño-4 combined model is optimal to predict the annual number of TCs forming in the FST region, as it has substantially small root-mean-squared error associated with its hindcasts. Similarly, the all-parameter-combined model (i.e., Niño-4 index, ES-SST, ES-GHT, and ES-UWNDS) appears appropriate for predicting TCs entering the FST region.

Lastly, the limitations of these models are determined from their hindcast skill in different ENSO conditions. While the all-parameter-combined ENT model appears to have substantially good skill over all years, the FORM model has the following limitations. First, the model may underestimate (overestimate) the formation for years where the onset of El Niño (La Niña) events is after the May–July preseason or where the previous La Niña (El Niño) event continued through the May–July season during its decaying phase. Accordingly, issuing updates of such cases using the October–December early-season predictors shows substantial improvement in the hindcast skill. The other limitation is that the model performance in neutral conditions is quite variable. Overall, no significant skill can be achieved for neutral conditions—even after an October–December update. This is contrary to the skill obtained during El Niño or La Niña events where model performance is improved substantially after an October–December early-cyclone-season update.

The present investigation has obvious benefits for the FST region. In addition to improved understanding of ENSO–TC relationships, the possible advance warning of a TC strike using the prediction models developed here can have great socioeconomic benefits in terms of more effective and coordinated preparation for cyclone events. An initial prediction of TC activity over the FST region during the May–July preseason and a subsequent update during the October–December early cyclone season can be found at online (at http://www.earthsci.unimelb.edu.au/~schand/). Future work is recommended to further evaluate the forecast skill of our FORM and ENT models and to test other possible lead times for their inclusion in these models.

Acknowledgments

The authors are thankful to the Fiji Meteorological Service for providing cyclone data for the Fiji region. The first author also acknowledges the Australian government–sponsored Endeavour Postgraduate Award for funding his doctorate degree at the University of Melbourne. We appreciate the constructive comments by the three reviewers.

REFERENCES

  • Addison, P. S., 2002: The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance. Institute of Physics Publishing Ltd., 353 pp.

    • Search Google Scholar
    • Export Citation
  • Basher, R. E., and X. Zheng, 1995: Tropical cyclones in the southwest Pacific: Spatial patterns and relationships to Southern Oscillation and sea surface temperature. J. Climate, 8 , 12491260.

    • Search Google Scholar
    • Export Citation
  • Bernardo, J. M., and A. F. M. Smith, 2000: Bayesian Theory. John Wiley and Sons, 586 pp.

  • Bolton, D., 1980: The computation of equivalent potential temperature. Mon. Wea. Rev., 108 , 10461053.

  • Briegel, L. M., and W. M. Frank, 1997: Large-scale influences on tropical cyclogenesis in the western North Pacific. Mon. Wea. Rev., 125 , 13971413.

    • Search Google Scholar
    • Export Citation
  • Camargo, S. J., and A. H. Sobel, 2005: Western North Pacific tropical cyclone intensity and ENSO. J. Climate, 18 , 29963006.

  • Camargo, S. J., A. G. Barnston, P. J. Klotzbach, and C. W. Landsea, 2007: Seasonal tropical cyclone forecasts. WMO Bull., 56 , 297309.

    • Search Google Scholar
    • Export Citation
  • Chan, J. C. L., 1985: Tropical cyclone activity in the northwest Pacific in relation to the El Niño/Southern Oscillation phenomenon. Mon. Wea. Rev., 113 , 599606.

    • Search Google Scholar
    • Export Citation
  • Chan, J. C. L., 2000: Tropical cyclone activity over the western North Pacific associated with El Niño and La Niña events. J. Climate, 13 , 29602972.

    • Search Google Scholar
    • Export Citation
  • Chan, J. C. L., 2007: Interannual variations of intense typhoon activity. Tellus, 59A , 455460.

  • Chan, J. C. L., and W. M. Gray, 1982: Tropical cyclone movement and surrounding flow relationships. Mon. Wea. Rev., 110 , 13541374.

  • Chan, J. C. L., J. Shi, and C. M. Lam, 1998: Seasonal forecasting of tropical cyclone activity over the western North Pacific and the South China Sea. Wea. Forecasting, 13 , 9971004.

    • Search Google Scholar
    • Export Citation
  • Chan, J. C. L., J. E. Shi, and C. M. Liu, 2001: Improvements in the seasonal forecasting of tropical cyclone activity over the western North Pacific. Wea. Forecasting, 16 , 491498.

    • Search Google Scholar
    • Export Citation
  • Chand, S. S., and K. J. E. Walsh, 2009: Tropical cyclone activity in the Fiji region: Spatial patterns and relationship to large-scale circulation. J. Climate, 22 , 38773893.

    • Search Google Scholar
    • Export Citation
  • Chia, H. H., and C. F. Ropelewski, 2002: The interannual variability in the genesis location of tropical cyclones in the northwest Pacific. J. Climate, 15 , 29342944.

    • Search Google Scholar
    • Export Citation
  • Chu, P-S., and X. Zhao, 2007: A Bayesian regression approach for predicting tropical cyclone activity over the central North Pacific. J. Climate, 20 , 40024013.

    • Search Google Scholar
    • Export Citation
  • Courtney, J., and J. A. Knaff, 2009: Adapting the Knaff and Zehr wind-pressure relationship for operational use in Tropical Cyclone Warning Centres. Aust. Meteor. Mag., 58 , 167179.

    • Search Google Scholar
    • Export Citation
  • Elsner, J. B., and C. P. Schmertmann, 1993: Improving extended-range seasonal predictions of intense Atlantic hurricane activity. Wea. Forecasting, 8 , 345351.

    • Search Google Scholar
    • Export Citation
  • Elsner, J. B., and C. P. Schmertmann, 1994: Assessing forecast skill through cross validation. Wea. Forecasting, 9 , 619624.

  • Elsner, J. B., and T. H. Jagger, 2004: A hierarchical Bayesian approach to seasonal hurricane modeling. J. Climate, 17 , 28132827.

  • Elsner, J. B., and T. H. Jagger, 2006: Prediction models for annual U.S hurricane counts. J. Climate, 19 , 29352952.

  • Evans, J. L., and R. L. Allan, 1992: El Niño/Southern Oscillation modification to the structure of the monsoon and tropical cyclone activity in the Australasian region. Int. J. Climatol., 12 , 611623.

    • Search Google Scholar
    • Export Citation
  • Flay, S., and J. Nott, 2007: Effect of ENSO on Queensland seasonal landfalling tropical cyclone activity. Int. J. Climatol., 27 , 13271334.

    • Search Google Scholar
    • Export Citation
  • Friedman, J. H., and W. Stuetzle, 1981: Projection pursuit regression. J. Amer. Stat. Assoc., 76 , 817823.

  • Gelfand, A. E., and A. F. M. Smith, 1990: Sampling based approaches to calculating marginal densities. J. Amer. Stat. Assoc., 85 , 398409.

    • Search Google Scholar
    • Export Citation
  • Goh, A. Z-C., and J. C. L. Chan, 2010: Interannual and interdecadal variations of tropical cyclone activity in the South China Sea. Int. J. Climatol., 30 , 827843.

    • Search Google Scholar
    • Export Citation
  • Gouriou, Y., and T. Delcroix, 2002: Seasonal and ENSO variations of sea surface salinity and temperature in the South Pacific Convergence Zone during 1976–2000. J. Geophys. Res., 107 , 3185. doi:10.1029/2001JC000830.

    • Search Google Scholar
    • Export Citation
  • Gray, W. M., 1968: Global view of the origin of tropical disturbances and storms. Mon. Wea. Rev., 96 , 669700.

  • Gray, W. M., 1975: Tropical cyclone genesis. Department of Atmospheric Science Paper 234, Colorado State University, Fort Collins, CO, 121 pp. [Available from the Department of Atmospheric Sciences, Colorado State University, Ft. Collins, CO 80523].

    • Search Google Scholar
    • Export Citation
  • Gray, W. M., 1979: Hurricanes: Their formation structure, and likely role in the tropical circulation. Meteorology Over Tropical Oceans, D. B. Shaw, Ed., Royal Meteorological Society, 155–218.

    • Search Google Scholar
    • Export Citation
  • Gray, W. M., 1984: Atlantic seasonal hurricane frequency. Part II: Forecasting its variability. Mon. Wea. Rev., 112 , 16691683.

  • Gray, W. M., 1994: Tropical cyclone propagation: Final report A092723. Department of Atmospheric Science, Colorado State University, 36 pp.

    • Search Google Scholar
    • Export Citation
  • Harr, A. P., 2004: Monsoon impacts on tropical cyclone variability. Review Topic B3e: Tropical cyclones. Naval Postgraduate School Press, 31 pp.

    • Search Google Scholar
    • Export Citation
  • Hastings, P. A., 1990: Southern Oscillation influence on tropical cyclone activity in the Australian/south-west Pacific region. Int. J. Climatol., 10 , 291298.

    • Search Google Scholar
    • Export Citation
  • Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77 , 437471.

  • Liu, K. S., and J. C. L. Chan, 2003: Climatological characteristics and seasonal forecasting of tropical cyclones making landfall along the South China coast. Mon. Wea. Rev., 131 , 16501662.

    • Search Google Scholar
    • Export Citation
  • Mantua, N. J., and S. R. Hare, 2002: The Pacific decadal oscillation. J. Oceanogr., 58 , 3544.

  • Marriott, J. M., N. M. Spencer, and A. N. Pettitt, 2001: A Bayesian approach to selecting covariates for prediction. Scand. J. Stat., 28 , 8797.

    • Search Google Scholar
    • Export Citation
  • McDonnell, K. A., and N. J. Holbrook, 2004: A Poisson regression model of tropical cyclogenesis for the Australian–southwest Pacific Ocean region. Wea. Forecasting, 19 , 440455.

    • Search Google Scholar
    • Export Citation
  • Mimura, N., L. Nurse, R. F. McLean, J. Agard, L. Briguglio, P. Lefale, R. Payet, and G. Sem, 2007: Small islands. Climate Change 2007: Impacts, Adaptation and Vulnerability, M. L. Parry et al., Eds., Cambridge University Press, 687–716.

    • Search Google Scholar
    • Export Citation
  • Nicholls, N., 1979: A possible method for predicting seasonal tropical cyclone activity in the Australian region. Mon. Wea. Rev., 107 , 12211224.

    • Search Google Scholar
    • Export Citation
  • Nicholls, N., 1984: The Southern Oscillation, sea-surface temperature, and interannual fluctuations in Australian tropical cyclone activity. Int. J. Climatol., 4 , 661670.

    • Search Google Scholar
    • Export Citation
  • Nicholls, N., 1985: Predictability of interannual variations of Australian seasonal tropical cyclone activity. Mon. Wea. Rev., 113 , 11441149.

    • Search Google Scholar
    • Export Citation
  • Nicholls, N., 1992: Recent performance of a method for forecasting Australian seasonal tropical cyclone activity. Aust. Meteor. Mag., 40 , 105110.

    • Search Google Scholar
    • Export Citation
  • North, G. R., T. L. Bell, and R. F. Chalan, 1982: Sampling errors in the estimation of empirical orthogonal functions. Mon. Wea. Rev., 110 , 699706.

    • Search Google Scholar
    • Export Citation
  • Preisendorfer, R. W., 1988: Principal Component Analysis in Meteorology and Oceanography. Elsevier, 425 pp.

  • Ramsay, H. A., L. M. Leslie, P. J. Lamb, M. B. Richman, and M. Leplastrier, 2008: Interannual variability of tropical cyclones in the Australian region: Role of large-scale environment. J. Climate, 21 , 10831103.

    • Search Google Scholar
    • Export Citation
  • Revell, C. G., 1981: Tropical cyclones in the southwest Pacific, Nov. 1969 to April 1979. New Zealand Meteorological Service Miscellaneous Publ. 170, 53 pp.

    • Search Google Scholar
    • Export Citation
  • Revell, C. G., and S. W. Goulter, 1986a: Lagged relationships between Southern Oscillation and numbers of tropical cyclones in the South Pacific region. Mon. Wea. Rev., 114 , 26692670.

    • Search Google Scholar
    • Export Citation
  • Revell, C. G., and S. W. Goulter, 1986b: South Pacific tropical cyclones and the Southern Oscillation. Mon. Wea. Rev., 114 , 11381145.

    • Search Google Scholar
    • Export Citation
  • Sheskin, D. J., 2007: Handbook of Parametric and Nonparametric Statistical Procedures. 4th ed. Chapman and Hall/CRC, 1736 pp.

  • Sinclair, M. R., 2002: Extratropical transition of southwest Pacific tropical cyclones. Part I: Climatology and mean structure changes. Mon. Wea. Rev., 130 , 590609.

    • Search Google Scholar
    • Export Citation
  • Smith, T. M., and R. W. Reynolds, 2004: Improved extended reconstruction of SST (1854–1997). J. Climate, 17 , 24662477.

  • Smyth, G. K., cited. 2009: StatBox 4.1: A statistics toolbox for Matlab, Walter and Eliza Hall Institute of Medical Research, Melbourne, Australia. [Available online at http://www.statsci.org/matlab/statbox.html].

    • Search Google Scholar
    • Export Citation
  • Solow, A., and N. Nicholls, 1990: The relationship between the Southern Oscillation and tropical cyclone frequency in the Australian region. J. Climate, 3 , 10971101.

    • Search Google Scholar
    • Export Citation
  • Thompson, C. S., S. Ready, and X. Zheng, 1992: Tropical cyclones in the southwest Pacific: November 1979 to May 1989. New Zealand Meteorological Service, 35 pp.

    • Search Google Scholar
    • Export Citation
  • Torrence, C., and G. P. Compo, 1998: A practical guide to wavelet analysis. Bull. Amer. Meteor. Soc., 79 , 6178.

  • Torrence, C., and P. J. Webster, 1998: A practical guide to wavelet analysis. Quart. J. Roy. Meteor. Soc., 124 , 19852004.

  • Trenberth, K. E., 1976: Spatial and temporal variations of the Southern Oscillation. Quart. J. Roy. Meteor. Soc., 102 , 639653.

  • Trenberth, K. E., and D. J. Shea, 1987: On the evolution of the Southern Oscillation. Mon. Wea. Rev., 115 , 30783096.

  • Wang, B., and J. C. L. Chan, 2002: How strong ENSO events affect tropical storm activity over the western North Pacific. J. Climate, 15 , 16431658.

    • Search Google Scholar
    • Export Citation
  • Wilks, D. S., 2006: Statistical Methods in the Atmospheric Sciences. 2nd ed. Academic Press, 627 pp.

  • Zhang, C., 2005: Madden-Julian oscillation. Rev. Geophys., 43 , RG2003. doi:10.1029/2004RG000158.

APPENDIX

Poisson Model Formulation Using the Bayesian Approach

A Poisson regression is a special case of generalized linear model (GLM) in which linear predictors are related to the response variable via a canonical link function, such that
i1520-0442-23-13-3425-ea1
In context of building a Poisson model for cyclone events, yi represents observed cyclone counts in seasons i = 1, … , N and λi is the mean seasonal TC rates for the ith observation related to the linear predictor model (β0 + Xiβ) via a canonical link function (natural logarithm in this case). The regression parameter β0 represents the intercept and β represents the vector of predictor coefficients associated with a row vector of predictor variables Xi. In essence, values for regression parameters are assumed fixed but unknown and are determined through a maximum likelihood procedure by maximizing the likelihood function f (y|β) for the model (e.g., McDonnell and Holbrook 2004). An alternative strategy that is adopted here follows Elsner and Jagger (2004). This method uses a Bayesian approach in which parameter values are treated not as fixed but as random variables. Inferences concerning the parameters are then obtained by combining our prior belief f (β) with the most frequent likelihood f (y|β) using Bayes’s rule, such that
i1520-0442-23-13-3425-ea2

The posterior distribution f (β|y) is the probability density of β conditioned on the cyclone counts y, and f (β) refers to information about the values of parameters of interest without reference to the data. In practice, evaluating analytical solutions of Eq. (A2) is computationally difficult. Therefore, the Markov chain Monte Carlo (MCMC) simulation, as advocated by Elsner and Jagger (2004), is considered a desirable alternative to obtain the posterior distribution. A widely used MCMC simulation method is the Gibbs sampler. A brief summary of the Gibbs sampler is presented next. Details on its application to cyclone events can be found in, for example, Elsner and Jagger (2004) and Chu and Zhao (2007).

In principal, the Gibbs sampler involves an iterative procedure that generates samples from the posterior distribution by successively using updates from previous samples on the current conditional. Suppose, for example, there are k components of β, defined as β = [β1, β2, … , βk], and we have a set of complete conditional posterior densities [ f (βi|βji, y), i = 1, … , k] available for sampling given y. The Gibbs sampling (Gelfand and Smith 1990) then involves successive drawing from this set of complete conditional posterior densities for all values of k. Given an arbitrary set of starting values [β1(0), β2(0), … βk(0)], the algorithm proceeds as follows:
i1520-0442-23-13-3425-ea3
This completes the first iteration of the Gibbs sampler. After a number of such iterations, the posterior probability distribution of βi conditioned on the data y [i.e., f (βi|y)] can be obtained. The kernel density estimation (Wilks 2006) could then be applied to obtain a smoother distribution of each β. The Gibbs sampling here is performed using the Windows version of Bayesian Analysis Using Gibbs Sampling (WinBUGS) software freely available online (at http://www.mrc-bsu.cam.ac.uk/bugs/). This software was developed in the Medical Research Council (MRC) Biostatistics Unit and the Imperial College of Medicine in London.

Fig. 1.
Fig. 1.

Location map of the southwest Pacific region showing the boundary used to define the FST region between 5° and 25°S and 170°E and 170°W.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 2.
Fig. 2.

(a) Genesis locations of the annual number of TCs in the FST region associated with FORM and ENT and (b) time series of the annual number of TCs associated with FORM and ENT. A box in (a) encloses the FST region. Because TCs in the FST region are spread over two calendar years, the first of these years is used in (b) when referring to a particular TC season.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 3.
Fig. 3.

The Morlet wavelet power spectrums of the annual number of TCs in the FST region associated with (a) FORM and (b) ENT. Shaded contours, which are at normalized variances of 1 and 2, are significant at 90% and 95% significance levels, respectively, assuming a red-noise background spectrum with a lag-1 coefficient of 0.0 (e.g., Torrence and Compo 1998). The “cone of influence” is shown as a boldface curve. Values on the period axis are given as log scale. (bottom) Also shown are the respective time series of 2- to 8-yr scale-averaged variances. The dashed line indicates the 95% significance level. Because TCs in the FST region are spread over two calendar years, the first of these years is used when referring to a particular TC season.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 4.
Fig. 4.

Correlation coefficients of the annual number of TCs in the FST region associated with (a) FORM and (b) ENT for the November–April season with 3-monthly running mean values of various ENSO indices starting in January–March preceding the TC season and ending in October–December following the TC season. The dashed line indicates the critical values of the Pearson correlation coefficient of 0.38 at the 99% significance level. The core November–April TC season is shaded in gray.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 5.
Fig. 5.

Spatial distribution of correlation coefficients for 1970/71–2008/09 between the November–April TCs associated with FORM and the May–July preseason (a) EVWS, (b) VORT, (c) DIV, (d) SST, (e) ThetaE, and (f) Rhum. Stippling represents areas in which statistical correlation exceeds the 99% significance level. Solid boxes in (d) represent the Niño-4 and Niño-3.4 regions, and crosses in others denote the center of critical regions about which the 5° × 5° averages are obtained as predictors. Dashed boxes indicate the FST region.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for the October–December early cyclone season.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 7.
Fig. 7.

As in Fig. 5, but for the November–April cyclone season.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 8.
Fig. 8.

Primary-mode EOF spatial loadings of standardized anomalies of (a) EVWS (m s−1), (b) VORT (×10−6 s−1), (c) DIV (×10−6 s−1), (d) SST (°C), (e) ThetaE (°C), and (f) Rhum (%) for November–April cyclone seasons 1970/71–2008/09. Solid (broken) isopleths give positive (negative) loadings. The EOFs are multiplied by SDs of their respective principal components to yield appropriate dimensions. Dashed boxes indicate the FST region.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 9.
Fig. 9.

As in Fig. 5, but for the annual number of TCs associated with ENT during November–April season and (a) May–July SST anomalies, (b) November–April SST anomalies, (c) May–July midtropospheric steering zonal winds, (d) November–April midtropospheric steering zonal winds, (e) May–July 500-hPa geopotential height, and (f) November–April 500-hPa geopotential height.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 10.
Fig. 10.

Posterior density plots of Bayesian regression coefficients and their associated autocorrelation coefficients for FORM and ENT models using all-parameter-combined predictors, as well as climatology. The dashed line is a zero reference line.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 11.
Fig. 11.

Predictive distributions of the annual number of TCs associated with FORM and ENT obtained using the VORT and Niño-4 combined FORM model and the all-parameter-combined ENT model. For ease of interpretation, distributions are grouped into years associated with below-normal, normal, and above-normal TC activity. Examples of the two best hindcasts (worst hindcasts) associated with each group are shown on the left (right) side of each model. Asterisks indicate the actual number of TCs observed in that year.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Fig. 12.
Fig. 12.

Comparison of predictive distributions of TCs formed in the FST region using the (left) May–July preseason predictors and (right) October–December early-cyclone-season predictors. Only three cases associated with late onset of ENSO events are shown as examples. Asterisks indicate the actual number of TCs observed in that year.

Citation: Journal of Climate 23, 13; 10.1175/2010JCLI3521.1

Table 1.

Comparison of the mean Bayesian coefficients and the associated SDs with the MLEs and the associated SEs obtained by the frequentist approach. Statistics are associated with the VORT and Niño-4 combined FORM model and the all-parameter-combined ENT model.

Table 1.
Table 2.

Summary of cross-validated RMSE for various predictor models associated with FORM and ENT. The final Bayesian models selected for further investigations are indicated in boldface.

Table 2.
Table 3.

Summary of cross-validated RMSE associated with the VORT and Niño-4 combined FORM model for years grouped into below-normal, normal and above-normal observation categories. Each observation category is further subdivided into different ENSO conditions. The subscript MJJ denotes MJJ preseason predictors, and the subscript OND denotes OND early-cyclone-season predictors. During El Niño (La Niña) years, below-normal (above normal) cases are very few or none at all and are therefore not shown.

Table 3.
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  • Addison, P. S., 2002: The Illustrated Wavelet Transform Handbook: Introductory Theory and Applications in Science, Engineering, Medicine and Finance. Institute of Physics Publishing Ltd., 353 pp.

    • Search Google Scholar
    • Export Citation
  • Basher, R. E., and X. Zheng, 1995: Tropical cyclones in the southwest Pacific: Spatial patterns and relationships to Southern Oscillation and sea surface temperature. J. Climate, 8 , 12491260.

    • Search Google Scholar
    • Export Citation
  • Bernardo, J. M., and A. F. M. Smith, 2000: Bayesian Theory. John Wiley and Sons, 586 pp.

  • Bolton, D., 1980: The computation of equivalent potential temperature. Mon. Wea. Rev., 108 , 10461053.

  • Briegel, L. M., and W. M. Frank, 1997: Large-scale influences on tropical cyclogenesis in the western North Pacific. Mon. Wea. Rev., 125 , 13971413.

    • Search Google Scholar
    • Export Citation
  • Camargo, S. J., and A. H. Sobel, 2005: Western North Pacific tropical cyclone intensity and ENSO. J. Climate, 18 , 29963006.

  • Camargo, S. J., A. G. Barnston, P. J. Klotzbach, and C. W. Landsea, 2007: Seasonal tropical cyclone forecasts. WMO Bull., 56 , 297309.

    • Search Google Scholar
    • Export Citation
  • Chan, J. C. L., 1985: Tropical cyclone activity in the northwest Pacific in relation to the El Niño/Southern Oscillation phenomenon. Mon. Wea. Rev., 113 , 599606.

    • Search Google Scholar
    • Export Citation
  • Chan, J. C. L., 2000: Tropical cyclone activity over the western North Pacific associated with El Niño and La Niña events. J. Climate, 13 , 29602972.

    • Search Google Scholar
    • Export Citation
  • Chan, J. C. L., 2007: Interannual variations of intense typhoon activity. Tellus, 59A , 455460.

  • Chan, J. C. L., and W. M. Gray, 1982: Tropical cyclone movement and surrounding flow relationships. Mon. Wea. Rev., 110 , 13541374.

  • Chan, J. C. L., J. Shi, and C. M. Lam, 1998: Seasonal forecasting of tropical cyclone activity over the western North Pacific and the South China Sea. Wea. Forecasting, 13 , 9971004.

    • Search Google Scholar
    • Export Citation
  • Chan, J. C. L., J. E. Shi, and C. M. Liu, 2001: Improvements in the seasonal forecasting of tropical cyclone activity over the western North Pacific. Wea. Forecasting, 16 , 491498.

    • Search Google Scholar
    • Export Citation
  • Chand, S. S., and K. J. E. Walsh, 2009: Tropical cyclone activity in the Fiji region: Spatial patterns and relationship to large-scale circulation. J. Climate, 22 , 38773893.

    • Search Google Scholar
    • Export Citation
  • Chia, H. H., and C. F. Ropelewski, 2002: The interannual variability in the genesis location of tropical cyclones in the northwest Pacific. J. Climate, 15 , 29342944.

    • Search Google Scholar
    • Export Citation
  • Chu, P-S., and X. Zhao, 2007: A Bayesian regression approach for predicting tropical cyclone activity over the central North Pacific. J. Climate, 20 , 40024013.

    • Search Google Scholar
    • Export Citation
  • Courtney, J., and J. A. Knaff, 2009: Adapting the Knaff and Zehr wind-pressure relationship for operational use in Tropical Cyclone Warning Centres. Aust. Meteor. Mag., 58 , 167179.

    • Search Google Scholar
    • Export Citation
  • Elsner, J. B., and C. P. Schmertmann, 1993: Improving extended-range seasonal predictions of intense Atlantic hurricane activity. Wea. Forecasting, 8 , 345351.

    • Search Google Scholar
    • Export Citation
  • Elsner, J. B., and C. P. Schmertmann, 1994: Assessing forecast skill through cross validation. Wea. Forecasting, 9 , 619624.

  • Elsner, J. B., and T. H. Jagger, 2004: A hierarchical Bayesian approach to seasonal hurricane modeling. J. Climate, 17 , 28132827.

  • Elsner, J. B., and T. H. Jagger, 2006: Prediction models for annual U.S hurricane counts. J. Climate, 19 , 29352952.

  • Evans, J. L., and R. L. Allan, 1992: El Niño/Southern Oscillation modification to the structure of the monsoon and tropical cyclone activity in the Australasian region. Int. J. Climatol., 12 , 611623.

    • Search Google Scholar
    • Export Citation
  • Flay, S., and J. Nott, 2007: Effect of ENSO on Queensland seasonal landfalling tropical cyclone activity. Int. J. Climatol., 27 , 13271334.

    • Search Google Scholar
    • Export Citation
  • Friedman, J. H., and W. Stuetzle, 1981: Projection pursuit regression. J. Amer. Stat. Assoc., 76 , 817823.

  • Gelfand, A. E., and A. F. M. Smith, 1990: Sampling based approaches to calculating marginal densities. J. Amer. Stat. Assoc., 85 , 398409.

    • Search Google Scholar
    • Export Citation
  • Goh, A. Z-C., and J. C. L. Chan, 2010: Interannual and interdecadal variations of tropical cyclone activity in the South China Sea. Int. J. Climatol., 30 , 827843.

    • Search Google Scholar
    • Export Citation
  • Gouriou, Y., and T. Delcroix, 2002: Seasonal and ENSO variations of sea surface salinity and temperature in the South Pacific Convergence Zone during 1976–2000. J. Geophys. Res., 107 , 3185. doi:10.1029/2001JC000830.

    • Search Google Scholar
    • Export Citation
  • Gray, W. M., 1968: Global view of the origin of tropical disturbances and storms. Mon. Wea. Rev., 96 , 669700.

  • Gray, W. M., 1975: Tropical cyclone genesis. Department of Atmospheric Science Paper 234, Colorado State University, Fort Collins, CO, 121 pp. [Available from the Department of Atmospheric Sciences, Colorado State University, Ft. Collins, CO 80523].

    • Search Google Scholar
    • Export Citation
  • Gray, W. M., 1979: Hurricanes: Their formation structure, and likely role in the tropical circulation. Meteorology Over Tropical Oceans, D. B. Shaw, Ed., Royal Meteorological Society, 155–218.

    • Search Google Scholar
    • Export Citation
  • Gray, W. M., 1984: Atlantic seasonal hurricane frequency. Part II: Forecasting its variability. Mon. Wea. Rev., 112 , 16691683.

  • Gray, W. M., 1994: Tropical cyclone propagation: Final report A092723. Department of Atmospheric Science, Colorado State University, 36 pp.

    • Search Google Scholar
    • Export Citation
  • Harr, A. P., 2004: Monsoon impacts on tropical cyclone variability. Review Topic B3e: Tropical cyclones. Naval Postgraduate School Press, 31 pp.

    • Search Google Scholar
    • Export Citation
  • Hastings, P. A., 1990: Southern Oscillation influence on tropical cyclone activity in the Australian/south-west Pacific region. Int. J. Climatol., 10 , 291298.

    • Search Google Scholar
    • Export Citation
  • Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project. Bull. Amer. Meteor. Soc., 77 , 437471.

  • Liu, K. S., and J. C. L. Chan, 2003: Climatological characteristics and seasonal forecasting of tropical cyclones making landfall along the South China coast. Mon. Wea. Rev., 131 , 16501662.

    • Search Google Scholar
    • Export Citation
  • Mantua, N. J., and S. R. Hare, 2002: The Pacific decadal oscillation. J. Oceanogr., 58 , 3544.

  • Marriott, J. M., N. M. Spencer, and A. N. Pettitt, 2001: A Bayesian approach to selecting covariates for prediction. Scand. J. Stat., 28 , 8797.

    • Search Google Scholar
    • Export Citation
  • McDonnell, K. A., and N. J. Holbrook, 2004: A Poisson regression model of tropical cyclogenesis for the Australian–southwest Pacific Ocean region. Wea. Forecasting, 19 , 440455.

    • Search Google Scholar
    • Export Citation
  • Mimura, N., L. Nurse, R. F. McLean, J. Agard, L. Briguglio, P. Lefale, R. Payet, and G. Sem, 2007: Small islands. Climate Change 2007: Impacts, Adaptation and Vulnerability, M. L. Parry et al., Eds., Cambridge University Press, 687–716.

    • Search Google Scholar
    • Export Citation
  • Nicholls, N., 1979: A possible method for predicting seasonal tropical cyclone activity in the Australian region. Mon. Wea. Rev., 107 , 12211224.

    • Search Google Scholar
    • Export Citation
  • Nicholls, N., 1984: The Southern Oscillation, sea-surface temperature, and interannual fluctuations in Australian tropical cyclone activity. Int. J. Climatol., 4 , 661670.

    • Search Google Scholar
    • Export Citation
  • Nicholls, N., 1985: Predictability of interannual variations of Australian seasonal tropical cyclone activity. Mon. Wea. Rev., 113 , 11441149.

    • Search Google Scholar
    • Export Citation
  • Nicholls, N., 1992: Recent performance of a method for forecasting Australian seasonal tropical cyclone activity. Aust. Meteor. Mag., 40 , 105110.

    • Search Google Scholar
    • Export Citation
  • North, G. R., T. L. Bell, and R. F. Chalan, 1982: Sampling errors in the estimation of empirical orthogonal functions. Mon. Wea. Rev., 110 , 699706.

    • Search Google Scholar
    • Export Citation
  • Preisendorfer, R. W., 1988: Principal Component Analysis in Meteorology and Oceanography. Elsevier, 425 pp.

  • Ramsay, H. A., L. M. Leslie, P. J. Lamb, M. B. Richman, and M. Leplastrier, 2008: Interannual variability of tropical cyclones in the Australian region: Role of large-scale environment. J. Climate, 21 , 10831103.

    • Search Google Scholar
    • Export Citation
  • Revell, C. G., 1981: Tropical cyclones in the southwest Pacific, Nov. 1969 to April 1979. New Zealand Meteorological Service Miscellaneous Publ. 170, 53 pp.

    • Search Google Scholar
    • Export Citation
  • Revell, C. G., and S. W. Goulter, 1986a: Lagged relationships between Southern Oscillation and numbers of tropical cyclones in the South Pacific region. Mon. Wea. Rev., 114 , 26692670.

    • Search Google Scholar
    • Export Citation
  • Revell, C. G., and S. W. Goulter, 1986b: South Pacific tropical cyclones and the Southern Oscillation. Mon. Wea. Rev., 114 , 11381145.

    • Search Google Scholar
    • Export Citation
  • Sheskin, D. J., 2007: Handbook of Parametric and Nonparametric Statistical Procedures. 4th ed. Chapman and Hall/CRC, 1736 pp.

  • Sinclair, M. R., 2002: Extratropical transition of southwest Pacific tropical cyclones. Part I: Climatology and mean structure changes. Mon. Wea. Rev., 130 , 590609.

    • Search Google Scholar
    • Export Citation
  • Smith, T. M., and R. W. Reynolds, 2004: Improved extended reconstruction of SST (1854–1997). J. Climate, 17 , 24662477.

  • Smyth, G. K., cited. 2009: StatBox 4.1: A statistics toolbox for Matlab, Walter and Eliza Hall Institute of Medical Research, Melbourne, Australia. [Available online at http://www.statsci.org/matlab/statbox.html].

    • Search Google Scholar
    • Export Citation
  • Solow, A., and N. Nicholls, 1990: The relationship between the Southern Oscillation and tropical cyclone frequency in the Australian region. J. Climate, 3 , 10971101.

    • Search Google Scholar
    • Export Citation
  • Thompson, C. S., S. Ready, and X. Zheng, 1992: Tropical cyclones in the southwest Pacific: November 1979 to May 1989. New Zealand Meteorological Service, 35 pp.

    • Search Google Scholar
    • Export Citation
  • Torrence, C., and G. P. Compo, 1998: A practical guide to wavelet analysis. Bull. Amer. Meteor. Soc., 79 , 6178.

  • Torrence, C., and P. J. Webster, 1998: A practical guide to wavelet analysis. Quart. J. Roy. Meteor. Soc., 124 , 19852004.

  • Trenberth, K. E., 1976: Spatial and temporal variations of the Southern Oscillation. Quart. J. Roy. Meteor. Soc., 102 , 639653.

  • Trenberth, K. E., and D. J. Shea, 1987: On the evolution of the Southern Oscillation. Mon. Wea. Rev., 115 , 30783096.

  • Wang, B., and J. C. L. Chan, 2002: How strong ENSO events affect tropical storm activity over the western North Pacific. J. Climate, 15 , 16431658.

    • Search Google Scholar
    • Export Citation
  • Wilks, D. S., 2006: Statistical Methods in the Atmospheric Sciences. 2nd ed. Academic Press, 627 pp.

  • Zhang, C., 2005: Madden-Julian oscillation. Rev. Geophys., 43 , RG2003. doi:10.1029/2004RG000158.