GEOPHYSICAL RESEARCH LETTERS, VOL. 31, L20110, doi: /2004gl020843, 2004

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1 GEOPHYSICAL RESEARCH LETTERS, VOL. 31,, doi: /2004gl020843, 2004 A Poisson regression model approach to predicting tropical cyclogenesis in the Australian/southwest Pacific Ocean region using the SOI and saturated equivalent potential temperature gradient as predictors Katrina A. McDonnell and Neil J. Holbrook Department of Physical Geography, Division of Environmental and Life Sciences, Macquarie University, Sydney, New South Wales, Australia Received 25 June 2004; revised 1 September 2004; accepted 24 September 2004; published 27 October [1] This paper explores the potential of the Southern Oscillation index (SOI) in combination with the saturated equivalent potential temperature gradient (EPT) as predictors of tropical cyclogenesis in the Australian/southwest Pacific Ocean. This is undertaken using a series of Poisson regression models of tropical cyclogenesis developed on a 2 latitude 5 longitude and monthly grid. Links between tropical cyclogenesis and the predictors are investigated, with the most significant models cross-validated, and the skill of their hindcasts evaluated. The September lead SOI-only Poisson regression model provided skillful predictions of the temporal variability of tropical cyclogenesis across the entire region, with a root-mean-square error 22% better than climatology. The combination SOI and EPT model adds spatial skill and further improves temporal skill. Temporal skill is best in the Eastern subregion (western tropical Pacific) (significant correlations with observations at 99% level), while spatial skill is best elsewhere. INDEX TERMS: 3309 Meteorology and Atmospheric Dynamics: Climatology (1620); 3339 Meteorology and Atmospheric Dynamics: Ocean/atmosphere interactions (0312, 4504); 3374 Meteorology and Atmospheric Dynamics: Tropical meteorology. Citation: McDonnell, K. A., and N. J. Holbrook (2004), A Poisson regression model approach to predicting tropical cyclogenesis in the Australian/southwest Pacific Ocean region using the SOI and saturated equivalent potential temperature gradient as predictors, Geophys. Res. Lett., 31,, doi: /2004gl Introduction [2] There has been considerable observational and statistical research investigating links between the Southern Oscillation index (SOI) and tropical cyclone frequency in the Australian/southwest Pacific Ocean region [e.g., Solow and Nicholls, 1990; Nicholls, 1992; Basher and Zheng, 1995]. Neville Nicholls from the Australian Bureau of Meteorology (BoM) has routinely conducted a SOI-based forecast of seasonal tropical cyclone activity, using a linear regression on first differences between the October lead SOI and tropical cyclone numbers [e.g., Nicholls, 1999]. This linear statistical model has been shown to perform very well in the Australian region. Nevertheless, like most statistical models incorporating SOI as a predictor of tropical cyclogenesis, it provides basin-wide, seasonal forecasts. Copyright 2004 by the American Geophysical Union /04/2004GL [3] This study seeks to explore the forecasting potential of tropical cyclogenesis at smaller spatial (sub-basin) and temporal (sub-seasonal) grid scales in the Australian/southwest Pacific Ocean region. McDonnell and Holbrook [2004] found that a Poisson regression model incorporating a combination of saturated equivalent potential temperature gradient (EPT) at different leads showed significant skill in hindcasting the temporal and spatial variability of tropical cyclogenesis for the Australian/southwest Pacific Ocean region. The EPT depends on the vertical temperature profile and is a measure of the potential for cumulonimbus convection from a lapse-rate stability viewpoint. The pointwise nature of the predictors (monthly SOI and EPT, and spatially resolved EPT) offers the potential for forecasting not only the number of tropical cyclones formed, but also in what part of the season and in what location they form. [4] A series of Poisson regression models of tropical cyclogenesis is developed here on a 2 latitude 5 longitude spatial grid and monthly grid in time. Links between tropical cyclogenesis and SOI, at various monthly leads, are investigated. The most significant Poisson regression models are cross-validated and the skill of their hindcasts are evaluated. In addition, EPT is combined with SOI in the Poisson regression model predictor set to see whether it adds value. Based on our analyses, we demonstrate that both the SOI-only and combination SOI and EPT Poisson regression models provide skillful predictions temporally and spatially. 2. Data [5] The Australian/southwest Pacific Ocean region (6 20 S, E) tropical storm data were obtained from BoM ( Observations associated with the cyclone seasons 1960/ /93 are used to develop the Poisson regression models over 33 years of data in this study. Tropical cyclone genesis is identified to occur at the time and location where the wind speed first exceeds 34 knots (17.5 m s 1 ) [e.g., Bureau of Meteorology, 1978]. These cyclone genesis occurrence points were binned into monthly (NovemberMay) cells in time and 2 latitude 5 longitude boxes in space across the entire region, as described by McDonnell and Holbrook [2004]. [6] Monthly SOI values were obtained from BoM ( The temperature fields at 1000 hpa and 500 hpa used to calculate the EPT [see McDonnell and Holbrook, 2004] 1of5

2 were obtained from the National Center for Environmental Prediction/National Center for Atmospheric Research Reanalysis Project monthly mean subsets ( edu/datasets/ds090.2/). The calculated monthly parameter values were averaged into 2 latitude 5 longitude boxes. 3. Poisson Regression Modelling and Analysis [7] In a Poisson model, the probability distribution, i.e., the probability of occurrence of exactly y tropical cyclones, is given by where m i is PrðY i ¼ yþ ¼ my i exp m i y! m i ¼ exp b 0þ P j b jx ij ; y ¼ 0; 1; 2;...; 1 ð1þ and where X ij is the data value for predictor j on observation i, and b j is the corresponding Poisson regression coefficient for predictor j. It can be shown theoretically that the expected number of tropical cyclones E(Y) = Var(Y) = m, when Y has a Poisson distribution [Kleinbaum et al., 1988]. [8] The general method of fitting a Poisson regression model is to use the Poisson model formulation to derive a maximum likelihood function. This means that for a given b (vector of Poisson regression coefficients), m is calculated for each set of predictors, and the likelihood of the observed number of tropical cyclones is estimated (equation (1)). The b that is then used for the forecast (or hindcast) is the one that maximises the product of the probabilities (in equation (1)) over all time [Elsner and Schmertmann, 1993]. The solution to the maximum likelihood equations is obtained through iteratively reweighted least squares [Kleinbaum et al., 1988]. [9] The advantage of Poisson regression over linear regression is that it is more applicable for modelling the occurrence of rare, discrete events such as the occurrences of tropical cyclones [Wilks, 1995]. Solow and Nicholls [1990] constructed a statistical model of the relationship between tropical cyclone frequency for the Australian region (as a whole) and the SOI using Poisson regression. Elsner and Schmertmann [1993] have also shown that a Poisson model provides a large increase in skill over linear statistical models in their study of seasonal numbers of intense tropical cyclones in the Atlantic. [10] Our Poisson regression models of tropical cyclogenesis are developed on a 2 latitude 5 longitude spatial grid and monthly grid in time. Links between tropical cyclogenesis and SOI, at various monthly leads of up to five months, are first investigated, followed by the inclusion of EPT at various leads. Three-fold cross-validation was performed on the selected Poisson regression model by dividing the data set into three equal subsets. One of the three subsets is used each time as the verification (prediction) set and the other two subsets are put together to form a developmental (training) set. This procedure was performed three times so that each year of observations fell in the verification subset once and hence was independently predicted or hindcast, simulating an actual forecast. The hindcast skill, and forecasting potential, of the model is ð2þ investigated using correlation, root-mean-square errors (rmses) and spatial distribution maps. The Poisson regression modelling methods and analysis techniques are described fully by McDonnell and Holbrook [2004]. 4. Results 4.1. SOI Predictor Model [11] As stated previously, a linear regression SOI-based forecast of Australian region tropical cyclone activity has been routinely conducted in the past [e.g., Nicholls, 1999]. Over the period , the linear model, with zero intercept, is defined as DðTCsÞ ¼ 0:15DðSOI October Þ ð3þ where D(TCs) is the predicted difference in tropical cyclone numbers between the coming and previous tropical cyclone season, and D(SOI October ) is the observed difference between October of the current year and the previous year. [12] As first differences are used, only 32 years of tropical cyclogenesis are hindcast (Figure 1a). The correlation coefficient between the observed and cross-validated hindcasts is 0.63 (significant at the 99% level), with a rmse of Significance of the correlation coefficient(s) is tested against the (reduced) effective number of degrees of freedom, following the method of Davis [1976]. Ten extra tropical cyclone seasons from 1993/ /03 were forecast using this model. The observed and forecast tropical cyclone numbers are also shown in Figure 1a. Only the 1999/2000 and 2001/02 seasons are forecast well, with all other seasons under-/over-predicted by between two and five tropical cyclones. [13] A total of 24 Poisson regression models were generated using SOI at various lead times. The SOI predictor model with the lowest deviance incorporates the September lead SOI (similar to the model of Solow and Nicholls [1990]). The Poisson regression coefficients and standard error estimates for each model parameter can be found in Table 1. Figure 1b shows the SOI model cross-validated hindcasts (until 1992/93) against observed seasonal tropical cyclone numbers. The most obvious failures are the out-ofphase period, the underpredictions from and the overprediction during the strong La Niña of The correlation coefficient between the observed and cross-validated hindcasts is 0.5 (significant at the 99% level), with a rmse of 2.33, representing a 22% improvement over the climatology hindcasts (i.e., the number of tropical cyclones predicted using only month, latitude and longitude as predictors). Figure 2a shows the total number of tropical cyclones formed in the Australian/ southwest Pacific region in each 2 latitude 5 longitude grid box, between 1960/61 and 1992/93. Three observed maxima can be clearly seen: off the northwest shelf of Australia; in the Gulf of Carpentaria; and in the Coral Sea. Figure 2b shows the spatial distribution of the total number of tropical cyclogenesis occurrences predicted by the model cross-validated hindcasts. The spatial hindcast skill of this model is obviously poor, with the base spatial structure being zonally oriented as a function of the latitude. This is not surprising as the SOI is a space-independent index. [14] As with the linear regression model, 10 extra tropical cyclone seasons were forecast using the September lead SOI 2of5

3 McDONNELL AND HOLBROOK: PREDICTION OF TROPICAL CYCLOGENESIS Table 1. Poisson Regression Coefficients (bj) and Standard Errors (SE) Estimated Over the Period 1960/ /93 for Both the SOI-Only Model and Combination SOI and EPT Model SOI-Only Figure 1. (a) The actual (solid line) and cross-validated hindcasts (dashed line) seasonal numbers of Australian/ southwest Pacific tropical cyclones formed during the 32-year period Hindcasts are predicted using Nicholls linear regression on first differences between the October SOI and tropical cyclone numbers. (b c) The actual (solid line) and cross-validated hindcasts (dashed line) seasonal numbers of Australian/southwest Pacific tropical cyclones formed during the 33-year period Predictors used are (b) SOI at the September lead, and (c) a combination of SOI and EPT at the September lead and the EPT 5 1-month lead gradient. Month, cyclone year, latitude, and longitude are also included as predictors in (b) and (c). Independent tropical cyclone observations (o) for 1993/ /03 are also forecast (x) for all three models. SOI/EPT Parameter bj SE bj SE Intercept SOISep EPTSep EPT5 1month Month Cyclone Year Latitude Longitude cross-validated model hindcast number of tropical cyclones are shown in Figure 1c. The correlation coefficient between the observed and hindcast tropical cyclone occurrences is 0.6 (significant at the 99% level) over the entire record. The rmse of the seasonal hindcasts is 2.15, a 34% improvement over the climatology hindcasts. The addition of the EPT predictors adds spatial hindcast skill (Figure 2c), with a reasonably realistic representation of the maxima off the northwest shelf and in the Gulf of Carpentaria. The overall spatial pattern is broadly displayed but the diminished spatial structure in the western Pacific is perhaps not that surprising as Poisson regression tends to smooth the predicted values over the spatial area. Both temporally and spatially, the combination SOI and EPT model shows greater hindcast skill than the SOI-only model. Poisson regression model (Figure 1b). Five of the 10 seasons are predicted extremely well, to within one tropical cyclone. However, 1993/94 and 1997/98 are underpredicted by up to three tropical cyclones. 1996/97, 2000/01 and 2001/02 are overpredicted by three to five tropical cyclones. Only the 1999/2000, 2000/01 and 2001/02 seasons are forecast better by the linear regression SOI model, the first two by only one tropical cyclone Combination SOI and EPT Predictor Model [15] A Poisson regression model combining EPT at leads of September and the 5 1-month lead gradient (identified by McDonnell and Holbrook [2004]) with the September SOI lead (see Table 1) was produced to determine whether EPT adds value to the hindcasts. The 5 1-month lead gradient reflects the rate of change in the EPT before and throughout the tropical cyclone season, e.g., for the prediction of December cyclogenesis, the July November gradient is used, and so on. The observed seasonal number and Figure 2. (a) Total number of tropical cyclones formed in the Australian/southwest Pacific region in each 2 latitude 5 longitude grid box, between 1960/61 and 1992/93. (b c) Cross-validated hindcasts of the total number of tropical cyclones formed. Predictors used are (b) SOI at the September lead, and (c) SOI and EPT at the September lead, plus the EPT 5 1-month lead gradient. Month, cyclone year, latitude and longitude are also included as predictors in (b) and (c). 3 of 5

4 Table 2. Statistical Summary of the Significant Poisson Regression Model Hindcasts for the Defined Subseasons/Subregions Over the Period 1960/611992/93 a Season/Region Corr. Coef. RMSE Improv. Over Climatology SEPTEMBER SOI Entire/Eastern % Mid/Eastern % Late/Eastern % SOI and EPT Combination Entire/Eastern 0.47* % Early/All % Mid/All % Early/Northern 0.39* % Mid/Eastern % Late/Eastern % a For example, Entire/Eastern identifies the seasonal number of tropical cyclones in the Eastern region, Mid/Eastern refers to mid-season tropical cyclone numbers over the Eastern region, and Early/All means the earlyseason numbers over the entire basin. The correlation coefficient and the rmse between the observed and cross-validated hindcasts are shown, together with the percentage improvement over climatology. Values of the correlation coefficient are given in bold if they are significant at greater than the 99% (98% marked with an asterisk) level, italicised if significant at greater than the 95% level, and plain text if significant at the 90% level. [16] The 10 tropical cyclone seasons from 1993/ /03 were again forecast, this time using the combination Poisson regression model of SOI and EPT. The observed and forecast tropical cyclone numbers are shown in Figure 1c. There are improvements in forecast skill by up to one tropical cyclone over the SOI-only model for forecasts in 1996/97, 1997/98 and 1999/2000. The combination Poisson model also forecasts all seasons except 2000/01 and 2001/02 better than the linear regression model. 5. Discussion [17] Previous studies [e.g., Dong, 1988; Solow and Nicholls, 1990], have shown that the relationship between the SOI and tropical cyclone occurrences is strongest for the Australian/southwest Pacific Ocean region. More tropical cyclones are observed to occur in this region when the SOI is positive rather than negative. [18] The correlation coefficient between the observed and cross-validated hindcasts from 1960/ /93 using the linear regression equation (0.63) is higher than for either of the Poisson regression models. The rmse is nevertheless equivalent to that of the SOI Poisson regression model, and higher than that of the combination SOI and EPT model. Further, the linear model forecasts of tropical cyclones formed for 1993/942002/03 were poorer overall than those forecast by the Poisson regression models (although 10 sample years is insufficient to undertake a proper statistical analysis). Only the 2000/01 and 2001/02 seasons are better predicted by the linear model. The rmse between the linear regression model forecasts and observed seasonal numbers was 3.2, compared to 2.6 for the SOI-only Poisson regression model. [19] It is important to remember that for each Poisson regression model hindcast, expected occurrences are determined within every 2 latitude 5 longitude cell in the region, and for every month in the tropical cyclone season from 1960/611992/93 (a total of possible observation events). Hence, the cross-validated hindcasts generated by each model can be readily summed across different subregions and subseasons, and evaluated against the corresponding observations to determine the potential Poisson regression model hindcast skill at subregional and subseasonal scales. Here, our investigation is extended by further dividing the Australian/southwest Pacific region into three subregions (based on the BoM forecast regions): the Western ( E), Northern ( E) and Eastern ( E) regions. The tropical cyclone season is also divided further into early (NovemberDecember), mid- (January March) and late (April May) seasons. [20] Significant correlation coefficients and rmses between the observed and cross-validated hindcasts of tropical cyclogenesis on the subseason/subregion scale, together with the percentage improvement over climatology, are presented in Table 2. The temporal variability of the seasonal number of tropical cyclones formed in the Eastern region is hindcast with the most skill. This is interesting to note as the actual spatial structure within the Eastern region is poorly represented by the models. It would appear that the SOI, the metric so often used to identify the phases of the Pacific-focused El Niño-Southern Oscillation (ENSO) phenomenon, is correspondingly the best predictor of the total number of tropical cyclones forming in the Eastern region (i.e., western tropical Pacific). The time variability of cyclones formed in the Western region is the most poorly hindcast by the Poisson regression models. 6. Conclusions [21] Our findings show that the September SOI lead is the best predictor of the temporal variability of tropical cyclogenesis for the Australian/southwest Pacific Ocean region, with the Poisson regression model cross-validated hindcast phase variability significantly correlated at the 99% level. The rmse for the model cross-validated hindcast is a 22% improvement over climatology. Combining spatial and temporal variations of EPT with the SOI in the Poisson regression model showed an increase in the correlation coefficient to 0.6 and an rmse 34% better than climatology. The model independent forecasts of the 10 tropical cyclone seasons 1993/ /03 showed greater skill than the linear first differences SOI model used in the past for basin-wide forecasts. These models performed best in the Eastern subregion where ENSO influences may be expected to be strongest. [22] Acknowledgments. The authors gratefully acknowledge Professor Don McNeil, Macquarie University, for his assistance with learning about Poisson regression techniques. We are also grateful to Dr. Neville Nicholls and the anonymous reviewer for their helpful comments. References Basher, R. E., and X. Zheng (1995), Tropical cyclones in the southwest Pacific: Spatial patterns and relationships to Southern Oscillation and seasurface temperature, J. Clim., 8, Bureau of Meteorology (1978), Australian Tropical Cyclone Forecasting Manual, 274 pp., Bur. of Meteorol., Melbourne, Victoria, Australia. Davis, R. E. (1976), Predictability of sea surface temperature and sea level pressure anomalies over the North Pacific Ocean, J. Phys. Oceanogr., 6, Dong, K. (1988), El Niño and tropical cyclone frequency in the Australian region and the northwest Pacific, Aust. Meteorol. Mag., 36, Elsner, J. B., and C. P. Schmertmann (1993), Improving extended-range seasonal predictions of intense Atlantic hurricane activity, Weather Forecasting, 8, of5

5 Kleinbaum, D. G., L. L. Kupper, and K. E. Muller (1988), Applied Regression Analysis and Other Multivariate Methods, 2nd ed., 718 pp., PWS-Kent, Boston, Mass. McDonnell, K. A., and N. J. Holbrook (2004), A Poisson regression model of tropical cyclogenesis for the Australian-southwest Pacific Ocean region, Weather Forecasting, 19, Nicholls, N. (1992), Recent performance of a method for forecasting Australian seasonal tropical cyclone activity, Aust. Meteorol. Mag., 40, Nicholls, N. (1999), SOI-based forecast of Australian region tropical cyclone activity, Exp. Long Lead Forecast Bull., 8(4), Solow, A., and N. Nicholls (1990), The relationship between the Southern Oscillation and tropical cyclone frequency in the Australian region, J. Clim., 3, Wilks, D. S. (1995), Statistical Methods in the Atmospheric Sciences, 467 pp., Academic, San Diego, Calif. N. J. Holbrook and K. A. McDonnell, Department of Physical Geography, Division of Environmental and Life Sciences, Macquarie University, Sydney, NSW 2109, Australia. (neil.holbrook@mq.edu.au) 5of5

A Poisson Regression Model of Tropical Cyclogenesis for the Australian Southwest Pacific Ocean Region

440 WEATHER AND FORECASTING VOLUME 19 A Poisson Regression Model of Tropical Cyclogenesis for the Australian Southwest Pacific Ocean Region KATRINA A. MCDONNELL AND NEIL J. HOLBROOK Division of Environmental