## Abstract

A real-time statistical model based on the work of Leroy and Wheeler is developed via multiple logistic regression to predict weekly tropical cyclone activity over the Atlantic and east Pacific basins. The predictors used in the model include a climatology of tropical cyclone genesis for each ocean basin, an El Niño–Southern Oscillation (ENSO) index, and two indices representing the propagating Madden–Julian oscillation (MJO). The Atlantic model also includes a predictor representing the variability of sea surface temperature (SST) in the Main Development Region (MDR). These predictors are suggested as useful for the prediction of tropical cyclogenesis based on previous work in the literature and are further confirmed in this study using basic statistics. Univariate logistic regression models are generated for each predictor in each region to ensure the choice of prediction scheme. Using all predictors, cross-validated hindcasts are developed out to a seven-week forecast lead. A formal stepwise predictor selection procedure is implemented to select the predictors used in each region at each forecast lead.

Brier skill scores and reliability diagrams are used to assess the skill and dependability of the models. Results show an increase in model skill over the time-varying climatology at predicting tropical cyclogenesis by the inclusion of the MJO out to a three-week forecast lead for the east Pacific and a two-week forecast lead for the Atlantic. The importance of ENSO and MDR SST for Atlantic genesis prediction is highlighted, and the uncertain effects of ENSO on east Pacific tropical cyclogenesis are revisited.

## 1. Introduction

Forecasting schemes on the seasonal time scale have been used since the early work of Gray (1984b). As noted by Lehmiller et al. (1997), a seasonal forecast informs of the overall amount of seasonal activity expected, knowing only that it will occur sometime during the hurricane season. To provide higher temporal resolution forecasts of tropical cyclone activity within a season, intraseasonal prediction models (from 10–60 days; Vitart et al. 2010) are an ideal tool. Prediction in the intraseasonal time scale allows for the use of predictors of shorter time-scale variability as well as those of interannual and longer time scales. These predictors are based on environmental changes observed to affect hurricane-favorable conditions.

Gray (1979) statistically related cyclogenesis to six climatological environmental conditions: adequate surface to 400-mb vorticity, sufficient Coriolis parameter for large-scale rotation, low vertical shear of the horizontal winds, surface to 60-m ocean temperatures above 26°C, deep atmospheric conditional instability, and high values of low- to midlevel tropospheric relative humidity. The presence of several of these components depends on the large-scale flow, which is known to be modulated by climate variability on various time scales (Landsea 2000).

On interannual time scales, El Niño–Southern Oscillation (ENSO) has been shown to affect tropical cyclogenesis in the Atlantic. During El Niño events, enhanced upper-level wind shear over the Atlantic basin results in unfavorable conditions for tropical cyclogenesis (Goldenberg and Shapiro 1996; Gray 1984a; Gray and Sheaffer 1991; Landsea et al. 1999; Shapiro 1987). In contrast, during La Niña events an enhancement of convection exists over the western Atlantic that aids in tropical cyclone development (Wyrtki 1982; Chu 2004). Although the relationship between ENSO phase and cyclogenesis has some variations depending on the time of the year as well as location within the Atlantic basin, generally an overall decrease in Atlantic tropical cyclones occurs during the El Niño phase (Shaman et al. 2009). While the relationship between east Pacific cyclone numbers and ENSO phase is not as clear as for the Atlantic basin, dividing the east Pacific region into subbasins has been shown to be helpful in clarifying the relationship. Collins and Mason (2000) noted that east of 116°W in the east Pacific, environmental conditions tend to be climatologically favorable for cyclogenesis regardless of ENSO phase, while west of 116°W ENSO produces a stronger effect (Collins 2007; Collins and Mason 2000). However, approximately twice as many intense east Pacific (full basin) hurricanes are observed during El Niño years compared to La Niña years (Gray and Sheaffer 1991; Whitney and Hobgood 1997). Furthermore, a change in tropical cyclone (TC) genesis average longitude is evident; tropical storms forming during El Niño events tend to form 5.7°W of the mean origin location when compared to La Niña storm genesis (Irwin and Davis 1999).

A more recent topic of interest is El Niño Modoki, characterized by central Pacific warming (CPW). El Niño Modoki events are thought to be becoming more likely due to recent global warming, a relationship hypothesized because these events were first observed in the 1990s (Yeh et al. 2009). Kim et al. (2009) found that during these CPW events, an increase in Atlantic basin tropical cyclone activity occurs. However, studies such as Lee et al. (2010) find that it is premature to draw strong conclusions about a possible relationship between CPW and Atlantic tropical cyclones.

On interannual and longer time scales, variations in sea surface temperatures (SSTs) in the Atlantic Main Development Region (MDR; 6°–18°N, 20°–60°W) have been linked to variations in Atlantic basin tropical cyclone activity (Mann and Emanuel 2006; Elsner 2006; Sabbatelli and Mann 2007). Anomalously high (low) SSTs in the MDR tend to concur with increased (decreased) TC activity. One form of variability that can modulate SSTs in the MDR, the Atlantic multidecadal oscillation (AMO), affects TC counts with a time scale of approximately 60–80 yr (Schlesinger and Ramankutty 1994). Indices representing the AMO are typically derived from North Atlantic SST anomalies north of the equator; however, deficiencies in the statistical representation of the AMO signal in indices have been observed (Goldenberg et al. 2001). Rather than using an AMO index, we examine MDR SST, which includes the influence of the AMO as well as other processes that modulate year-to-year variability in SST. Furthermore, MDR SSTs have been shown to be a proxy of potential TC intensity, particularly when considering MDR SST relative to the tropical mean SST (Emanuel 2005; Vecchi et al. 2008).

On the intraseasonal time scale, the Madden–Julian oscillation (MJO) has been observed to modulate TC activity in both the Atlantic and east Pacific basins. Maloney and Hartmann (2000a,b) found that MJO-induced low-level wind anomalies over the east Pacific results in circulation changes over the east Pacific, Gulf of Mexico, and Caribbean regions. These circulation changes have been observed to affect the development of TCs; over the Gulf of Mexico and Caribbean regions, cyclone genesis during low-level east Pacific westerly anomalies outnumbers that during easterly anomalies by about 4 to 1 (Maloney and Hartmann 2000a). MJO-induced changes over the Atlantic MDR have also been observed (Mo 2000; Maloney and Shaman 2008; Klotzbach 2010).

Various statistical models have been developed with the goal of tropical cyclogenesis prediction that exploits modes of atmospheric variability. An empirical method of intraseasonal statistical prediction of TC probabilities for all major basins was developed by P. Roundy (see http://www.atmos.albany.edu/facstaff/roundy/tcforecast/tcforecast.html) based on the relationship between TCs and various wave modes. A caveat in generating the predictors from outgoing longwave radiation (OLR), as noted by Frank and Roundy (2006), is that there are instances in which waves will affect TC genesis without significantly altering the OLR pattern but rather through fluctuations in low-level vorticity and/or vertical shear (e.g., MJO effects on Atlantic and east Pacific basins).

Leroy and Wheeler (2008) generated an intraseasonal multiple logistic regression model to predict Southern Hemisphere weekly TC activity. This model includes five predictors: two real-time multivariate MJO indices developed by Wheeler and Hendon (2004), the two leading modes of interannual SST variability over the Indo-Pacific region, and the climatological seasonal cycle of tropical cyclone activity. A comparison between the Leroy and Wheeler model, recently improved to include a gridded framework that allows spatial variability, and Southern Hemisphere TC forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF) forecast model is provided in Vitart et al. (2010), justifying the usefulness of such a statistical approach. While the study’s results are dependent on the skill score calculation used, the comparison found that the statistical model was more reliable and showed higher skill past a week-1 forecast lead than the dynamical ECMWF model. The study demonstrated that after calibrating the dynamical model to reduce error and then using this improved model in combination with the statistical model, better results were obtained than using either model separately, further justifying the development of such a statistical tool.

Based on the aforementioned physical relationships between climate variability on various time scales and Atlantic and east Pacific TC genesis, as well as motivated by the success of the Leroy and Wheeler (2008) Southern Hemisphere cyclone prediction model, a real-time multiple logistic regression model for the prediction of weekly tropical cyclone genesis probabilities is created for the Atlantic and east Pacific basins. Section 2 describes the development and justification of the predictors chosen. Section 3 explains the forecast scheme and details the forward selection scheme utilized by the model. Section 4 provides and discusses example hindcasts. Section 5 focuses on the skill and reliability of the model. Last, section 6 summarizes the model results, and future work is discussed.

## 2. Predictor development and justification

### a. A cyclogenesis index

A binary cyclogenesis index is developed for both the Atlantic and east Pacific basins. Since weekly probability forecasts are the desired outcome, a 1 is assigned if at least one TC formed within a given week and a 0 if cyclogenesis was not observed. In the development of this index, TC observations are obtained from the National Hurricane Center (NHC) Atlantic and east Pacific hurricane best-track database (HURDAT), using the first recorded observation of the storm as the cyclogenesis time (Davis et al. 1984; Jarvinen et al. 1984). Only cyclones reaching at least tropical storm strength (34 kt, 1 kt = 0.5144 m s^{−1}) are considered. Figure 1 illustrates the location of cyclogenesis during 1975–2009, excluding the year 1978. The model is developed using data from 1975–2009, with the exception of 1978 because of missing MJO index data during that year (discussed in section 2c). Note that we bound the east Pacific basin at 120°W, similar to the Collins and Mason (2000) subregion boundary of the east Pacific discussed in the introduction.

### b. Basin climatology predictor

The climatology of TC genesis is based on historical data and represents the average occurrence of TC formation. A climatology of tropical cyclogenesis is developed for each basin as a predictor to represent the TC seasonal cycle (Fig. 2). A raw climatology is calculated by averaging the stratified cyclogenesis index at every week for each year (dashed curve). The raw climatology is then smoothed with a 1–2–1 filter four times for the Atlantic and twice for the east Pacific (solid curve). The smoothed climatology of each basin is used as a predictor in the final models. The east Pacific climatology is smoothed less than the Atlantic in order to preserve the probability minimum in late July and early August. This minimum is likely due to the midsummer drought, shown to be associated with a decrease in TC activity in the east Pacific basin (Magaña et al. 1999). Figure 2 also shows the time period used for model development for each basin, with 15 May–30 November used for the east Pacific and 1 July–31 October used for the Atlantic. In the case of the Atlantic, better model performance was obtained with a time domain centered over only the most active part of the hurricane season, primarily because of the MJO predictor. We believe this is because the higher climatological probability of TC genesis appears to be associated with a stronger statistical relationship between the MJO and genesis. We are not entirely sure why this should be the case, although one possible link may be through the MJO modulation of African easterly wave activity (e.g., Alaka and Maloney 2012). African easterly wave activity peaks in the middle of the Atlantic hurricane season (e.g., Thorncroft and Hodges 2001).

### c. The MJO as an intraseasonal predictor of tropical cyclogenesis

To represent the eastward propagation of the MJO in the model, the Wheeler and Hendon (2004) Real-time Multivariate MJO (RMM) indices, known as RMM1 and RMM2, are used as predictors in both the Atlantic and east Pacific models. These indices are derived from the first two empirical orthogonal functions (EOFs) of the equatorially averaged (15°S–15°N) and normalized 200- and 850-mb zonal wind fields and satellite OLR. Values from 17 March to 31 December 1978 are missing from the RMM datasets because of missing OLR data; for this reason the year 1978 is excluded from all calculations in this study. To illustrate the relationship between tropical cyclogenesis and the MJO, genesis locations are binned according to eight MJO phases as defined by Wheeler and Hendon (2004). If the RMM amplitude, defined as , is less than one, the MJO event and corresponding TCs are excluded. Results are shown in Fig. 3 for both basins.

Over the Atlantic basin, most TCs initiate genesis during phases 1 and 2 of the MJO, agreeing with results shown by previous studies (Klotzbach 2010; Kossin et al. 2010; Ventrice et al. 2011). Cyclogenesis is least frequent during phase 7. Normalized to the number of days in each phase, approximately one-third more cyclones form during phases 1 and 2 than phases 6 and 7. In the Gulf of Mexico and northwest Caribbean regions alone, we find that approximately 4 times more cyclones are observed during phases 1 and 2 than phases 6 and 7, in agreement with Maloney and Hartmann (2000a).

For the east Pacific, cyclogenesis is more likely when convection is suppressed over the east Indian Ocean and Maritime Continent (Maloney and Hartmann 2000b). This occurs approximately around phases 8 and 1 of the RMM MJO indices. In contrast, cyclones are less likely to occur when convection is enhanced over the east Indian Ocean and Maritime Continent. This is observed during RMM MJO phase 4. Approximately 4 times more cyclones are observed in phase 1 than phases 3 or 4. Normalized to the number of days in each phase, approximately twice as many storms form during phases 8 and 1 than during phases 3 and 4. Given physical reasoning from previous literature and the basic relationships just shown between TC genesis and the MJO, RMM1 and RMM2 are both used in the final model.

### d. Equatorial Pacific SST patterns

Two other predictors considered in this study are the leading modes of tropical east Pacific SST. To formulate the predictor indices, near-equatorial Pacific SST data are used from the 1° gridded, global Hadley Centre Sea Ice and Sea Surface Temperature (HadISST) reanalysis dataset (Rayner et al. 2003). A three-point running mean average is applied to the monthly SST data to reduce intraseasonal influence. Furthermore, the annual cycle and long-term mean are removed. The first two EOFs are calculated via the covariance matrix in the region 30°S–30°N, 70°W–110°E. The predictor indices are represented by the standardized principal components (PCs) of the EOFs. Figure 4 displays the regression of the unweighted SST data onto the two leading EOFs.

The first leading pattern is representative of ENSO, characterized by the equatorial east Pacific warming of SSTs coinciding with SST cooling over the western Pacific. The second leading mode of variability is representative of the central Pacific El Niño. This mode is characterized by CPW events and, as mentioned in the introduction, Kim et al. (2009) hypothesized that during these events, an increase in Atlantic basin TC activity occurs. However, since 1950 only 5 yr were representative of CPW events (Kim et al. 2009). Furthermore, Lee et al. (2010) concluded that some of the increases in Atlantic storms during CPW events were likely influenced by other sources, such as a larger Atlantic warm pool. The study noted that it is too early to associate CPW events with an increase in TCs in the Caribbean and Gulf of Mexico regions. Because of the small sample size of such events within the full time frame of this study, as well as the uncertain relationship between CPW events and Atlantic TCs, the second EOF is not used in the final statistical model developed in this study. However, for thoroughness we test the relationship between CPW events and Atlantic TCs as we do for all predictors (section 3b).

To show the relationship between the El Niño index developed and TC genesis in both basins, the El Niño index is interpolated to weekly resolution and probability curves are generated. Warm (cold) events are defined when the El Niño index is greater (less) than 1 (−1) standard deviations. The cyclogenesis index is separated between warm and cold events and weekly probabilities are calculated. To remove high-frequency variance, the resulting weekly values are smoothed by applying a 1–2–1 filter 9 times. These are shown as a percentage in Figs. 5 and 6 for the Atlantic and east Pacific basins, respectively. Higher probabilities signify that a higher number of TC genesis events were observed in a given week from 1975–2009 in one phase of ENSO relative to the other phase.

In the Atlantic basin, warm ENSO events are associated with an overall reduction in TC activity (Gray 1984a; Shaman et al. 2009). Figure 5 demonstrates this relationship and supports the use of ENSO as a predictor in the finalized model. For the east Pacific (Fig. 6), however, a clear relationship between ENSO phase and tropical cyclogenesis is not evident, in agreement with Collins and Mason (2000). This implies that ENSO may not be useful as a predictor. Model results using only storms that become hurricanes are later discussed, and ENSO may be an important factor for the east Pacific with such conditional sampling. Regardless, if ENSO does not benefit the finalized model for the east Pacific, the forward selection scheme (discussed in section 4) will not select it; therefore, retaining it in the model will not adversely affect the results.

### e. MDR SST as a predictor of tropical cyclogenesis in the Atlantic basin

The HadISST dataset is used to develop an index to represent SST anomalies in the MDR region (6°–18°N, 20°–60°W). As was done for the El Niño index, the annual cycle and long-term mean are removed from the SST monthly dataset. The El Niño index and the MDR SST index generated are found to be minimally correlated at lag 0 (*r* = 0.16) for July–October, ensuring the independence of the predictors in the Atlantic model. To demonstrate the relationship between Atlantic tropical cyclogenesis and MDR SST, anomalously cold and warm events are used to calculate weekly probability curves in the same way as for ENSO. Anomalously cold (warm) events are defined when the MDR SST index is below (above) −1 (+1) standard deviations. Results are shown in Fig. 7.

The figure shows that during anomalously warm MDR SSTs more TCs tend to form in the Atlantic basin than during anomalously cold MDR SSTs, especially in July and early August. The difference in TC probability between the early and late season agrees with the results of DeMaria et al. (2001). Climatologically, the beginning of a hurricane season is restricted by thermodynamic effects. Since favorable SSTs are important for sufficient vertical instability to generate TCs, early season anomalously cold SSTs would inhibit instability and therefore TC formation. Later in the season, however, instability tends to be climatologically favorable and dynamical effects are what limit TC activity.

Based on the results shown in this simple analysis as well as the literature discussed in the introduction, MDR SST is considered a reasonable predictor in the Atlantic model.

## 3. A logistic regression model

### a. Background

As used by Leroy and Wheeler (2008), logistic regression is an ideal choice of prediction scheme as it allows the input predictand to be dichotomous (Hosmer and Lemeshow 2000). A dichotomous index represents “yes” or “no” events, denoted by a 1 (yes the event was observed) or a 0 (the event did not occur), as in the case of the cyclogenesis index developed. Logistic regression is defined as follows:

where represents the regression coefficients, or the relationship between the dichotomous predictand and the historical predictor values, and are the current predictor values for *m* number of predictors. The equation results in the conditional probability that the variable *Y* occurs given the value of *x*. The calculated probability is a value between 0 and 1, which can be easily interpreted as a percent probability of occurrence (Hosmer and Lemeshow 2000).

Forecasts using (1) are generated out to a seven-week lead (e.g., a four-week lead would use the most current predictor data to predict the probability of a storm developing four weeks from now). To eliminate any possible bias in the model with a storm forming within a certain day of the week, a week is defined beginning on every day within the cyclogenesis index, resulting in overlapping weekly probabilities as done by Leroy and Wheeler (2008). Furthermore, we lag all predictors to represent what is available in real time. Therefore, all datasets utilizing SSTs are lagged by one month and the RMM indices are lagged by one day.

### b. Univariate curves

While physical reasoning between the predictors chosen and TC genesis is discussed in the literature and by some simple analysis in this study, it is possible that a predictor will either not show a sufficiently strong relationship with TC genesis to be used for prediction or not be a good fit for logistic regression. To determine if the predictors are a good fit for the finalized model, univariate logistic regression models are calculated for each predictor at each forecast week lead while taking into account the appropriate lags based on when the data are available in real time. As an example, univariate curves for a two-week forecast lead for both basins are shown in Figs. 8 and 9. The logistic regression models (solid curves) are calculated independently of the averaged probabilities (dots). The fit appears to support the choice of model when the logistic curves match the averaged probabilities, while the relationship is represented by the slope of the curve.

To calculate the averaged probabilities (dots), the dichotomous Atlantic (east Pacific) TC observations are binned according to predictor value into 17 (20) bins of approximately 246 (340) values each and averaged for all predictors. The *x* axis of the climatology predictor in Figs. 8 and 9 represent the smoothed, stratified climatology values from Fig. 2; however, in the model they are cross validated in the sense that the hindcast year is excluded from each calculation of climatology. The *y* axis is the probability of a TC observation occurring within a certain climatological probability range. For example, the probability of an east Pacific cyclogenesis event in June based on only the smoothed climatology is between 25% and 45% according to Fig. 2, therefore, that event would be binned according to its smoothed climatological probability (between 25 and 45 in the *x* axis of Fig. 9). Climatology is the only predictor not lagged at each forecast lead. Since it is based on all other years with the exception of the forecasted year, it is always known and is therefore the same at all forecast leads (e.g., the first week of June climatology for a week-1 forecast lead will be the same as the first week of June climatology for a week-7 forecast lead). To better quantify the fit of the models shown in Figs. 8 and 9, a correlation between the logistic curve and the averaged probabilities for each predictor is provided in each plot. High correlations (e.g., climatology *r* = 0.97 for both basins) indicate a good logistic fit, while low correlations (e.g., east Pacific RMM2 *r* = 0.06) indicate a poor fit or a weak relationship. As previously mentioned, we include a logistic regression curve for CPW events (Fig. 8, bottom right, shown as PC2). With a low correlation at all forecast leads (only week 2 shown), we are further justified to not use CPW as a predictor for Atlantic genesis in the finalized model. We find that using a correlation coefficient to quantify the goodness of fit in the figures shown is justified since the relationships are mostly linear, although a more objective quantitative method is used to decide which predictors are useful in the finalized multivariate model. This method will be discussed next.

### c. Stepwise selection scheme

As a means of quantifying the usefulness of each predictor per forecast lead, a forward stepwise selection scheme is implemented to both the Atlantic and east Pacific models for every lag, as was done by the Leroy and Wheeler (2008) Southern Hemisphere model. Stepwise selection schemes are commonly used in regression as a basis for determining the “importance” of a variable, including it only if significant given a specific criterion (Hosmer and Lemeshow 2000). In a forward selection scheme, variables are sequentially added to an initially empty set until the addition of further variables no longer improves prediction, meaning it no longer decreases the criterion. Typically this test for significance, called the likelihood ratio test, is done by comparing the log likelihood of the model with and without a particular variable. It is represented by

where denotes the likelihood of the fitted model and is the likelihood of the saturated model, or the model that includes all the predictors. The result is a value known as the deviance, used as the criterion in this study. The deviance is analogous to the residual sum of squares in linear regression and follows a chi-square distribution (Hosmer and Lemeshow 2000; Cheng et al. 2006). The larger the deviance, the worse is the fit of the observed values to the expected values. The forward selection scheme stops once deviance can no longer be reduced, so no advantage is gained by including the other predictors. Using an inverse chi-square cumulative distribution function at 1 degree of freedom, the 95% significance critical value is found. One degree of freedom is used because a single variable is being added at a time in stepwise regression. This critical value is the minimum deviance difference value allowed in the selection scheme; if the value becomes less than the critical level the iterations terminate (Cheng et al. 2006).

An advantage of using a forward selection scheme is that it provides a ranking of the selected predictors, which shows their relative level of importance based on the significance criterion (Hosmer and Lemeshow 2000). This provides information on which predictors have the greatest statistical influence on tropical cyclogenesis in each basin for each lead in the models. Using the full dataset, Table 1 lists the ranks given by the forward selection scheme for each basin at every lead. A box is left blank if the predictor is not selected.

Climatology is the first chosen predictor for both basins at every lead, meaning it accounts for the most variability in the tropical cyclogenesis observations. In the Atlantic basin, MDR SST was chosen second at all forecast leads while ENSO was chosen third. Interestingly, ENSO was always chosen as a predictor in the east Pacific basin. Although always chosen last with the exception of week 4, it shows ENSO may have some statistically significant influence, although minor, in east Pacific cyclogenesis. One of the MJO indices was typically chosen as the second predictor for the east Pacific out to week 4. The selection between RMM1 and RMM2 for both basins agrees with the univariate curves (Fig. 9; only week 2 shown); the selection scheme tends to first choose the RMM index with the steepest curve, or not at all if there appears to be no relationship (i.e., has a “flat” logistic curve, as the case with RMM2 in Fig. 9 which was not chosen by the selection scheme). It also “catches” and excludes those predictors that appear to have a useful relationship in the univariate logistic regression curves (i.e., the curve has steepness) but have no significant relationship when compared to the TC observations. This includes not only cases when the averaged observed bin averages (i.e., solid circles in Figs. 8 and 9) appear as a scatterplot with no clear relationship about a steep logistic curve and hence have low correlation, but also cases such as RMM2 in Fig. 8 where the correlation appears erroneously useful because of an outlier point that is largely responsible for the high correlation.

## 4. Hindcasted probabilities

Probability hindcasts using multiple logistic regression are generated independently for each year from 1975 to 2009, with the exception of 1978, taking into account the real-time availability of each dataset used. Our hindcasts are generated in a cross-validated way whereby we exclude the year being hindcast from the computation of the regression coefficients. The regression coefficients are then used to “forecast” the excluded year (Elsner and Schmertmann 1994). Using (1), a relationship is found between the tropical cyclogenesis dichotomous index and each predictor. Although week 0 provides no predictability, it is shown as a means of comparison. The probability that a cyclone will form, based on the hindcasted year’s predictor values, is found for TCs reaching at least tropical storm strength.

Examples of the hindcasts out to a three-week forecast lead are shown in Figs. 10 and 11 for the Atlantic and east Pacific basins, respectively. The hindcasted probability curves generated by the logistic regression model using a forward selection scheme are shown by the solid curve. An independent logistic regression model and forward selection scheme are used for each forecast lead. Overlaid for comparison is a dashed curve representing the climatology of each basin. The calculation of the climatology shown excludes the year hindcasted. The gray bars along the *x* axis represent a week in which cyclogenesis occurred. Because of the overlapping weeks previously defined, each gray bar is at least one week long. The RMM indices provide the short-term variability on the order of days to a week in the probability hindcasts. ENSO and MDR SST, on the other hand, vary slowly on the order of months and longer; they have the tendency to shift most if not all of the full season hindcast curve.

Atlantic probability hindcasts are shown for the years 1982 and 2005 from 1 July to 31 October (Fig. 10). The unfavorable conditions throughout the 1982 season resulted in the formation of only three tropical storms and one hurricane in the seasonal range analyzed in this study according to the HURDAT dataset. A strong El Niño event along with anomalously low values of MDR SST decreased the forecasted probability of cyclogenesis by up to 25% at the shortest forecast lead times. Probabilities below climatology occur at all forecast leads during 1982 (week leads 4–7 not shown). Weak MJO conditions provided little intraseasonal variability in the probability forecasts.

The year 2005 is naturally of high interest because of its highly active Atlantic hurricane season. In the seasonal time range of this study, 8 tropical storms and 14 hurricanes formed. An increase in genesis probabilities above a seasonal climatology is observed at all forecast leads due to anomalously high values of MDR SST (Fig. 10). The intraseasonal variations in the forecasts are due to moderate MJO activity (amplitude > 1; Wheeler and Hendon 2004). At a week-1 forecast lead, probabilities increase from ~35% to ~65% in only a week and a half in the month of August. Late August and early September probability forecasts peak at approximately 15%–20% above the known seasonal climatology, coinciding with an observed peak in TC activity. Increased probabilities are also shown for the month of October, corresponding with observed high TC activity.

The east Pacific hindcasts range from 15 May (shown as M15 in the *x* axis) to 30 November. East Pacific hindcast probability curves are shown in Fig. 11 only for the year 2002 since the forecasts are dominated by the MJO predictors, variations in which provide all of the interesting cases. The 2002 season consisted of moderate El Niño conditions as well as strong MJO variability, with RMM amplitudes (as previously defined) between 1.5 and 2.9 from the beginning of the season through August (with the exception of the last week of July), and RMM amplitudes greater than 2 in November. Most of the variability in the probability hindcasts occurs during the periods of high-amplitude MJO activity. The week-1 forecast lead shows a probability increase of ~15% in mid-July, escalating the probability of cyclogenesis to almost 80%. Observations (gray bars) tend to coincide with high peaks in probability. The moderate El Niño had little effect on the probability curves, only slightly increasing probabilities.

## 5. Model skill and reliability

### a. Brier skill scores

Qualitatively it can be observed from the hindcasts if an increase or decrease in probability corresponds to a cyclogenesis event or lack thereof. To assess the skill of the hindcasts over all years, a quantitative statistical approach must be used. The most common method for verification of dichotomous events is the Brier score (BS), as used in the Leroy and Wheeler (2008) model, and is defined by the equation:

where represents the forecasted probability between 0 and 1 for event *i* out of a total of *n* events (Wilks 2006). The observations are dichotomous where if the event occurred and if it did not occur. The Brier score calculation in essence calculates the mean-squared error of the forecasted probability. Since both the forecasts and observations are bounded by 0 and 1, so is the Brier score. A Brier score of 0 denotes a perfect forecast, meaning always equals . In contrast, a Brier score of 1 indicates that the forecast is wrong for every event. Using the calculated Brier score, a Brier skill score (BSS) is then computed:

where BS is the Brier score calculated from the forecasts and represents a reference forecast Brier score. A Brier score using all hindcasted values and corresponding observational values of TCs is generated per basin and forecast lead using (3). A logistic regression model using only a seasonal mean climatology is used to generate reference hindcasted probabilities, as in Leroy and Wheeler (2008); hence, the same value is predicted for every day representing the seasonal average probability of cyclogenesis within that basin. These reference probabilities are then used to calculate the reference Brier score. The Brier skill scores are multiplied by 100 to denote a percentage. This is ideal as it allows the Brier skill score to represent a percent decrease in mean squared error (by the forecasts) over a mean seasonal climatology. A Brier skill score of 100% is achieved when every forecasted probability is perfect, meaning only a 0% or 100% is forecasted every time in perfect accordance with observations. Statistical models tend to have a lower sharpness (the capability of the model to deviate from its mean climatology; Vitart et al. 2010). For this reason, statistical models rarely predict very low or very high probabilities and a 100% Brier skill score is not possible with the model developed in this study.

Five (four) separate sets of Brier scores are calculated for the Atlantic (east Pacific) basin per week lead and plotted in Fig. 12. The solid black curve denotes the Brier skill scores for the hindcast probabilities generated with all available predictors using the selection scheme. Individual hindcast probabilities are also calculated using only a set of predictors in conjunction with the time-varying climatology: MJO + climatology, ENSO + climatology, MDR SST + climatology, as well as the time-varying climatology alone, represented as shown in the legend. A comparison can then be made to represent the relative importance of the predictors and the skill improvement when compared to a mean seasonal climatology as well as to the time-varying climatology. Furthermore, we generated Brier skill scores using the time-varying climatology as the reference climatology (not shown). This results in a shift in the skill of all predictors so that the time-varying climatology BSS becomes zero. The change it makes in skill difference between the predictors is miniscule. We decided to use the seasonal mean-climatology to provide the additional information on the skill of the time-varying climatology relative to a mean climatology.

The Atlantic climatological time-varying Brier skill score shows an improvement of approximately 8.5% over the model using only a seasonal mean climatology. The addition of the MJO alone to the time-varying climatology shows a skill increase out to forecast lead week 2 of around 0.5%. The Brier skill score of the model using ENSO + climatology generates an improvement greater than 1% over the time-varying climatology alone, decreasing only slightly at the longest lead times. The MDR SST + climatology model improves skill by approximately 2.5% from the time-varying climatology alone with a decrease in skill at longer forecast leads. Including all selected predictors, there is a skill improvement of over 4.5% in the BSS over the model using only the time-varying climatology, with this skill generally decreasing over forecast lead. Note that the BSS of individual models do not add linearly to result in the model including all selected predictors. This is due to the nature of (3) as well as the forward selection scheme included in the complete model. Overall, the model of the selected predictors improves the skill of the model by over 13% at the shortest leads relative to the mean seasonal climatology.

The east Pacific varying climatology BSS alone shows almost a 15.5% improvement over using a mean seasonal climatology. Not surprisingly given the discussion above, including ENSO as a predictor adds only an additional small fraction of a percent improvement without much change over week lead. Including the MJO with climatology improves the skill by over 1% during weeks 1–2, with further slight improvement given the predictor selection scheme; these tend to decrease toward the climatology model skill over longer forecast leads. Overall, including all selected predictors improves the skill of the model by almost 17% at the shortest leads relative to the mean seasonal climatology. The Brier skill score values found in this study are of a similar magnitude to those in Leroy and Wheeler (2008) for Southern Hemisphere TC activity.

### b. Reliability diagrams

Testing the reliability of a model is commonly done via reliability diagrams (e.g., Leroy and Wheeler 2008). This is done by binning the dichotomous genesis observations and the hindcasted probabilities according to the hindcasted probability. For the Atlantic, observations and hindcasted probabilities are binned into 17 groups of roughly 246 values each. For the east Pacific the bins consist of 20 groups of approximately 340 values each. For each group the bins are averaged and plotted to form the reliability curve.

The perfect hindcast, shown by the solid diagonal line in Figs. 13 and 14, occurs when the hindcasted probability equals the observed probability. A 10% interval about that perfect curve is shown by the dashed diagonal lines. When the reliability curve lies above (below) the perfect diagonal, the hindcast underestimates (overestimates) cyclone genesis relative to observations. The mean observed probability is shown by the horizontal line.

Atlantic basin reliability curves out to a week-2 forecast lead are shown in Fig. 13. The first average category (first circle) greater than 0.1 probability reflects that we use only 1 July–31 October in the Atlantic model. In the case of the east Pacific (Fig. 14), probability of TC genesis is slightly overestimated by 0.05 for the first two bins (the two lowest probability groups) at every forecast lead. Sensitivity tests on the smoothing of climatology found the overestimation is not due to the smoothing; rather, Fig. 9 suggests the logistic fit of climatology causes the overestimation of the lowest probabilities. Overall the models are shown to be fairly reliable, with reliability curves near the diagonal line.

### c. Hurricane-strength *BSS*

Various studies have analyzed the relationship between hurricane strength storms (≥64 kt) and the MJO for the Atlantic (Barrett and Leslie 2009; Maloney and Hartmann 2000a; Klotzbach 2010) and the east Pacific (Maloney and Hartmann 2000b; Chu 2004; Collins and Mason 2000). Hindcast probabilities are calculated using cyclogenesis observations of only storms that eventually reach hurricane strength using the same method as for the broader analysis of the tropical cyclones. Brier skill scores are also found in the same manner as for those of at least tropical storm strength, and the results of this analysis are shown in Fig. 15. A hindcast model of mean seasonal climatology of genesis of storms that eventually reach hurricane strength is used as a reference Brier score.

The Atlantic hurricane-strength Brier scores show an improvement in the MJO predictor skill, increasing the skill of the MJO + climatology model out to a week-3 forecast lead when compared to the model including tropical storms. At week 1 the MJO appears as equally skilled as ENSO. The importance of MDR SST to Atlantic hurricanes in shown by the MDR SST + climatology model, with the highest skill improvement over a time-varying climatology when compared to the other individual predictor models. Improvements in forecast lead in the hurricane model when compared to the tropical cyclone model are likely due to favorable conditions related to the predictors. A storm is more likely to strengthen to hurricane strength if conditions remain favorable; however, in unfavorable conditions, a storm may dissipate before it has a chance to strengthen. At the shortest forecast lead times, a skill improvement of approximately 13.5% above a mean seasonal climatology of hurricanes is shown in the full predictor model.

Unlike the Atlantic model, the east Pacific model showed no major improvement in skill when restricting the cyclogenesis index to include only hurricane-strength storms. The skill provided by the ENSO + climatology model is just slightly greater than for the time-varying climatology model alone. Restricting this analysis to only storms that reach major hurricane strength (category 3+) would likely improve the results based on the literature discussed in the introduction. The skill above a time-varying climatology appears to be due to the MJO predictor.

## 6. Summary and conclusions

This study describes the development of a real-time intraseasonal prediction model for tropical cyclogenesis in the Atlantic and east Pacific Ocean basins based on multiple logistic regression. Predictors used include ENSO, the MJO, and a climatology of genesis for each basin, as well as an index to represent MDR SST in the Atlantic. Each dataset is lagged appropriately corresponding to its availability in real time. An independent model for each basin is generated out to a 7-week forecast lead. After undergoing a forward selection process, the predictors selected are used to calculate regression coefficients that are input into the regression model to generate cross-validated hindcasts for each year from 1975 to 2009 (1978 excepted).

Brier skill scores and reliability diagrams are generated to determine the skill and dependability of the models. Results show an increase in model skill at predicting tropical cyclogenesis by the inclusion of the MJO out to a three-week forecast lead for the east Pacific and a two-week forecast lead for the Atlantic. Furthermore, the importance of MDR SST is shown in the Atlantic Brier skill scores, with a higher skill improvement over a time-varying climatology than for all other predictors out to a week-6 forecast lead. The inclusion of ENSO only slightly improved the skill of the east Pacific model. The model generated by the forward selection scheme showed improvements above a mean seasonal climatology of almost 17% for the east Pacific and over 13% for the Atlantic.

When only considering storms that reach hurricane strength, the inclusion of the MJO in the Atlantic models show a further increase in skill out to a three-week forecast lead, with similar skill improvements above a time-varying climatology as produced by ENSO for a week-1 forecast lead. Including ENSO and MDR SST increased the skill of the Atlantic model over a time-varying climatology out to a seven-week forecast lead. For the east Pacific hurricane model no real improvement in skill was observed by including ENSO. The modest effect of ENSO on the east Pacific basin models agree with the results of Collins and Mason (2000).

The results found in this study are similar to those in the Southern Hemisphere tropical cyclone statistical model of Leroy and Wheeler (2008). The primary differences between the models developed here and the Leroy and Wheeler (2008) model are a result of the strength of the predictor for each basin (e.g., the MJO has a stronger influence over the Indian–west Pacific Ocean region than over the Atlantic basin), as well as the inclusion of MDR SST in the Atlantic model. However, skill improvements due to the inclusion of the selected predictors are of the same order of magnitude as in Leroy and Wheeler (2008) and reliability curves show a dependable model.

One drawback to statistical models is the lower sharpness that comes with such models (Vitart et al. 2010). As previously mentioned, statistical models rarely predict very low or very high probabilities largely because of the constraints provided by climatology; for this reason, a BSS of 100% is not possible with such a statistical model. Furthermore, the models currently require ENSO, MJO, or MDR SST activity to be present to improve skill over the time-varying climatology. ENSO-neutral seasons with weak MJO activity as well as average MDR SST conditions (in the Atlantic model) follow the known climatological probabilities in the forecasts. In addition, the model currently lacks other forms of variability that have been shown to affect TC activity, some of which we hope to implement in future model versions and are discussed below.

Future work will focus on improving forecasting skill and extending the lead time of useful skill. A means of working toward the improvement of the models is the inclusion of other predictors. Some predictors under consideration include an index of West African monsoon intraseasonal variability (Bunting et al. 1975; Gray 1990; Gray et al. 1993; Landsea and Gray 1992; Maloney and Shaman 2008), the North Atlantic Oscillation (NAO; Elsner 2003; Elsner et al. 2000, 2001; Molinari and Mestas-Nuñez 2003), and the Pacific decadal oscillation (PDO; Lupo et al. 2008). Such phenomena have been shown to have robust relationships with Atlantic tropical cyclone activity.

Improvements of current predictors are also being pursued. Variations in the MDR SST predictor were considered. Motivated by recent works such as Vecchi et al. (2008), an index of MDR SST relative to the tropical mean SST (SST_{mean} − SST_{MDR}) was tested as a predictor. While it contributed positively relative to an individual model of MDR SST + climatology, little change was observed in the full model. Removing the MDR SST signal from the mean tropical SST revealed a signature strongly influenced by ENSO, which is already implemented in the model. For this reason MDR SST alone was used as a predictor. While the mean tropical SST contains variability distinct from ENSO and MDR SST, a separate index would have to be examined in order to avoid high correlations between predictors. Additionally, ways of utilizing differences within predictors as a function of time of the year will be explored. Figure 6 demonstrates differences between El Niño and La Niña east Pacific TC probabilities during the months of September and October, and Fig. 7 shows that Atlantic TC probability differences with MDR SST are strongest from the early season through August. Currently the model is not dependent on day of the year when considering contributions from ENSO and MDR SST. Future work will include ways to further utilize predictor seasonal variations in order to improve forecast skill.

Furthermore, additional spatial improvements can be considered to engender model improvements. Differences in tropical cyclogenesis mechanisms have been observed in subregions of the Atlantic basin, therefore, regional-scale prediction might benefit from different predictors (Ballenzweig 1959; Goldenberg and Shapiro 1996; Hess and Elsner 1994; Hess et al. 1995). Sensitivity tests conducted in this study by dividing the Atlantic basin into two and three subregions found that the logistic fitted MJO signal was strongest for the full basin. Further sensitivity tests may reveal a better definition of subregions that may improve forecast skill. While spatial improvements can benefit predictor selection, it can also benefit preparedness. Knowing more specifically where a cyclone is likely to form will benefit those living nearby who may be affected (Lehmiller et al. 1997).

## Acknowledgments

We thank Matthew Wheeler and one anonymous reviewer whose constructive and thorough comments greatly improved the manuscript. We would also like to thank Anne Leroy, David Thompson, Edwin Chong, Phillip Chapman, and Philip Klotzbach for discussions that improved this work. Furthermore, we are grateful for the financial support of the Center for Multiscale Modeling of Atmospheric Processes (CMMAP) and the Alliance for Graduate Education and the Professoriate (AGEP). This work was supported by the Climate and Large-Scale Dynamics Program of the National Science Foundation under Grants AGS-0946911 and AGS-1025584, and by the NOAA/MAPP Program under Contracts NA08OAR4320893 and NA12OAR4310077. The statements, findings, conclusions, and recommendations do not necessarily reflect the views of NSF, NOAA, or the Department of Commerce.

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