**Objective Partitioning of 5-day Ensemble Forecasts of Tropical Cyclones**

Jenni L. Evans and Alex M. Kowaleski

Tropical cyclone (TC) track forecasts pose challenges if substantial inter-model forecast spread or bifurcation exists. Clustering of ensemble prediction system (EPS) forecasts provides an objective method of grouping ensemble members with related features, achieving maximum within-cluster similarity and among-cluster difference.

We implement and test a method for path clustering of ensemble track forecasts using probabilistic regression mixture models (Gaffney 2007). We apply this method to TC track forecasts from the 50-member European Centre for Medium-Range Weather Forecasting (ECMWF) Integrated Forecast System (IFS). We consider a set of thirty initialization times from three TCs: Western Pacific Typhoons Sinlaku (2008) and Noul (2015) and Atlantic Hurricane Ike (2008). Clustering is performed on forecast storm tracks out to 120 hours (Kuruppumullage Don et al. 2015).

In mixture-model path clustering, the polynomial order of each model and number of models (clusters) must be selected to define the mixture model. Therefore, for each initialization time we perform clustering using two through seven clusters and 1^{st}-through 5^{th}-order polynomials. We use the Bayesian Information Criterion (BIC) and fraction of cluster assignments with assignment probability below 0.95 (F_{0.95}) to determine the optimal model specification. Low BIC values indicate a balance between goodness-of-fit and model simplicity, while low F_{0.95} values indicate unambiguous cluster assignments.

We also formulate variants of BIC and F_{0.95 }using cross-validation with resampling. In cross-validation diagnostics the 50 ensemble forecasts are split into a training set (40 members) and a test set (10 members). Mixture-model parameters are calculated from the training set, and posterior BIC and F_{0.95 }values_{ }are calculated on the test set. This procedure is repeated 100 times for each model specification, providing additional model selection criteria (Kuruppumullage Don et al. 2015). Mixture-model selection is illustrated using aggregate data from all 30 initialization times.

Both traditional and cross-validation F_{0.95} (not shown) indicate that 1^{st}-order trajectories typically produce the largest fraction of cluster assignments below 0.95. However, neither F_{0.95} diagnostic provides substantial discrimination beyond eliminating 1^{st}-order solutions.

BIC diagnostics provide more information. For traditional BIC values there are substantial improvements between 1^{st}-order and 2^{nd}-order trajectories throughout the forecasts. For some forecasts there is an additional advantage in increasing polynomial order to 3^{rd}. Importantly, traditional BIC exhibits no substantial advantage beyond 3^{rd}-order for any of the forecasts, suggesting that 3^{rd}-order suffices to capture the complexity of all forecast tracks.

Figure 1. Traditional BIC with polynomial order and number of clusters for all forecasts. Polynomial order is indicated by line color.

The cross-validation BIC (not shown) favors more parsimonious model specifications, at times indicating use of 1^{st} or 2^{nd}-order polynomials. Importantly, cross-validation BIC increases (deteriorates) for polynomial orders beyond 3^{rd} for most forecasts. This reinforces the evidence from traditional BIC that 3^{rd}-order polynomials suffice.

The cross-validation BIC also supports a smaller number of clusters that the traditional BIC. For 3^{rd}-order polynomials (red curves), whereas traditional BIC decreases monotonically with number of clusters, cross-validation BIC usually reaches a minimum between three and five clusters or *increases* monotonically with cluster number.

The 30 ECMWF IFS forecasts considered here differ substantially in complexity. Nevertheless, we can conclude from traditional and cross-validation diagnostics that mixture models with 5 clusters and 3^{rd}-order trajectories suffice to capture the variation in all forecasts among the three storms examined.

Six tracks of Hurricane Ike produced by the 3^{rd}-order, 5-cluster mixture model are shown to illustrate the trajectories produced by path clustering (Fig. 2). This specification yields mean trajectories that are distinct and meaningful as potential paths for the storm. This is true throughout initialization times, but an especially effective use of clustering is demonstrated for forecasts initialized at 12 UTC 5 September (Fig. 2b).

For this initialization time, in the red mean trajectory Ike moves the farthest south, before turning westward and making landfall on the Yucatan Peninsula. The other mean trajectories are progressively farther to the northeast, with Ike moving through the Yucatan Channel (magenta), over western Cuba (blue), into the Straits of Florida (cyan), and into South Florida (green). The blue cluster is the most populous (16 members). In contrast, the red and green clusters are the smallest (6 members each). The 3^{rd}-order, 5-cluster specification effectively captures the variation in IFS ensemble members, providing information on the potential tracks of Ike (clusters' mean trajectories) and the likelihood of each such track (clusters' populations).

The five mean trajectories from 12 UTC 5 September show substantial spread in direction of motion, but less difference in speed, indicating that at this time direction was the primary uncertainty in the forecast track of Ike. At other initialization times, the five mean trajectories show substantial spread in speed of motion. For example, while the magenta and blue trajectories from the 00 UTC 7 September initialization (Fig. 2d) are similar in direction, the ensemble tracks are primarily partitioned between these two clusters based on speed: at 120 hours, the magenta cluster is substantially closer to landfall.

Figure 2. Ike mean cluster paths from (a) 00 UTC 0904, (b) 12 UTC 0905, (c) 12 UTC 0906, (d) 00UTC 0907, (e) 00 UTC 0908, and (f) 12 UTC 0908. The most populous cluster is represented by a thicker line. The best track is in black.

The 3^{rd}-order, five-cluster mixture model solutions chosen capture the track variability in both speed and direction of motion, synthesizing 50 IFS ensemble forecasts into a small number of representative and distinct trajectories. We are in the process of examining this method for a larger set of TCs to evaluate its potential as an operational forecast tool.

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