1) Global organization

The characterization of global organization utilizes the deviation angle variance technique (Pineros et al. 2008). We first compute the 2D image gradient of T_b at each position s = (x, y) [denoted ∇T_b(s)]. Consider the core structure of Edouard (2014) in Fig. 8: the image gradient of a perfectly axisymmetric TC is expected to point either directly toward or away from the TC center. The deviation angle compares the direction of ∇T_b(s) to the gradient directions expected of such an idealized storm. We denote the deviation angle of the image gradient of T_b at a point s as ψ(s); values near ψ(s) = 0 indicate local axisymmetry.

We summarize the level of organization by taking the variance of ψ(s) over circular regions centered on the TC. We will denote the ORB statistic for global organization over a region of fixed radius r as

We will consider the value of DAV over a range of radii from r = 50 km to r = 400 km, resulting in ORB functions for global organization. Figure 1 shows DAV(r) as ORB functions for the two infrared images of Fig. 8.

2) Radial structure

The characterization of radial structures is based on radial profiles (e.g., Sanabia et al. 2014), the angular averages of T_b(s). At a fixed distance r from the TC center, define the ORB statistic for radial structure as

The primary complication comes from centering the coordinate system (i.e., defining the location of r = 0). Sanabia et al. (2014) solve this by manually identifying the center of each image, whereas we automate image centering by maximizing $\overline{T}\left(r\right)$ at low r when an eye is present (McNeely et al. 2019). In this work, we use the best track center when an eye is not detected. This automated image centering is also used in the computation of DAV(r).

Our treatment of the radial profile also differs from earlier work in its use of functional features, defined in section 3b, rather than designed features along the curve. Sanabia et al. (2014) identify several critical points along this curve for use as features, such as the radial location of the minimum T_b within 200 km. In lieu of such designed features on $\overline{T}\left(r\right)$, section 3b details a method for the recovery of data-driven functional features from general ORB functions; these radial profiles serve as ORB functions for radial structure.

Figure 10 demonstrates the relationship between the radial profile and TC intensity. Radial profiles discard angular information, making them informative primarily when bulk symmetry is high, as determined from bulk morphology ORB functions defined in the next section. In such cases, radial profile temperatures can help discriminate between eye–eyewall structures and uniform central dense overcasts. In an eye–eyewall structure, the temperature at the center r = 0 will be high, while the minimum of the radial profile, ${\min }_r\overline{T}\left(r\right)$, will be low; this typically indicates a strong TC. Conversely, a uniform central dense overcast is marked by low temperatures near the center r = 0 due to the absence of a clear, warm eye and generally indicates a TC is below hurricane strength (<64 kt). The slope of the profile also provides insight into the extent of the storm’s convection: gentle slopes indicate deep convection sustained over large regions, and sharp slopes indicate more localized deep convection most often associated with an eyewall.

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Hourly radial profiles: Example hourly eye-centered radial profiles for Hurricane Nicole (2016). Lighter colors indicate higher intensities. The inset depicts Nicole’s intensity trajectory. The high-intensity phase of the TC’s evolution corresponds to the clearest eye–eyewall structure.

Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-19-0286.1

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3) Bulk morphology

The characterization of the bulk morphology of T_b utilizes sublevel sets. A sublevel set (hereinafter “level set”) on T_b(s) identifies the points s = (x, y) for which T_b is at or below a certain threshold. Formally, define a level set as

As the temperature threshold c is reduced, the level set shrinks to a region of lower temperatures and typically stronger convection. This definition of $\mathcal{L}$(c) gives rise to a variety of ORB statistics for bulk morphology.

ORB currently uses four summaries for the structure of $\mathcal{L}$(c); see McNeely et al. (2019) for the full mathematical definitions. SIZE(c), a function of the level set $\mathcal{L}$(c), gives the area covered by $\mathcal{L}$(c) in kilometers squared (the black region in Fig. 2), providing insight into the coverage of various levels of convection in a TC. Similarly, SKEW(c) is a function of the level set $\mathcal{L}$(c) and gives the displacement of the center of mass of $\mathcal{L}$(c), normalized by the mean radius of points in $\mathcal{L}$(c), where “radius” is the distance from the TC center. This reveals whether the TC cloud pattern is biased in a particular direction (Fig. 2). The remaining two are measures of the raggedness (SHAPE) and stretching or eccentricity (ECC) of the level set $\mathcal{L}$(c). Note that any functions of level sets (such as our SIZE, SKEW, SHAPE, and ECC) can themselves be regarded as functions of the level set threshold c. Hence, we refer to their continuous approximations as ORB functions for bulk morphology.

1) Dimensionality reduction via PCA

When we densely sample the ORB statistics, we obtain approximations of continuous ORB functions. Now, we will switch to a different representation (coordinate system) in which we can reduce the dimensionality of the ORB functions while retaining most of the information in the original functions. (Here, “dimension” refers to the number of values we use to represent each function; thus, the original dimension of the ORB functions would be the number of threshold/sampling values.) A more compact representation simplifies the analysis and enables us to visualize the evolution of TC convective structure over time in a low-dimensional phase diagram defined by the new representation.

Given a suitable orthogonal basis, we can write any continuous function, such as the radial temperature profiles in Eq. (2), as a linear combination of basis functions. More specifically, after subtracting the average over the basin, the function f(x) describing deviations from the basin average is

where f_i(x) denotes the ith basis function and α_i = ⟨f, f_i⟩ is the scalar orthogonal projection of f(x) onto the ith basis function, representing how much of the shape of f_i(x) is present in f(x). We call the value of f_i(x) the loading of the basis function at x.

For continuous functions, the number of elements in a basis is infinite. However, provided that a proper basis is chosen, one can reasonably approximate functions using a finite linear combination of basis functions. In this work, we will use a data-driven approach called principal component analysis to find a low-dimensional basis representation of our ORB functions. PCA returns an ordered set of orthogonal and normalized vectors whose linear combinations can fully reconstruct the original vectors—that is, the densely sampled ORB functions. Henceforth we will refer to the basis functions f₁, f₂, … from PCA as empirical orthogonal functions, to emphasize that these basis functions are empirical (data driven) and orthogonal. We compute EOFs {f_i} separately for each basin and each ORB feature. The EOFs are constant in time, whereas the projections α_i of the stamps onto respective EOFs f_i are scalars varying with time. We refer to the latter α_i-values as ORB coefficients; these coefficients serve as a time-varying representation of TC convective structure.

The first few EOFs provide the most information, and in our case, these tend to capture larger-scale structures of an ORB function. We can project an observed ORB function (a single function computed from a stamp) onto the first K EOFs of that feature to obtain the best K-dimensional approximation. [Figure A1, described in more detail in appendix A, demonstrates such a reconstruction for radial (RAD); appendix A also includes details on PCA.] In our study, we use the first K coefficients {α₁, α₂, …, α_K} of each ORB function at each time stamp as both inputs (predictors) to machine-learning methods as well as a means to visualize the evolution of TC convective structure with time.

2) Analysis of TC structure and evolution

We compute basin-specific EOFs separately on 14 470 NAL stamps and 9818 ENP stamps (section 2). It only takes K = 2 or 3 EOFs f_i(x) to explain most of the variability in the six current ORB functions (i.e., DAV, RAD, SIZE, SKEW, SHAPE, and ECC). While objective methods (such as cross validation) exist for selecting the number of basis functions, we here use a heuristic 90% variance threshold. A larger set may prove useful in forecasting-only settings, but some of the higher-order basis functions lack the interpretability of the first two–three EOFs.

Reconstruction using the first three EOFs of radial structure explains 90% of the variance of observed radial profiles (Fig. 4). EOF 1 (solid purple) roughly measures the overall temperature anomaly of the profile, with higher loadings in the TC interior; f₁(r) is positive, so generally the corresponding ORB coefficient is positive (α₁ > 0) when the radial profile $\overline{T}\left(r\right)$ has warmer-than-average cloud tops. EOF 2 (dotted blue) measures the core temperature relative to outer-band temperatures. Since f₂(r) changes sign from positive to negative around 200 km, generally α₂ > 0 when the radial profile shows a warmer core; hence, α₂ < 0 could indicate a uniform central dense overcast or an obscured eye. EOF 3 (dashed green) indicates the presence of an eye–eyewall structure, where f₃(r) > 0 for r < 50 km and r > 300 km and f₃(r) < 0 elsewhere; hence generally α₃ > 0 indicates an IR-visible eye. These three EOFs alone can construct close approximations to the wide range of the profiles apparent in Fig. 10.

In Fig. 5, we apply the same ORB framework to the SIZE function for Hurricane Katrina (2005). For SIZE, we only need K = 2 EOFs to capture 90% of the variance in either basin. Because EOF 1 (solid purple) of SIZE measures the rate at which $\mathcal{L}$(c) accumulates area with increasing temperature thresholds c, α₁ for SIZE indicates colder overall T_b. EOF 2 (dashed green) flattens the middle of the SIZE function when α₂ > 0, resulting in more coverage at high temperatures (the exposed sea surface) and low temperatures (deep convection).

Although this study’s classification methods use 6-h values to match predictors from the SHIPS developmental database (section 4), one benefit of T_b observations is high temporal resolution. It is straightforward to track the structural evolution of TCs with ORB coefficients. The EOFs f_i(x) for each ORB feature are constant within a basin, but the ORB coefficients are implicitly functions of time; that is, α_i = α_i(t). One temporal phenomenon that can be observed in TCs is the diurnal cycle (e.g., Dunion et al. 2014), where deeper convection occurs near the TC core around local sunset and then spreads radially outward through the following afternoon. In Fig 11, the time evolution of the first two ORB coefficients for SIZE exhibits a 24-h periodic clockwise oscillation for Hurricane Katrina (2005). Though SIZE is insensitive to the location of convection and thus cannot measure the outward motion of such pulses, the observed oscillations in this phase space may be a manifestation of the TC diurnal cycle. In the phase diagram, we see two primary deviations from this cycle: when the TC turns north and accelerates [near point D, when it becomes a major hurricane (100 kt)] and when the TC begins to interact with land (the rightward translation toward point E).

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Visualizing SIZE evolution with ORB: (a) The spatial trajectory shows the path of Hurricane Katrina (2005) through the Gulf of Mexico from 1800 UTC 25 Aug through 0000 UTC 30 Aug, with the labels marking the locations of displayed ORB functions. (b) At six points (A–F, labeled) in time, we plot the SIZE function (black) and the sample mean SIZE function for the NAL basin (gray). (c) The trajectory of the ORB coefficients of the SIZE function in phase space during the storm’s path. The clockwise cyclic path in (c) matches the local night/day cycle as well as the storm’s path in (a); see the text for details.

Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-19-0286.1

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1) Logistic lasso

Logistic regression is similar to traditional linear regression but bounds the mean response $\mathbb{E}$(Y|x) = $\mathbb{P}$(Y= 1|x), or equivalently the probabilities p(x) ≡ $\mathbb{P}$(Y = 1|x), to values between 0 and 1. We also assume a different distribution for the response, in this case the binary response or “labels” Y_i, than in traditional linear regression. Rather than directly fitting Y_i using linear regression with independent and identically distributed (iid) errors ε_i from a normal distribution, we instead assume that the labels Y_i (conditional on x_i) are iid observations from a Bernoulli distribution with parameter p_i. We then fit a linear function to logit transformations of p_i; this results in the generalized linear model

where x_ij represents the jth predictor or the jth component of the input vector x_i. As in standard linear regression, we can use maximum likelihood estimation (MLE) to find the best-fitting β = (β₁, …, β_d) coefficients. Note that our data come from time series of intensities; due to temporal correlations, the assumption of conditional independence Y_i|x_i may not hold for events close in time. Nevertheless, our logistic lasso model can still provide useful diagnosis.

The advantage of a linear model, or in this case a generalized linear model, is that it is easy to directly relate predictors to the mean response $\mathbb{E}$(Y_i|x_i). Recall that the coefficients in a traditional linear regression model Y_i = β₀ + β₁x_i1 +$\cdots$ + β_dx_id + ε_i with normally distributed errors ε_i reflect how important a predictor is relative to the other variables in the model. In a linear model, the regression coefficient β_j tells us that an increase in the jth variable x_ij by one unit, while holding all the other d–1 variables constant, is on average associated with an increase (if β_j > 0) or decrease (if β_j < 0) in the response Y_i by |β_j| units. The linearity of logistic regression admits a similarly straightforward interpretation that relates the probabilities p_i to the predictors. From Eq. (5) we have that an increase of x_ij by one unit, holding all other variables constant, is associated with a change of the log odds [i.e., the logarithm of the odds p_i/(1 − p_i)] of the event Y_i = 1 by β_j, which is equivalent to multiplying the odds by exp(β_j).

Unfortunately, when d is large (i.e., when we have many predictors), the estimate $\widehat{\mathit{\beta }}$ is highly variable. The statistical model also becomes difficult to interpret. To reduce the variance of the statistical model and to help interpretability, one typically reduces the size of the model by excluding variables that contribute little to the fit from the predictor set. The logistic lasso is able to sift through a large number of predictors by maximizing a penalized MLE problem

where L(β; $\mathcal{D}$) denotes the likelihood of the coefficients β = (β₁, …, β_d) when observing data $\mathcal{D}$ = {(x₁, Y₁), …, {(x_N, Y_N)}, the term $\lambda {\displaystyle {\sum }_{j=1}^d\left|{\beta }_j\right|}$ is the lasso penalty, and λ ≥ 0 is a tuning parameter. Adding this penalty for λ > 0 discourages large (positive or negative) coefficient values and results in all coefficients shrinking toward 0 as λ increases. To penalize all predictors equally, the predictors x are scaled (to standard deviation 1) and centered (to mean 0). For sufficiently large λ, small regression coefficients are set to 0. Increasing λ leads to smaller statistical models with fewer predictors. We fit our logistic lasso model to TCs between 1998 and 2009 via 10-fold cross validation; see appendix B for details. Note that we later in sections 4b and 4c assess the final statistical models (with tuned parameters) on an independent test set not used in the cross validation; in our case, all TCs from 2010 to 2016.

2) Assessing classification performance

To evaluate the performance of the fitted statistical models in an objective way, we convert the predicted probabilities ${\widehat{p}}_i$ to binary classes C_i, for which we have a ground truth. Here, C_i = 1 denotes an ongoing RI (or RW) event and C_i = 0 denotes the absence of such an event. We perform this conversion by choosing a probability cutoff p^★; for observations x_i for which the probability ${\widehat{p}}_i>p^{\star }$, the classifier predicts onset of or ongoing rapid change (C_i = 1); if ${\widehat{p}}_i<p^{\star }$, the classifier predicts that there is no rapid change (C_i = 0). The cutoff is often fixed to a default value of p^★ = 0.5, which makes sense in the case of balanced data (i.e., when the two classes occur with the same proportions) but not when one class represents rare events such as RI and RW. As we change the value of p^★ in the range [0, 1], we can make the resulting classifier more or less sensitive to the event, trading between higher true positive rates at lower p^★ (TPR; the fraction of events correctly identified as such) and higher true negative rates at higher p^★ (TNR; the fraction of nonevents correctly identified as such). A so-called receiver operating characteristic (ROC) curve describes the properties of a classifier by showing its TPR and TNR values at different cutoffs p^★ (Fig. 13). An ideal classifier would have a ROC curve hugging the top-left corner, which represents a 100% true positive and true negative rate. (The trivial SHIPS + Persistence model tends to perform best, as expected). A poor classifier would be near the gray diagonal, which corresponds to no better than chance. Hence, a common way of quantifying the performance of a classifier for all possible choices of thresholds is to compute the area under the ROC curve (AUC). In our study, we include both ROC curves and AUC values. In addition, to address the issue of unbalanced data, we choose the threshold p^★ of the binary classifiers in section 4b to maximize the so-called balanced accuracy (equivalent to half of the Peirce score) defined as BA = (TPR + TNR)/2 (Brodersen et al. 2010).

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Classification by logistic lasso: ROC curve comparison across all four predictor sets, by basin and by type of rapid change. See Fig. 6 and Table 1 for AUC metrics with bootstrap confidence intervals and significance tests of differences in AUC between statistical models.

Citation: Journal of Applied Meteorology and Climatology 59, 10; 10.1175/JAMC-D-19-0286.1

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1) Classification by logistic lasso

We fit the lasso coefficients $\widehat{\beta }$ in Eq. (5) for all four predictor sets in Table 2. Figure 13 and Fig. 6 (top) summarize the final lasso classification results on test data by type of rapid change (RI or RW) and by basin (ENP or NAL); with four settings and four predictor sets there are 16 different models. In Fig. 6, the square markers represent the AUC on TCs between 2010 and 2016 (the test sample) for statistical models fitted on TCs between 1998 and 2009 (the train sample). We assess the uncertainty in the AUC estimates (conditional on the train sample) by resampling the test sample 250 times with replacement. The figure shows a 95% pivotal bootstrap confidence interval of the AUC for each model fitted with logistic regression or one of the nonlinear classifiers discussed in section 4c.

According to a permutation test² of differences in AUC, the statistical models based on the satellite-derived ORB-only predictor set achieve a classification accuracy comparable to the models based on the SHIPS-only environmental predictor set. This result suggests that ORB is able to adequately assess aspects of evolving convective structure relevant to intensity change by directly capturing the structure of clouds (rather than the environment in which those structures evolve). Furthermore, ORB predictors used as a complement to traditional environmental predictors, as in the SHIPS + ORB predictor set, may further improve accuracy. The significance tests for RI/non-RI in the NAL basin in particular underscore the potential for the ORB framework to complement environmental predictors.

In summary, our analysis implies that comprehensive study of GOES IR can provide a high-resolution view of the evolution of TC internal structure, and that information complements traditional NWP-derived environmental predictors. These results appear robust across widely different classification methods (see section 4c and Fig. 6).

2) Regression coefficients in the fitted logistic lasso

As mentioned, the logistic lasso model [Eq. (5)] allows for fast variable selection and straightforward interpretation of the regression coefficients $\widehat{\beta }$. Figure 7 shows the fitted nonzero lasso coefficients (20 for RI/non-RI and 37 for RW/non-RW) in the NAL basin; the corresponding “selected” variables form a subset of the 116 SHIPS + ORB predictors.

The regression coefficient for RAD3 in this model is −0.53, meaning that a 1σ increase in RAD3 (the clarity of the eye–eyewall structure) is associated with a 0.53 decrease in the predicted log odds of the rapid change event, or a decrease of a factor of exp(−0.53) = 0.59 in the predicted odds, given that all other predictors are held constant. Likewise, the coefficient for U200 in that same fit is −0.75; the same 1σ increase in U200 is associated with a decrease in the log odds by 0.75, and the odds by a factor of exp(−0.75) = 0.47. We center the predictors (in addition to scaling to σ = 1) so that a predictor value of zero corresponds to the sample mean value in that basin.

When predictors with high collinearity are present, a traditional multiple regression will suffer from inflated coefficient variance. Lasso suffers from a different version of the problem, since it will often select one variable out of a group with strongly correlated variables, returning regression coefficients of 0 for the rest. For this reason, the recovered support (the set of predictors with nonzero regression coefficients in the fit) of the final fitted classifier may exclude predictors expected to appear; for example, OHC does not appear in the logistic lasso for ENP RI, but that does not mean it is not important to the intensity-change process. Instead, other variables (such as RSST) may contain similar information with respect to intensity change as OHC. This effect is reversed for ENP RW, where RSST is dropped from the model while OHC remains.

A positive regression coefficient means that above-average values of a predictor are associated with increased log odds of rapid intensity change, while negative coefficients mean that above-average values of a predictor are associated with reduced log odds. Hence, rapid intensity change is more likely when the sign of the predictor matches the sign of the coefficient. In the left panel, the classifier for RI tends to predict ongoing RI events when (i) current eye–eyewall structure is weak (negative RAD3 regression coefficient), (ii) eye–eyewall structure is strengthening (positive Δ₂₄RAD3 regression coefficient), (iii) current 200-hPa zonal winds are low (negative U200 regression coefficient), (iv) current wind shear is low (negative SHDC regression coefficient), and (v) current midlevel humidity is elevated (positive RHMD regression coefficient). In the center panel, the RW classifier uses a wider range of weaker effects, predicting RW for cases of (i) higher current interior symmetry than exterior symmetry (negative SHAPE2 regression coefficient), (ii) recent decay of interior symmetry (positive Δ₆SHAPE2 and Δ₁₂SHAPE2 regression coefficients), (iii) decaying eye–eyewall structure (negative Δ₂₄RAD3 regression coefficient), and (iv) strong current eye–eyewall structure (high RAD3 regression coefficient).

The SHIPS + ORB results suggest a more complex relationship between structure, environment, and intensity in the case of RW than in the case of RI (via a large number of small to medium-sized regression coefficients as opposed to a few dominant effects). RI is thought to be strongly driven by internal processes, while RW results from a more complex relationship between the TC and its environment. This is reflected in the logistic lasso fitted under the ORB-only setting (shown in the online supplemental material). RI is dominated by radial profile predictors (for core structure), SHAPE2 (which reflects the core symmetry), and SKEW1 (for overall skew of the cloud tops, which is influenced by upper-level winds and shear). RW encompasses a wider range of regression coefficients, including global measures such as SIZE, various DAV predictors, and change in overall T_b (Δ₂₄RAD1), alongside the core structure measurements.

The logistic lasso models fitted for the ENP basin (see the online supplemental material) show similar results, with differences largely explained by the narrow region of TC activity. Outside of this region, SSTs decrease and become hostile to TCs. The logistic lasso for RI fitted under the SHIPS + ORB predictor set predicts RI when (i) SST is high (RSST), (ii) shear is low (SHDC), (iii) the TC is farther south (LAT), (iv) T_b is low and dropping, particularly in the core (RAD2; RAD1; Δ₂₄RAD1), (v) the eye–eyewall structure is intensifying (Δ₂₄RAD3), and (vi) the storm has access to ample upper-level moisture (RHHI).